ANALIZA CIRCUITELOR ELECTRICE CU PSPICE. LUCRARI DE LABORATOR/ CONSTANTINESCU FLORIN, IONESCU ANISOARA, MARIN CONSTANTIN-VIOREL, NITESCU MIRUNA ✽✴✲✯✷✓✳✯✮✺✹ ❄ ✂✁☎✄☎✆✞✝✠✟✡✝ ☛✌☞✎✍✏✁✑✄✓✒✕✔✖✝✗✁✏✟✘✄✚✙✗✁☎✄✑✟✞✔✜✛✢☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✤☞✥☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞ ✦✧☞ ✒★✟✞✔✪✩✜✔✪✫✠✝✗✁☎✄☎✝✤✬✭✁☎✙✗✮✯✁☎✝✠✰✲✱✳✩✖✱✴✔✵✂✶✷✬✭✔✜✛☎✄ ☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞ ➓ ➓◗➓ ✸✧☞ ✖✏✱✭✛✑✁✹✝✠✁✑✔✳✺✭✄✻✩✼✝✠✽✭✙✕✁☎✝✠✟✞✙✕✁✢☞✣☞✣☞✣☞✣☞✣☞✾☞✿☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞❁❀ ➂ ❂✧☞ ✆✵✙✗✩✖✱✳✟❃✔❄✔✪✩✖✄✏✬✭✁☎✙✗✽✳✩✖✄✑✰✲✄❅✩✖✙✌✁ ☞✣☞✣☞✣☞✣☞✣☞✤☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞ ➎✠❆ ❇✴☞✎✍✻✔✜✽✳✩✜✔❄✙✗✮✕✁☎✝✌✆❃✔✖✄ ☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✤☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞✣☞ ➂✏➓❏❿ CAP. 1. BREVIAR TEORETIC ✂✁☎✄✆✞✝✟✄✡✠☛✌☞✎✍✏✝✡✍✒✑✓✂✔✕☞✖✞✗✘✍ ✙✌✚✜✛✣✢✥✤✧✦✩★✫✪✭✬☎✮✫✯✱✰✲✤✴✳☎✬✵✤✧✶✷✤✸★✫✯✌★✫✪✞✯✱✦✩✹✻✺✼✳✽✮✾✤✸✤✿✺✼✹❁❀ ✙✌✚✸✙❂✚❄❃❅✤✸❀❂✶❆✳❇✤❈★✩✪✣✺❈✤✼✬✵✤✧✯❄❀❂✪ ✞❉ ❊✒❋✼●✷❍❏■✞❊✱❑▼▲◆❍✷■✘❑P❖❘◗❙❑✫❍✞❖❯❚❱◗❲❊▼❳❨❖✿❋❩❚❈❖❘❬❪❭✲◗✧❊✒❋✧❫✷❴❯❍✷❵❈❖❛❊◆❍✷■❆❖❛❑P▲✱❜✷❭✩❑P▲✱❊✩❑✫❍✞❭✩❵✿❖❘❖❝❴❞❊✱❭▼❑✲❊❡❋✧❚❞❍✭❖❞❭❢❋✧❍✷■❏❚❣❋❙❭✫❚✸❖✿❋✐❤✸❭P❑✲❍✷❚❈❊ ❋✐❖❯❥❦❍✞❴❘❚❛❭✲■❧❜✷❊♠❫✕❥♥❍✞❴❘❵❈❖❯❥♦❊♣❍❏■✞❖❛❑P▲♠❜✷❊♣❚❛❊✲■❂❋✐❖q❍✷■✞❖❄r❙❖✌❑✫❍✷◗❙❊✫■✷❵✸❖✸s❡t❁✉❆❖✿❋✼❚✸▲♠❑P❖❘◗❲❑✫❍✞❖q❚✸❊✕❋✐❖❯❥♥●✞❴❯❊✕❑✲❭✲◗❙❊♣■❏❍✈❭✫❍♦❋❙❫✷❴❯❍✷❵❈❖❛❊✕❋✐❭✫❍❧❭✲❍ ❍❏■✣■✷❍❏❥♦▲✫◗✵❖❘■✞❤✸❖❯■✞❖❯❚✥❜✷❊✕❋✐❫✷❴q❍✷❵✸❖❞❖❈s❏✇①❖❯◗❲❑✫❍✞❖❘❚❞❍✞❴❂❜✷❖❯■♥❤✼❖❞②❡❍✷◗✧❭❱③❧❭✫◗❙❊♠❫♥❋④❫❏❴❘❍✷❵❈❖❞❊♣❍✷■✞❖❞❑▼▲ ⑤⑥⑨ ⑦ ❋✐❊☎●✞❫❡❚✂❜✷❖✿❋✸❚✸❖❯■✞②✷❊❱❜✷❫❡❍❆❭⑩❋✐❖q❚❛❍❆❭✲❵❛❖❛❖✿❶ E − E2 ❜✷❭❨❑P▲⑥⑤ s❡⑧✕❭✩❑❨▲ ≠⑦ I= 1 R ⑨ r✐❖❂❑✲❖❘◗❙❑✫❍✞❖q❚❛❍✞❴✞❭✩◗❩❊♠❫❣❖❻■❆❤✼❖❘■❆❖❻❚❈❭❾❚❈❊♣❜✷❊♠❋✐❫✷❴❯❍✷❵❛❖❝❖✞❿✼❫❡◗❙❖❞❑▼❊ ⑤ ❸ ⑦ •❽ ⑦ ❽∈ r✐❖✞❑▼❖❯◗❲❑✩❍✭❖❘❚❘❍✞❴❆■✷❍❦❭✩◗❲❊♠❋✐❫✷❴❯❍✷❵✿❖❞❊❡s ❞❷ ❷❝❸ ❜✷❭✩❑P▲⑥t ❹ ✿❺ t ❼ ◗❲❊✲❳▼❍✞❴❯❚✿▲ ≠⑦ ⑦ •❽ ≠⑦ ❷❘❸ ❜❏❭▼❑P▲⑥t❇❹❻❺✸t✴❼❞⑨ ⑦ ◗❲❊▼❳✩❍✞❴❯❚✸▲ ✇⑥❖❻◗✧❑✲❍✞❖q❚❛❍❆❴❄❜✷❖❯■♦❤✸❖❞②❡❍✷◗✧❭♠➀✈❭✲◗❙❊⑩❫➁❋❙❫✷❴❯❍✷❵❛❖❛❊➂❍❏■✞❖❛❑P▲ ❋✐❖q❚❛❍❆❭✲❵❛❖❛❖❈❶ I +I s❏⑧✕❭P❑✩▲✕➃➂⑨ ❋✐❊❱●✞❫❡❚✥❜✷❖✿❋✼❚✸❖q■✞②✷❊✕❜✷❫❨❍✞▲ ❜✷❭❨❑▼▲✕➃ U = s1 s2 ≠⑦ ⑦ G ❼ ⑨ ◗❙❊✲❳P❍✞❴q❚✸▲ ⑨ r④❖❂❑✲❖❘◗❙❑✾❍✞❖❯❚❛❍✞❴❂❭✫◗❩❊♠❫✕❖❘■❆❤✸❖❘■✞❖❯❚❈❭❾❚❈❊♣❜✷❊♠❋✐❫✷❴❯❍✷❵❛❖❛❖✭❿✼❫❡◗❙❖❞❑▼❊ ⑤ ❸ ❽✸➄❘➅➇➆❄❽✧➈ ⑦ ⑦ •➉ ⑦ ➉ ∈ r✐❖✜❑P❖❯◗✼❑✩❍✞❖q❚❛❍❆❴✞■❏❍❦❭✲◗✧❊⑩❋✐❫❏❴❘❍✷❵❛❖❝❊➊s ❷❞❷❝❸ ❜✷❭✩❑P▲ ❹ ❼ ◗❙❊✲❳✩❍✞❴❻❚❈▲ ❽✧➈ ➆❄❽✼➈ ≠ ⑦ ⑦ •➉ ≠ ⑦ ✇⑥❫❡■✞❜✷❖q❵✸❖❞❭✣❜❏❊✣❊✲✉✷❖❈❋✧❚❛❊✩■➊❚✸▲♥r❙❖✥❍✷■❆❖❛❑✲❖❘❚❛❭✲❚❛❊✣❭♥❋❙❫✷❴❘❍❏❵✸❖❞❊✩❖✌❍✷■✷❍❂❖➋❋✐❖❛❋✧❚❈❊❾❥➌❴❝❖❯■✞❖❘❭✲◗⑥❜❏❊➍■✎❊P❑✫❍✞❭✲❵❛❖❛❖➋❭✩❴❞②✷❊✫➎✷◗❙❖❞❑▼❊♠➏➍✉✷⑨✌➐ ❷❘❸ ❜❏❭▼❑P▲ ❊❨❋✧❚❈❊♠➑❆➒P➓➇➔ ≠→ s➊➏❦❑✩❊❨❭❨❋❩❚❈▲✕❑P❫❡■✞❜✷❖❯❵✸❖❘❊❣❊✲✉❏●✷◗❙❖❯❥❧▲❣❤✼❭✫●✷❚❞❍✞❴✌❑❨▲✕❊▼❑✫❍✞❭✾❵✸❖❘❖❛❴❘❊✣❋❙❖❈❋✼❚❈❊✲❥♥❍✭❴❯❍✞❖❁❋✧❍✷■❏❚➋❴❝❖q■✞❖❛❭✫◗➣❖❯■✞❜✷❊✫●✞❊✲■❆❜✷❊✲■❏❚✸❊✕❖❯■✷❚❞◗❙❊ ❊P❴❞❊❏sP⑧➍❭✩❑P▲➂❜❏❊✲❚❻➏❦⑨ ❊✲✉✜❖❛❋✧❚❛▲❱❜✷❫❡❍✞❭♠❋✐❖❻❚❘❍✞❭✩❵❞❖❞❖❆❜✷❊❱❖❯■✷❚✸❊✫◗❲❊❨❋✴●✷◗❲❭❨❑❾❚❈❖❝❑❏❶ ⑦ ❷❘❸ ◗❙❭✫■✞②❡❍✞❴↔❴❯❍✞❖✂➏↕❊❨❋✧❚✿❊❣■✞❺✾➙❦r❙❖✵❜❏❊❾❚❈❊✲◗✼❥➁❖❯■✭❭✲■❏❚❛❍✞❴❄❥➁❖❘■❆❫❡◗✼❍✞❴❯❍✞❖❄●✷◗❙❖❯■✞❑✩❖❯●✞❭P❴➋➎❆❫❡◗❙❜✷❭✫❚↔❑✩❍✎❑P❫❆❴❯❫❏❭P■❆❭❦❚❛❊✩◗✸❥✈❊✫■✞❖❞❴❛❫❡◗ ❴❞❖❯➎✞❊✲◗✧❖❁❊➊❋✼❚✿❊❣■✷❍✞❴✸➛❂➜❝■➝❭P❑✩❊➊❋✼❚✽❑P❭✫❳✣❍✷■❆❭❧❜❏❖❘■✷❚❞◗❩❊♥❊▼❑✩❍✞❭✾❵✸❖❘❖✴❊❨❋✼❚✸❊❦❴❛❖q■✞❖❝❭✩◗⑥❜❏❊✲●✞❊✫■✞❜✷❊✩■✷❚❈▲❦❜✷❊✣❑❨❊✲❴❛❊P❴❝❭P❴❻❚❈❊✈r✧❖✽❋❙❖❈❋✼❚❛❊P❥❦❍✞❴➋❭✲◗✧❊❧❫ ❖❯■✞❤✼❖❯■✞❖❯❚✿❭✫❚✿❊❱❜✷❊♠❋❙❫✷❴❯❍✷❵✿❖❞❖❛➞ ❷❞❷❝❸ ◗❙❭✫■✞②❡❍✞❴✥❴❘❍❆❖✻➏➟❊❏❋✼❚❈❊⑩■✞❺❾➙⑩r✐❖✥❜✷❊✫❚✸❊✫◗✼❥❧❖❘■✞❭✫■✷❚❘❍✞❴✂❥➁❖❯■✞❫❡◗✼❍✞❴q❍✞❖✂●❏◗❙❖❘■❆❑▼❖❯●✞❭P❴✂➎✞❫❡◗❲❜❏❭P❚❁❑✲❍❅❑P❫✷❴❞❫✷❭✩■✞❭⑩❚❛❊P◗✸❥➁❊✫■✞❖❝❴❞❫❡◗ ❴❞❖❯➎✞❊✲◗✧❖✜❊❨❋❩❚❈❊①■✞❊✩■✷❍✞❴✿➛➊➜❝■❦❭❨❑✩❊❨❋✧❚❂❑❨❭✫❳✕❋❙❖❛❋✧❚❛❊✲❥♥❍✞❴✷■✷❍✣❭✲◗❙❊♠❋❙❫✷❴❯❍✷❵✿❖❞❊❡s ➠ ■✣❑▼❖❯◗❙❑✫❍✞❖❻❚❈❊▼❴❘❊♠❋✧❖❯❥♥●✞❴❛❊☎●✷◗❙❊✲❳❨❊P■➊❚❈❭✫❚✿❊①❥✈❭P❖✞❖❘■✞❭✲❖❘■❏❚✸❊♠❋❲❊♠❫❡➎❂❋❙❊✲◗❩❬✞▲✷❶ ❷ ➡ ➜❝■✡❤❩❖❛②❡❍❏◗❲❭✱③✜➞✽●✞❊✲■❏❚❛◗✼❍✘⑤➣⑨ ❭✩❬✞❊✫❥➢❜✷❊✫❚❯➏➍⑨ r✐❖♠❊✲✉❆❖✿❋❲❚❈▲✒❫✒➎✷❍✞❑P❴❛❭✱❤❩❫❡◗✼❥➁❭✲❚❈▲✖■✷❍✷❥❧❭P❖✕❜✷❖❻■➤❋✼❍✷◗✐❋❙❊✒❜❏❊ ⑦ ⑦ ❚❈❊✫■❂❋✐❖❘❍❏■✞❊ 3 ❷❞❷ ➡ ➜❞■❢❤✸❖❛②❨❍✷◗❙❭♥➀✜➞❂●✞❊✩■✷❚❝◗✼❍❢➃❱⑨ ⑦ ❭✲❬✜❊✲❥ ❜✷❊✫❚❞➏➍⑨ r④❖✽❊✲✉❆❖✿❋✼❚✸▲✈❫✎❋✐❊▼❑✾❵✸❖❯❍✷■✞❊✈❤✼❫❨◗✸❥♦❭✫❚❈▲✣■✷❍➊❥➁❭▼❖✽❜✷❖❯■✱❋✧❍❏◗❙❋✐❊✈❜✷❊ ⑦ ✫❑ ❍✷◗❙❊✫■✷❚❲s ➏❦❑❨❊❨❋✼❚✿❊♥❫❡➎❂❋❙❊✩◗✸❬✞❭✩❵✿❖❘❖➋●❆❫❡❚✽❤❩❖✵②❏❊✲■✞❊✫◗❙❭P❴❝❖❘❳▼❭✫❚✿❊✕●✞❊✫■✷❚❛◗❩❍➝❫❡◗❲❖❞❑▼❊♥❑P❖❘◗✧❑✲❍✭❖❯❚❇❤✸❫❡◗✼❥➁❭✲❚❇❜✷❖q■◆◗❙❊❾❳❡❖✿❋✼❚✿❫✷❭✫◗❙❊❦❜✷❖❯●✞❫✷❴❞❭✲◗❙❊♥❑✫❍✎⑤ ✁ r❙❖ ⑦ ❋✧❍❏◗❙❋✐❊❢❖❯■✞❜✷❊✫●✞❊✩■✞❜✷❊✫■✷❚❈❊❡➞❇❑▼❫❨■✞❜✷❖❯❵✿❖❞❖❝❴❞❊❢❜❏❊✱❊✩✉❆❖❞❋✧❚✿❊✩■✷❵❛▲✒r④❖①❍✷■❆❖❛❑✲❖❘❚❈❭❾❚❈❊➝❤✸❖❛❖❯■✞❜ ❊✲✉❏●✷◗✧❖❘❥➁❭✫❚✿❊✎➜❞■❪❤❛❍✷■✞❑✩❵✿❖❞❊✱❜✷❊◆●✞❭✫◗❙❭✫❥✈❊✩❚❝◗✧❖❝❖ ❑P❖❯◗❙❑✲❍❆❖❻❚❞❍✞❴❘❍❆❖❩s ✂ ☎➒ ✄✝✆✫➒✟✞♠③ ■ ❑▼❖❯◗❲❑✩❍✞❖❻❚➤❴❘❖❘■❆❖❝❭✩◗ ❤✸❫❨◗✼❥➁❭❾❚ ❜✷❖❯■ ◗❲❊✲❳➊❖❛❋✼❚❈❫✷❭✲◗✧❊ ❜✷❖❘●❆❫✷❴❝❭✫◗❙❊ ❴❝❖❯■✞❖❞❭✲◗❲❊➌❑✲❍ ⑤ ✁ r✐❖✡❋✧❍✷◗❙❋✐❊ ➉ ⑦ ❖❯■✞❜✷❊✫●✞❊✩■✞❜✷❊✫■✷❚❈❊✕❭✫◗❲❊✕❫✈❋④❫❏❴❘❍✷❵❛❖❛❊❱❍✷■✞❖❝❑P▲➂●✭❊✫■✷❚❞◗✼❍➁❫❡◗❲❖❛❑P❊➂❬✜❭✩❴❞❫❡◗❲❖❄❭✩❴❞❊♣❚❈❊✲■❂❋❙❖❯❍✷■✞❖❞❴❛❫❨◗✽❊✲❴❛❊P❑✲❚❞◗❲❫❡❥❧❫❡❚❈❫✷❭✲◗✧❊❣r❙❖✌❭▼❴❘❊♠❑P❍❏◗❲❊P■❏❵✿❖❝❴❞❫❡◗ ❊P❴❞❊▼❑✫❚❛◗❲❫❨❥♦❫❨❚✸❫❡◗✧❖✻❜❏❭✩❑P▲⑩r❙❖❆■✷❍✷❥➁❭✲❖✻❜❏❭▼❑✲▲⑩❋✧❍❏■✷❚❄❋❙❭✫❚✿❖✿❋❙❤✼❭▼❑✩❍➊❚❛❊♣❍✷◗✸❥➁❭✲❚❛❫✷❭✩◗❲❊▼❴❘❊➂❑P❫❡■✞❜❏❖❘❵❈❖❝❖❈❶ ❺✌■✷❍♥❊✲✉❆❖✿❋✼❚✸▲①■✞❖❘❑▼❖✞❫⑩➎✭❍✞❑▼❴❘▲❱❤✼❫❡◗✸❥➁❭✲❚❛▲①■✷❍✷❥➁❭❨❖❆❜✷❖❯■✈❋✧❍✷◗❙❋✐❊❱❜✷❊☎❚❈❊✲■❂❋✐❖❯❍✷■✞❊❨➞ ❺✌■✷❍♥❊✲✉❆❖✿❋✼❚✸▲①■✞❖❘❑▼❖❂❫✣❋❙❊✲❑✲❵❛❖❘❍✷■❆❊➂❤❩❫❡◗✼❥➁❭✲❚❛▲①■✷❍✷❥➁❭▼❖❂❜✷❖❯■♥❋✼❍✷◗✐❋❙❊♠❜✷❊⑥❑✫❍✷◗❙❊✫■✷❚✧s ✠ ☎➒ ✞✡✄✝☛✝☞✫➓✌✆✐③❆➓ ❷ ➒✝✍✏✎✕➒✒✑✩➒✓☞ ❷ ➓❻③❆➓❞➒✩③ ❋✧❊✆❜✷❊✩❥✈❫❨■❂❋✼❚❛◗❙❊P❭✲❳▼▲✒●✷◗❙❖❯■ ◗❙❊P❜❡❍✞❑P❊✲◗❙❊ ❴❛❭✆❭✫➎❂❋✼❍✷◗❙❜✜s ✷■ ❚❛◗❲❺❩❫✘➎✷❍✞❑P❴❛▲✆❜❆❊☛❋✼❍✷◗✐❋❙❊☛❜✷❊ ❽ ❚❈❊✫■❂❋✐❖❘❍❏■✞❊✈❜✷❖❯■✖❚❈❊▼❫❨◗❙❊✲❥❧❭♦❭ ❺✸❭➁❭✈❴❘❍✞❖✕✕ ✔ ❖❯◗❲✗❑ ✖✘✖✞❫✷❤✼❤☎◗❙❊✲❳P❍✞❴❯❚❈▲✚✙♠✕ t ✛✖⑨ s✌⑧✕❭P❑P▲♥✕ t ✛ ❋✼❍✷■✷❚➣❭➊❋✼❚✸❤✸❊P❴➋➜❞■❆✢❑ ✜✫✣ ❚ ✙♠✕ t ✛ ≠⑦ ❽✼❽ ⑦ ❑P❖❯◗❙❑✲❍❆❖❻❚❞❍✞❴✻■✷❍➁❭✫◗❲❊❣❋❙❫✷❴❘❍❏❵✿❖❞❊❏s❡⑧✕❭✩❑P▲♠t ✛ ❋✼❍✷■❏❚✥❭❡❋✧❚❈❤✸❊▼❴❂➜❝■✭☎❑ ✜P✤ ❚ ✙♠t ✛ ⑨ ➞✭❭✲❚❞❍✷■✞❑P❖✜●❆◗❲❖❯■✈❭P❑P❊▼❭❨❋✼❚✿▲➂➎✷❍❆❑▼❴❞▲➂●✞❫✷❭✫❚❈❊♠❑✩❖❯◗❙❑✲❍❆❴❛❭ ⑦ ❍❏■❅❑✲❍✷◗✧❊✲■✷❚✴❜✷❊⑩❬❆❭▼❴❞❫✷❭✫◗❙❊✕❭✫◗✼➎✞❖❯❚❛◗❲❭✫◗❲▲ ❑P❭✲◗✧❊✣❋❲❭❾❚❈❖❈❋✐❤✸❭P❑▼❊⑩❊▼❑✫❍✞❭❾❵❈❖❞❭ ⑨ r✐❖✌❑P❖❘◗❙❑✫❍✭❖❯❚❛❍✞❴✂❭✲◗❙❊⑩❫❧❖❯■✞❤✼❖❯■✞❖❯❚✸❭✾❚✸❊➍❜✷❊❣❋❙❫✷❴❯❍✷❵❈❖❝❖✸s Ι ⑦ ⋅❽ ⑦ ■➍◗❙❭✫❵✸❖❞❫❡■✞❭✫❥➁❊✲■✷❚✂❋✐❖❻❥❧❖❞❴❝❭✫◗✽❋❙❊①●✞❫✷❭✫❚✸❊⑥❤✼❭P❑✩❊☎●✞❊✩■✷❚❛◗✸❍✣❫✣❋✐❊P❑✲❵❛❖❘❍❏■✞❊❱❤✼❫❡◗✸❥➁❭✩❚❞▲♣■✷❍✷❥➁❭P❖✜❜✷❖❯■✈❋✸❍✷◗✐❋❙❊❱❜✷❊♠❑✫❍✷◗❲❊✩■✷❚❲s ➉ ✥✧✦✩★ ❷ ✑ ❷ ➒✗☛✢✪✼③◆❋❙❊➍❜✷❊✩❥♦❫❨■❂❋✼❚❛◗❙❊✲❭✩❳P▲❣❑P❫❡■❂❋✐❖❞❜✷❊✫◗✫❾✜ ■✞❜✈❑P▲❣❊✲✉❆❖✿❋✧❚❞▲❣❜❏❫❡❍❂❭❣❋✐❫❏❴❘❍✷❵❈❖❝❖✂❿✼❑▼❭✫◗❲❊❣❑P❫❡◗❙❊❨❋✧●➊❍✷■✞❜✈❭P❑P❊▼❴❞❫❡◗❙❭❨r❙❖✂❬✞❭✲❴❛❫❡◗✧❖❁❭✩❴❞❊ ❚❈❊✫■❂❋✐❖❘❍❏■✞❖❝❴❞❫❡◗➍r✧❖❇❑✲❍✷◗❲❊✩■✷❵❛❖❝❴❞❫❡◗⑩❊✩❴❞❊✩❑✫❚❛◗✧❫❡❥➁❫❡❚✿❫❡◗✧❖ ❸ r✐❖✽❭✲◗❙▲✾❚✸▲✫■✞❜➝❑✩▲➁❊✩❴❞❊✈❑▼❫✷❖❯■✞❑P❖❛❜✞s✭✬✌❖❞❊ ❋✐❫❏❴❘❍✷❵❛❖❛❖❄r❙❖❂❤✼❖❘❊ ❑P❖❯◗❙❑✲❍❆❖❻❚❞❍✞❴❘❍❆❖ (E dk u d = u1 − u 2 ●✞❊✫■✷❚❞◗✼❍ r❙❖ id = i1 − i2 ❬✞❭P❴❞❫❡◗❙❖ ■✷❍✞❴❞❊ r❙❖ (u1 , i1 ) (u 2 ,i2 ) ❑P❊▼❴❞❊♦❜✷❫❨❍✞▲ ❋✐❫✷❴❘❍❏❵✿❖❞❭♠❜✷❖❛❤❩❊✲◗❲❊✫■✷❵❈▲➂❑P❭✩◗❲❊❡➞❏❜✷❭✫❚✸❫❡◗❲❖❯❚❈▲➂❴❘❖❘■❆❖❛❭✫◗❙❖❻❚❈▲❾❵❈❖❝❖❞➞✜❋❙❭✫❚✿❖✿❋❙❤✼❭▼❑✲❊♠❊▼❑✫❍✞❭✫❵✿❖❞❖❝❴❞❊ ❭❨❴❝❊ = E k − E k = 0 si I dsk = I sk − I sk = 0 ) ❚❈❊✲■❂❋❙❖❯❍✷■✞❖❘❴❛❫❡◗ s↔✇⑥❫❡■✞❤✸❫❨◗✼❥ r✐❖ ❑✫❍✷◗❙❊✾■✷❵✸❖❞❴❝❫❨◗ ❊❨❴❝❊P❑❾❚❞◗❙❫❡❥❧❫❡❚✸❫❨◗❙❖ ❚✸❊P❫❡◗❙❊✫❥➁❊▼❖⑥❴❘❍❆✯ ❖ ✮✥❊▼❴❞❴❞❊▼②➊❊✩■ ✰ u dk i dk = 0 ➞✽❜✭❭✲◗ toate laturile ❜❏❊▼❫✷❭✩◗❲❊❨❑P❊ ●✞❊✫■✷❚❞◗✼❍ ❋✧❍✷◗✐❋❙❊ u dk ❋❙❭P❍ idk ❋✧❍✷■✷❚❅■✷❍✞❴❞❖❛➞❣◗❙❊✲❳P❍❆❴❻❚❈▲ ✱ Rk idk2 = 0 ❜✷❊P❑▼❖ idk = 0 ●✷◗❙❖❯■ ❚✸❫❏❭✲❚❈❊ toate rezistoarele ◗✧❊✩❳❨❖❈❋✼❚❈❫✷❭✫◗❲❊❡❴❘❊❧r❙❖↔❋✧❍✷◗✐❋❙❊P❴❝❊♥❜✷❊❦❑✲❍✷◗✧❊✲■✷❚❲s ➠ ■✞❴❞❫✷❑✫❍✞❖❯■✞❜◆❴❞❭❾❚❞❍✷◗❙❖❞❴❝❊❦❑✩❍ id = 0 ❴❞❭✫❚❛❍✷◗✧❖❝❴❞❊❢❋✐❑✫❍✷◗✸❚❈❑▼❖q◗❙❑✲❍❆❖❻❚⑩❑▼❫❨◗❲❊❏❋✼●✷❍❏■✭❳❡▲✩❚❛❫✷❭✩◗❲❊❢❋✼❍✷◗✐❋✐❊P❴❞❫❡◗❦❜✷❊♦❚✸❊✫■❂❋✐❖❯❍✷■✞❊✎❑✲❍ ❑✩❍♦◗❙❊✲❳▼❖✿❋✧❚❈❫➊❭✲◗❲❊♥❑✩❍➁⑤♣⑨ Ed = 0 ∞ ◗❙▲✫❥✲✲✜ ■❅■✷❍✷❥➁❭✩❖ s↔⑧✕❭P❑▼▲➝❫✒❭➊❋✼❚✿❤✼❊P❴⑥❜✷❊✱❋❲❍❏◗✐❋❙▲❢❜❏❊ ❚❈❊✫■❂❋✐❖❘❍❏■✞❊✓■✷❍➤❤❩❭▼❑✲❊ ●✞❭✫◗✼❚❞❊✆❜✷❖❯■✷❚❞◗❲❺✼❫❪➎✷❍✞❑P❴❝▲ ❜✷❊❪❋❙❑✩❍✷◗✸❚✿❑P❖❯◗❙❑✲❍❆❖❻❚❈❊✒❭✩❚❝❍✷■✞❑P❖❣❑✫❍✷◗❙❊✫■✷❚❞❍✭❴⑩●✷◗❲❖q■ ❋✼❍✷◗✐❋❙▲✆❊❡❋✼❚✿❊✱■✷❍✞❴✸s➣⑧✕❊▼❑✲❖ ❊✲✉❆❖✿❋✧❚✸❊✫■➊❵❈❭✈❴❻❍✞❖ id ≠ 0 ❖❘❥♥●✞❴❞❖❛❑✲▲✈❊✩✉✭❖❈❋✼❚✿❊✫■✷❵❈❭✣➎✷❍✞❑P❴❝❊P❖↔❜✷❊♦❋✼❍✷◗✐❋❙❊✈❜✷❊✣❚❈❊✲■❂❋✐❖❯❍✷■✞❊❨➞❁r✐❖➋➜❝■✱❑P❫❡■❂❋❙❊❨❑P❖❘■➊❵❈▲ ❋✧❍❏◗❙❋✐❊▼❴❞❊✕❜✷❊♣❚❛❊✲■❂❋✐❖q❍✷■❂❊➊s✞❉❆❖❯❥❧❖❛❴❘❭✲◗➣❋❙❊♠❜✷❊✩❥✈❫❨■❂❋✧❚❛◗❲❊❨❭❾❳❡▲♠❑P▲✕❊✲✉❆❖❈❋✸❚❈❊✲■✷❵❛❭♣❚✸❊✫■❂❋✐❖❯❍✷■✞❖❘❴❝❫❡◗ ud ≠ 0 ❑✫❍✷◗❙❊✫■✷❚✵❖❘❥♥●✞❴❞❖❯❑❨▲❦❊✩✉✷❖✿❋✼❚✿❊✫■✷❵✸❭ ❋❙❊▼❑✫❵❈❖❻❍✷■✞❖❘❴❛❫❨◗⑥❤✸❫❡◗✼❥➁❭✫❚✸❊✕■✷❍✷❥➁❭P❖✴❜✷❖❯■✎❋✧❍✷◗❙❋✐❊❦❜✷❊❦❑✲❍✷◗✧❊✲■✷❚❩➞✌❜✷❊✲❑▼❖ 4 id = 0 r✐❖❁●✷◗❙❖❯■ ❴❝❭❱➎✞❫❡◗✸■✞❊P❴❛❊✕❋✧❍❏◗✐❋❙❊▼❴❞❫❡◗❇❜✷❊ ud = 0 r✐❖✥❴❛❭✕➎✞❫❡◗✼■❆❊▼❴❞❊ ❋✧❍❏◗❙❋✐❊▼❴❞❫❡◗➂❜✷❊♦❑✫❍✷◗❙❊✫■✷❚✧s ➠ ■✓❑▼❫❨■❂❋❙❊❡❑✲❖❘■❏❵✸❭♥❑❨❊✩❴❞❊✈❜✷❫❡❍✞❭♦❋✐❫❏❴❘❍✷❵❈❖❝❖ (u1 , i1 ) r❙❖ (u 2 ,i2 ) ❑❨❫✷❖❻■✜❑▼❖❞❜❆➞❄❜❏❊▼❑P❖☎❋✼❍❂❤✸❖❞❑✩❖❝❊✫■✷❵❈❭✈❊❡❋✧❚❛❊ ❜❏❊✲❥➁❫❡■❂❋✧❚❞◗❲❭✩❚✿▲➊s✁❣s t✽s ⑧✣s ✂☎✄✁✆✞✝✠✟☛✡✌☞✎✍✑✏✒✏✔✓ ➎❏❍✞❑▼❴❞▲➊➛ ✏❸ ❜❏❫❡❍✞▲❣❋✼❍✷◗✐❋❙❊✕❜❏❊➂❚✸❊✫■❂❋✐❖❯❍✷■✞❊➂■❏❍◆❋❙❊➂●❆❫❡❚➋❑P❫❏■✞❊P❑✩❚✿❭♠➜❝■✣●❆❭P◗❲❭P❴❞❊▼❴✌❜✷❊❨❫✷❭✫◗❲❊❨❑✩❊✕❤❩❫❡◗✸❥➁❊▼❭P❳▼▲⑩❭❡❋✼❚❈❤✸❊▼❴✌➜❯❥♥●✷◗❙❊✫❍✷■✞▲✕❫ ✏✕✏ ❸ ❜✷❫❡❍❆❭✖❋✼❍✷◗✐❋❙❊◆❜❏❊◆❑✫❍✷◗❙❊✫■✷❚☎■❆❍✆❋❙❊✈●✞❫❡❚⑥❑❨❫❏■❆❊▼❑✫❚✸❭❧➜❞■✆❋❙❊✫◗❙❖❝❊❅❜✷❊▼❫❏❭✲◗❙❊P❑✩❊❅❤✼❫❡◗✸❥➁❊❨❭✩❳❨▲◆❭❨❋✸❚✸❤✼❊P❴✽➜❝❥♥●✷◗❙❊✫❍✷■✞▲◆❫ ❋✐❊P❑✫❵✿❖❯❍✷■✞❊➊➛ ❨❑ ❫❡■✞❜✷❖❯❵✿❖❞❖❝❴❞❊❱❊✩✉➊●✷◗❙❖❯❥♦❭✫❚✿❊❱➜❝■❧❤❈❍✷■✞❑✫❵✸❖❞❊♠❜✷❊①●✞❭✫◗❙❭✫❥♦❊✫❚❞◗❲❖❛❖❂❑P❖❻◗❙❑✫❍✞❖❻❚❞❍✞❴❯❍✞❖❄❋❙❊♣●✞❫❡❚✌❚❛❊❡❋✧❚❛❭♣❥❧❍❆❴❻❚❂❥➁❭▼❖✜❍❂r✐❫❨◗↔❜✷❊❨☎❑ ✜❾❚ ❑P❫❡■✞❜❏❖❘❵❈❖❛❭⑥②✷❊✫■✞❊✩◗❲❭❨❴❯▲✘✗✙✝✠✚ ✛✢✜✤✣✷➛ ✏✒✡ ❸ ➜❞■❧❑▼❭❾❳P❍❆❴✜❍✷■✷❍✞❖❂❑P❖❘◗❙❑✫❍✭❖❯❚❄❑✫❍➁❋❲❍❏◗✐❋❙❊♠❑▼❫❡❥➁❭✾■✞❜✷❭✫❚✸❊✕❋❙❊♣●✞❫✷❭✫❚✿❊♠❑✲✥ ❭ ✗✙✝✦✚ ✛✏❋❙▲✕❋✧❊♠❭✩■✷❍✞❴❻❊▼❳❨❊☎●✞❊✲■❏❚❛◗✼❍❧❭✫■✷❍✷❥➁❖❘❚❈❊ ❬❆❭▼❴❞❫❡◗❙❖☎❭❨❴❯❊✈●✞❭✫◗❙❭✲❥➁❊✾❚❛◗❙❖❞❴❞❫❡◗⑩❜✷❊◆❑P❫❡❥➁❭✲■❆❜✷▲❡➞➋❭P❑▼❊❨❋✼❚✸❊♥❬✞❭❨❴❝❫❡◗❙❖❇❤✼❖❝❖❯■✞❜◆◗❙▲P❜✷▲❨❑▼❖❯■✞❖❝❴❘❊◆❊✲❑P❍❆❭✲❵❛❖❛❊✲✧❖ ✗✙✝✌✚ ✛☎✜★✣✎❑✫❍➝●✞❭✩◗❲❭P❥❧❊✩❚❝◗❙❖❘❖ ❋✧❍❏◗❙❋✐❊▼❴❞❫❡◗↔❑P❫❡❥➁❭✫■✞❜✷❭✫❚✸❊❱❑P❫❡■❂❋✐❖❞❜✷❊✲◗✧❭✲❵❛❖✜❜➊◗❲❊✲●✷❚❂■✞❊P❑✩❍✷■✞❫❆❋❙❑✫❍✷❚❈❊❏➛✭❜✷❊✩❚❞❊✩◗✸❥➁❖❘■❆❭✲◗❙❊✲❭①❍✷■✞❫❡◗✴❑▼❫➊■✞❜✷❖❯❵✸❖❞❖✌❋✧❖❯❥♥●✞❴❛❊P➞✷❊✲✉❏●✷◗❙❖❯❥➁❭✲❚❛❊⑥➜❞■ ❤✿❍✷■✞❑✫❵✸❖❞❊ ❜✷❊♥●✞❭✲◗✧❭✲❥❧❊✲❚❞◗❙❖❘❖✵❑P❖❯◗❙❑✲❍✭❖❯❚❞❍✞❴❘❍❆❖❛➞✌●✞❊✲■❏❚❝◗✼❍✓❊✲✉❆❖❈❋✼❚❈❊✲■➊❵❈❭ r④❖✴❍❏■❂❖❯❑❨❖❻❚❛❭✩❚✿❊P❭♦❋④❫❏❴❘❍✷❵❈❖❝❊P❖❁❍✷■✷❍✞❖❇❭❡❋✼❚❈❤✼❊✩❴☎❜❏❊❧❑❨❖❯◗❩❑✲❍✜❖❻❚☎■✷❍ ❊❨❋✧❚❈❊➣●✞❫✭❋④❖❯➎✞❖❞❴❝▲➊➛ ✡ ❸ ❍❏■♥❑P❖❘◗❙❑✫❍✞❖q❚✌❴❞❖❘■❆❖❝❭✫◗↔❭✫◗❲❊⑩❋❙❭✫❍♥❫❦❋④❖❯■✞②❡❍❏◗❲▲✕❋④❫❏❴❘❍✷❵❈❖❞❊▼➞❆❋❲❭✩❍❣■✞❖❘❑▼❖❂❫❦❋④❫❏❴❘❍✷❵❛❖❛❊P➞❆❋❙❭✩❍♥❫✕❖q■✞❤✼❖❯■✞❖❯❚✿❭✫❚✿❊❱❜✷❊♠❋✐❫✷❴❯❍➊❵✸❖❘❖❈s ✏✕✏✖✏ ❸ ✩✫✪✒✬✭✪✯✮✱✰✳✲✵✴✙✶✷✰✹✸✠✺✼✻✢✺✾✽✳✰✳✻✢✰✑✿❀✲❁✺ ➐ ❍✞❑▼❴❞❊✩❴❞❊♥❜✷❊♥❋❲❍❏◗✐❋❙❊❧❜❏❊❦❚❛❊✲■✜❋④❖❯❍✷■✞❊➊➞❄❋✐❊✲❑✲❵❛❖❘❍❏■✞❖❝❴❞❊♥❜✷❊❧❋✧❍❏◗✐❋❙❊❧❜❏❊✣❑✫❍✷◗❙❊✫■✷❚➣r❙❖➋●❏◗❲❊✩❳❨❊✲■❏❵✿❭✈❋✼❍✷◗✐❋❙❊❨❴❝❫❡◗♣❑▼❫❡❥➁❭✫■✞❜✷❭✫❚✿❊ ① ❖❯■✞❤✼❴❯❍✞❊✫■✷❵❈❊▼❭❾❳❡▲❦❊✩✉✞❖❛❋✼❚❈❊✲■✷❵❈❭ r✐❖✥❍✷■✜❖❯❑❨❖❻❚❛❭✲❚❈❊▼❭♥❋❲❫✷❴❘❍❏❵✸❖❞❊✩❖❁❍✷■✷❍❆❖✴❑P❖❘◗❲❑✫❍✞❖❯❚↔◗✼❊✲❳➊❖❈❋✸❚❈❖❘❬♦■❆❊▼❴❞❖❻■✞❖❘❭✩◗➣➜❞■➁❥✈❫✭❜◆❋✐❖❘❥❧❖❞❴❝❭✫◗❱❑✫❍➝❭✕❍✷■✷❍✞❖ ❑P❖❯◗❙❑✲❍❆❖❻❚❅❴❝❖❯■✞❖❞❭✲◗✐s ➠ ■ ●✞❴❯❍❂❋✱❖❯■✷❚✸❊✫◗✼❬✞❖❯■ ❑▼❫❡■❆❜✷❖❻❵❈❖❛❖♥❴❘❊❡②❏❭❾❚❈❊✡❜✷❊✡❤✼❫❡◗✼❥❧❭ ❑✲❭✲◗❙❭P❑❾❚❈❊✫◗❙❖❈❋✸❚❈❖❝❑P❖❝❖✣◗❙❊❾❳❡❖✿❋✼❚✸❫❡◗❩❍✞❴❘❍❆❖❈s ❭P❚❛▲✡✢❑ ✾✜ ❚✸❊✫❬✞❭ ❽ ❊✲✉❆❊✩❥♥●✞❴❝❊➊s ✇⑥❖❻◗❙❑✫❍✞❖❻❚❞❍✞❴↔❜✷❖q■➝❤✼❖❞②✜s↔➙▼s ❭✷s✭➜❝■➝❑P❭✲◗❲❊♥❜✷❖❞❫✷❜✷❃ ❭ ➋❂ ❊P■❆❊✲◗⑥❭✫◗❙❊❦❑▼❭✫◗❙❭✲❑✲❚❈❊✩◗✼❖✿❋✧❚✿❖❘❑▼❭♥❜✷❖q■➝❤✼❖❞②✜s↔➙Ps ➎✌s✜■❏❍➝❭✫◗❙❊❧❋✐❫❏❴❘❍✷❵❛❖❛❊❦❭❨r④❭ ❑✫❍✷❥ ◗❙❊✲❳✩❍✞❴❘❚❛▲❱❜✷❖❘■✣❤✼❖❞②✜s✌➙▼s➇❑❡s➊❿❩❑▼❭✫◗❙❭✩❑✫❚✿❊✫◗❙❖✿❋✼❚✿❖❞❑✩❭♠❜✷❖❘❫✷❜✷❊❨✙❖ ❂↔❊✫■✞❊✲◗✵r✐❖✜❑P❊P❭➂❭♠❋✧❍✷◗❙❋✐❊✲❖❂❜✷❊①❚✿❊✫■❂❋✐❖❻❍✷■❆❊⑥■❏❍✈❋❙❊❱❖❘■✷❚❈❊✫◗✐❋✐❊▼❑✾❚✸❊P❭✫❳➊▲ ❸ s ⑧✕❭❨❑P▲♠❋✧❊❱❖❻■✭❚❝◗❙❫❏❜❡❍✞❑P❊⑥◗✧❊✩❳❨❖❛❋✧❚❈❫❡◗✸❍❂❴✞❜✷❊♥➙ ✔ ❿✿❤✼❖❞②✜s✌➙Ps ❜✜s ❸ ❑✩❖❯◗❲❑P❍❆❖❻❚❞❍✭❴✻❭✫◗❲❊❱❫✣❋✐❫✷❴❯❍✷❵❛❖❝❊♠❜➊❊✫❚✸❊✫◗✼❥❧❖❻■✞❭✩❚❈▲♣②❡◗❲❭❨❤✸❖❞❑①➜❝■♥❤❩❖❛②✜s✜➙▼s ❊➊s Ω ✸❤ ❖❛②✜s❛➙ ✇⑥❖❻◗❙❑✫❍✞❖❻❚❞❍✞❴✥❜✷❖❘■➁❤✼❖❞②✜s❅❄✻s ❭➊s❏➜❞■❅❑✲❭✲◗✧❊❣❜✷❖❞❫✷❜✷❭➍❜✷❊✕❑✫❍✷◗❙❊✫■✷❚➋❑❨❫❡■❂❋✼❚✿❭✫■✷❚✴❭✲◗❙❊⑩❑✩❭✫◗❙❭▼❑✾❚✸❊✩◗❲❖✿❋✼❚✿❖❝❑❨❭➍❜✷❖❻■➁❤✼❖❞②✜❆s ❄✻s ➎✻s❏■❏❍❅❭✲◗❙❊ ❋✐❫❏❴❘❍✷❵❛❖❛❊➝❭❨r❙❭❢❑✫❍✷❥ ◗❲❊✲❳▼❍✞❴❻❚❛▲❢❜❏❖❘■❪❤✼❖❛②✞❇s ✻❄ s ❑➊s✽❿✸❑❨❭✫◗❲❭▼❑✫❚✿❊✫◗❙❖✿❋❲❚❛❖❛❑✲❭✱❜✷❖❝❫✭❜✷❊P❖♣❜✷❊➝❑✫❍✷◗❲❊✩■✷❚➂❑▼❫❡■✜❋✧❚✿❭✩■✷❚⑩r④❖❱❑P❊✩❭➝❭✱❋✧❍✷◗❙❋✐❊✩❖❱❜✷❊ ❑✫❍✷◗❙❊✫■✷❚↔■✷❍❢❋④❊♥❖❯■✷❚❈❊✲◗✐❋❙❊P❑❾❚❈❊▼❭✲❳▼▲ ❸ s✜⑧✕❭▼❑✲▲➁❋❙❊♥❖❻■❏❚❛◗❙❫❏❜❡❍✞❑P❊❦◗✧❊✩❳❨❖❈❋✼❚❈❫❡◗✼❍✞❴➋❜✷❊➝➙ ✔ ❿✼❤✸❖❞②✜❈s ❄✻s ❜✜s ❸ ❑❨❖q◗❙❑✲❍❆❖❘❚❞❍✞❴➋❭✲◗❙❊❦❫✖❋✐❫❏❴❻❍✷❵❈❖❝❊ Ω ❜❏❊✲❚❈❊✲◗❩❥♦❖❯■✞❭✾❚✸▲⑥②❡◗❙❭P❤✼❖❞❑①➜❝■♥❤✸❖❞②✜s✦❄✻s➇❊✭s 5 ✸❤ ❖❛②✜s ❄ ✇⑥❖❻◗❙❑✫❍✞❖❻❚❞❍✞❴❂❜✷❖❯■♥❤✸❖❘②✜s✁✜s➇❭❏sP➜❯■♥❑❨❭✫◗❲❊➂❜❏❖❛❫✷❜❏❭➣❚❞❍✷■✞❊P❴✜❭✫◗❙❊⑥❑▼❭✫◗❙❭✲❑P❚❛❊✩◗✼❖✿❋✧❚✿❖❞❑✩❭❱❜✷❖❯■♥❤❩❖❝②✜s✂✞s ➎✌s➊❭✩◗❩❊♣❚❝◗❲❊P❖✌❋✐❫✷❴❻❍❏❵✿❖❝❖ ❤✸❖❛②✜✄s ❜❏❊✲❚❈❊✲◗❩❥♦❖❯■✞❭✾❚✸❊⑥②❡◗❙❭P❤✼❖❞❑①➜❝■♥❤✸❖❞②✜s☎✜s ❑✷s ⑧✕❖❯■♥❭P❑▼❊❨❋✼❚✿❊❱❊P✉❆❊✲❥♥●✭❴❞❊♠❋✧❊①◗❙❊✾❥♦❭✫◗❙❑✲▲♣❚❝◗❲❊P❖✜❭➊❋✸●✞❊▼❑✫❚❈❊✷❶ P❊ ✉❆❖✿❋✼❚✸❊✫■➊❵✸❭✡❍✷■✞❊✲❖✜➎✷❍✞❑P❴❛❊❱❤✼❫❡◗❩❥♦❭✫❚❈❊♣■✷❍✷❥➁❭✩❖✌❜➊❖❯■✣◗❲❊P❳❨❖❈❋✸❚❈❫✷❭✲◗✧❊➂❑P❫❡■✷❚❞◗❙❫✷❴❞❭✫❚✿❊❱➜❝■♥❑✫❍✷◗❙❊✫■✷❚✥❑✩❭✫◗❙❊①■✷❍➁❋❲❍❏■✷❚✥❑▼❫❨■✷❚❛◗✧❫✷❴❛❭✫❚❈❊✕r④❖ ➜❝■✈❚✸❊✫■❂❋✐❖❯❍✷■✞❊➍❿✸❑▼❭♠➜❝■❅❤✼❖❞②✜s✿➙▼s➇❭❡s ❸ ●✞❫✷❭✩❚❈❊❣❑P❫❡■✞❜❡❍❆❑▼❊➍❴❛❭✕❖❯■✞❊✲✉❆❖✿❋❲❚❛❊✲■❏❵✸❭ ❋✐❫➊❴❯❍✷❵❛❖❝❊❨❖❈➛✂❋④❫❏❴❻❍✷❵❈❖❝❭➍❊✩✉✷❖✿❋✧❚✸▲➍❜✷❭✩❑P▲♠➜❝■❅❭▼❑P❊❨❋❲❚✥❚✿❖q● • ❜✷❊♣➎❏❍✞❑▼❴❞▲♠❋✧❊❱❖❯■✷❚❝◗✧❫✷❜❡❍✞❑P❊♣❍✷■❣◗❲❊✲❳❨❖❈❋✼❚❈❫❡◗❁●✞❊✫■✷❚❞◗✿❍♥❑❨❭✲◗❲❊♣❍✷⑨ ❊P✉❆❖✿❋✼❚✸❊✫■➊❵✸❭ • ❖❻⑨ ±∞⇔ ±∞ ❿✸❤✼❖❞②❄➙▼s ❜✞s ❸ ➞ ❍✷■❆❊▼❖◆❋❙❊❨❑✫❵✿❖❯❍✷■✞❖✈❤✼❫❡◗❩❥♦❭✫❚❈❊✘■✷❍✷❥➁❭P❖♦❜✷❖❯■ ◗❙❊✲❳➊❖❛❋✼❚❈❫✷❭✲◗✧❊ ❑▼❫❨■✷❚❛◗✧❫✷❴❝❭✫❚✸❊✘➜❞■ ❚✸❊✩■✞❋❙❖❯❍✷■✞❊ ❑P❭✲◗✧❊✘■✷❍ ❋✧❍✷■✷❚ ❑❨❫❡■✷❚❞◗❲❫✷❴❞❭✲❚❈❊❱r✐❖❆➜❝■♥❑✫❍✷◗❲❊✩■✷❚✂❿✸❑▼▲➣➜❝■♥❤✸❖❞②✜s ❄✻s ❭➊s ❸ ●✭❫✷❭✲❚❈❊⑥❑❡❫❨■✞❜❡❍✞❑P❊❱❴❛❭⑥❖❘■❆❊✩✉❆❖✿❋✧❚❞❊✩■✷❵❈❭✡❋④❫❏❴❘❍✷❵❛❖❛❊P❖❈➛✞❋❲❫✷❴❯❍✷❵✸❖❞❭⑥❊✩✉❆❖✿❋✧❚✿▲❱❜✷❭P❑✩▲⑥➜❯■ ❭❨❑P❊❡❋✼❚❆❚❈❖❯●♥❜✷❊❱❋✐❊▼❑✾❵✸❖❯❍✷■✞❊♠❋❙❊⑥❖❘■✷❚❞◗❲❫✷❜❡❍❆❑❡❊①❍✷■➍◗❙❊✲❳❡❖✿❋❩❚❈❫❡◗✥●✞❊✲■❏❚❛◗✼❍✣❑▼❭✫◗❲❊①❍✭⑨ ❖❯⑨ ±∞⇔ ±∞ ❿❩❤✸❖❛②✜s ❄❂s ❜✜s ❸ ➞ ◗❙❊✲❳➊❖❛❋✼❚❈❫✷❭✲◗✧❊▼❴❘❊♣❑✫❍♥❑P❭✩◗✼❭❨❑✫❚✿❊✫◗❙❖❛❋✼❚❈❖❝❑P❖❆■✞❊❾❥➁❫❡■✞❫❨❚✸❫❡■❆❊➂❤❩❭✲❬✞❫❨◗❙❖❝❳❨❊P❭✫❳❨▲➂❭✫●✞❭✫◗❙❖❻❵❛❖❛❭♠❋❙❫✷❴❯❍✷❵✿❖❞❖❞❴❝❫❡◗✥❥❧❍✭❴q❚✸❖❯●✞❴❞❊♣❿❩❤✸❖❛②✜✄s ✜s ❭❡s ❸ s ➠ ■✞❭P❖❯■✷❚✸❊♠❜❏❊♠❭♠❤✼❫❡◗✼❥❦❍✞❴❞❭♠❑❡❫❨■✞❜✷❖❯❵✿❖❝❖❘❴❛❊♠❜❏❊♠❊▼✉✭❖❛❋✼❚✸❊✫■✷❵❈▲ r❲❖✜❍✷■✞❖❞❑▼❖q❚✸❭✾❚✸❊♠❭✕❋✐❫❏❴❘❍➊❵❈❖❛❊✲❖✜❬✞❫❡❥↕❜❏❊▼❤✸❖❯■✞❖✌❭✫■✷❍✷❥➁❖❘❚❛❊♣❚✸❖❯●✷❍✷◗✧❖ ❜❏❊⑥◗✧❊✩❳❨❖❛❋✧❚❈❫✷❭✫◗❙❊♣❖❯❥♥●✞❫❡◗✸❚❈❭✲■❏❚✿❊⑥➜❘■✣❭❡❑✲❊❡❋✧❚✜❑❨❫❡■✷❚✿❊✲✉❏❚❲s • ✕ ❭P❜✷❖❘❑▼▲♣●✞❊✫■✷❚❞◗✼❍❧❫❡◗❙❖❝❑❨❭✫◗❲❊♠❜✷❫❡❍✞▲♣●✭❍✷■✞❑✫❚✸❊♠❿✿❍✎✗❞➞ ✆ ✝✞✝✦✏ ✆✞✠✚ ✟✁✟☛✡✌☞✎✞✍ ✟ ✝✠☛✆ ✍✑✏✙✚✒✟✁✟♣❭✫◗❙❊♠❫❦❑▼❭✫◗❙❭P❑✲❚❛❊✲◗❙❖✿❋✼❚✿❖❘❑▼▲✔❨✓ ❑✫◗❲❊❏❋❲❑P▲✲❚❛❫✷❭✩◗❲❊✖♠ ❖✘✗ ❸ ➞♠❿❛❍✚✙✸➞♠❖✛✙ ❸ ❜✷❊ ●❆❊❪❑P❭✩◗❲❭❡❑✫❚✿❊✫◗❙❖❞❋✧❚✿❖❞❑✩▲✆❭✲❬✞❊✫❥ ❍ ❖✁ s ❍ ∆ •∆ ❖➍⑨❁❿❛❍✚✙♥❺❩❍✎✗ ❸ ❿✼❖✛✙❧❺❩❖✜✗ ❸ ≥⑦ s ✆ ✝☛✝✎✏ ✆ ✚✠✟ ✟✢✡✌★ ☞ ✆ ✚✒✟ ✏✠✍✞✚✣✍✦✟☛✝✞✑✆ ✍✑✏✙✚✒✟✁✟✱❭✲◗❙❊ ∆ •∆ ⑦ ✆ ✝✞✝✦✏ ✆✞✠✚ ✁✟ ✟☛✡✌☞ ✗✙✝ ✚✕✏ ✤✦✥ ✼❿ ❜✷❊⑥❴❛❭✧✓✫❍✷■✷➎✞❫❡❍❏■❂❜✷❊❨❜✌✕✩❣ ★ ■❆❊✲❥➁▲✫◗❙②✷❖❯■✞❖❻❚ ❸ ❭✫◗❙❊♣❫➍❑✩❭✩◗❲❭❨❑❾❚❈❊✲◗✧❖❛❋✼❚✸❖❘❑▼▲⑥❑❾❍➍●✷◗❙❫❡●❏◗❙❖❛❊✫❚❈▲✲❵❈❖❝❴❞❊➊❶ ❺✼❜✷❭P❑P▲➂❊❨❋✼❚✸❊⑥❑P❫❡■✷❚❞◗❙❫✷❴❞❭❾❚✜➜❞■✣❑✲❍✷◗✧❊✲■✷❚✻❭✫❚❛❍✷■❆❑▼❖ ❺✼❜✷❭P❑P▲➂❊❨❋✼❚✸❊⑥❑P❫❡■✷❚❞◗❙❫✷❴❞❭❾❚✜➜❞■➍❚✸❊✫■❂❋❙❖❻❍✷■❆❊♠❭❾❚❞❍✷■✞❑P❖ ✆ ✝✞✝✦✏ ✆✞✠✚ ✟✁✟☛✡✌☞ ✗✙✝ ✚✕✏ ✤✫✪ lim u( i) = ±∞ i→ ± ∞ lim i( u) = ±∞ u→ ±∞ ❿✸❜✷❊✱❴❛✬ ❭ ✓✓✖✞❭❨❴❛❤✸❺✿❍✷■❆➎✞❫❨❍✷■❂❜✷❊P✌❜ ✱ ✕ ❺❦❋④❊✫❥➁❖❛❺✿■✞❊✲❥➁▲✫◗❙②✷❖❯■✞❖❻❚ ❸ ❭✫◗❙❊✱❫ ❑▼❭✫◗❙❭P❑✲❚❛❊P◗❩❖✿❋✧❚✿❖❘❑❡▲➝❑✫❍ ●❏◗❙❫❡●✷◗❙❖❘❊✩❚✸❭✫❚✸❊✲❭①◗❙❊✫❳❨❖❈❋✼❚❛❫❡◗✼❍✞❴❯❍✞❖❂❜✷❊☎❚✸❖❯● ■❏❍✷❥✈❭P❖❆●✞❊✫■✷❚❛◗❩❍❣❍✷■✞❭❱❜✷❖❯■➊❚❞◗❲❊➂❊✲✉❏❚❞◗❲❊✩❥✈❖q❚✸▲✫❵✿❖✞❿✧❋✼●✷◗❙❊ ❋❙❭✫❍✈❋✼●✷◗❙❊❱❺ ❸ s ∞ ➉ ➆ ∞ ✭➍❫❡❥ ❊✩■✷❍✷■✷❵❛❭ ➜❝■➤❑▼❫❨■✷❚✸❖❯■✷❍✞❭✫◗❙❊❨➞❱❤✼❭✫◗❲▲ ❭✒❴❘❊❪❜✷❊✫❥➁❫❡■❂❋✼❚❞◗❙❭❡➞❱❫❪❚❛❊▼❫❨◗❙❊✲❥➁▲ ❜❆❊✒❊✲✉❆❖✿❋❲❚❛❊✩■✷❵❞▲☛➞q❫❪❚✸❊P❫❡◗❙❊✫❥❧▲ ❜✷❊ ❊✲✉❆❖✿❋✧❚✸❊✫■➊❵❈▲⑩r❲❖❆❍✷■✞❖❞❑✩❖❯❚✿❭✫❚✿❊♠r❙❖❂❫⑩❚✿❊P❫❡◗❙❊✫❥➁▲♣❜❏❊⑥❍➊■❆❖❛❑P❖❘❚❛❭✲❚❛❊❏s 6 ✝ ✁ ✟ ✟ ✝✂✁ ☞☎✄✝✆✕✗❆✝ ✝✟✞✾✏ ✆✞✚ ✝✡✠✎✍✔☞☞☛ ✬❄❖❞❊♣❍✷■❧❑✩❖❯◗❙❑✫❍✞❖❻❚✥❤✸❫❡◗❩❥♦❭✫❚✂❜✷❖❯■♦❋✼❍✷◗✐❋❙❊♠❖q■✞❜✷❊✩●✞❊❾■❆❜✷❊✲■❏❚✸❊✕r✐❖✷◗❙❊✲❳❡❖❞❋✧❚✸❫✷❭✫◗❲❊♠❜✷❖❯●✞❫✷❴❞❭✲◗❩❊➂●❆❭❡❋❲❖❘❬❆❊♠❑❡❫▼■❏❚❛◗❲❫✷❴❞❭✫❚✸❊⑥➜❝■❣❚❈❊✲■❂❋✐❖❯❍✷■✞❊♠❋❙❭✫❍ ➜❝■✎❑✫❍✷◗❙❊✫■✷❚↔❭❨❋✼❚✸❤❩❊✩❴❄➜❝■✞❑☎✜❾❚ ❬✭❭P◗❲❖❞❭✫➎✞❖❝❴❞❭❣❜✷❊➍❑▼❫❨■✷❚❛◗✧❫✷❴✥●❆❫✷❭✲❚❈❊❣❴❯❍✞❭❦❫❡◗❲❖❝❑P❊⑩❬❆❭❡❴❘❫✷❭✩◗❲❊✕➜❝■✷❚❞◗❲❊✣❺ ∞ r✐❖ ➆ ∞ s➊➏❦❑P❊➊❋❩❚✴❑▼❖❯◗❙❑✫❍✞❖❻❚➋❭✩◗❲❊ ❍❏■❣■✷❍✷❥➁▲✫◗↔❖❯❥♥●✞❭✲◗✴❜✷❊♠❋❙❫✷❴❘❍❏❵✿❖❞❖✜❜✷❭P❑P▲⑩❋✼❍✷■❏❚❁❋❲❭✩❚✿❖✿❋✐❤✿▲❡❑✫❍✷❚❛❊⑥❍❏◗✼❥✈❭✩❚❈❫✷❭✫◗❲❊▼❴❘❊➂❑P❫❡■✞❜❏❖❘❵❛❖❛❖✂❶ ✏ ❸ ❫❡◗❙❖❝❑❨❊✱➎✷❍✞❑P❴❛▲ ❤✼❫❡◗✼❥➁❭✫❚✸▲ ❜✷❖❘■✡◗✧❊✩❳❨❖❛❋✧❚❛❫✷❭✩◗✼❊✆❑P❫❡■✷❚❞◗❲❫✷❴❛❭✩❚❛❊✒➜❝■➤❑✲❍✷◗✧❊✲■✷❚✣❑▼❭✫◗❙❊✱■✷❍ ❋✼❍✷■✷❚❧❑▼❫❨■✷❚❛◗✧❫✷❴❛❭✫❚❈❊❪r❙❖✕➜❯■ ❚❈❊✫■❂❋✐❖❘❍❏■✞❊♦❑P❫❏■✷❵❛❖❘■✞❊❅❑✩❊P❴❇●✷❍❏❵✸❖❯■ ❍❏■❢◗✧❊✩❳❨❖❈❋✧❚❛❫❡◗➍❜✷❊❧❚❈❖❯● ❋✐❭✫❍✓❜❆❫❡❍❆▲➁◗❙❊✲❳❡❖✿❋✼❚✸❫❏❭✲◗❙❊❅❜✷❊➁❚❛❖❻✍ ● ✌ ❫❡◗❙❖❞❊✫■✷❚✸❭✫❚✿❊❅❜✷❖❞❤✸❊P◗❲❖❯❚➣➜❝■ ➉ ◗✧❭✲●✞❫❨◗✼❚✂❑✲❍❧❋④❊✫■✌❋✼❍✞❴❂❜✷❊☎●✞❭✫◗❙❑✫❍✷◗❙②✷❊✫◗❙❊❱❭✩❴❆➎✷❍✞❑✲❴❝❊P❖❈➞ ✏✕✏ ❸ ❫❡◗✧❖❛❑✲❊❧❋✐❊✲❑✲❵❛❖❘❍✷■❆❊✣❤✸❫❡◗❩❥♦❭✫❚❈▲❣❜✷❖❯■♦◗❲❊P❳▼❖✿❋✼❚✸❫✷❭✾◗❙❊❦❑▼❫❡■❏❚❛◗❙❫❏❴❛❭✫❚✿❊⑩➜❞■✈❚❈❊✲■✜❋④❖❯❍✷■✞❊❦❑P❭✲◗❲❊✕■✷❍➝❋✼❍✷■✷❚✵❑P❫❡■✷❚❞◗❲❫✷❴❛❭✩❚❞❊♥r✐❖✌➜❝■ ❑✫❍✷◗❙❊✫■✷❚➋❑P❫❡■✷❵❈❖❯■✭❊✕❑✩❊❨❴❆●✷❍✷❵❈❖❘■ ❍✷■♥◗❲❊✩❳❨❖✿❋❩❚❈❫❡◗❇❜✷❊➂❚❛❖❘● ❋❙❭✫❍➁❜✷❫❡❍✞▲➂◗✧❊✩❳❨❖❛❋✧❚❛❫✷❭✩◗❲❊♠❜✷❊➂❚✿❖❯✎ ● ✌ ❫▼◗✧❖❛❊✫■✷❚❈❭❾❚❈❊✕❜➊❖❛❤❩❊✲◗❙❖ ❚✥➜❞■♥◗❲❭✫●✞❫❡◗✼❚ ➉ ❑✫❍✈❋✐❊✫■❂❋✧❍✞❴✞❜✷❊①●✞❭✫◗❙❑✫❍✷◗❲②✷❊✩◗❲❊♠❭✩❴✌❋❲❊▼❑✫❵✿❖❯❍✷■✞❖❞❖❈s ✂☎✄✁✆✞✝✠✟☛✡✌☞✎✍✑✏✒✏✯✓ ✏ ❸ ❭✫❬ ✜✫■✞❜❣❍✷■♥■✷❍✷❥➁❭✫◗✽❖❯❥♥●✞❭✲◗❇❜✷❊➍❋④❫❏❴❘❍✷❵❈❖❝❖✂❑✩❖❻◗✧❑✲❍✞❖❯❚❞❍✞❴✌❭✲◗❲❊⑩❑▼❊P❴✻●✷❍✷❵❛❖❻■➁❫✈❋✐❫✷❴❯❍✷❵✿❖❞❊❏➛✞❜❏❊♠❊▼✉✭❊❾❥♥●✞❴❯❍♦❜❏❖❝❫✷❜✷❭❱❚❛❍✷■❆❊❡❴ ❜❏❖❘■✣❑▼❖❯◗❙❑✫❍✞❖❻❚❞❍✞❴❂❜✷❖❯■♥❤✿❖❛②❨❍✷◗❙❭ ❣❊❨❋✼❚✸❊①❍✷■➍◗❲❊✩❳❨❖❈❋✸❚✿❫❨◗❁●✞❭❨❋✐❖❻❬✣❑❡❫❨■✷❚❛◗❲❫❏❴❛❭✫❚❂➜❝■✕❚❈❊✲■✜❋✐❖❻❍✷■✜❊♣❖❞❭✲◗❁❑▼❖❯◗❩❑P❍❆❖❻❚❞❍✞❴✜❭✫◗❙❊☎❚❛◗✧❊✩❖✌❋❙❫✷❴❘❍❏❵✿❖❞❖❩s ✏✕✏ ❸ ◗❙❊❨❋✼❚❛◗✧❖❝❑✫❵✿❖❞❭◆❑✩❍✱●✷◗❙❖❯❬✞❖❯◗❲❊✖❴❞❭➁➎✷❍❆❑❡❴❻❭✖❤✼❫❨◗✼❥✈❭✩❚❛▲❅■✷❍❏❥♦❭P❖⑥❜✷❖❻■☛❋✼❍✷◗④❋④❊✎❜✷❊♦❚✸❊✫■❂❋❙❖❯❍✷■✞❊✎❜✷❖❘■✱❚❈❊P❫❡◗❙❊✫❥♦❭♦●✞❊✫■✷❚❞◗✸❍ P❑ ❖❯◗❙❑✲❍❆❖❻❚❈❊✩❴❞❊ ❴❝❖❯■✞❖❞❭✲◗❲❊ ❊❨❋✼❚✿❊✡❍✷■ ❑❨❭✲❳ ●✞❭✫◗✸❚✸❖❘❑❾❍✞❴❞❭✫◗✒❭P❴✈❑P❫❡■✞❜✷❖❯❵✸❖❘❊▼❖ ✏➁❿✧❋✼❍✷◗✐❋✐❊P❴❝❊ ❜✷❊☛❚✿❊✫■❂❋❙❖❻❍✷■✞❊☛●✞❫❏❚❅❤✼❖➁❑▼❫❡■✜❋④❖❞❜✷❊✩◗❲❭✫❚✸❊ ◗✧❊✩❳❨❖❈❋✼❚❈❫✷❭✫◗❲❊♣■✞❊✩❴❞❖❘■❆❖❝❭✫◗❲❊➂❑P❫❡■✷❚❞◗❲❫✷❴❞❭✩❚❞❊⑥➜❞■✣❑✲❍✷◗❲❊✫■✷❚✂❑P❭✲◗❙❊☎■✷❍❧❋❲❍❏■✷❚✂❑▼❫❡■❏❚❛◗❙❫❏❴❛❭✾❚✸❊❱r✐❖❆➜❯■❣❚❛❊P■✜❋④❖❯❍✷■✞❊ ❸ ✏✕✏✖✏ ❸ ◗❙❊❡❋✼❚❞◗❲❖❝❑✫❵✸❖❘❭ ❑✫❍✈●✷◗❙❖❻❬❆❖❘◗❙❊➍❴❝❭❦❋④❊❨❑✾❵✸❖❯❍✷■✞❊P❭❣❤✸❫❡◗✸❥➁❭✫❚✸▲♠■✷❍✷❥➁❭P❖❁❜✷❖❯■✎❋✼❍✷◗✐❋❙❊❣❜✷❊➍❑✲❍❏◗❙❊✲■❏❚↔❜✷❖❯■♥❚✸❊P❫❡◗❙❊✫❥➁❭⑩●✞❊✫■✷❚❞◗✸❍ ❑P❖❯◗❙❑✲❍❆❖❻❚❈❊✩❴❞❊➂❴❞❖❯■✞❖❝❭✫◗❙❊♠❊❨❋❲❚❛❊➂❍✷■❧❑✩❭✲❳➂●✞❭✫◗✼❚✿❖❞❑✲❍❆❴❝❭✫◗↔❭P❴✌❑▼❫❡■❆❜✷❖❘❵❛❖❛❊✲✫❖ ✏ ✏❂❿✧❋✧❍✷◗❙❋✐❊P❴❝❊♠❜❏❊♠❑✲❍✷◗✧❊✲■✷❚✌●❆❫❡❚✥❤✿❖✌❑▼❫❡■✻❋❙❖❛❜✷❊✫◗❙❭✾❚✸❊♣◗❙❊❾❳❡❖✿❋✸❚✸❫✷❭✩◗❲❊ ■❆❊▼❴❞❖❘■❆❖❝❭✫◗❙❊♣❑P❫❡■✷❚❞◗❙❫✷❴❞❭✫❚✿❊⑥➜❞■⑩❚✸❊✫■❂❋✐❖❻❍❏■❂❊⑥❑▼❭✫◗❲❊①■➊❍✈❋✧❍✷■✷❚✻❑❨❫❡■✷❚❞◗❲❫✷❴❞❭✲❚❛❊⑩r❙❖✞➜❝■❦❑✲❍❏◗❙❊✲■❏❚ ❸ s ✇⑥❫❡■✞❜❏❖❻❵❈❖❛❖❘❴❛❊♥❜✷❊♥❊✲✉❆❖❈❋✼❚❛❊✲■✷❵❈▲ r❲❖✴❍✷■✞❖❞❑✩❖❯❚✿❭✫❚✸❊♥❭♥❋✐❫✷❴❯❍✷❵✸❖❘❊▼❖❄❍✷■✷❍❆❖✽❑▼❖❯◗❲❑✲❍❆❖❘❚➋■✞❊P❴❝❖❯■✞❖❛❭✫◗❱❑✫❍✎◗❲❊✩❳❨❖❈❋✸❚❈❫✷❭✲◗✧❊❧❑✫◗❙❊❨❋❙❑✩▲✩❚✸❫❏❭✲◗❲❊❣●✞❫❡❚➣❤✸❖ ❤❩❫❡◗✼❥❦❍✞❴❛❭✩❚❛❊➂❑✫❍❣◗✧❊▼❤✸❊✩◗❲❖❯◗❙❊♣❴❘❭❱❊✩✉✷❖✿❋✧❚✿❊✫■✷❵❈❭ r❙❖✷❍✷■✞❖❞❑▼❖❯❚✿❭✫❚✿❊P❭①❍✷■✞❫❡◗❁❑❡❖q◗❙❑✫❍✞❖❘❚❛❊♣❴❞❖❯■✞❖❝❭✫◗❙❊➊s ✝ ✁ ✟ ✟ ✝✂✁ ☞✑✏✒✆✕✗❆✝ ✝✟✞✾✏ ✆✞✚ ✝✡✠✎✍ ✔ ✏ ✓ ✏ ✕ ✡ ✠✙✏✒✠ ✍ ✏✕✚ ☞❆✚ ✝✖☛✕✗ ✬❄❖❞❊✕❍✷■❅❑▼❖❯◗❲❑✩❍✞❖❻✙ ❚ ✘ ✼❤ ❫❡◗✸❥➁❭✲❚✵❜➊❖❯■➝❋✼❍✷◗✐❋❙❊❣❖❯■✞❜✷❊✫●✞❊✩■✞❜✷❊✫■✷❚❈❊❧r❲❖✥◗❲❊✲❳❨❖❈❋✸❚✸❫❏❭✲◗❙❊➍❑✲◗✧❊❡❋❙❑P▲❾❚❈❫✷❭✲◗✧❊❣❑P❫▼■✷❚❛◗✧❫✷❴❛❭✫❚❈❊✕➜❝■❧❚❈❊✫■❂❋❙❖❘❍✷■❆❊❧❋❙❭✫❍ r✐❖ s❄✇➣❫❡■❂❋✐❖❘❜✷❊✩◗❲▲P❥ ➜❝■❢❑✲❍✷◗✧❊✲■✷❚❇❭❨❋✧❚✸❤✸❊P❴✥➜❞■✞❑☎✜✫❚ ❬✞❭✫◗❙❖❝❭✩➎✞❖❘❴❝❭✈❜✷❊♥❑P❫❡■✷❚❞◗❙❫✷❴❁●❂❫✷❭✫❚❈❊❧❴❯❍✞❭✈❫❡◗✧❖❝❑P❊✣❬✞❭▼❴❞❫➊❭✫◗❙❊♥➜❘■✷❚❞◗❲❊ −∞ +∞ ❑P❖❯◗❙❑✲❍❆❖❻❚❞❍✞❴✚✘✜✛✣❫❡➎✷❵❈❖❻■❏❍✷❚♥●✷◗❙❖❯■ ➜❝■✞❴❛❫❏❑✲❍✞❖❯◗❲❊❨❭ ◗✧❊✩❳❨❖❈❋✸❚❈❫✷❭✲◗✧❊▼❴❞❫❡◗◆■❆❊▼❴❘❖❘■✞❖❞❭✫◗❙❊❪❑✫❍✟◗✧❊✩❳❨❖❈❋✼❚❛❫✷❭✩◗❲❊✘❴❞❖❻■❆❖❛❭✫◗❲❊☛❑P❍➤◗❲❊P❳▼❖✿❋❲❚❈❊✫■➊❵✸❊P❴❝❊ Rk > 0 s❨⑧✕❭P❑P▲✢✘✜✞✛ ❭✲◗✧❊♣❫❦❋④❫❏❴❘❍✷❵❈❖❝❊♠r❙❖✷■✷❍✷❥➁❭❨❖✭❍✷■❆❭♣●✭❊✲■✷❚❞◗✸❍♥❫❨◗❙❖❛❑P❊①❬✞❭P❴❛❫❡◗❲❖ Rk r❙❖✞➜❞■✣✘✜✛❆❊✩✉✭❖❈❋✼❚✿▲❱❫♠●✞❊✩◗✼❊❨❑✓✖✞❊⑥❭✲◗✼➎❆❫❡◗❙❊P❺ ❑P❫✷❭✩◗✸➎✞❫❡◗✧❊❣❭❨❋✧❚✸❤✸❊✲❴❄➜❝■✞❑✟✜✫❚✒◗❲❊✲❳❡❖✿❋❩❚❈❫✷❭✫◗❙❊✩❴❞❫❡◗➣❑P❫❡■✷❚❞◗❙❫✷❴❘❭P❚❛❊✕➜❯■❅❑✲❍✭◗❲❊✲■✷❚✴❜✷❖❯■✤✘ ❴❞❊✕❑P❫❡◗❙❊❨❋❲●❏❍✷■✞❜❧◗❙❊✲❳❨❖❈❋✸❚✸❫❏❭✲◗❙❊➍❜✷❖❯■❅❭✲◗✼➎❆❫❡◗❙❊♠➜❝■ ✘✜✛❂r❙❖✞◗✧❊✩❳❨❖❈❋✼❚❈❫✷❭✫◗❲❊❡❴❞❫❡◗➋❑P❫❡■✷❚❞◗❙❫✷❴❞❭❾❚❈❊⑥➜❞■➍❚✿❊✫■❂❋✐❖❻❍❏■✞❊❱❜✷❖❘■✣✘ ❴❞❊➂❑P❫❡◗❙❊❨❋✼●✷❍✷■✞❜♠◗❙❊✲❳❡❖✿❋✧❚✸❫❏❭✲◗❲❊♠❜✷❖❯■♥❑✲❫✷❭✫◗✼➎✞❫❡◗❲❊❱➜❯✚ ■ ✘✥✛❝➞❏❭✲❚❞❍✷■✞❑❨❖✕✘ ❭✫◗❙❊❱❫✣❋✐❫❏❴❘❍✷❵❛❖❝❊♠r❙❖❆■✷❍✷❥➁❭✩❖❆❍✷■✞❭➊s ✂☎✄✁✆✞✝✠✟☛✡✌☞✎✍✑✏✒✏ ✏ ☛❪❑P❫❡■✞❜❏❖❘❵❈❖❛❖❘❴❛❊✓❜✷❊✒❊P✉❆❖❛❋✼❚❈❊✲■❏❵✿▲ r❙❖➂❍✷■✞❖❞❑P❖❘❚❛❭✲❚❛❊✒❭❪❋❙❫✷❴❯❍✷❵❈❖❛❊✲❖➂●✷◗❙❖❯❬✞❖❯❚✸❫✷❭✫◗❲❊✒❴❞❭ ❋✧❍✷◗❙❋✐❊P❴❝❊✒❖❯■✞❜✷❊✫●✞❊✩■✞❜✷❊✫■✷❚❈❊ ❋✧❍✷■✷❚ ❭P❑❨❊✩❴❞❊▼❭Pr✐❖♥❑✩❭❱❴❝❭①❍✷■✣❑✩❖❯◗❙❑✲❍❆❖❘❚✻❴❝❖❯■✞❖❞❭✲◗❙➛ ✏✕✏ ☛❦❋✼●✷◗❙❊❱❜✷❊P❫❆❋✐❊✲➎❆❖❘◗❲❊❱❜✷❊➂❚❛❊▼❫❨◗❲❊P❥❧❭♣●✷◗❲❊❨❑P❊▼❜✷❊✾■✷❚❈▲❡➞❏❑❡❫❨■✞❜✷❖❻❵❈❖❝❖❞❴❞❊⑩❋✧❍❏■✷❚✥❋❲❚❞◗❲❖❞❑❾❚❂❚✸❫❨●✞❫✷❴❞❫✷②✷❖❞❑▼❊♠❿❈■➊❍❧❊✲✉❆❖❈❋✧❚❝▲✕❑▼❫❨■✞❜✷❖❯❵✿❖❞❖ ❴❞❊P②✷❭✲❚❛❊❪❜❏❊ ❤✼❫❡◗✸❥➁❭✆❑▼❭✫◗❙❭P❑❾❚❈❊✲◗✧❖❈❋✸❚✸❖❘❑▼❖❘❴❛❫❨◗✈◗❙❊✩❳❨❖❛❋✼❚❈❫✷❭✫◗❙❊▼❴❞❫❡◗✈■✞❊P❴❞❖❘■✭❖❞❭✲◗✧❊ ❸ ➛♣❑▼❭✱❍✷◗✼❥➁❭✲◗✧❊ ❫❡➎❂❋✐❊✫◗✸❬✞▲✩❥ ❑▼▲✱❍✷■➤❑❨❖❻◗❙❑✫❍✞❖❯❚♥❑✫❍ ◗✧❊✩❳❨❖❈❋✼❚❈❫✷❭✫◗❲❊✕❑✲◗❲❊➊❋❙❑✩▲✫❚❈❫✷❭✩◗✼❊➍❋✐❊♠❑▼❫❡❥♥●✞❫❨◗✼❚✸▲➍❋✧❖q❥♦❖❞❴❞❭✲◗✴❑P❍♥❍✷■❧❑P❖❘◗❲❑✫❍✞❖❯❚✥❑✲❍♥◗❙❊❾❳❡❖✿❋✼❚✸❫✷❭✫◗❲❊✕❴❘❖❘■✞❖❘❭✲◗❲❊⑩❜✷❖❘■♥●✷❍❏■✞❑✲❚❁❜✷❊♣❬✞❊P❜✷❊✩◗❲❊♠❭❡❴ 7 ❊✲✉❆❖✿❋✧❚✸❊✫■➊❵❈❊✩❖✌r❙❖✭❍✷■❆❖❛❑P❖❻❚❈❭❾❵❈❖❝❖♦❋❙❫➊❴❯❍✷❵✸❖❘❊▼❖✸s ✝ ✁ ✟ ✟ ✝✂✁ ☞✁✎✆✕✗❆✝ ✡ ✠✙✠✏ ✦ ✍ ✏ ✚✒☞✙✚✕✝ ☛ ✬❄❖❞❊✕❍✷■✎❑P❖❘◗❩❑✩❍✞❖❻❚✴❤✼❫❡◗✸❥➁❭✩❚↔❜✷❖❯■♦◗❲❊✲❳❡❖✿❋✸❚✸❫✷❭✫◗❙❊❦❜✷❖❻●❆❫✷❴❛❭✫◗❙❊❦❑✲◗✧❊❡❋❲❑▼▲✫❚❈❫✷❭✲◗❲❊➍❑❨❫❡■✷❚❞◗❲❫✷❴❛❭✫❚✿❊➍➜❝■✈❚✸❊✫■❂❋❙❖❻❍✷■❆❊❧❋✐❭✫❍❅➜❝■❅❑✲❍✭◗❲❊✲■✷❚✴❭❡❋✧❚❈❤✿❊❨❴ ➜❝■✞❑☎✜✲❚✈❬❆❭✲◗❙❖❞❭✫➎✭❖❛❴❞❭❱❜✷❊❱❑❡❫❨■✷❚❛◗❲❫❏❴✞●❆❫✷❭✲❚❈❊❱❴❘❍❆❭➂❫❨◗❲❖❛❑P❊①❬✞❭❨❴❝❫✷❭✫◗❲❊♠❖❻■✷❚❞◗❙❊❱❺ ∞ r❙❖ ➆ ∞ s❨⑧✕❭P❑P▲➂❊✲✉❆❖✿❋❩❚❛▲➊➞❆❋❙❫✷❴❯❍✷❵✸❖❘❭❱❭▼❑P❊❡❋✼❚❞❍✞❖❂❑✩❖❯◗❲❑✩❍✞❖❘❚ ❊❨❋✧❚❈❊➣❍✷■❆❖❛❑P▲♣❜❏❭▼❑P▲♠❋❲❍❏■✷❚✥❋④❭✫❚✿❖✿❋❙❤✼❭P❑✲❍✷❚❛❊①❍✷◗✼❥➁▲❾❚❛❫✷❭✲◗✧❊▼❴❞❊♣❑❨❫❡■❆❜✷❖❻❵❈❖❝❖✂❶ ❖ ❸ ❫❡◗❲❖❞❑❡❊✒➎✷❍✞❑✲❴❛▲✆❤✼❫❡◗❩❥♦❭✾❚✸▲✆❜✷❖❯■ ◗❲❊✲❳❡❖✿❋❩❚❈❫✷❭✫◗❲❊☛❑▼❫❡■❏❚❛◗❲❫✷❴❞❭❾❚❈❊ ➜❞■ ❑✫❍✷◗❲❊✩■✷❚✣❑▼❫❡■❏❵✸❖❯■✞❊ ❑P❊P❴⑩●✷❍✷❚❛❖❘■ ❍✷■ ◗❙❊✲❳➊❖❛❋✼❚✸❫❨◗ ❑P❫❡■✷❚❞◗❙❫✷❴❞❭✫❚❂➜❝■❣❚❛❊✲■❂❋❙❖❯❍✷■✞❊ ❖❞❖ ❸ ❫❡◗❙❖❞❑P❊✣❋✐❊✩❑✫❵✿❖❯❍✷■✞❊➍❤✼❫❡◗✸❥➁❭✫❚✸▲➍❜✷❖❘■✈◗✧❊✩❳❨❖❈❋✼❚❈❫✷❭❾◗✧❊❣❑P❫❡■✷❚❞◗❙❫✷❴❞❭✲❚❈❊✕➜❝■♥❚✿❊✩■❂❋❙❖❘❍❏■✞❊❣❑P❫❡■✷❵❈❖❘■❆❊❣❑P❊✩❴✂●❏❍✷❚✸❖❯■✈❍✷■♥◗❙❊✲❳➊❖❛❋✧❚❈❫❡◗ ❑P❫❡■✷❚❞◗❙❫✷❴❞❭✫❚❂➜❝■♥❑✫❍✷◗❙❊✫■✷❚ ✂☎✄❅✆✞✝✞✟☛✡ ☞✦✍✑✏ ✏✯✓ ✏ ❸ ◗❲❊➊❋✼❚❛◗✧❖❯❑✩❵✿❖❝❖❘❴❛❊❱❑✩❍❣●❏◗❲❖❘❬✞❖❯◗❙❊⑥❴❛❭①➎✷❍❆❑▼❴❘❭➂❤❩❫❡◗✼❥➁❭❾❚❈▲①■✷❍✷❥➁❭▼❖❂❜✷❖❯■✈❋✼❍✷◗④❋④❊♠❜❏❊➣❚❈❊✲■✜❋④❖❯❍✷■✞❊♠r✐❖✞❴❛❭♠❋❲❊▼❑✫❵❈❖❻❍✷■✞❊❨❭❱❤✼❫❡◗✸❥➁❭✾❚✸▲ ■❏❍✷❥➁❭▼❖❂❜✷❖❯■✈❋✼❍✷◗✐❋✐❊❱❜✷❊⑥❑✫❍✷◗❙❊✫■✷❚✥❋❲❍❏■✷❚✌❑❨❭❾❳P❍✷◗✧❖✭●✞❭✫◗✼❚❛❖❛❑✫❍✞❴❝❭✩◗❲❊❱❭P❴❝❊❱❑▼❫❨■✞❜✷❖❯❵✿❖❞❖❝❴❞❫❡◗❇✏ ❸ ❲r ❖❁✏✒✏ ❸ ➞ ✏✕✏ ❸ ❚❛❊▼❫❨◗❲❊P❥➁❭✒■✷❍ ❭❨❋❙❖❛②❡❍❏◗❲▲✘❊✲✉❆❖✿❋❩❚❈❊✫■✷❵✸❭ ❋✐❫✷❴❯❍✷❵✸❖❞❊✩❖✿➞♠❜➊❊✆❊✩✉✭❊✲❥♥●✞❴❯❍ ✷ ❜ ❭P❑▼▲✓➜❞■✟❑▼❖❯◗❲❑✲❍❆❖❘❚❞❍✞❴❣❜❏❖❘■ ❤❩❖❝②✜s✿➙Ps ❭✷s♠❋✧❊ ➜❝■✞❴❞❫✷❑✫❍✞❖❛❊❨r✼❚✸❊❦❜❆❖❘❫✷❜✷❭❃❂↔❊✲■✞❊✫◗❱❖❞❜➊❊P❭▼❴❘▲❧❑✫❍➝❫➁❜➊❖❛❫✷❜❏▲ ❂↔❊✫■✞❊✲◗➣◗❲❊❨❭▼❴❘▲❧❿✸❑P❭✩◗❲❊✣❊❨❋✼❚✸❊❣❍✷■➁◗❙❊✩❳❨❖❛❋✧❚❈❫▼◗⑥❑P❫❡■✷❚❞◗❙❫✷❴❞❭✲❚✴➜❘■❅❚✸❊✫■❂❋✐❖❯❍✷■✞❊ ❸ ❭✫❚❞❍✷■✞❑❨❖❆❭❨❑✩❊❨❋✧❚❂❑❨❖q◗❲❑✲❍✞❖❯❚✂❋✐❭❾❚❈❖❈❋❙❤✼❭✲❑▼❊⑥❑❨❫❡■✞❜✷❖❯❵✿❖❘❖❛❴❘❊➣❚❈❊▼❫❨◗❙❊✲❥❧❊P❖✻❜➊❭✩◗❄■❏❍♥❭✫◗❙❊♠❋✐❫➊❴❘❍✷❵❈❖❯❊➊➞ ✏✕✏✖✏ ❸ ❍✷■✈◗❙❊✲❳❡❖✿❋✼❚✿❫❡◗⑥❋✼❚❞◗❙❖❝❑✫❚↔❑✫◗❙❊▼❋❙❑P▲✲❚❈❫❡◗↔●✞❫✷❭✫❚❈❊❣❤✼❖✥❑▼❫❡■✜❋④❖❞❜✷❊✫◗❙❭✫❚➋❑▼❫❨■✷❚❛◗✧❫✷❴❛❭✫❚✥❭✩❚ ✜✫❚✥➜❝■✈❚✿❊✫■❂❋✐❖❘❍❏■✞❊✕☎❑ ✜✲❚✵r✧❖✂➜❞■❅❑✫❍✷◗❲❊✩■✷❚✼➛ ❍❏■ ❑P❖❯◗❙❑✲❍❆❖❘❚✓❤✼❫❡◗❩❥✈❭✩❚✓❜✷❖❯■ ❋✼❍✷◗④❋④❊ ❖❯■✞❜✷❊✫●✞❊✾■✞❜❆❊✫■✷❚❈❊ r✐❖✎◗✧❊✩❳❨❖❛❋✧❚❈❫✷❭✫◗❲❊ ❋❲❚❞◗❲❖❞❑✲❚✓❑✫◗❙❊❡❋✐❑✲▲✲❚❈❫✷❭✾◗❙❊ ❋❙❭❾❚❈❖❈❋✐❤✸❭P❑P❊ ❑▼❫❡■❆❜✷❖❘❵❈❖❯❖❞❴❝❊ ❚❈❊P❫❡◗❙❊✫❥♦❊✲❴❛❫❨◗✕➙✲✙➞ ❄✈r✐✎❖ ♥❜✷❭P❑▼▲✕❋✧❍✷■❏❚➋❋✐❭❾❚❈❖❈❋❙❤✼▲✲❑❾❍✷❚❈❊➂◗❙❊❨❋✼❚❛◗❲❖❞❑❾❵❈❖❝❖❞❴❝❊✕❑✫❍♥●✷◗❙❖❯❬✞❖❻◗✧❊♠❴❝❭➂➎✷❍❆❑❡❴❘❭♠❤✼❫❡◗✼❥➁❭✫❚✸▲♣■❏❍✷❥➁❭▼❖❄❜❏❖❘■❅❋✼❍✷◗✐❋❙❊♠❜✷❊ ❚❈❊✫■❂❋✐❖❘❍❏■✞❊♠r✧❖✌❋❙❊▼❑✫❵✿❖❯❍✷■✞❊P❭❱❤✼❫❡◗✼❥❧❭✫❚✸▲①■✷❍✷❥➁❭✲❖✻❜➊❖❯■✈❋✼❍✷◗④❋④❊♠❜❏❊♣❑✫❍✷◗❙❊✫■✷❚❩➞✷❜✷❊✲❑▼❖♥❭✫◗❲❊➂❫❦❋✐❫✷❴❯❍✷❵❞❖❞❊⑩r❲❖❆■✷❍✷❥➁❭P❖❦❍❏■✞❭❏s ✂☎✄✝✆✟✞✡✠☞☛✍✌✡✠✎✌✑✏✒✌✎✞✔✓✕✏✗✖✘☛✙☛✙✚✙✛✜✠✣✢✤✛✦✥✧✌✔★✟✖✩☛✪✏✫✚✪✌✬✚✍✛✭✠✮★✯✛✜✰✱✓✟✠☞☛✙✚✙✛✜✠ ✂☎✄✙✲✫✄✭✳✴☛✙✠✫✞✡✓✯☛✵✥✶✌✷✚✵☛✍★✸☛✪✏✭✠✫✌ ✬❄❖❞❊⑥❍❏■♥❑P❖❘◗❙❑✫❍✞❖q❚❄❑P❭✫◗❲❊➂❑P❫❡■✷❵❈❖❯■✞❊①◗❙❊✫❳❨❖❈❋✼❚❈❫✷❭✫◗❲❊♠❴❝❖❻■❆❖❛❭✩◗❲❊❱❜✷❖❯●✞❫✷❴❞❭✲◗❲❊❨➞❡◗❙❊❾❳❡❖✿❋✼❚✸❫❏❭✲◗❙❊❱❴❘❖❻■✞❖❞❭✩◗❲❊⑥❥♥❍✭❴❯❚✿❖❯●✞❫✷❴❞❭✲◗✧❊➂❭✫❬ ✜✫■✞❜ ❫ ◗✧❊✲●✷◗✧❊✩❳❨❊✲■❏❚✸❭✫◗❲❊❱❑❡❫❨■✷❚❝◗❙❫❏❴❝❭✫❚✸▲➣➜❞■⑩❚✸❊✫■❂❋✐❖❯❍✷■✞❊♠r✧❖❂❋✧❍✷◗❙❋✐❊❱❖❯■✞❜✷❊✫●✞❊✲■❆❜✷❊✩■✷❚✸❊❱❜✷❊⑥❑✫❍✷◗❙❊✫■✷❚✧s ✹❄❊✫■✷❚❛◗❩❍✓➜❞■✞❑P❊✫●✷❍✷❚⑩❋④❊✎❑P❫❡■❂❋✐❖❝❜❏❊P◗❲▲✎❑✩❖❯◗❙❑✫❍✞❖❻❚❞❍✞❴✵●❆❭❡❋✐❖❯❬✞❖❘❳❡❭✾❚✧s ✬❄❖❘❊ ✏✽❬✞❊▼❑✾❚✸❫❨◗✼❍✞❴①❑✫❍❆◗❲❊✫■✷❵❈❖❝❴❞❫❡◗❦r✐❖①❤❩❖❝❊ ❢ ✡ ❬✞❊P❑✲❚❈❫❡◗✼❍✞❴ ❚❈❊✫■❂❋✐❖❘❍❏■✞❖❝❴❞❫❡◗❢❴❛❭✾❚❛❍✷◗✧❖❛❴❘❫❡◗✱②❨◗❙❭▼❤✿❍✞❴❘❍❆❖❧❭❨❑P❊❡❋✸❚❞❍✞❖♥❑▼❖❯◗❙❑✫❍✞❖❘❚❩✡ s ✮➋❫✷❭✫❚✸❊✡❊P❴❞❊❾❥➁❊✲■❏❚✿❊P❴❛❊☛❭✩❍ ❫ ◗❲❊✫●❆◗❙❊✲❳❡❊✩■✷❚❛❭✲◗❲❊ ❑▼❫❡■❏❚❛◗❲❫✷❴❞❭✫❚✿▲✡➜❯■ ➃ ✺❆❊❨❋✼❚✸❊✕❥➁❭✲❚❞◗❲❖❛❑P❊✩❭❦❑P❫❡■✞❜❡❍✞❑✩❚❛❭P■❏❵✿❊P❴❛❫▼◗➣❴❞❭✲❚❞❍✷◗❙❖❞❴❝❫❨◗✾s✞⑧✕❭P❑✩▲ ✛ ❊❨❋✼❚✸❊ ❚❈❊✫■❂❋✐❖❘❍❏■✞❊❡➞✻❜✷❊▼❑✲❖✵❋❙❊✕●✞❫✷❭✫❚✸❊♥❋❙❑✩◗❲❖❝❊❦❖✻✺➊⑨ ➃✼✺ ❍♦❍✷■✞❜✷❊❦✼ ⋅ ❥➁❭✫❚❛◗✧❖❝❑P❊✩❭❱❜✷❊❱❖❘■❆❑▼❖❞❜✷❊✫■✷❵✿▲⑥❴❛❭✾❚❛❍❏◗❙❖❛❺❛■❆❫✷❜❡❍✷◗❙❖❆❚❈❊▼❫❨◗❙❊✲❥➁❭ ❭❱❴❯❍✭❖ ✔✕❖q◗❙❑✓✖ ✖✞❫✷❤✼❤➋❋❙❊⑩❋❙❑✫◗❙❖❞❊❇➏ ❖✭⑨ ❋④❭✫❍✕➏❦➃✽✺ ❍✭⑨ ✗ ⋅ ⋅ ⑦ ❽ ⑦ ➠ ■✖❑▼❖❯◗❲❑✩❍✞❖❻❚❞❍✞❴↔❖❘■❆❖❘❵❞❖❞❭P❴✴❴❞❭♥❑▼❭✫◗❲❊✈❋④❊♥❑P❫❡■✞❊❨❑✫❚✿❊P❭✫❳❨▲➁r✐❖↔❋✧❍✷◗❙❋✐❊P❴❛❊♥❖❯■✞❜✷❊✫●✞❊✫■✞❜✷❊✫■✷❚✸❊♥❜✷❊♥❑✫❍✷◗❙❊✫■✷❚❩➞✜❚✸❊P❫❡◗❲❊✩❥➁❭ ❭♥❴❘❍✞❖ ❽ ✔✕❖❯◗❙❑✓✖ ✖✞❫✷❤❩❤✕❋④❊➝❋❙❑✩◗❲❖❛❊♥➏❦❖✽⑨✟❖✻✾❁❍✷■❆❜✷❊◆❖✿✾✴❊❨❋✼❚✸❊✈❍✷■✒❬✞❊P❑❾❚❈❫❡◗➍❭P❴❝❊✎❑✩▲✫◗✼❍✞❖➣❑❡❫❨❥❧●❆❫❡■✞❊✫■✷❚✸❊✎❋✼❍✷■✷❚♠❋✧❍❏❥♦❊❅❭▼❴❘❊◆❑✫❍✷◗❙❊✫■✷❵❈❖❝❴❞❫❡◗ ❋✧❍❏◗❙❋✐❊▼❴❞❫❡◗➁❖❘■❆❜✷❊✲●❆❊✲■✞❜❏❊✲■✷❚❈❊ ❑P❫❡■✞❊P❑❾❚❈❭✫❚✿❊ ❴❘❭✒❤✼❖❞❊▼❑✲❭✲◗✧❊❢■✞❫✷❜✜s❇⑤❱❊✫❳✩❍✞❴❯❚✿▲✖➏ ➃ ✺ ✭❍ ⑨❄❖✿✾♣r❙❖✕❜❆❊P❫➊❭✫◗❙❊▼❑P❊❢❍✭⑨❂❁ ➏ ❀ ⋅ ⋅ ⋅ ✭ ❿✭ ❤✼❖❞❖❻■❆❜ ✔❂ ✄➝■✞❫✷❜❡❍✭◗❲❖✕❖❝❭✩◗ ❄❃✿⑨ ⑦ ❸ ◗❙❊✲❳P❍❆❴❘❚❈▲✖➏ ⋅➃ ✺ ⋅➏❅❀ ⋅ ⑨❁❖✿✾①❋❙❭P❍✆■✞❫❡❚ ✜✫■✞❜ ❥➁❭✫❚❛◗✧❖❝❑P❊▼❭ ❑P❫❡■✞❜❨❍✞❑✩❚✸❭✫■✷❵❈❊▼❴❞❫❡◗❄■✞❫❏❜❡❍✷◗❙❖❞❴❛❫❨◗➋❑✩❍♥❆ ➃ ❃❈❇❊❉ ⑨✻➏❦➃ ✺ ➏ ❀ ◗❙❊✲❳P❍✞❴q❚✸▲❱➃❆❃❈❇❊❉ ⑩⑨❄❖✿✾④❑❨❭✫◗✼❊♠❊❨❋✸❚✸❊⑥❊▼❑✫❍✞❭✫❵✿❖❞❭①●✭❫❡❚✸❊✫■✷❵❈❖❝❭P❴❛❊✲❴❛❫❨◗❁■✞❫✷❜❨❍✷◗❙❖❞❴❝❫❡◗✾s ⋅ ❬❆❊▼❑✩❚✿❫➊◗✸❍✞❴♣●✞❫❡❚❈❊✫■✷❵✸❖❘❭▼❴❘❊❨❴❝❫❡◗❦●✷◗❙❖❯❥➁❊▼❴❞❫❡◗ ✠ ✭ ✭ ✭ ➏✕❚ ✜✲❚✞●✞❊✫■❆❚❝◗❩❍✈❋❙❑✩◗❲❖❝❊✫◗❙❊▼❭❱❊✲❑✲❍✞❭✫❵✿❖❞❖❞❴❛❫❡◗❁❜❆❊⑥❑▼▲✾❚❛◗❙❊☎❍✷■✣❫❡●✞❊✫◗❙❭✲❚❈❫❡◗✥❍✷❥➁❭✲■❂➞❏✟❑ ✜✫❚❄r✐❖✷●✞❊✩■➊❚❞◗✸❍♥❑P❫❡■❂❋✧❚❝◗❩❍✞❖❻◗❙❊P❭①❥➁❭✲❚❘◗❙❖❝❑P❊▼❖ 8 ➃ ❃❈❇❊❉ ➜❝■✷❚❞◗❙❺❛❍✷■ ●❆◗❙❫❏②❡◗❙❭✫❥ ❜✷❊✡❑P❭▼❴❞❑✲❍❆❴❛➞❦❋✐❊✆●❆❫❡❚✎❋✼❚✸❭✫➎✞❖❝❴❞❖✣◗❙❊P②❡❍✞❴❘❖✣❥✣❍✞❴q❚✈❥➁❭✩❖♦❋✐❖❻❥❦●✞❴❞❊☛❜✷❊❨✢❑ ✜✫❚♥❍✷❚✸❖❘❴❛❖❻❳❡❭✫◗❙❊P❭☛❤✼❫❡◗✸❥➁❊❨❖ ❥➁❭✫❚❛◗✧❖❝❑P❊✩❭P❴❛❊➂●❏◗❙❊✩❳❨❊✩■➊❚✸❭✫❚✿❊⑩❭✲■✷❚❈❊✫◗❙❖❛❫❨◗✐s✻❉✞▲⑩❑▼❫❡■✜❋✐❖❝❜✷❊✩◗❲▲✩❥ ❍✷■❅❊✲✉❆❊✫❥✣●❆❴❘❍✎❋✐❖q❥❧●✭❴❯❍✈●✞❊✫■✷❚❞◗✼❍❅❑▼❭✾◗❙❊⑩❬❆❫❡❥ ❋❙❑✫◗❙❖❝❊♠❚✿❊P❫❡◗❙❊✫❥➁❭ ❽ ❭❱❴❯❍✞❖ ✔✕❖❘◗❙✓❑ ✖ ✖✞❫❏❤✼❤✧s✩➏♥❴❘❊❨②✷❊❾❥ ✁❞⑨ s ⑦ ✭ ➜❝■❣■❆❫✷❜❡❍✞❴❂❜✷❊①●✞❫❡❚❈❊✫■✷❵✸❖❘❭▼❴ ➜❝■❣■❆❫✷❜❡❍✞❴❂❜✷❊①●✞❫❡❚❈❊✫■✷❵✸❖❘❭▼❴ ❶ ❉ ✭ ✁✂❲❶ (V1 − V2 ) G2 + (V1 − V2 )G3 + (V1 − V3 ) G1 = G (V1 − V2 ) (V2 − V1 )G2 + (V2 − V1 )G3 = +1 ✭ ✝ s❆❉✞❭✫❍ ✄ ✆ V−1V(G(1G+ G+2G+ )G+3V− G(G) − V+2G(G)3=++G12 − G ) = 0 ✄☎ 1 2 3 2 2 3 V3 = 0 ✮❄◗❲❊❡❑☎✜✫■✞❜✕➜❝■❣❥➁❊✫❥❧➎❏◗✼❍✞❴✞❜❏◗❲❊✫●✷❚✜❚❈❊✫◗✼❥✈❊✫■✷❍✞❴❆●➊◗❙❫❏❜❡❍❂❋↔❜✷❊✕❋✼❍✷◗✐❋❙❭❱❑P❫❡❥➁❭P■❆❜✷❭✲❚❈▲⑥❊▼❑✫❍✞❭✫❵✿❖❝❖❞❴❝❊❱❜✷❊✫❬✞❖q■✂❶ ✠ ✡ ✞ ✞✟ V1(G1 + G2 + G3 ) − V2 (G3 + G2 ) = G (V1 − V2 ) − V1(G2 + G3 ) + V2 (G2 + G3 ) = +1 V3 = 0 ✞❉ ❊♦●✞❫✷❭✫❚✸❊❅❖❘❥➁❊P❜✷❖❝❭✫❚♠❜❏❊▼❜❡❍❆❑▼❊✎❫✖◗❙❊P②❡❍✞❴❞▲❢❋❙❖❻❥♥●✞❴❞▲◆❜❏❊☞☛✍✌✍✎✑✏✓✒✍✎✔✒✖✕✗✎✔✏✙✘✚✏✓✘✛☛✜✕✢✒✣✌✥✤✦✏✜✒◆❿❩❜✷❊✖❑✲▲✲❚❞◗❙❊✈❍✷■✆❫❡●✞❊✫◗❙❭✫❚✸❫❨◗ ❍❏❥♦❭✫■ ❸ ❭➂❊✲❑✩❍✞❭✲❵❛❖❛❖❘❴❝❫❡◗✥●✞❫❡❚❈❊✲■❏❵✸❖❘❭▼❴❞❊✩❴❞❫❡◗❁■❆❫✷❜❡❍✷◗④❖❝❴❞❫❡◗✾s❨t↔❑✫❍✞❭✲❵❛❖❛❭➣➜❝■❣■✞❫✷❜❨❍✞❴★✧♠❊❨❋✼❚✿❊❱❜✷❊♠❤✸❫❡◗✼❥➁❭❡❶ ✩ ✩ ✩ G − Vi G = I k k ∈ j sk k ∈ j k k ∈i , j Vj ❍❏■✞❜✷❊❢●✷◗❲❖❯❥➁❭ ❋✼❍✷❥➁▲✱❑❨❫❡■✷❵❈❖❻■✞❊❢❚❈❫✷❭✾❚✸❊✱❑❨❫❡■✭❜❡❍❆❑✲❚❈❭✲■❏❵✸❊✲❴❛❊✱❑❨❫❡■✞❊✲❑✲❚❈❭❾❚❈❊➝➜❘■☛■✞❫❏❜❡❍✞✪❴ ✧❏➞①❭✱❜✭❫❡❍✞❭✒❋✼❍✷❥➁▲✒❑P❫❡■✷❵❈❖❘■✞❊✖❚❈❫✷❭✾❚✸❊ ✁✫✼➃✖✬✌❍✷■❂❜❏❊♠➃✖✥✬ ❊❡❋✧❚❈❊♠❑▼❫❨■✞❊▼❑✫❚❈❭✲❚❛▲❱➜❞■✷❚❘◗❙❊♣■✞❫✷❜❨❍✷◗❙❖❞❴❛✪❊ ✧❦r❙✭❖ ✏✌❖❯❭✩◗✽❋✧❍❏❥♦❭♠❜❏❖❘■❦❥✈❊✩❥♥➎✷◗✸❍✞❴✌❜❨◗❙❊✲●❏❚✥❑▼❫❡■❏❵✸❖❯■✞❊♣❚❈❫❡❵✸❖ ❑✫❍✷◗❙❊✫■✷❵❈❖❛❖➋❋✼❍✷◗✐❋❙❊▼❴❘❫❡◗➣❿✼❖q■✞❜✷❊✩●✞❊✫■✞❜✷❊✫■✷❚✸❊❦❋❙❭P❍➁❑❨❫❡❥➁❭✫■✞❜✷❭✫❚✸❊ ❸ ❑▼❫❨■✞❊▼❑✫❚✿❭✫❚✿❊➍❴❛❭♠■✞❫✷❜❡❍❆❴✮✧ ❿✼❴❯❍✞❭✫❵✸❖❄❑✫❍✎❋❙❊✲❥♥■✷❍❆❴ ❜✷❭❨❑▼▲➍❖❘■❏❚❛◗❲▲ ➆ ➜❝■❣■❆❫✷❜❡❍✞❴★➍ ✧ r❙❖✜❑✫❍✈❋❙❊✫❥❧■❏❍✞❴❂❺❄❜✷❭❨❑✩▲⑥❖❛❊❨❋✽❜➊❖❯■♥❭✲❑▼❊❨❋✧❚✿❭ ❸ s ❉✞❊✕❫❡➎✜❋✐❊✲◗✸❬✞▲⑩❑▼▲✕❫♥❑P❫❡■✞❜❡❍❆❑✲❚❈❭✲■➊❵❈✰ ❭ ✘ ✯ ❑▼❫❡■✜❊▼❑✾❚✿❭✫❚✸▲❱➜❝■✷❚❞◗❲❊➂■✞❫✷❜❨❍✷◗❲❖❛❴❞✱❊ ❁✏ r❙❖✲➍ ✧ ❭✫●✞❭✲◗✧❊❱➜❞■♥❥❧❭✲❚❞◗❙❖❯❑P❊▼❭⑩➃ ❃ ❇ ❉ ➜❞■♥●✞❭✾❚❛◗✸❍ ●❏◗❙❫✷❜❏❍✜❋④❊❨❴❘❊ ✭ ●❆❫➊❳❡❖❯❵❈❖❝❖♥❑✲❍❧❋✐❊✲❥♥■✞❊✲❴❛❊❡❶ coloana linie i j i + − j − + ❋✐❖❯❥❧❖❝❴❞❭✲◗✵❋✧❊♣●✞❫❏❭✲❚❛❊➂❫❨➎❂❋❙❊P◗✸❬❆❭♠❑✩▲❱❫✕❑P❫❡■✞❜❡❍✞❑✩❚❛❭P■❏❵✸▲❱❜✷❊☎❚❛◗❲❭✩■❂❋❙❤✼❊✫◗➋❭①❍✷■✞❊P❖❂❋❲❍✭◗❙❋✐❊♣❑P❫❡❥➁❭✫■✞❜✷❭✫❚✿❊♠❑✩❭⑥➜❝■♥❤✸❖❘②❡❍✷◗❙▲❱❭✩●✞❭✫◗❲❊♣➜❝■ ✯✴✳✶✵✸❆✷ ❚❈❫❡❚✻➜❞■➍●✞❭✫❚❛◗✼❍➍●✞❫➊❳❨❖❻❵❈❖❝❖❦❑✩❍✈❋❙❊✲❥♥■✭❊P❴❛❊⑥❜✷❖q■❣❚❈❭✲➎✞❊❨❴✿s 9 coloana i j linie k l + − − + ✹❄❊✫■✷❚❞◗✼❍✈❍✷■♥●✷◗❙❫❏②❡◗❙❭✫❥ ❜✷❊➍❑▼❭P❴❝❑✫❍✞❴❄❊❨❋✧❚✿❊♠❥✣❍❆❴❻❚❁❥➁❭✲❖✴❋❲❖❘❥♥●✞❴❯❍✎❋✧▲➍❖❘■➊❚❞◗❙❫✷❜❨❍✞❑▼▲➍❤✼❖❞❊✩❑P❭✲◗❙❊⑩❑▼❫❡■❆❜❡❍✞❑✫❚✸❭✫■✷❵❈▲➂●✷◗❙❫❡●❏◗❙❖❛❊❦❋❙❭✲❍❅❜✷❊ ❚❞◗❙❭✫■❂❋❙❤✼❊✫◗✽➜❝■❧❑▼❊✲❴❛❊➂●❆❭❾❚❞◗✼❍♥●✞❫➊❳❨❖❘❵❛❖❛❖✈●✷◗❙❊P❑✩❖❘❳▼❭✩❚✸❊♠❜❏❊✕❭✩❑P❊❡❋✼❚❈❊➂◗❙❊P②❡❍✞❴❘❖✌❜✷❊▼☎❑ ✜✫❚➋❋④▲♠❑❨❫❡■✜❋❲❚❞◗✼❍✞❖❘❭➊❋✧❑P▲➂❥❧❭P❚❘◗❲❖❛❑P❊P❴❝❊✁ r❙❖❄➃✼➊✺ r❙❖❄❋④▲ ❫❨●✞❊✲◗✧❊✩❳➊❊⑥❑✲❍✣❭▼❑✲❊❏❋✼❚❛❊▼❭➊s ✹ ✜✫■✞▲♦❭P❑✲❍✷❥ ❭✲❥ ❑❡❫❨■❂❋❙❖❛❜✷❊✫◗❙❭✾❚⑥❑▼▲➁❑✩❖❯◗❲❑P❍❆❖❻❚❞❍✭❴✵■✷❍❢❑❡❫❨■✷❵✿❖❯■✞❊◆❋✧❍✷◗❙❋✐❊✈❖❞❜✷❊❨❭P❴❛❊✈❜✷❊♥❚✸❊✫■❂❋❙❖❘❍❏■✞❊✈❿✧❋❙❭✩❍❂➞❁❜❏❭▼❑❨▲✈❴❞❊▼❺✼❭ ❭✫❬✷❍✷❚❩➞✞❊▼❴❘❊✕❭✩❍✈❤❩❫❆❋✧❚✌❚❛◗✧❭✲■❂❋❙❤✼❫❨◗✼❥➁❭✲❚❈❊♠➜❝■◆❋✼❍✷◗④❋④❊♠❖❞❜✷❊P❭❨❴❯❊❣❜✷❊⑩❑✲❍✷◗✧❊✲■✷❚ ❸ s❏t✴✉✭❖❈❋✼❚❈▲♠➜❞■✜❋④▲➍r④❖✥❋❲❍❏◗✐❋❙❊✕❜✷❊➂❚❈❊✲■✜❋✐❖❘❍✷■❆❊✕❑▼❭✫◗❙❊➂■❏❍◆❋❙❊ ●❆❫❡❚✌❚❛◗✧❭✲■❂❋❙❤✼❫❨◗✼❥✈❭➂➜❘■➁❋✧❍✷◗✐❋❙❊♠❜✷❊♠❑✫❍✷◗❙❊✫■✷❚✧s❡⑧✕❭P❑❡▲♠❑P❖❯◗❲❑✲❍✞❖❯❚❞❍✞❴❂❑▼❫❡■✭❵❞❖❯■✞❊♠❜✷❫❆❭✫◗↔❫♥❋✐❖❯■✞②❡❍❆◗❲▲✕❋✧❍❏◗✐❋✐▲♠❜✷❊♠❭P❑▼❊➊❋✸❚✌❚✸❖❯●✈❭✾❚❛❍❏■✞❑▼❖❄❋❙❊ ❭P❴❞❊▼②✷❊✩✭ ④❉ ⑨ ⑦ r❙❖✷◗❙❊✩❳✩❍✞❴q❚✸▲ ✭ ✂ ⑨✌t✱❿❈■❏❍✈❋❙❊①❥♦❭✲❖✌❋✐❑✲◗❲❖❘❊⑥❚❈❊P❫❡◗❲❊✩❥✈❭ ❽ ❭❱❴❯❍✞❖ ❣ ✔ ❖q◗❙❑✓✖ ✞ ✖ ❫✷❤✼❤✂➜❞■➍■✞❫✷❜❡❍❆❴✻❜❏❊⑥●❆❫❡❚✸❊✫■✷❵❈❖❝❭P❴✚✭ ✂ ❸ s t❁❬✞❖❞❜✷❊✫■✷❚✼➞✵❜✷❭▼❑❨▲❅❊✩✉❆❖✿❋✼❚✸▲✈❥➁❭✩❖①❥♥❍✞❴ ❚❈❊❢❋✼❍✷◗✐❋✐❊❅❜✷❊➝❭✩❑P❊❡❋✼❚➣❚✸❖❯●✆❑P❭✲◗❲❊✎❭✲❍ ❍✷■✒■✞❫❏❜ ❑P❫❡❥♥❍✷■☛❋✐❭✫❍✆❑✩❭✩◗❲❊➝❤✼❫❡◗✸❥➁❊✩❭P❳❨▲➁❍✷■ ✓✫❥➁❖❘■❆❖❛❺✼❭✫◗✼➎✞❫❨◗❲❊✖♠ ✕ ■✞❫✷❜❡❍❆❴✴❜✷❊✕●✞❫❡❚❈❊✫■✷❚✸❖❞❭✩❴❄■✷❍✞❴✌❬✞❭❦❭P●❆❭✲◗✸❵❈❖❘■❆❊✣❭✲❑❡❊❨❋✸❚✸❊✲❖✵❋✼❚❞◗✼❍✞❑✫❚❛❍✷◗✧❖❄❖❛❭✩◗✽●✞❫❨❚✸❊✫■➊❵✸❖❞❭✩❴❞❊✩❴❞❊❦❑✩❊P❴❛❫❡◗✧❴❝❭P❴❻❚❈❊➍■✞❫❏❜❡❍✷◗❙❖ ❭P❴❞❊❣❋✧❚❛◗✸❍❆❑✲❚❘❍✷◗❙❖❞❖✻◗❲❊➊❋❲●❆❊✩❑✩❚❞❖❯❬✞❊❣❋✐❊➂●✞❫❨❚➋❊✩✉➊●✷◗❙❖❻❥➁❭✕❑❨❭❣❋✼❍✷❥➁❊✕❜❏❊➂❚✸❊✫■❂❋✐❖q❍✷■❂❖✌❊P❴❝❊❨❑❾❚❞◗❲❫❡❥➁❫❡❚❈❫✷❭✫◗❙❊❏s❡⑧✕❭P❑❡▲⑩❑▼❖❯◗❲❑✩❍✭❖❘❚❘❍✞❴❄❑P❫❡■✷❵❈❖❘■✭❊ ❜❏❫❡❍✞▲✒❭❨❋✼❚✸❤✼❊P❴♠❜✷❊✒❋✼❍✷◗✐❋❙❊✒❑✲❭✲◗✧❊✖■✷❍ ❭✲❍✆❍✷■✆■❂❫✷❜✆❑▼❫❨❥❧❍❏■ ❋✐❊✱❭P❴❛❊P②✷❊ ✭ ❉ ⑨ ➞✽◗❲❊✲❳P❍✞❴❯❚✿▲ ✭ ✂ ⑨❄t r❙❖✕❋✐❊✱❖❘■❏❚❛◗❙❫❏❜❡❍✞❑P❊✱❫ ⑦ ■❆❊▼❑✩❍✷■✞❫✭❋④❑✫❍✷❚❈▲✣❋✧❍✷●❆❴❛❖q❥♦❊✫■❆❚❝❭P◗❲✄ ▲ ✂✩s✞✇ ✜✫■✞❜♦❋✐❊❦❋④❑✫◗❙❖❯❍❅❊▼❑✫❍✞❭✫❵✿❖❞❖❝❴❞❊✕➜❝■♥■✞❫✷❜❡❍✷◗✧❖❛❴❞❊❣❜✷❊♠●✞❫❡❚❈❊✲■❏❵✸❖❞❭✩❴❞❊ ✁ ✭ ↔r④❖✎✆ ✭ ☎↔❴❘❭P❚❘❍✷◗❙❭➍❑✲❍❅t ✂ ❋✐❊ ❑▼❫❡■✜❋④❖❞❜✷❊✩◗❲▲✘●✞❭✫◗❙❑✫❍✷◗✐❋❙▲✟❜❏❊✡❑✲❍✷◗✧❊✲■✷❚❞❍✞✝ ❴ ✂✾s❱➏❦❑P❊❏❋✸❚❈❊▼❖❧■❆❊✩❑✫❍❆■✞❫❆❋❙❑✫❍✷❚❈❊✟❋✧❍✷●✞❴❞❖❻❥❧❊✩■✷❚✿❭✩◗❩❊✡➜❛❖➁❑▼❫❡◗✧❊❡❋✧●✷❍❏■✞❜✷❊ ❊P❑✲❍❆❭✲❵❛❖❝❭ ❋✧❍❏●✞❴❛❖q❥➁❊✲■✭❚✿❭✩◗❩▲ ✭ ☎ ❺✜✭ ⑨❄t ✂❲s ✹ ✜✫■✞▲♠❭▼❑✫❍✷❥↕❭✫❥ ❑P❫❡■❂❋❙❖❛❜❏❊✲◗❙❭✫❚✂❑✩▲✕❭✫❬✞❊✫❥ ❜✷❫✷❭✫◗☎❋✼❍✷◗④❋④❊♠❑❨❫❡❥❧❭✩■✞❜✷❭✾❚✸❊⑥➜❝■✣❚❛❊✲■❂❋❙❖❯❍✷■✞❊➊s➊✇⑥❫❡❥➁❭❾■✞❜❏❭❱➜❞■❧❑✲❍❏◗❙❊✲■❏❚➋❋✧❊ ●❆❫✷❭✲❚❈❊①❚❛◗❲❭✫■❂❋✐❤✸❫❡◗❩❥♦❭⑥➜❝■♥❑P❫❡❥❧❭✲■✞❜❏▲❱➜❘■❣❚❛❊✩■❂❋❙❖❘❍➊■✞❊❏❶ ⑧✕❭❨❑P▲❱❭▼❑P❊▼❭❨❋✼❚✿▲♣❚❞◗❙❭✫■❂❋❙❤✼❫❡◗✼❥➁❭✫◗❙❊♣■✷❍❧❊❡❋✼❚✿❊♣●❆❫❆❋✐❖❘➎❆❖❛❴❘▲♠❿✸❑▼❭❱➜❝■♥❤✼❖❞②❡❍✷◗✧❭♠❜✷❊♣❥➁❭▼✟❖ ✞ ❫❆❋ ❸ ❋❙❊♠❑❨❫❡■❂❋❙❖❝❜❏❊✲◗❙▲♠❫♥❋✼❍✷◗✐❋✐▲♠❑✫✝ ❍ ✠☛✡✝☞♠➜❯■ ❴❞❭✫❚❛❍✷◗✧❭❣❜✷❊➍❑▼❫❡❥➁❭✫■✞❜✷▲♥r❙❖✴❋❙❊➍●❏◗❲❫✷❑❨❊▼❜✷❊✲❭✩❳❨▲❣❑P❭➍➜❞■➁❑❨❭✫❳✩❍✞❴↔❋❲❍❏◗✐❋❙❊✩❖❁❖❞❜✷❊P❭✩❴❞❊❣❜✷❊♠❚✸❊✫■❂❋✐❖❯❍✷■✞❊➍❿✿➜❘■✎❊P❑✲❍✜❭❾❵❈❖❛❖❻❴❛❊♠■✞❫✷❜❡❍✷◗✧❖❛❴❞❫❡◗➣❜✷❊ ●❆❫❡❚✸❊✫■✷❵❈❖❛❭✲❴❛❊✩✭ ❉ r❙❖ ✭ ✂ ❘❴ ❭✲❚❞❍✷◗❲❭❱❜✷❊❱❑P❫❡❥➁❭✲■✞❜❏▲①❭✩●✞❭✫◗❲❊♣●✷◗❲❖❘■✝✌★✏❂r④❖✎✍✖✏✌r❙❖✌❋✧❊❱❭P❜✷❭✩❍✞②✷▲⑥❊▼❑✫❍✞❭✫❵✸❖❻✧❭ ✭ ❉ ⑨ ✭ ✂ ❸ s 10 ⑧✕❊❨❑▼❖✞❭P❴❛②✷❫❨◗❲❖❘❚❘❥❧❍❆❴✻❜❏❊➂❋✐❑✫◗❙❖❯❊✩◗❲❊❱❭❱❊✩❑✫❍✞❭✲❵❛❖❛❖❘❴❛❫❨◗❁●✞❫❡❚❈❊✫■✷❵✸❖❘❭▼❴❞❊✩❴❞❫❡◗❄■❆❫✷❜❡❍✷◗❙❖❞❴❞❫❡◗↔❊❨❋✼❚✸❊❡❶ ✐❋ ❊✕❤❩❭▼❑➂❚❈❫✷❭✫❚✿❊➂❚❞◗❙❭✫■❂❋❙❤✼❫❡◗✼❥❧▲✩◗❲❖❝❴❞❊➂●✞❫❆❋✐❖q➎✞❖❞❴❝❊✕❭❨❴❯❊❦❋❲❍❏◗✐❋❙❊▼❴❞❫❡◗❇❜✷❊➂❚❈❊✲■✜❋④❖❯❍✷■✞❊➂➜❞■❅❋✼❍✷◗✐❋✐❊✕❜❏❊✕❑✲❍❏◗✐❊✫■✷❚↔r❙❖❄❭P❴❝❊✕❑P❫❡❥➁❊✲■✷❳▼❖❞❴❛❫❡◗ ➜❝■♥❑✫❍❆◗❲❊✫■✷❚✜➜❞■✣❑▼❫❡❥➁❊✩■✭❳P❖❆➜❝■❣❚❞❊✩■❂❋❙❖❯❍✷■✞❊❨➞ • ❋✐❊♥❭P❴❛❊P②➊❊❣●✞❫❡❚❈❊✲■✷❵❈❖❝❭P❴❯❍✞❴↔❜✷❊➍◗✧❊▼❤✼❊✫◗❙❖❯■✷❵✿▲♥❭❨❋❩❚❈❤✼❊✲❴✴➜❯■✞❑✒✜❾❚ ❑☎✜P❚➋❥❧❭▼❖❁❥♥❍✞❴❻❚❛❊❣●✞❫❡❚❈❊✲■✷❵❈❖❝❭P❴❝❊♥❭P❴❝❊❣■✞❫✷❜❨❍❆◗❙❖❞❴❛❫❨◗❱❋❙▲❦●❆❫✷❭✩❚✿▲♥❤✼❖ • ❊P✉❏●✷◗❲❖❯❥➁❭✲❚❈❊⑥❑▼❭♠❋✼❍✷❥➁❊♣❜❏❊♣❚✿❊✫■❂❋❙❖❘❍✷■❆❖✜❊P❴❛❊✲❑✩❚❝◗❙❫▼❥➁❫❡❚❈❫✷❭✫◗❙❊❡➞ ❨❑ ❫❡■❂❋❙❖❞❜✷❊✫◗❙❭✲■❆❜◆r❙❖✥■✞❊P❑✫❍✷■✞❫❆❋✐❑✫❍✷❚❈❊✩❴❞❊❧❋✼❍✷●✞❴❞❖❯❥✈❊✾■✷❚✸❭✫◗❙❊❦❿✼❑✫❍✷◗❙❊✫■✷❵✸❖❘❖✂❍✷■❆❫❡◗❱❋✼❍✷◗✐❋❙❊❦❜✷❊➍❚❈❊✫■❂❋✐❖❘❍❏■✞❊❣❑P❫❡■✞❊P❑✲❚❈❭❾❚❈❊✕➜❝■✷❚❞◗❙❊❣❭P❴❻❚❈❊ ■✞❫✭❜❡❍✷◗❙❖❂❜✷❊P❑☎✜✲❚✻❑▼❊P❴❝❊❱❜✷❊⑥❴❞❭➣●✷❍❏■✞❑✩❚❛❍✞❴✷●✷◗❙❊P❑P❊▼❜✷❊✫■✷❚✥r❙❖✜❑✫❍✷◗❙❊✫■✷❵❛❖✻❜❏❊♣❑❨❫❡❥❧❭✩■✞❜✷▲ ❸ ❋❙❊➂❋✐❑✫◗❙❖❯❊✕❋❲❖✿❋❩❚❈❊✩❥✣❍✞❴❂❜➊❊♠❊✲❑✲❍✞❭✫❵✿❖❞❖✸❶ • Vj G − Vi G k = I k ∈ j k k ∈i , j k ∈ j sk ❴❞❭⑥❑▼❭✩◗❲❊♠❋❙❊♣❭❨❜✷❭✫❍✞②✷▲❱❊✩❑✫❍✞❭✩❵✿❖❘❖❝❴❞❊➂❋✧❍❏●✞❴❛❖❯❥➁❊✲■❏❚✿❭❾◗✧❊➂❜❏❊➂❤❩❫❡◗✸❥➁❭ ✠✂✁ ☎✄ ✝✆✟✞✡✠ ✒ Vl − Vm = E α s ✕ ✍ ☛ ☛ ☛ ✔ V4 = 0 V1 = E ✑ 1 ✎ 1 1 1 ✎ − V1 = I2 + ✏ R1 R3 R1 ✒ ☛☛ V ✒✓ ✌ 2✔ ✑ ☛ ✒ ✎ ☛ V ✒ 1 + 1 ✎ − V 1 = −I ☛ 3✓ R R ✏ 1 R 2 2 2 4 ☛ V −V ☛ V2 − V3 = 2 1 2 ☛☞ ☛ ⑧➂❍❏●✞▲ ❑▼❊ ❋❙❺✼❭✫❍ R1 ❜❏❊✲❚❈❊✲◗✸❥❧❖❯■✞❭✲❚☛●✞❫❨❚✸❊✫■✷❵❈❖❝❭P❴❝❊P❴❛❊ ■✞❫✷❜❡❍❏◗❙❖❛❴❞❫❡◗ ❋④❊ ●❆❫❡❚ ❑✩❭❨❴❛❑✫❍✭❴❞❭ ❚❈❊✲■✜❋✐❖❻❍✷■✞❖❞❴❝❊ ❑✲❭ ❜✷❖❞❤❈❊✩◗❲❊✩■✷❵❛❊ ❭P❴❝❊ ●❆❫❡❚✸❊✫■✷❵❈❖❛❭✲❴❛❊✲❴❛❫❨◗✽r❙❖✜❭✩●✞❫❏❖✜❑✩❍✷◗❲❊✫■✷❵✿❖❞❖❈s ✕✗✖✘✕✙✖✛✚✢✜✤✣✦✥★✧✩✜✫✪☎✬✮✭✯✬✱✰✤✜✤✭✯✜✳✲✴✣✵✬ ✬❄❖❞❊❦❍❏■✎❑❨❖❘◗❲❑✫❍✭❖❘❚❇❑✫❍◆◗❲❊✲❳❡❖✿❋✸❚✸❫✷❭✫◗❙❊❦❑▼❫❨■✷❚❛◗✧❫✷❴❛❭✫❚✿❊➍➜❝■♦❚✸❊✫■❂❋✐❖❯❍✷■✞❊✈r❲❖❇❋✼❍✷◗④❋④❊♥❖❯■✞❜✷❊✫●✞❊✫■✞❜✷❊✩■✷❚✿❊❦❜✷❊❦❑✩❍✷◗❲❊✲■✭❚✼s✥❉✞❑✲◗❲❖❞❖❻■✞❜ ❊P❑✩❍✞❭✫❵✿❖❞❖❝❴❝❊❱❑P❫❡■❂❋✼❚❈❖❻❚❞❍✷❚✸❖❯❬✞❊⑥❭✩❴❞❊①◗❙❊✫❳❨❖❈❋✼❚❈❫✷❭✫◗❲❊▼❴❞❫❡◗ ✡✷✶✴✸✹✞✻✺✽✼✷❚❈❊P❫❡◗❙❊✫❥♦❭ ❭⑥❴❯❍✞❖ ✔⑩❖❯◗❲❑✓✖ ✖✞❫❆❤✸❤✎ ✡ ☞✕r❙❖❆❚✿❊P❫❡◗❙❊✫❥♦❭⑥❭ ❺✸❭⑥❭♠❴❻❍✜❖ ⋅ ❽ ❽✼❽ ✔✕❖❯◗❙❑✓✖ ✖✞❫✷❤❩✾ ❤ ✞ ✡ ✩✿❁❧ ❀ ◗❙❊✲❳✩❍✞❴❘❚❈▲❣❊✲❑✲❍❆❭✲❵❛❖❛❖❞❴❝❊♠●✞❫❡❚✿❊✫■✷❵❈❖❛❭✲❴❛❊✲❴❛❫❨◗❪■✞❫❏❜❡❍✷◗❙❖❞❴❛❫❨◗ ✶✛✸ ✯✿❂❀❃✺ ✡ ✘❄❆❅❁❉➊❍❆◗❙❋❙❊P❴❛❊➍❜➊❊⑩❚❛❊✲■❂❋❙❖❯❍✷■✞❊➍❑▼❭✩◗❩❊⑩■✷❍ ✏ ✏ ⋅ ✏ ❋✐❊①●✞❫❨❚✻❚❞◗❩❭✩■❂❋❙❤✼❫❡◗✸❥➁❭⑥➜❞■❧❋✧❍✷◗❙❋✐❊❱❜✷❊♣❑✩❍✷◗❲❊✫■✷❚✂r✐❖✜❑P❫❡❥➁❊✫■✭❳▼❖❞❴❛❊➣➜❝■❦❑✲❍❏◗❙❊✲■❏❚❁❋❙❊①❚❞◗❲❭✲❚❛❊▼❭✲❳▼▲♠❋✐❖❻❥➁❖❯❴❞❭✫◗➋❑✲❍✣❑❨❖❻◗❲❑✫❍✞❖❘❚❛❊▼❴❘❊♠❴❯❖❯■✞❖❛❭✫◗❲❊➊s 11 ✾✖✂✁ ✬ ✪☎✬✱✣☎✄ ✜✳✭ ✲✦✣✦✬✡✲ ✧✝✭✯✬ ✜✝✆✟✞✴✰✤✧✡✠ ✜✤☞ ✜ ☛✷✣★✜✳✌ ✭ ✄ ✬ ✪✍✞☞✎✝✲✑✏ ✬❄❖❞❊✎❍❏■ ✬✓✒ ✪✔✞✛✭✖✕✘✗ ✲☎☛✖✙✡✆✚✞✙✭ ❙❋ ❖✿❋❲❚❛❊✩❥ ❜❏❊❢❊P❑✫❍✞❭✲❵❛❖❛❖⑥❭P❴❛②✷❊✫➎✷◗❙❖❘❑▼❊✖■✭❊P❴❞❖❘■✞❖❘❭✲◗✧✜ ❊ ✛ ✸ ✁❃✺ ✎❍✷■✞❜✷❊ ✁ ❊❡❋✧❚❛❊◆❬✞❊❨❑❾❚❈❫❡◗✸❍✞❴①■✞❊P❑✲❍✷■❆❫❆❋✐❑✲❍❏❚✸❊P❴❛❫❨◗✐s ⑤❱❊P❴❝❭✫❵✿❖❞❭✕❑✩❭✫◗❙❊✕❜✷❊P❤✼❖❯■✞❊❨r❩❚❈❊➂❥❧❊✲❚❈❫✷❜✷❭✕❖q❚✸❊✫◗❙❭✾❚✸❖❯❬✭▲ ✘❦❊✔✕ ✢ ❚✿❫❡■❆❺✸⑤♣❭✩● ✖❂❋❙❫❡■❅❋❙❊❣❫❨➎✷❵✿❖❯■✞❊✕❜✷❊P❳✩❬✞❫✷❴❯❚ ✜✾■✞❜❣●✞❊✕❤✓➜❞■❅❋❙❊✲◗✧❖❛❊✡✮➋❭✔✣✞❴❛❫❨◗ ✡ ☞ ➜❝■ ❙❍✷◗✼❍❆❴✜❴❯❍✭❖ ✥✁ ✤ ✦ ✧➂❿ ✁❦❴❯❭♠❖q❚✿❊✩◗❲❭✲❵❛❖❝❭ ❸ ❶ ✞ ✧ ✫✬✭ ★ ✫✬✭ ★ ✫✬✭ f x ( j + 1) ✩✪ = f x ( j ) ✩✪ + J x ( j ) ✩✪ ★ ✫✬✭ ★ x ( j + 1) − x ( j ) ✩✪ + ... ✯ ❍❏■✞❜✷❊①❥➁❭P❚❘◗❙❖❞❑✩❊P❭✜✮✦✸ ✁ ✤ ✦ ✧ ✺ ✲ ✱ ∂ f1 ∂ fn ✴ ... ✱ ✴ ∂ x1 ✴ ✱ ∂ x1 ✱ ✡ . . . ✴ ✱ ∂ f1 ∂ fn ✴ ... ✱✰ ✴ ∂ xn ∂ xn ✳ x= x( j) ❋✐❊✕■✷❍✷❥➁❊❨r✼❚✸❊✶✵ ❭❨❑P❫❡➎✞❖❘❭✲■✷❍❆❴✵❋❙❖✿❋❲❚❛❊✲❥♥❍✞❴❯❍✞❖↔r✧❖↔❋❲❊✣❑❨❭✩❴❞❑✫❍✞❴❝❊P❭✫❳➊▲✕➜❝■♦●✷❍❏■✞❑✲❚❞❍✞❴ ✁✥✤ ✦✷❂✧ ❅ ❥♥●✷❍❏■ ✜✲■❆❜♦❑▼❫❨■✞❜✷❖❯❵✸❖❞❭❣❑P❭ ✁✸✤ ✦✺✹✼✻ ✧✵❋✐▲❦❤✸❖❛❊ ❽ ❋✐❫❏❴❘❍✷❵❛❖❛❊❦❭❦❊✩❑✩❍✞❭✫❵✿❖❞❊✩❖✽✛ ✸ ✁❃✺ ❧r④❖❄■✞❊P②✷❴❞❖ ✜✲■✞❜♥❚✸❊✫◗✼❥❧❊P■❆❖❝❖❁❜❏❊✣❫❡◗✧❜✷❖❘■✎❋✧❍❏●✞❊✲◗✧❖❛❫❡◗①❜✷❖❘■◆❜✷❊✩❳✩❬✞❫✷❴❝❚❈❭✲◗✧❊▼❭⑩➜❞■➝❋❙❊✫◗❙❖❞❊ ✮✥❭✼✞✣ ❴❞❫❡◗❱❋❙❊ ✡ ☞ ✞ ❫❨➎✷❵✸❖❯■✞❊①◗❙❊P❴❝❭✫❵✿❖❞❭❱❜➊❊⑥◗✧❊▼❑✫❍✷◗❙❊✫■➊❵✸▲❱❭①❥❧❊✲❚❈❫✷❜✷❊P❖✜❖q❚✸❊✫◗❲❭✫❚✸❖q❬✞❊ ➍ ✘ ❊✼✢♠❚❈❫❡■✞❺✸⑤⑥❭✲✘● ✖❂❋✐❫❡■✂❶ ( j+1) ( j) ( j) ( j) x = x − J −1( x ) f ( x ). ➏❦❴❞②✷❫❡◗✧❖❘❚❞❥♥❍✞❴✂●✞❴❞❊P❭✩❑❨▲✣❜❏❊✣❴❘❭✣❫♦❭✫●✷◗❙❫❡✉❆❖❘❥➁❭✫❵✸❖❘❊✣❖❯■✞❖q❵✸❖❘❭▼❴❘▲ ✸✁ ✤✷✾ ✧✵r❙❖➋❊▼❤✸❊❨❑✫❚❛❍✭❊P❭✲❳▼▲✕❍✷■♦■✷❍✷❥➁▲✫◗♠❤✸❖❘✉◆❜✷❊❦❖❯❚✸❊✫◗❲❭✩❵✿❖❝❀ ❖ ✿ s✞⑧✕❭P❑▼▲ ✾ ❊✫◗❙❫✷❭✫◗❙❊P❭❦➜❞■➊❚❞◗❙❊❦❜✷❫❡❍✞▲♥❖❯❚✿❊✫◗❙❭✲❵❛❖❛❖✡❋✧❍✞❑✲❑▼❊❨❋❙❖❘❬❆❊ x( N) − x( N −1) ❊❨❋✧❚✸❊♠❥♦❭❨❖✌❥♦❖❞❑✩▲♥❜✷❊P❑✟✜✩❚✵❫◆❊✫◗❙❫✷❭✫◗❙❊❦❖❘❥❦●✷❍❂❋✐▲ ❑P❫❡■❂❋✐❖❞❜✷❊✫◗❙▲♥❑▼▲❣❥❧❊✲❚❈❫✷❜✷❭♥❊❨❋✼❚✿❊♥❑▼❫❨■✷❬✞❊✫◗❙②✷❊✫■✷❚✸▲♥❖❞❭✲◗♣❋✐❫✷❴❘❍❏❵✸❖❘❭❧❊❨❋✼❚✿❊❦✉❂❁ ❃✚❄✜❿ ❊✫◗❙❫✷❭✫◗❙❊P❭①■✷❍✈❋✐❑✲❭▼❜✷❊♠❋✧❍✷➎➍❬✞❭✩❴❞❫✷❭✫◗❙❊P❭ ε ❅n x = x k =1 k 2 ❸ s❂⑧✕❭P❑P▲❧❜❨❍✷●✞▲❆✿ ✾ ε ❋❙❊ ❖❯❚✸❊✫◗❙❭✫❵✿❖❞❖ ❋✐❊♣❑P❫▼■❂❋❙❖❞❜✷❊✩◗❲▲➂❑P▲①❥✈❊✫❚❈❫✷❜✷❭❱❊▼❋✧❚❈❊♣❜❏❖❘❬✞❊✫◗❙②✷❊✫■✷❚❈▲❡s ⑧ ❭❨❑▼▲❱❤✼❿✿✉ ❸ ⑨ ❭✫◗❙❊①❥➁❭▼❖❆❥♥❍✞❴q❚✸❊♠❋✐❫✷❴❯❍✷❵❈❖❝❖❛➞❨●✷◗❙❖q■♥❭P❑▼❊❨❭❨❋❲❚❛▲①❥♦❊✫❚❈❫✷❜✷▲✕❋❙❊❱❜✷❊✲❚❈❊✫◗✼❥✈❖❯■✞▲①■✷❍✷❥➁❭P❖❆❍✷■✞❭❱❜✷❖❯■✷❚❛◗✧❊➂❊P❴❞❊⑩r✐❖ ✕ ⑦ ❭✫■✷❍✷❥➁❊♠❑P❊✩❭⑥❑▼❭✩◗❲❊❱❊❨❋❲❚❛❊①❥✈❭P❖ ✓❨❭✫●✷◗❲❫❡●✞❖❞❭✫❚✿▲✂✕♣❜❏❊➂❭✫●✷◗❙❫❡✉❆❖❘❥➁❭✫❵✿❖❞❭♣❖❯■✞❖q❵✸❖❘❭▼❴❞▲①✉ ❁ ❇❈❄ s ⑧✕❭❨❑▼▲♥✉✎❭✲◗✧❊✈❫❢❋❙❖❯■✞②❡❍✷◗❙▲♥❑❨❫❡❥❦●❂❫❡■❆❊P■❏❚✸▲❣❥➁❊✲❚❈❫✷❜✷✝ ❭ ✘❦❉❊ ✢♠❚❈❫❡■✞❺✿⑤❱❭✲●✘✖❂❋❙❫❡■✱❭✫◗❙❊✈❫❅❖❘■✷❚❈❊✫◗✼●✷◗❙❊✩❚❈❭✲◗❲❊✈②✷❊P❫❡❥➁❊✫❚❝◗❙❖❘❑▼▲ ❋✐❖❯❥❦●✞❴❛▲➊s ❄ ✬ ❖❘❊✈❤✸❿✸✉ ❸ ❭✲❬ ✲✜ ■✞❜ ✭➍❭❨❴❘❫✷❭✩◗❲❊▼❭⑥✉ ❁❋❊❋❄ tgα = f ' ( x ◗❙❊✫●✷◗❙❊✲❳❨❊✲■➊❚❈❭✲◗✧❊✩❭♥②❡◗❙❭P❤✼❖❞❑▼▲✈❜➊❖❯■➝❤✼❖❞②❡❍✷◗❲❭✈❜❏❊✣❥♦❭✲❖ ✾❫❆❋⑩r④❖✽❤✿❖❞❊♥✉ ❁ ❇❈❄ ❭✲●✷◗✧❫➊✉❆❖❯❥♦❭✫❵❈❖❝❭✈❖❻■❆❖❻❵❈❖❝❭P❴❝▲➊s ❋❙❊♠❫❡➎✷❵✿❖q■✞❊♣●✷◗❲❖❯■ (0) ✞ x (1) =x (0) (0) (0) ❭▼❜✷❖❞❑P▲❡➞❏❜✷❊▼❫➊❭✫◗❙❊P❑▼❊ − f '−1 ( x ) ⋅ f ( x ) (0) f ( x ) ➞➊✉❂❁❋❊❋❄➊❋✐❊①●✞❫✷❭✫❚✸❊❱❜✷❊✫❚❈❊✲◗✼❥❧❖❯■✞❭❱❖❘■✷❚❛❊✲◗④❋④❊P❑✫❚ ✜✩■✞❜⑩❚❛❭✲■❆②✷❊✲■❏❚✸❭❱❴❞❭●✛ ✸ ✁❃✺♠➜❘■❣✉❂❁ ❇❍❄➊❑✲❍✣❭✩✉✭❭❏■❱✉❂s )= (0) (1) x −x 12 ➠ ■♥❑P❫❡■✷❚❈❖❘■✷❍❆❭✲◗❙❊♠❋❙❊❱❫❡➎✷❵❈❖❻■❣✉ ❁✁ ❄ ➞❡✉ ❁ ✂❈❄ r✐❖❂❋❙❫✷❴❯❍✷❵✿❖❘❭⑥■❏❍✷❥➁❊✲◗❙❖❞❑✩▲☎❚❈❖❘■✞❜❏❊➂❑✲▲✩❚❝◗❲❊♠❋✐❫✷❴❯❍✷❵✿❖❞❭♣❊✲✉❆❭P❑❾❚❈▲⑥✉ ✷s ☎✄ ❪❊✾❚✸❫❏❜✷❭ ♥❊✔✢♠❚❈❫❡■✞❺✿⑤❱❭✲● ✖✜❋④❫❨■ ■✷❍ ❊➊❋✼❚✸❊ ❑▼❫❡■❏❬✞❊✲◗✧②✷❊✲■❏❚✸▲ ●✞❊✫■✷❚❛◗✸❍ ❫❡◗❙❖❝❑❨❊ ❭P●❏◗❲❫➊✉❆❖❯❥♦❭✾❵✸❖❞❊ ❖❯■✞❖q❵✿❖❞❭▼❴❘▲❏s❅⑧✕❊ ❊✲✉❆❊✩❥♥●✞❴❻❍✜➞✜❤❩❖❛❊✕❤❩❿✿✉ ❸ ❜✷❖❘■➁❤✸❖❞②❡❍✷◗❙❭➍❜✷❊➂❥➁❭P❖ ✾❫✭❋①r❙❖❄❭✫●✷◗❙❫➊✉❆❖❯❥➁❭✲❵❈❖❝❭⑩❖❘■✞❖q❵✿❖❞❭✩❴❞▲♠✉ ❁ ❇❍❄ s❨➏✕●✞❴❛❖❘❑✒✜✲■✞❜❦❭P❑▼❊P❭➊❋❩❚❈▲➂❥➁❊❾❚❈❫✷❜✷▲❣❋✐❫❏❴❻❍✷❵❈❖❝❭ ■❏❍✷❥➁❊✲◗❙❖❞❑P▲⑥❬❆❭➂❫✭❋④❑P❖❝❴❞❭➣➜❞■✷❚❞◗❙❊☎❬✞❭P❴❛❫❡◗✧❖❝❴❝❊❱✉ ❁✺❊ ❄ r❲❖❆✉ ❁ ❄ r❙❖❆■✷❍❧❋④❊①❬✞❭❱❭✫●✷◗❙❫❡●❆❖❛❭❱❜✷❊❱❋❙❫✷❴❘❍❏❵✸❖❘❭♣❊P✉❆❭P❑❾❚❈✾ ▲ ✁ ❅ ✆ ✘ ✞ ✝✄ ✁ ❉✞❊ ❫❡➎❂❋❙❊✫◗✼❬✞▲ ❑P▲ ❜✷❭P❑✩▲ J ( x ( j) ) ⋅ x ( j+ 1) = J ( x ( j) ) x ( j) − f ( x ( j) ) ❊P❑✩❍✞❭✫❵✿❖❞❖✕❴❝❖q■✞❖❛❭✩◗❲❊ ❭✒❑✩▲✫◗✼❍✞❖♠❥✈❭✫❚❛◗✧❖❝❑P❊ ❋❙❊ ❋✐❑✲◗❲❖❞❊ ◗❙❊✩❴❞❭❾❵❈❖❝❭ ◗✧❊▼❑✫❍✷◗❙❊✫■✷❵✿▲ ❋✼❍✷➎ ❤✼❫❡◗✸❥➁❭ ❊❨❋✧❚✸❊✆❊✫❬✭❖❛❜✷❊✫■✷❚❧❑❨▲ ❴❞❭❪❤✼❖❘❊▼❑✲❭✲◗✧❊❪❖q❚✸❊✫◗❙❭❾❵❈❖❞❊❪❋✐❊✒◗❙❊✩❳❨❫✷❴❯❬✞▲✒❍✷■ ❋✐❖✿❋❲❚❛❊✲❥ ✸ ✁ ✺✡r✐❖➂❚❛❊P◗✸❥❧❊✫■ ❴❝❖q➎✞❊✩◗ ✮ ❜❏❊ ✟✞ ✠ ✡ J (x ( j) )x ( j) − f (x ( j) ) ❜✷❊ ❚❞◗❲❊✲➎❏❍✞❖❝❊✒◗❲❊▼❑P❭✩❴❞❑✫❍✞❴❝❭✫❚✸❊❨s ✇➣❭❨❴❝❑✫❍✞❴❘❍❆❴ ✵✫❭✩❑P❫❡➎✞❖❞❭✫■✷❍✞❴❯❍✞❖✴❖❯❥♥●✞❴❝❖❘❑▼▲❦❜✷❊✫❚✸❊✫◗✼❥✣❖❘■✞❭✩◗❲❊P❭✣❭✕■ ❬✞❭P❴❛❫❡◗✧❖❁❭❨❴❝❊❦❜✷❊✩◗❲❖❘❬❆❭❾❚❈❊✩❴❞❫❡◗➣➜❝■ ✁ ❑P❊▼❊❨❭❦❑✩❊✕■✷❍✎❊❨❋✼❚✸❊❦❜✷❊P❴❛❫✷❑ ❋✐❖❯❥❦●✞❴❘❍✻✭ s ✹❄❊✫■✷❚❞◗✸❍➝❍✷■✓❑▼❖❯◗❲❑✲❍❆❖❘❚☎◗❲❊P❳❨❖❈❋✸❚❈❖❘❬❢❴❛❖❯■✭❖❝❭✫◗⑩❊✩✉✭❖❈❋✼❚❈▲▼➞✥➜❞■✜❋④▲❨➞➋❫❅●❏◗❙❫✷❑P❊❡❜❨❍✷◗❲▲❧❥♥❍✞❴❯❚✽❥♦❭P❖①❋❙❖❯❥❧●❆❴❛▲♦❜❏❊✈❊❡❤✸❊P❑✩❚❛❍✭❭✫◗❲❊♦❭ ✞✠ ✡ ❖❯❚❈❊✲◗❲❭✫❵✸❖❘❖❛❴❞❫❡◗✐s❨✇⑥❫❡■❂❋❙❖❞❜✷❊✫◗❙▲✲❥➟❴❘❭♣➜❝■✭❑P❊✲●✷❍❏❚✻❍❏■♥❊✲✉❆❊✲❥♥●✞❴❯❍✂❶ ↔❭♥❤✸❖❛❊P❑❨❭✲◗❲❊❦❖❻❚❈❊✫◗❙❭❾❵❈❖❝❊❣◗❙❊✲❳▼❖✿❋✼❚✸❫❡◗❩❍✞❴❁■✞❊✩❴❞❖❘■❆❖❝❭✫◗✽●✞❫✷❭✫❚✸❊♥❤✼❖✂➜❞■✞❴❞❫✷❑✫❍✞❖❻❚❇❑✫❍◆❍✷■◆❑❡❖❯◗❩❑✲❍✜❖❻❚✵❊▼❑✓✖✞❖❯❬✞❭P❴❯❊✩■✷❚☎❋✐❊✫◗❙❖❯❊♥❤✼❫❨◗✼❥➁❭✲❚✵❜➊❖❘■❏❚❛◗❙❺✸❫ ❋✧❍❏◗❙❋✐❭♠❜✷❊♣❚✸❊✫■❂❋✐❖q❍✷■✞❊♠➜❯■♦❋❙❊✫◗❙❖❛❊❱❑✫❍✣❍❏■✣◗❙❊✲❳❨❖❈❋✼❚✿❫❨◗✾s ➋❭♣●❆◗❲❖❯❥➁❭♠❖❘❚❈❊✲◗❲❭✫❵❈❖❝❊✕❋✼❍✷◗✐❋❙❭✕❭✫◗❲❊♣❚✿❊✩■❂❋❲❖❘❍❏■✞❊▼❭♠❊P❴❞❊▼❑✾❚❛◗❙❫❨❥✈❫❡❚❈❫✷❭✩◗❲❊❱t ❁ ❇❈❄ r✐❖ ☛ ☞☛ ◗✧❊✩❳❨❖❈❋✼❚❈❫❡◗✸❍✞❴✌❭✫◗❙❊♣◗❙❊✲❳❡❖✿❋✸❚✸❊✩■✷❵❞❭➂⑤ ❁ ❇❈❄ ⑨✌❚✸② ❁❇ ❄ α s ✹✥❊✲■✷❚❞◗✼❍❧❭♠❜✷❫❡❍✞❭♠❖❯❚❈❊✲◗❙❭✫❵❈❖❝❊♠❭✫❬✞❊✲❥ ❄r❙❖ ✌✞ ✍ ✡ ✠ ✟✎✌✞ ✍ ✡ ✑✏ ✡ ❆✶ α ❄❊✲❚❈❑❡sP➏❦❑P❊❡❋✼❚✥❑P❖❘◗❲❑✩❍✞❖q❚ ✞✁✍ ✡ ✐❋ ❊ ■✷❍❏❥♦❊❨r✼❚✿❊✡❑✲❖❘◗❲❑✩❍✞❖ ❚❞❍✞❴❦❊❨❑✗✖✞❖❯❬✞❭✲❴❛❊✫■✷❚➁❜➊❖❈❋❙❑✾◗❙❊✲❚❲s ↔❭☛❤✸❖❛❊P❑✩❭✫◗❙❊✆❖❘❚❛❊✲◗✧❭✲❵❛❖❛❊☛❋✐❊❪❑P❭✩❴❞❑✫❍✞❴❝❊❨❭✲❳▼▲✒●✞❭✩◗❲❭✲❥➁❊✫❚❝◗❙❖❞❖✣❑✲❖❘◗✧❑✲❍✞❖❯❚❞❍✭❴❘❍✞❖ ❊P❑✟✞✖ ❖❻❬✜❭▼❴❘❊✲■❏❚❄❜➊❖✿❋❙❑✲◗✧❊❾❚✥r❙❖✻❋✐❊☎◗❙❊✲❳❡❫✷❴❯❬✭▲①❍✷■♥❑P❖❯◗❲❑❾❍✞❖❯❚✂❴❝❖❯■✞❖❛❭✫◗✐s ➏❦❑❨❊▼❭❨❋✸❚✸▲◆●✷◗❙❫❏❑▼❊P❜❡❍✷◗❙▲➝❋✐❊♦●✞❫✷❭✩❚❞❊➝❭✲●❆❴❝❖❞❑✩❭◆❍✷■✷❍✞❖⑥❑P❖❘◗❙❑✫❍✞❖q❚♣◗❙❊✲❳▼❖✿❋✧❚✿❖q❬ ■✞❊P❴❞❖❘■✭❖❞❭✲◗❣❫❡◗❙❖❞❑✟✜✫❚♠❜✷❊✎❑P❫❡❥♥●✞❴❛❖❞❑✩❭✫❚➂❭✲◗❦❤❩❖ ❭P❑❨❊❡❋✸❚❈❭❏s✎✇➣❖❘◗❙❑✫❍✞❖q❚❛❍✞❴ ❊▼❑✓✖✞❖q❬✞❭❨❴❝❊✫■✷❚☛❜✷❖✿❋❙❑✩◗❲❊✲❚✆❴❞❭↕❖❻❚❈❊✲◗❲❭✫❚✿❖❞❭ ❭▼❴✱❍✷■✷❍✞❖✒◗❲❊P❳❡❖✿❋✼❚✸❫❡◗➤❑P❫❡■✷❚❞◗❙❫✷❴❞❭✫❚✆➜❝■ ❑✲❍❏◗❙❊✲■❏❚☛❑✫❍ ✑☛ ✓✒ ✕✔ ✍ ❑P❭✩◗❲❭▼❑✫❚❈❊✲◗❲❖❞❋✧❚✿❖❞❑▼❭ ✞ ✡✸ ✺♠❑▼❫❨■✷❵✸❖❯■✞❊①◗❲❊✲❳❡❖❞❋❲❚❈❫❡◗✼❍❆❴✻❑✫❍❣◗✧❊✫❳❨❖❈❋✸❚✸❊✫■✷❵❛❭ ✑✖ ✡ ✏ R (k ) = ∂u ∂i ➜❞■❧❋✐❊✲◗❲❖❘❊♠❑✲❍❧❋✼❍✷◗✐❋❙❭❱❜✷❊①❚✿❊✫■✞❋④❖❯❍✷■✞❊ ❫❨➎✷❵✸❖❯■✷❍✷❚❈▲❣❑P❭✕➜❝■❅❊✲✉❆❊✲❥♥●✞❴❯❍✞❴✌●✷◗❲❊▼❑❨❊P❜✷❊✫■✷❚✼☞s ✹❄❊✾■✷❚❛◗✸❍✈❍✭■✈◗❙❊✲❳❡❖✿❋✼❚✸❫❡◗➣❑P❫❡■✷❚❞◗❙❫✷❴❘❭✩❚✥➜❞■✈❚❈❊✫■❂❋❙❖❘❍✷■❆❊❣❑✩❍➁❑❡❭✫◗❲❭❨❑✲❚❛❊✲◗✧❖❈❋✸❚✸❖❘❑▼▲ ❑P❖❯◗❙❑✲❍❆❖❻❚❞❍✞❴✽❊❨✗❑ ✖✭❖❯❬✞❭P❴❝❊✫■✷❚⑥❜✷❖❞❋✐❑✲◗❙❊✾❚➣❭✩◗❲❊❧●✞❭✫◗❙❭✲❥❧❊✫❚❛◗❙❖❘❖ G (k ) = ∂i ∂u r❙❖ ✚✙ ✞ ✗ ✡ ✂ ✞✗ ✡ ✏ ✑✘ ✡ ✸❂✞✻✺ ❅✥⑧❣❭P❑✩▲❧◗❲❊P❳❨❖❈❋✸❚✸❫❨◗✼❍✞❴✽❭✫◗❙❊♦❑✩❭✫◗❙❭P❑❾❚❛❊P◗❲❖✿❋✧❚❞❖❞❑▼❖ u( k ) ❴❞❖❯■✞❖❛❭✫◗❲❊①●✞❊①●✞❫❡◗✼❵❛❖❘❍❏■✞❖❈➞❨●✞❭✫◗❲❭P❥❧❊✫❚❛◗❩❖❞❖✜❑P❖❘◗✧❑✲❍✭❖❯❚❛❍✭❴❯❍✞❖❂❊✩❑✓✖✞❖❯❬✞❭P❴❝❊✫■✷❚✂❜✷❖❛❋✐❑✫◗❲❊✲❚✥❋❙❊⑥❜✷❊✲❚❛❊✩◗✸❥♦❖❯■✞▲⑥❑▼❭⑥➜❯■♥❤✸❖❞②❡❍✷◗❙▲❏❶ 13 ✠ i( k ) ✹✻◗❙❖❯■ ✷❍ ❚❈❖❝❴❘❖❝❳❨❭✩◗✼❊❨❭ ❑✩❖❯◗❙❑✫❍✞❖❻❚❞❍✞❴❯❍✞❖✟❊P❑✗✖❆❖❘❬✞❭✲❴❛❊✾■✷❚ ❜✷❖❈❋✐❑✫◗❲❊✫❚ ❭✫●✞❴❝❖❞❑P❭✲◗❲❊P❭ ❋✐❖❯❥❦●✞❴❛❖❞❤✸❖❞❑▼▲⑥❑▼❫❨■❂❋✐❖❛❜❏❊✲◗❙❭✫➎✞❖❘❴❩s ❥❧❊✫❚✸❫✷❜❏❊❡❖ ➍❊✼✢♠❚❛❫❡■✞❺✸⑤⑥❭✲✘● ✖❂❋✐❫❡■ ❋❙❊ ✘ ✂✁☎✄✝✆✟✞✡✠☞☛✌✆✎✍✑✏✒✠✔✓✕✞✖✏✘✗✙☛✛✚✜✠✢✍✣✆✟✤✥✚✜✏✖✓✔✦✥✓★✧✟✓✪✩✫✏✭✬✮✚✯✓✡✧✰✏✘✬✱✆✰✠✲✆ ✬❄❖❞❊✖❑❨❖❻◗✧❑✲❍✭❖❯❚❛❍❆❴➣◗❙❊P❜❡◗❙❊❨❋✐❫❡◗✕❥➁❫❡■❆❫✷❭▼❴❯❚❈❊✲◗✸■✞❭✩■✷❵❈▲◆❤❩▲✲◗❙▲✎❤✼❖❞❴❘❚❞◗✸❍☛❜❏❖❘■✆❤❩❖❛②❡❍❏◗❲❭✱❿✸❭ ❸ ❍❆■✞❜✷❊✎❑P❭P◗❲❭❨❑✫❚✸❊✫◗❲❖✿❋❲❚❈❖❝❑P❭✎❜✷❖❛❫❏❜✷❊❨❖ ❊❨❋✧❚❈❊♣❜✷❭✫❚✿▲⑥➜❝■♥❤✸❖❞②❡❍✷◗❲❭♠❿❛➎ ❸ r❙❖✜❑❨❭✫◗❲❭▼❑✫❚✸❊✫◗❲❖✿❋❲❚❛❖❛❑✲❭♣❜❏❊⑥❚❞◗❙❭✫■❂❋✐❤✸❊✩◗✂➜❞■➍❚✿❊✫■❂❋✐❖❯❍✷■✞❊❨➞➊❊❡❋✼❚❛❊♠❜✷❊❡❋❙❊✫■✞❭✫❚✸▲➣➜❘■✣❤✸❖❛②❡❍❏◗❙❭❱❿✼❑ ❸ s ⑧✕❭❨❑P▲ u1 ( t ) = sin ω t ❿✼❤❩❖❝②✜s↔❜ ❸ ❭✲❚❞❍✷■✞❑P❖⑥❜✷❖❻■✆❑P❭✫◗❙❭▼❑✫❚❈❊✲◗❲❖✿❋✼❚✿❖❝❑P❭✎❜✷❊❅❚❞◗❲❭✩■❂❋❙❤✼❊✫◗⑩➜❞■✱❚❈❊✲■✜❋✐❖❘❍✷■❆❊➁◗❙❊✲❳P❍❆❴❻❚❈▲✖❑P▲ ✢✞ ✳ ✸ ✺✖❬✞❭ ✏ ❭✫❬✞❊❨❭❱❤✸❫❏◗✸❥➁❭❱❜➊❖❘■✣❤✼❖❞②❡❍✷◗❲❭♠❿✸❊ ❸ s ➠ ■❣❭▼❑P❊❡❋✼❚✻❑▼❭❾❳✕❋④❊✫❥♥■✞❭✩❴❯❍✞❴❂❜✷❊⑥❖❛❊❨r❙❖❘◗❲❊❱❭✫◗❙❊❱❫❣❤✸❫❡◗❩❥♦❭❱❜➊❖❞❤✼❊✫◗❙❖❻❚❛▲➂❤✸❭✩❵❛▲➂❜❏❊♣❑P❊✩❴❂❜✷❊❱❖❻■❏❚❛◗❙❭✫◗❲❊ ❜❏❭✲❚❈❫❡◗❙❖❯❚✸▲❱■✞❊▼❴❘❖❘■❆❖❛❭✫◗❲❖❻❚❈❭✲❵❛❖❝❖✱❑P❭✲◗❲❭P❑❾❚❈❊✩◗❲❖❈❋✼❚❛❖❝❑❨❖❯❖❁❜➊❊♠❚❛◗✧❭✲■❂❋❙❤✼❊✫◗✐s✭⑧✕❭✩❑P▲ u1 ( t ) = 1 + sin ω t ❿✸❤✼❖❞②✜s✞❤ ❸ ❭✲❚❞❍✷■✞❑P❖❁❋❙❊✫❥❧■❆❭▼❴❯❍✞❴❄❜✷❊ ❖❞❊❨r✐❖❘◗❩❊❱❊➊❋❲❚❛❊♠❋✐❖❘■✷❍✜❋④❫❏❖❛❜✷❭P❴❂❿✼❤✸❖❞②✜s➊② ❸ ❑▼❭♠r✐❖✞❑▼❊✲❴❂❜✷❊❱❖❘■✷❚❞◗❲❭✩◗❲❊✷s✷❉✷●✷❍✷■✞❊✩❥➟❑❨▲ ✞✛✘✰✌ ✏✓✎✥✌ ✞ ✏ ✎✔✒✵✣✴ ✏ ☛ ✓✷✏ ✶ ✘ ✒✽✙✆ ✏✓✘ ✏✹✸ ✎☎❑✫❍♥❫❦❋✐❖❘■❆②❡❍✷◗❲▲ ❋✧❍❏◗❙❋✐▲✕❜✷❊❣❋❙❊✫❥✣■✜❭▼❴❂❿✼❑✫❍➁❊✕❋✐❭✲❍❧❖✻✺☎❬✭❭✫◗❙❖❛❭✫➎✞❖❘❴❂➜❘■♥❚✿❖❯❥♥● ❸ ✛ ✞✛✘ ✌✥✤ ✏✻✛✼ ✘ ✒✽✸✾✴✑✿ ✆✷✸ ☛ ✒ ✄✱✘❀★✸ ✆✜✒ ✄✱✏✶✌✍✡✏ ❁❀✸ ✌✑✿✘✼✛✎✑✏✶✌ ✒ ✎✙✿ ☛✙✕✝✞✛✘✛☛❂✸ ✎✑✒ ✛❃✼✛✎ ✄❄✸ ☛✍✒ ✄✱✘❀✸★✆✟✞✱✆✟✞ ✏❅❁ ✒ ✏ ✘ ✓✎✣✸ ✎✑✒ ❅ ➠ ■✎❑✲❭✩❳✩❍✞❴❄➜❝■✎❑✲❭✲◗❙❊✕❍✷■✎❋❙❖❯■✞②❡❍✷◗✽◗❙▲❨❋✧●✷❍✷■✜❋➣■❏❍✎❭✲◗❲❊❦❭❨❑✩❊✲❊▼❭➊r❙❖❁❤✸❫❡◗❩❥♦❭❦❑✫❍✎❋④❊✩❥❦■✞❭P❴❘❍✞❴ ✏ ✏ ✏ ❜❏❊✣❖❯■✷❚❞◗❙❭✲◗❲❊❦❋✧●✷❍✷■✞❊✫❥➌❑✩▲ ✌ ✏✙✎✔✌✽✞✛✏ ✟✞✱✆ ✛ ✞✛✘ ✌✥✤✦❆✏ ✼✛✘ ✒✑✸✾✴❇✿ ✆❈✸ ☛✍✒☎✄✱✘❀✸ ✆✶✒ ✄❄✸ ✎✑✏❈s✎✹❄❊✫■➊❚❞◗✼❍✈❍✷■✎❑P❖❘◗❲❑✫❍✞❖❯❚✽❜❏❭❾❚↔❤✿❍✷■✞❑✫❵✿❖❞❫❡■✞❭✫◗❙❊▼❭➍❴❛❭ ❋✐❊✫❥♥■✞❭P❴❝❊①❥➁❭✲◗❲❖✌❋❲❭✲❍➍❥♦❖❘❑▼❖❂❜✷❊✫●✞❖q■✞❜✷❊♠❜✷❊①●✞❭✫◗❲❭✲❥➁❊✫❚❛◗❲❖❞❖✜❋✐❊✫❥♥■✞❭✩❴❯❍✞❴❯❍✞❖❂❜✷❊⑥❖❘■✷❚❞◗❲❭✫◗❲❊✭s ■✈◗❙❊✲❳❡❖✿❋✧❚✿❫❨◗✽■✞❊▼❴❞❖❯■✞❖❛❭✫◗❇❜✷❖❯●✞❫✷❴❞❭✲◗➣❤❈❍❏■✞❑✲❵❈❖❝❫❨■✞❊▼❭✲❳❨▲❣❴❝❭❦❋✐❊✲❥♥■✞❭P❴❝❊♠❥➁❖❝❑P❖❄❜✷❭❨❑▼▲♠●✷❍✷❚❈❊✲❥ ❭✫●✷◗❙❫❨✉❆❖q❥♦❭➍❜✷❊✩●✞❴❝❭➊❋❙❭✫◗❲❊▼❭ ➉ ✏ 14 ●❏❍✷■✞❑✫❚❛❍✞❴❯❍✞❖☛❜✷❊ ❤❈❍❏■✞❑✲❵❈❖❞❫❡■✞❭✫◗❙❊ ●❆❊ ❑✩❭✫◗❙❭▼❑✾❚✿❊✫◗❙❖✿❋❲❚❛❖❛❑P▲ ❑✲❍ ❜✷❊✫●✞❴❝❭❨❋❙❭✩◗✼❊❨❭ ●✞❊ ❚✿❭✫■✞②✷❊✫■✷❚❈❭↕➜❞■ ●✷❍✷■✞❑✫❚❛❍❆❴ ❋✼❚❈❭❾❚❈❖❛❑↕❜✷❊ ❤✿❍✷■✞❑✫❵✸❖❞❫❡■✞❭✫◗❙❊➊s➣⑤♣❊✲❳❡❖✿❋❩❚❈❫❡◗✼❍❆❴✣❑P❫❡■✷❚❞◗❙❫✷❴❘❭✲❚❣➜❞■✡❚❈❊✲■✜❋④❖❯❍✷■✞❊✆❜✷❖❯■ ❤✸❖❞②❡❍✷◗❙❭✆❜✷❊✒❥✈❭P❖ ❫❆❋✎❑✲❍✟❑✩❭✩◗❲❭▼❑✩❚❛❊✲◗❲❖✿❋✼❚✸❖❘❑▼❭ ✯✶✛✸✹✞✻✺ ❋❙❊ ❑P❫❡❥♥●✞❫❡◗❩❚✸▲❱❜✷❖❯■❣●✷❍❏■✞❑✲❚✂❜✷❊①❬✞❊P❜✷❊✫◗❲❊♠❭✩❴✌❋❲❊✲❥♥■✭❭❨❴❻❍✞❴❯❍✞❖❂❿❈❬✞❭✫◗❲❖❞❭❾❵❈❖❛❖❘❴❝❊❱❜✷❊☎❚✸❊✫■❂❋❙❖❘❍❏■✞❊➊r④❖✞❑✫❍✷◗❙❊✫■✷❚❂➜❯■ ❙❍✷◗✸❍✞❴❆●✷❍✷■❆❑P❚❘❍✞❴❯❍✞❖✌❋✧❚❞❭✩❚✿❖❝❑ ✞ ✏ ✡ ✞ ❜❏❊✘❤✿❍✷■✞❑✫❵✸❖❞❫❡■✞❭✫◗❙❊ ❸ ❑P❭ ❍✷■➤◗❙❊✲❳❏❖❞❋✼❚✸❫❡◗◆❴❛❖❯■✞❖❝❭✫◗✎❑✫❍ ❑❨❫❡■✞❜❨❍✞❑✲❚❈❭✫■✷❵✸❭ ❑▼❭✫◗❲❊✡❋④❊❪■❏❍✷❥➁❊❡r✸❚✸❊ dg( u) = tg α du p. s. f . ❑P❫❡■✞❜❨❍✞❑✩❚✸❭✫■✷❵❈❭✱❊▼❑✓✖✞❖❯❬✞❭▼❴❘❊✩■✷❚✿▲✒❜❏❊ ❋❙❊✲❥♥■✞❭✲❴♣❥♦❖❞❑❪r❲❖✕❜✷❊✫●✭❖❯■✞❜✷❊ ❜❏❊❢●✷❍✷■✞❑✫❚❞❍✞❴✕❋✧❚✸❭✾❚✸❖❘❑✒❜✷❊✒❤❛❍❏■✞❑✩❵✿❖❞❫❡■✞❭✩◗❲❊❏✯ s ✌✹ ❊✩■✷❚❞◗✸❍✘❍❏■ G= ◗✧❊✩❳❨❖❈❋✼❚❈❫❡◗➍❑P❫❡■✷❚❞◗❲❫✷❴❛❭✩❚①➜❞■✓❑✫❍✷◗❙❊✫■✷❚♠❑✫❍ ❑P❭P◗❩❭❨❑✫❚✿❊✫◗❙❖❛❋✧❚❛❖❛❑✲▲❅❍✭⑨❄❤❩❿✼❖ ❸ ❋❙❊✎❜✷❊P❤✼❖❘■❆❊❡r✸❚✸❊♦◗❲❊✲❳❡❖✿❋❩❚❈❊✫■✷❵✿❭❅❊▼✓❑ ✖✞❖❯❬✞❭▼❴❘❊✩■✷❚✿▲✎❜✷❊➝❋✧❊✫❥♥■✞❭▼❴ ❥➁❖❞❑ R= d f (i ) s . . . p s f di ✬❄❖❞❊ ❍✷■ ❚❞◗❲❭✩■✭❳❡❖✿❋✸❚✸❫❡◗ ), u BE = uˆ BE (i , u B CE ❋✐❊✫❥♥■✞❭P❴↕❥❧❖❝❑ u ❋✐❊ i ) = iˆ (i , u C C B CE ❜✷❊✲❳P❬❆❫✷❴❘❚❈▲ ,i ,i ,u ✂✁ BE 0 B 0 C 0 CE 0 ⑤❱❊✲❳P❍✞❴q❚✸▲❡❶ BE =u û r❙❖ ❑❨❭ ❥✣❍✜❴❻❚❛❖❘●✞❫❏❴❛❭✫◗ ❑✫❍ ❑P❭✫◗❲❭▼❑✫❚✿❊✩◗❲❖✿❋❲❚❛❖❛❑P❖❝❴❘❊✭❶ s✔✹✌❊✩■✷❚❞◗✸❍➁❭✕❜✷❊✫❚✸❊✫◗✼❥➁❖❘■❆❭➂●✞❭✫◗❙❭✲❥❧❊✫❚❛◗❙❖❘❖✌❑❨❖❻◗❲❑✫❍✞❖❻❚❞❍✞❴❯❍✞❖❄❊P✗❑ ✖✞❖❯❬✞❭✲❴❛❊✫■✷❚➋❜✷❊ iˆ C BE ◗❲❊P❳▼❖✿❋✼❚✸❫❡◗ ➜❯■ ❋✐❊✫◗❲❖❛❊ ✮➋✼ ❭ ✣❆❴❛❫❡◗ ➜❞■ ④❍❏◗✼❍✞❴ ●✌sq❋✫s ❤✧s ❿✧❜✷❊✫❚✸❊✫◗✼❥➁❖❻■❆❭✲❚ ❜✷❊ ✞ ✬❄❖❝❊❨❑P❭✲◗✧❊✎❚❈❊✫■❂❋❙❖❘❍✷■❆❊✓r✐❖❱❑✫❍✷◗❙❊✫■✷❚⑩❭✲◗✧❊❢❫✆❑P❫❡❥♥●✞❫❡■✞❊✫■✷❚❈▲✱❑▼❫❡■✜❋❲❚❈❭✫■✷❚✸▲✱❿✸❑❨❫❡◗❲❊➊❋✼●✷❍✷■✷❳❡❭✫❚✸❫❏❭✲◗❲❊ ●✻sq❋✫s ❤✧s✷r✐❖✷■✞❫❡❚❛❭✩❚✿▲⑥❑✲❍✣❖❘■✞❜❏❖❛❑✲❊▼❴❘❊ u ❥➁❫✷❜✷❊❨❴❝❭✫❚ BE 0 ⑦ ❸ r❙❖✜❫➍❑❨❫❡❥✣●❆❫❡■✞❊✫■✷❚✸▲①❬✞❭✫◗❙❖❞❭✲➎❆❖❝❴❞▲①➜❝■❣❚❛❖❻❥♥●♥❿✼❜❏❊⑩❋❙❊✫❥♥■✞❭▼❴ ❸ ➞➊❭P❜✷❖❞❑✩▲❏❶ +u be , i B =i +i , B0 b i C =i +i , C0 c u CE =u CE 0 +u s ce ∂ uˆ BE BE ⋅ + i p.s. f . p.s. f .u ce BE BE 0 ∂i b ∂u B CE ∂ iˆ ∂ iˆ C + C p.s. f .i + i =i p.s. f .u ce C C0 ∂ i b ∂u B CE u sau u =u = r i + αu , ce be b b ∂ uˆ BE r = p.s. f . , ∂i b B α= + i ∂ uˆ ∂u ∂ uˆ u = β ⋅ i + ce c r b c BE p.s. f . , CE ⑤❱❊✩❳✩❍✞❴❻❚❈▲⑥❑✩❖❯◗❙❑✲❍❆❖❘❚❞❍✭❴❂❊✩✓❑ ✖✞❖❘❬❆❭▼❴❘❊✲■✷❚✂❜✷❊❱❋✐❊✫❥❧■❆❭✩❴✷❥♦❖❞❑➊❶ 15 β = unde ∂ iˆ ∂i C B p.s. f . , 1 r = ˆ c ∂i C p.s. f . ∂u CE ➏ ❑❨❊❡❋✼❚➂❑❨❖❻◗❲❑✫❍✞❖❘❚❱●✞❫✭❭❾❚❈❊➝❤✼❖➣❍❏❚✸❖❘❴❛❖❻❳❡❭✾❚✕❜❏❭▼❑P▲◆●✷❍✷■✞❑✩❚❝❍❆❴♠❜✷❊➝❤❈❍✷■✞❑✫❵✸❖❞❫❡■✞❭✫◗❙❊➝❋✐❊➝❜✷❊✲●❆❴❝❭➊❋❙❊✩❭✲❳❡▲❅➜❞■❏❚❛◗❙❺✸❫ ❳❨❫❡■✞▲✱❜✷❖q■ ❦ ●❆❴❛❭✫■✷❍✞❴❦❤✼❖❞❊▼❑✲▲✲◗❙❊✲❖✣❑❨❭✫◗❩❭❡❑✫❚✿❊✫◗❙❖✿❋❩❚❈❖❝❑P❖✕●✞❊✲■❏❚❛◗✼❍➤❑P❭✫◗❙❊ ●❆❭P◗❲❭✫❥❧❊P❚❘◗❲❖❛❖ ●✞❫❨❚✈❤✼❖❦❑P❫❡■❂❋❙❖❛❜❏❊✲◗❙❭✫❚✿❖➍❑▼❫❨■❂❋✧❚✸❭✫■✷❵❈❖❈s ➠ ■ rb , α , β , rc P❭ ❑❨❊✩❊P❭❡r❙❖♠❥♦❭✫■✞❖❞❊✲◗❲▲✆❋❙❊ ●✞❫❏❭✲❚❈❊ ❜✷❊✾❚✸❊✩◗✸❥➁❖❯■✭❭❢❍✷■ ❑P❖❘◗❲❑✩❍✞❖q❚♥❊❨❑✓✖✞❖❘❬❆❭✩❴❞❊✲■✷❚✣❜✷❊❪❋❙❊✫❥♥■✞❭▼❴♠❥✈❖❘❑✓●✞❊✩■✷❚❞◗✸❍➤❫❡◗❙❖❞❑P❊✓◗❙❊✲❳➊❖❛❋✼❚✸❫❨◗ ■❆❊▼❴❞❖❘■❆❖❝❭✫◗✾s ➏✕■✞❭❨❴❝❖❘❳❨❭ ❍❏■✷❍✞❖✱❑✩❖❯◗❙❑✫❍✞❖❻❚✓❑P❭✲◗✧❊ ❤✿❍✷■✞❑✫❵✿❖❞❫❡■✞❊✩❭P❳❨▲ ❴❛❭ ❋❙❊✲❥♥■✭❭P❴❛❊✟❥➁❖❝❑P❖✱❋❙❊ ●❆❫✷❭✲❚❈❊ ❤✼❭P❑✩❊✟●✞❊ ❍✷■ ❑▼❖❯◗❲❑✫❍✞❖❘❚ ❊P❑✟✞✖ ❖❻❬✜❭▼❴❘❊✲■❏❚✌❴❞❖❻■✞❖❘❭✩◗➋❑▼❫❨■❂❤✸❫❡◗✼❥ ❭P❴❞②✷❫❡◗❙❖q❚❝❥♥❍✞❴❯❍✞❖✸❶ ➙Ps ✞❉ ❊❅❜✷❊✲❚❈❊✫◗✼❥➁❖❻■✞▲✈●✷❍✷■✞❑✫❚❛❍❆❴♣❋✼❚✸❭✾❚✸❖❘❑◆❜✷❊➁❤❈❍✷■✞❑✫❵✸❖❞❫❡■✞❭✫◗❙❊❅❑▼❫❡■✜❋④❖❞❜✷❊✫✫◗ ✫✜ ■✞❜✱❑▼❖❯◗❙❑✫❍✞❖❯❚❛❍✞❴❇❊✩✉❆❑P❖❻❚❈❭❾❚♣❜✷❊◆❋✧❍✷◗④❋④❊P❴❞❊◆❜✷❊♦❑✩❍✷◗❲❊✩■✷❚ ❑P❫❡■✷❚❈❖❘■❏❍✷❍➌❿✧❋✼❍✷◗✐❋❙❊ ❖❯■✞❜✷❊✫●✞❊✲■✜❜✷❊✫■✷❚✸❊➟❑✲❍ ●❂❭✫◗❙❭✫❥➁❊❾❚❞◗❙❖❝❖✒❑P❫❡■❂❋✧❚❛❭✲■❏❵✸❖➝➜❝■ ❚❞❖❘❥♥● ❸ ➞✖❋✧❍✷◗④❋④❊P❴❝❊➟❜✷❊ ❋❙❊✲❥♥■✞❭❨❴✱❤✼❖❞❖❻■❆❜ ●✞❭❨❋✐❖❯❬✞❖❝❳P❭✲❚❈❊❡s ❂s ❉✞❊♦❜❏❊✲❚❈❊✲◗❩❥✈❖q■❂▲✈❑P❖❻◗❙❑✫❍✞❖❻❚❞❍✞❴☎❊✲✗❑ ✖❆❖❘❬✞❭P❴❝❊✫■✷❚⑥❜❏❊➁❋✐❊✫❥♥■✞❭P❴✵❥✈❖❘❑✈❭❡❋✐❫❏❑▼❖❘❭✲❚☎●✻sq❋ s➇❤④s❄❜❏❊P❚❝❊P◗✸❥➁❖❯■✞❭❾❚➣❴❞❭✣●❆❍❏■✞❑✲❚❞❍✞❴♠➙❣●✞❊✫■✷❚❞◗✸❍ ❄ ❤✼❖❞❊P❑▼❭✫◗❲❊①◗❙❊✫❳❨❖❈❋✼❚❈❫❡◗❄■✞❊❨❴❘❖❘■✭❖❞❭✲◗ s ✞s ❉✞❊➝❤✸❭P❑❡❊✎❭✫■✞❭❨❴❝❖❯❳❨❭✎❑▼❖❯◗❙❑✫❍✞❖❻❚❞❍✞❴❯❍✞❖♣❊✲✟❑ ✖✞❖❻❬✜❭▼❴❘❊✲■✷❚➂❜✷❊✱❋❙❊✲❥♥■✞❭P❴①❥➁❖❝❑➝❫❡➎✷❵❈❖❘■❏❍✷❚♣●❏◗❙❖❘■ ❖❘■✷❚❈❊✫◗❙❑▼❫▼■❆❊▼❑✫❚✸❭✫◗❲❊P❭❢❑P❖❯◗❲❑✲❍✞❖❯❚✿❊P❴❞❫❡◗ ❊P✟❑ ✖✞❖❻❬✜❭▼❴❘❊✲■✷❚❈❊✕❜✷❊❦❋❙❊✲❥♥■✭❭❨❴✻❥❧❖❛❑❦r④❖✂❭➂◗❙❊✲❳▼❖✿❋✧❚✿❫✷❭✫◗❙❊P❴❛❫❡◗❇❴❞❖❘■❆❖❝❭✫◗❙❊❏s❨➏❦❑P❊❡❋✼❚➋❑P❖❻◗✧❑✲❍✞❖q❚↔❊❨❋✼❚✸❊✕❊✲✉❆❑P❖❻❚❈❭✲❚✥■✷❍✷❥➁❭✩❖✂❜✷❊✣❋✧❍✷◗④❋④❊✲❴❛❊ ❜✷❊➤❋✐❊✲❥♥■✞❭P❴❛➞♥❋✼❍✷◗✐❋❙❊❨❴❝❊ ❜❏❊✡❑✲❍✷◗✧❊✲■✷❚◆❑▼❫❡■❏❚✿❖❻■❏❍✷❍➟❤✸❖❛❖❯■✞❜✟●✞❭❨❋✐❖❘❬✞❖❻❳➊❭✲❚❛❊❏s♥❉✞❫✷❴❯❍➊❵✿❖❞❭✡❫❏➎✷❵❛❖❘■❏❍✷❚✸▲ ❊❡❋✼❚❈❊✡❑▼❫❡❥♥●✞❫❨■✞❊✲■❏❚✸❭ ❬✞❭✫◗❙❖❞❭✲➎❆❖❛❴❘▲①➜❝■❣❚❛❖❘❥♥●♥❭①◗❲▲❨❋✼●✷❍✷■❂❋✧❍✞❴❯❍✞❖❂❑P❖❘◗❲❑✫❍✞❖❻❚❘❍✞❴❯❍✞❖❩s ❂s ❉✞❊ ❬✞❊✫◗❙❖❝❤✼❖❞❑✩▲↕❜✷❭P❑▼▲ ❑▼❫❨■✞❜✷❖❯❵✸❖❘❖❛❴❞❊ ❜✷❊↕❤❈❍✷■❆❑✲❵❈❖❛❫❨■✞❭✲◗✧❊ ❴❛❭ ❋❙❊✲❥❦■❂❭P❴✓❥❧❖❞❑ ❋✼❍✷■✷❚✘❋✐❭❾❚❈❖❈❋❙❤✸▲P❑✲❍❏❚✸❊ ●✞❊✫■✷❚❛◗❩❍ ❚✸❫✷❭✾❚✸❊ ◗❙❊✲❳❨❖❈❋✼❚❈❫✷❭✲◗❲❊❨❴❘❊⑥■❆❊✩❴❞❖❘■✞❖❘❭✲◗✧❊❏sP⑧✕❭P❑✩▲❱❊P✉❆❖❞❋❲❚❈▲☎❍✷■✈❋✐❖q■✞②❏❍✷◗✂◗❙❊✩❳❨❖✿❋✧❚✿❫❡◗✥■✞❊✩❴❞❖❘■❆❖❝❭✫◗❁●✞❊✫■✷❚❞◗✼❍♥❑✲❭✲◗❙❊⑥❭P❑▼❊❨❋✼❚✸❊⑥❑▼❫➊■✞❜✷❖q❵✿❖❞❖❦■❏❍✈❋✧❍✷■✷❚ ➜❝■✞❜✷❊✫●✞❴❞❖❘■✞❖❯❚✿❊①◗❙❊❾❳P❍❆❴❘❚❛❭✲❚❞❍✞❴✞❭✲■✞❭✲❴❛❖❻❳❡❊❨❖✭❜✷❊❱❴❛❭①●✷❍➊■✞❑✫❚❞❍✞❴ ❣❊▼❋✧❚❛❊❱❖❻■✞❑❨❫❡◗❲❊P❑✲❚❲s 16 ✁✄✂✆☎✝✟✞✠☎☛✡☛✌☞✎✍✑✏✒✟✓✕✔✗✖✘✄☎✙✍ ✚✄✛ ✜✣✢✥✤✧✦✩★✫✪✭✬✯✮✱✰✳✲✱✤✵✴✶✬✷✤✹✸✺✤✻★✫✰✄★✱✪✼✰✽✦✩✾✌✿✧✴✯✮❀✤✻✤✻✿❁✾❃❂ t ✉❆❖✿❋❲❚❈❊✫■✷❵✸❭❦❋❙❖✂❍✷■✞❖❞❑✩❖❯❚✿❭✫❚❈❊✩❭✣❋✐❫❏❴❘❍✷❚❈❖❝❊P❖✻❍✷■❏❍✞❖❁❑P❖❻◗❙❑✫❍✞❖❯❚↔❜✷❖❯■✞❭✫❥➁❖❝❑❣❊❨❋✸❚✸❊➍❴❝❊P②✷❭✲❚❛❭❣❜✷❊➍❊P✉❆❖❛❋✧❚❛❊✲■✷❚❈❭✕❊P❑✲❍❆❭P❚❝❖❛❊P❖❄❜✷❊❦❋✧❚✸❭✫◗❲❊❣❖q■ ➋ ❊✲✉❆❖❈❋✼❚❈❭✕❭✫❚❛❍✷■❆❑▼❖❄❖❯■❅❴❞❖❻❚✿❊✩◗❩❭✩❚❝❍✭◗❲❭⑩❥❧❭✲❚❈❊❾❥➁❭✩❚❞❖❞❑▼❭❦❋④❊⑩❭✲◗❙❭✾❚✿❭➍❑▼❭❏❶✜❜✷❭❨❑✩❭➍❤ ❤❩❫❡◗✼❥❧❭⑩■✞❫➊◗✸❥♦❭P❴❞❭❏s✭⑧✕❭✲❑❡❭⑩❊▼❑✫❍✞❭❾❚❈❖❞❭❅❄ x = f ( x ,t ) ❊❨❋✧❚❈❊ ✴❖❯●❂❋✐✓❑ ✖✞❖❻❚❛❳❡❭✫■✞❭✕❿✿●✞❊✲■❏❚❛◗✼❍➁❫❡◗❲❖❞❑P❊♠❇✉ ❆✄❈❀❉❋❊✼●❍❈■❉❃❏▲❑■❉◆▼P❖❘◗ ❖❳❲✼❨❃❩❬❈❀❉ f ( x 1 , t ) − f ( x 2 , t ) ≤ k x 1 − x 2 ❙✭❚❱❯ • ❖❭❈❁◗❪❖ ❏▲❑✻❫❵❴✣❖ ▼✩❛❜❉ ❉◆❴ ❴P❝❘❈❁❉ ❴✱▼✩❴❬❞ ▼✫◗❡❉❢❴❬❞✻❣✧❩✁❤✐◗❥❝❦❖❭❈✧◗✻❖ ❉❜❞✧❏▲❑❧❫♠❫♥❴✫❑■♦✭❉ ❉♣◗✻❴❭❤❃❴✫◗ ▼✱❉✵❖❭▼ ❴✫◗❪❉❜❴ ❖♥❈✻◗✻❴✫❑■❖✣❴✫❑■❖✣❏ ❚ ❙ ❯ ❚ ❯ ❙✭❚ ❙✭❚ ❚ ❙✺❚ ❙ ❯ ❈❀❏✺❛ ◗❥❉❥❖ ❉❜▼❭❴rq✁❖ ◗◆❑ ❏▲❑✹❉❥▼✩❖s❈✧◗❡❴✫❑■❖❦❉ ❉t◗❡❉◆❴✱❛◆❴ ❙ ❙✭❚ ❚ ❙ ❚ x 0 = x( t 0 ). ✉✟❖✈▼P❏ ❈❁◗❜❑ ❉❥❖❭✇❁◗✻❖①❴✱❑✻②✟❏▲❑■❖✩❛❜❖ ❏▲❑✧❫❵❴✩❛❍❴✩❛✥▼❭❉♣❑✹▼ ❉t◗ ❛ ❉❍▼③❴③❑■❖✈▼✩❏ ❉ ❖ ❫♥⑤✫❑✶❫❵❴③❊✁❉❢❫ ❖✈▼❭❏ ❖ ❈■❴✱◗❡❏✭❴✱❑✹❖ ❚ ❙ ❚ ❙ ❙ ❙ ❚✭④ ❚ ❙✭❚❬❚✭❙ ❯ ❚✟❯ ❚ ✇❀❉ ❫❵⑤✫❑⑥❫❵❉ ❉t❫ ❖✽②✁❏▲②✟❉ ❖✺⑦r⑧⑨❉⑩❑■▼ ❉t◗ ❛ ❉ ❴③❫❵❉❢▼❶❈❀❖✽❑■❖✫q✭❑■❖③❷▲❉ ◗❥⑤❶▼✩❴ ❫ ❛❢◗❪❉tq✟❏▲❑✧◗❸❑✹❖③❷❹❉❪❈✻◗❪❉❢❺✗❴✫❺✟❻ ❙✭❚☛❚✁❙ ❚ ❯ ❚ ❙ ❙ ❯ ❚ ❚ ❙✭❚ ❙ ❚✁❯ ❖✩❛◆❖③❫⑥❖ ◗❡❖✩❛❥❖ ❉ ❴③❫⑥❉◆▼✱❖ ❉ ❴✫❑✻②✟❏✺❑✹❖❼❈ ②❋❈✧◗❪❉❢◗ ❉♣◗❡❖❼▼ ❈ ❑❀❈■❖ ❖✗◗❪❖ ❈❀❉ ❖❽✇■❉♥❖✩❛❜❖✫❫♥❖ ◗❪❖✱❛◆❖ ❉ ❴✫❫❬❉◆▼✩❖ ❉ ❚ ❯ ❚ ❯ ❚ ❙ ❙ ❙ ❙ ❯ ❚ ❙✭❚ ❚ ❯ ❚ ❯ ❚ ▼✩❏✭❴✱❑✻②✟❏▲❑❁❖❾❈ ②❋❈✧◗✻❉t◗ ❉⑩◗❪❖❦▼ ❈ ❑❀❈❀❖ ❖⑨▼ ❑■❖ ◗✧⑦ ❙ ❙ ❙ ❙ ❯ ❙ ❚ ☛ ❿➀❊✁❉❡❈✹◗❪❖ ✻◗ ❴❘❞✧❏❭❑✧❫❬❖✩❉ ❏▲❑✻❫❵❴✩❛❥❖⑨❴❘❖P▼ ❴➁◗❪❉❜❖✩❉ ❖s❈❁◗❡❴✫❑■❖❦❖❭❈❁◗❡❖❘❛❢❖P♦✭❴✫◗❡❴ ❖❾➂ ❚ ❚ ❙ ❯ ❯ ❖③❊✁❉❪❈❁◗❪❖ ◗✻❴➃❈❀❉ ❉❥▼✩❉⑩◗❪❴➁◗❪❖✱❴➃❈❀❏✭❛ ❉❥❖③❉❦❫ ❛❢◗❥❉⑩q✟❏❭❑✻◗ ❛ ❉❦❑■❖③❷❭❉❪❈✧◗❡❉t❺☛q✟❖ ◗❥❑ ❏▲❑✹❉◆▼✩❖✽❺✟❴✩❛❥❏▲❑❁❉s❴P❛◆❖✎q✟❴✫❑■❴③❫⑥❖✫◗❥❑■❉❢❛❥❏❭❑ ❚ ❙✭❚ ❙✭④ ❙ ❙ ❙ ❚ ❙ • ❈ ❑■❈❀❖✩❛❥❏❭❑✵❣✻▼✩❴✩❑✹❖❘❉ ❛❥❏✺▼ ❉t❖❹❈➄▼⑨❖❭❛t❖✱❫❵❖ ◗❥❖P❛❢❖ ❉ ❴✫❫❵❉❥▼✩❖③❝✥▼P❏ ❖P▼✫◗❪❴➁◗❪❖❦❛◆❴✆q✟❏▲❑✧◗❥❉❥❤ ❙ ❚ ❙ ❚ ❯ ❚ ❚ −1 ❖❭▼P❉✯❖✩❊✁❉◆❈❁◗❡❖ ◗✻❴ ❖③❊✁❉❪❈❁◗❪❖ ✻◗ ❴➅❫❵❴✫◗❜❑■❉◆▼✱❖✩❛❥❏❭❑ ❈■❉ ❏▲❑❳▼P❴✫q✟❴❭▼✩❉❢◗❥⑤ ❉✶❈■❉✶❉ ▼✫◗❡❉t❺✟❉❢◗❥❴③◗❥❉ ❉ ❴③❫❵❉◆▼P❖ ❚ ❚ ❙✭❚ ④ ❚✁❯▲❙ ❯ ❚ ∆−1 ∆* ❯ ❖ ❛◆❖❦q✟❖ ◗❥❑ ❏▲❑■❉◆▼P❖✶❺✟❴❭❛❜❏❭❑■❉❱❴✩❛❜❖✆q✟❴✫❑■❴➆❫♥❖✫◗t❑■❉◆❛◆❏▲❑ ❖☛▼✩❏ ◗◆❑■❏✭❛✟❴P❉❋❴③▼P❖❭❈✧◗✻❏▲❑✥❖P❛❢❖✩❫⑥❖ ◗✻❖❹⑦ ❚ ❚✭❙ ❚ ❙ ❯ ❚ ❚ ❖ ❉◆❴P♦✣➈❃▼✩❴✫q✟❴P▼③❉❢◗❪❴➁◗❪❉❜❛◆❖ ❉ ❴➁❫❵❉◆▼✱❖❘❴P❛❢❖❘▼P❏ ❖ ❈❀❴③◗❪❏✭❴✫❑■❖✱❛◆❏▲❑ ❉ ❴✱❑✻②✟❏❭❑■❖▲❤①❉ ▼✫◗✻❉t❺✟❉t◗✻❴➆◗✻❉❢❛❥❖ ❉ ❴➆❫♥❉◆▼✱❖❘❴③❛❥❖ ❙✺❚✟❯ ∆ ➇ ❯ ❯ ❚ ❚✁❯ ❚ ❯ ❚ ❚✟❯▲❙ ❯ ❚ ②✁❏▲②✟❉ ❖P❛◆❏▲❑ ❉ ▼P❏✺❴③❑✧②✁❏▲❑■❖♥➉❸❈❀❉ * ❉◆❴P♦❶➈r▼✩❴③q✁❴▲▼③❉❢◗❥❴③◗❥❉❥❛❢❖ ❉ ❴✫❫♥❉❢▼P❖❵❴P❛❢❖➊▼✩❏ ❖ ❈❀❴✫◗✻❏❹❴✫❑■❖✩❛❥❏▲❑ ❉ ▼❭❏✭❴③❑❧②✟❏▲❑✹❖❭❤ ❚ ❯ ❚ ❯ ❚ ❚✟❯ ❚ ❯ ❚ ∆ ➇ ❯ ❉ ▼③◗❪❉❢❺✁❉❢◗❥❴③◗❥❉❥❛❢❖ ❉ ❴✫❫⑥❉❥▼③❖❳❴③❛❥❖✆②✟❏❭②✟❉ ❖P❛◆❏▲❑ ❉ ❴✫❑✧②✟❏❭❑■❖P➉❀⑦ ❚✟❯▲❙ ❯ ❚ ❚ ❯ ❚ ⑧⑨❴✫❑✹❴P▼✫◗✻❖✫❑ ❛❘➋➀❉❢q❱❈❀▼③➌✟❉♣◗❡❷✩❉❥❴ ❴P❛❸❞ ▼③◗❥❉❥❖✩❉❸❞ q❱❏✭❴✫◗✻❖➍❞✧❉ ❖ ❈ ❉ ▼❭❴✫❑✹❴❭▼③◗❥❖③❑ ❛❘➋❍❉tq❋❈❀▼③➌✁❉❢◗❥❷P❉◆❴ ❴✩❛s❖P▼ ❴➁◗❪❉◆❉❜❛◆❏▲❑ ❙ ❚ ❙✺❚ ❚✭❙ ❯ ❯▲❙ ❯ ❚ ❙ ❚ ❙ ▼✩❏ ❈❁◗❪❉t◗ ◗❥❉❢❺✟❖①❴P❛❢❖➎❖✩❛❥❖✫❫⑥❖ ◗❪❖✱❛◆❏▲❑ ❖①▼▲❉♣❑■▼ ❉❢◗❧⑦✼➏❸❉ ❴✩▼✩❖▲❈✧◗❃❫❵❏▲◗❪❉❢❺➐❉ ◗❡❖❭❏▲❑✹❉❢❴➎▼✩❉❢❑✹▼ ❉t◗❡❖✩❛❥❏❭❑✶❉ ◗❥❖✩❑✹❖❹❈■❖✱❴③❷❭❴➎▼✩❏ ❉t◗✻❉❢❉ ❖ ❚ ❙ ❚ ❯ ❙ ❚ ❚ ❙ ❚ ❚✟❯ ❯ ❖③❊✁❉❡❈❁◗✻❖ ◗❪❴⑥❈■❉ ❉◆▼✱❉t◗❡❴✫◗✻❖①❖✱❊✺q✺❑■❉⑩❫❵❴✱◗◆❖✈❉ ❞ ▼✫◗✻❉◆❖ ❖➅❖③▼ ❴✫◗❡❉◆❉❜❛◆❖✈▼P❏ ❈✹◗❥❉❢◗ ◗❥❉❢❺✟❖✈❴✩❛❜❖✈❖P❛❢❖➁❫❵❖ ◗✻❖③❛❥❏▲❑ ❖➅▼✩❉t❑✹▼ ❉❢◗➑❈❀❉ ❖ ❚ ❙✭❚ ❚ ❙✭❚ ❯ ❙ ❚ ❙ ❚ ❯ ❙ ❯ ❫❵❏ ❛ ❖❘❉ ◗❪❖✫❑✹▼P❏ ❖P▼✱◗◆❴✱❑✹❖❘❴❘❴✱▼✩❖▲❈✧◗❪❏▲❑➀❖❭❛❜❖➆❫♥❖ ◗❪❖✺⑦ ❯❭❙ ❯ ❚ ❚ ❚ ➒ ▼P❴③❷ ❛❍▼✩❉❢❑✹▼ ❉t◗❡❖✩❛❥❏❭❑⑨❛❜❉ ❉◆❴③❑✹❖✣❖✱❛◆❖✫❫❬❖ ◗❪❖✱❛◆❖ ❉ ❴✫❫❵❉❜▼✩❖❵❈ ◗✯❛❢❉ ❉◆❴✫❑■❖ ❖❭▼③❉❍❖✩❊✁❉◆❈✹◗❪❴ −1 ❈■❉ ⑦❋➏❸❴✩▼✱❴ ❚ ❙ ❙ ❚ ❚ ❯ ❚ ❙✭❚ ❚ ❯ ∆ ∆ *−1 ❫ ❛t◗❪❉❢q✟❏❭❑✻◗ ❛❬❑■❖③❷P❉❡❈✹◗❥❉❢❺➓❴✫❑■❖➔❏→❈➄❏✺❛ ◗❪❉◆❖❽❈■❉ ❫❵❴✩❉ ❴✗q✟❖ ◗◆❑ ❏▲❑✹❉◆▼P❖✒❺✟❴❭❛❥❏❭❑✹❉➊❴✩❛❥❖✕q✁❴③❑■❴✫❫❵❖③◗◆❑■❉❢❛❥❏▲❑✙❈ ❑■❈❀❖✩❛◆❏▲❑ ❙ ❙ ❙ ❚✭❙ ❙✭❚ ❚ ❙ ❙ • 17 ❉ ▼P❏ P❖ ▼✫◗❡❴✫◗❪❖ ❛◆❴ q✟❏▲❑✧◗❪❉❥❤ ❴✫◗ ▼P❉ ❖③❊✁❉❪❈✻◗❪❴ ❖P▼ ③❴ ◗❪❉❜❴ ❚ ❙✭❚ ❙ ⑦ ✁ ✁ ❈➄❴ ❙ ✂ x = Ax + Bµ s + B * µ s x = f ( x, t ) ❖▲❈❁◗❪❖✆➋➀❉❢q❱❈❀▼③➌✟❉t◗❥❷P❉◆❴ ▲❴ ➂ ✉✟❖✆q✟❏✺❴③◗❪❖❘❴③❑✹❴✫◗✻❴⑨▼✩⑤☎✄ ❴③▼P❖❹❈✻◗✄▼✩❴✱❷ ❚ ❚ f ( x, t ) ✇■❉ ❖③❊✁❉❡❈❁◗❡❴ ❉ ❡◗ ❏❭◗ ❖❭❴ ❴ ❚ ❯ ✭❙ ❚ f ( x1 , t ) − f ( x2t ) = A( x1 − x2 ) ❚✟❯ ❖✫q✟❖ ❖ ◗❪❖ ❚✟❯ ❚ A( x1 − x2 ) ≤ k x1 − x2 ❖ ❯ ❏ ❈✹◗❥❴③❑❁❖ ❉ ❞✧❏▲❑✻❫❵❴ ❚ ▼✩❏ ❁❈ ◗❡❴ ◗❪❴ ❚ ❚ ❲ ❚ ❴▲❈❁◗❥❞✻❖P❛ ❏❹❑✻❫♥❴③❛❥❴ ❉ ✩▼ ❴✱◗ ❚ ⑦ ❖▲❈❁◗❪❖ ❉❥❞✧❏❭❑✻❫✑❫❵❴③❑❁♦✭❉ ❉❢◗❪❴➃❴✫◗ ▼✱❉➎▼✩❉t❑✹▼ ❉♣◗ ❛❸❴✱❑✹❖❶❏✒❈■❏✭❛ ◗✻❉◆❖ ❉◆▼✱❴➍q✟❖ ◗❥❑ ❏❭❑■❉❜▼❭❖❶❈✧◗❪❴③❑■❖ ❙✭❚ ❚ ❙✭❚ ❙ ❙ ❙ ❙✭❚ ❚ ❙ f (0, t ) ❉ ❉t◗❡❉◆❴P❛❢❴ ⑦ ❚ x (t 0 ) ✆ ▼P❏ ◗✻❉ ❴✫❑✹❖❬❈❀❖➅▼✩❏ ❈■❉ ❖③❑❁⑤ ❫♥❴✩❉➀▼P❴③❷ ❛✵▼✩❉❢❑✹▼ ❉t◗❡❖✩❛❥❏❭❑ ❖❭❛t❉ ❉❥❴✫❑✹❖✣❞✧❴✫❑■❴❾❫❵❴✱❑❧❉❢❫❵❉ ❖❬❈✹◗❥❴③❑❁❖➅❉ ❖③❊✁▼P❖❭❈➆⑦ ❚ ❚ ❚✭❙ ❚ ❯ ❚✭❙ ❙ ❙ ❚ ❚ ❯ ❚ ✝❸◗ ❙✭❚ ▼✩❉❋▼P❴ ❚✟❯ ❴③❺✟❖✫❫ ❫⑥❴③❑■❉t❫⑥❉ ❯ ❖❳❈✧◗❪❴③❑✹❖❘❉ ❚ ❖✱❊✁▼❭❖❭❈✥q✭❑■❏▲②✁❛❥❖✫❫❬❴s❈❀❖✶◗❥❑✹❴✫◗❡❖✩❴③❷▲❴❘❈❀❉❢❫⑥❉❜❛❢❴✱❑❀⑦ ➏❸❴❭▼P❴ ▼③❴③❑■❴✩▼➁◗❪❖✫❑✹❉❪❈✧◗❡❉◆▼✩❉❜❛◆❖❽❖✩❛❜❖✫❫❵❖ ◗◆❖❭❛❜❏✺❑ ❉ ❴✫❫❵❉❥▼✩❖ ❈ ◗✙❈❧◗◆❑■❉◆▼➁◗ ▼✫❑■❖P❈■▼❭❴➁◗❪❏✭❴✫❑■❖ ❈■❉ ❖✫❑✹❉❢❺✁❴✩②✁❉❜❛❜❖ ❴➆◗ ▼▲❉ ❚ ❯ ❚ ❙✭❚ ❯ ❙✭❚ ❴✩▼❭❖▲❈✻◗❪❖✱❴❸❴ q✟❴✫❑■❴✫❫❵❖➁◗◆❑ ❉ ❴✫❫❵❉❜▼s❣ ❈■❴ ❝ ❖ ❛✄❉ ❏▲❑✹❉◆▼P❖❳q ▼✫◗ ❖s❞ ▼✫◗❡❉❢❏ ❴✫❑✹❖❹❤ ❖P▼③❉✄❖✱❊✭❉❡❈✧◗❡❴ −1 ⑦ ❙➅❙✺❚ ❙❬❯ ❚ ❙✭❚ ❯ ❙✭❚ ❚ ❯ C d ❙ Ld ❚ ❚✭❙ ❚ ∆ ➏❸❴❭▼✩❴➊❑■❖✫❷❭❉❪❈✧◗❪❏✭❴✫❑■❖P❛❢❖➍❈ ◗①❈✧◗❥❑■❉❢▼③◗❳▼✫❑■❖▲❈❀▼③❴③◗❪❏❹❴✫❑■❖➍❈❀❉⑨❫ ❛♣◗✻❉tq✟❏▲❑✻◗ ❛✆❑■❖➁❷✺❉◆❈✧◗✻❉t❺ ❴✱❑✹❖✎② ▼✩❛❥❖ ❞✧❏▲❑❧❫♥❴➆◗✻❖ ❫♥❴✩❉ ❉ ❙✭❚ ❙ ❙ ❚✭❙ ❙ ❚✭❙ ❯ ❚ ❈ ❑■❈❀❖ ❖⑨◗❪❖ ❈❀❉ ❖s❈➄❉✄❈■❖✩▼③◗❥❉ ❉❇❞✻❏❭❑✧❫❵❴③◗❪❖ ❫❵❴P❉ ❉ ❈ ❑❀❈❀❖ ❖❘▼ ❑■❖ ◗❧❤✭❴➁◗ ▼✩❉❋❴✱▼❭❖❭❈✹◗✟❫ ❛t◗✻❉tq✟❏▲❑✻◗❋❑■❖③❷P❉❡❈❁◗❡❉♣❺✣❴✱❑✹❖❘❏ ❙ ❯ ❚ ❙✺❚ ❙✭❚ ❚❹❙ ❯ ❚ ❙ ❯ ❙ ❚ ❙✭❚ ❙ ✟⑦ ✡❃❖ ❚ ◗❥❑ ❙ ▼❭❴❸▼✩❉❢❑✹▼ ❙ ❉♣◗ ❙ ❛ ❈❀❏✺❛ ◗❥❉❥❖ ❉◆▼P❴❹✟⑦ ✞❦❖✫❷ ❛t◗❡❴❸▼✩❴❸❖✩❊✁❉◆❈✹◗❪❴❸❖✩▼ ❴✱◗❥❉❜❉◆❛❥❖ ❖➎❈❁◗✻❴✫❑❧❖➎❉ ❞❧❏▲❑✧❫❵❴ ❏❹❑✻❫♥❴✩❛◆❴ ✠ ❙ ❙✭❚ ❙ ❙ ❯ ❚ ❚ x = f ( x, t ) ❉ ❴✫❫❵❉❥▼ ❈■❴❽❴❭❉♣②❋❴❼❏❅❈❀❏❹❛ ◗❡❉❢❖ ❉❥▼③❴✠q✟❖ ◗❜❑ ❏▲❑■❉❢▼▲❖ ❈✻◗✻❴✫❑■❖❼❉ ❉⑩◗❡❉❢❴❭❛t❴ ❖▲❈❁◗❪❖ ❈ ❞✧❉❜▼✩❉❥❖ ◗✽▼✱❴ ❞❶❈❀❴❼❞✻❉❥❖ ❯ ❚ ❙ ❙✺❚ ❚ ❙ ❚ ❙ ❚ x (t 0 ) ➋✵❉tq❋❈■▼③➌✟❉t◗❥❷▲❉❢❴ ❴✭⑦ ❚ ✝❸❫☞☛ ❙ ❈❁◗◆❉◆❞✧❉◆▼✱❴✫◗ ❯ ❖✱▼❭❉✻➂ ✌✎✍✑✏✓✒✔✍✖✕✘✗✚✙✜✛ ❚ ▼③❉❢❑■▼ ❙ ❉t◗✄❞✻❴✫❑■❴✶❫❵❴③❑■❉t❫⑥❉ ❯ ❖❦❈✧◗✻❴✱❑✹❖⑨❉ ❚ ❖✩❊✁▼✱❖❭❈➀❴✩❺✁❴ ❚✟❯ ➂ ❖✩❛❥❖✫❫❬❖ ◗❥❖ ❉ ❴③❫⑥❉❥▼✩❖⑨▼ ▼❭❴✫❑✹❴P▼✫◗❪❖③❑✹❉❡❈✧◗❡❉◆▼✱❉✄❈✧◗❥❑✹❉◆▼③◗❇▼✫❑■❖▲❈✹▼❭❴➁◗❪❏✭❴✫❑■❖s❈❁❉ ❖✫❑■❉t❺✼❴③②✟❉◆❛❜❖ ❚ ❯ ❚ ❙ ❯ • ❖✩❛❥❖✫❫❬❖ ◗❥❖❵❑■❖❭❈❀❉❥❈✧◗❪❉❢❺✁❖♥▼ ▼✱❴✫❑■❴P▼✫◗❡❖✱❑❧❉❡❈❁◗❡❉❜▼❭❉✆❈✧◗❢❑■❉❥▼✫◗⑨▼✱❑✹❖❹❈■▼✱❴✫◗✻❏✭❴✱❑❧❖ ❴▲❈✧◗❡❞❧❖P❛r❉ ▼③❴③◗ ✢ ❖✎❞s❖❭❈✧◗✻❖ ❚ ❙ ❚ • x = f ( x, t ) ❙✭❚✁❯ ➋✵❉❢q❋❈■▼✫➌✟❉t◗❡❷❭❉❜❴ ❴ ❚ ❴✫❑■❖❘❏➅❈❀❏✺❛ ◗❥❉❜❖s❈■❉ ❫❵❴✱❉ ❴✆q✟❖ ◗◆❑ ❏❭❑■❉❜▼✩❖❳❈✧◗❪❴③❑❁❖❦❉ ❉⑩◗❪❉❜❴✩❛❜❴ ⑦ ❙ ❚✭❙ ❙✭❚ ❚ ❙ ❚ x (t 0 ) ⑧⑨❴③❑✹❴✩▼③◗❪❖✫❑ ❛ ➋❍❉tq❋❈❀▼③➌✁❉⑩◗❥❷P❉◆❴ ❖❭❈❁◗❡❖ ❉ ❈➄❴ ❖❭❈❁◗ ❛ ❖ ❑■❖❭❈✧◗❥❑✹❉◆▼➁◗❪❉t❺✄⑦ ➏❸❖ ❖✩❊✁❖✫❫✣q✟❛ ❴✩▼P❴ ❙ ❚ ❚ ❯ ❙ ❯ ❙ ❯ ❴✫◗ ▼✩❉ ❈➄❉ ❖✩❊✁❉◆❈❁◗✻❴ ❲✘❴▲❈❁◗❥❞✧❖P❛ ❉ ▼✩❴③◗ ❚✺❙ ❙✭❚ ❚ f ( x, t ) = x 2 + s (t ) ❙✭❚ f ( x1 , t ) − f ( x 2 , t ) = x12 − x 22 ➏❸❴❭▼✩❴ q✁❖ ◗❥❑ ❏▲❑■❉◆▼P❖ ❈■❉ ⑦✣✞⑨❖✱❷ ❛⑩◗❪❴❘▼P❴❦❑❁❖✫❷❹❉❥❈✧◗❪❏✭❴③❑✹❖✩❛◆❖❳▼ ❖P❛❢❉ ❉❥❴✫❑■❉⑩◗❪❴➁◗❪❉❱q✟❏✭❛◆❉ ❏▲❫❵❉❜❴✩❛❜❖❦q✟❏❹◗ ❚ ❙ ❙ ❙✈❚ ❚ ❚ x12 − x 22 ≤ k x1 − x 2 x1 x2 ▼✩❏ ▼❭❖❳❛◆❴❘❞ ▼✫◗❡❉◆❉✟❞❃▼✱❴✱❑✹❖ ❈ ◗❇➋❍❉tq❋❈❀▼➁➌✟❉t◗❥❷▲❉❢❖ ❖❹⑦ ❚✟❯❭❙ ❙✭❚ ❚✁❙ ❙✺❚ ❚ ✉✟❖⑥q✟❏✺❴③◗❪❖ ❖✫❫♥❏ ❈✧◗❥❑■❴❬▼③❴ ▼✱❉t❑■▼ ❉⑩◗⑨▼ ❖❭❛◆❖③❫⑥❖ ◗✻❖ ❉ ❴✱❫❬❉❢▼P❖➐❈❁❉❍❑■❖➁❷❹❉❥❈✧◗✻❉t❺✟❖✣▼ ▼❭❴✫❑✧❴❭▼✫◗❡❖✫❑■❉❡❈❧◗❪❉❥▼③❉✶❈✧◗❥❑✹❉❢▼✩◗ ❯ ❚ ❙✭❚ ❙ ❙ ❚ ❯ ❚ ❙ ▼✫❑■❖❭❈❀▼P❴➆◗✻❏✺❴③❑■❖⑨❴✫❑■❖❦❏✈❈❀❏✭❛ ◗✻❉⑩❖ ❉◆▼✱❴ ❖❭✇➄❉✟❞✌q✼❏✺❴③◗❪❖s❈➄❴ ❞✧❉◆❖✆➋✵❉⑩q❋❈❀▼✫➌✟❉t◗◆❷❭❉❜❴ ❴❹⑦ ❙ ❙✭❚ ❯ ❚✭❙ ❚ ✤✎✥✑✦✓✧✔✥✖★✘✩✫✪✬✪✮✭ ❉❜❖ ❙✼❚ ▼③❉❢❑■▼ ❙ ❉♣◗r▼ ❙ ❖P❛❢❖✱❫❬❖ ❚ ◗❪❖ ❯ ❉ ❚ ❴③❫⑥❉❥▼✩❖❬▼ ❙ ▼✩❴✱❑✧❴✩▼✱◗❡❖✫❑✹❉❪❈✧◗❪❉❜▼✩❉✯❈❁◗❥❑✹❉❢▼③◗r▼➆❑■❖▲❈■▼❭❴➁◗❪❏✭❴✫❑■❖♥❈■❉ ❯ ❖✱❑❧❉❢❺✟❴✫②✟❉◆❛❜❖❹⑦ ➏❸❴❭▼✩❴❵◗❪❏✭❴✫◗❡❖♥❑■❖✱❷✩❉❪❈✧◗❪❏✭❴✱❑❧❖P❛◆❖✽❈ ◗❸❈✧◗◆❑■❉❜▼✫◗❘▼✫❑✹❖▲❈❀▼✩❴➁◗❪❏✭❴✫❑■❖ ❈➄❉✯❫ ❛❢◗❥❉❢q✁❏▲❑✧◗ ❛r❑✹❖✩❷❭❉❪❈✻◗✻❉t❺✙❴✱❑✹❖➊❏➃❈❀❏✭❛ ◗❪❉❜❖ ❈❀❉ ❫♥❴✩❉ ❴❹❤ ❙✭❚ ❙ ❙ ❙ ❚✭❙ ❙✺❚ ❴✫◗ ▼❭❉✁▼❭❉⑩❑✹▼ ❉t◗ ❛ ❉ ❴✫❫❵❉❜▼⑨❴③❑■❖❘❏✈❈❀❏✭❛ ◗❡❉❜❖s❈✹❉ ❫❵❴P❉ ❴✆q✟❖ ◗◆❑ ❏❭❑■❉❥▼✩❖❘❈✹◗❥❴✱❑✹❖❦❉ ❉t◗❡❉◆❴✱❛◆❴ ⑦ ❙✭❚ ❙ ❙ ❯ ❚ ❙ ❚✭❙ ❙✭❚ ❚ ❙ ❚ x (t 0 ) 18 ✉✭◗❪❉❢❫❅▼❭⑤❾❑✹❖③❷▲❉❡❈❧◗❪❏✭❴✫❑■❖✩❛❜❖➎▼ ❖✱❫❬❏ ❏▲◗❪❏ ❖➎❞❧❴③❺✟❏❹❑■❉❢❷▲❖✩❴✫❷❹❴➎❴✫q✟❴✫❑■❉⑩◗❪❉t❴ ❏✺❑⑨❈■❏✭❛ ◗❪❉❥❉✄❫ ❛⑩◗❥❉❢q✟❛◆❖➎❴③❛❥❖ ❚ ❚ ❚ ❙✭❚ ❙ ❙ ▼✩❉t❑■▼ ❉⑩◗ ❛ ❉r❑■❖③❷P❉❡❈❁◗◆❉t❺✄⑦ ✡❱❑■❖③❷❹❖ ◗❡❴ ❴P▼❭❖❭❈❧◗❪❏▲❑❸❑■❖③❷❭❉❪❈✧◗❪❏✭❴③❑❁❖➐q✟❏✺❴③◗❪❖ ▼P❏ ▼✩❖✽❛◆❴ ❉ ❖③❊✁❉❡❈✧◗✻❖ ◗✻❴ ❖❭▼ ❴③◗❥❉❥❖③❉ ➍ ❖ ❈✧◗❪❴③❑■❖ ❉ ❙ ❙ ❙ ❚ ❚✁❯▲❙ ❚ ❚ ❙ ❯ ❚ ❞❧❏▲❑✧❫⑥❴ ❏▲❑✻❫❵❴P❛◆⑤❹⑦ ❚ ❖➅❑✹❖✩❷▲❉◆❈✹◗❪❏▲❑ ❛ ❖P❛❢❉ ❉◆❴✫❑❘❴✫❑■❖✣▼P❴✫❑■❴✱▼✫◗❪❖③❑✹❉❡❈✧◗❡❉◆▼P❴ ✁✄✂✆☎✞✝✠✟☛✡ ✭ ❉◆❖❬▼✱❉t❑✹▼ ❉⑩◗ ❛ ❖❬❏▲❑ ❉ ❛✯❉ ◗❡❴✩❉ ❉ ❞✧❉❢♦ ❑✹❴❬❴ ❙ ❙ ❯ ❯ ❚✭❙ ❚ ❯ ❚ ❙ ❙✭❚✁❯ ❙ ❚ ❚ ❉ ❞✧❉❥♦ ❑✹❴❦②✄⑦❭➏❸❖✩❏✭❴✫❑■❖✱▼✩❖ ❑■❖✱❷ ❛♣◗✻❴❘▼✩❴rq✟❖ ◗❢❑ ❨ ❩➎❉❋❈❀▼✱❴ ❖❘❉❜❴✱❑❃q✟❖ ◗◆❑ ❩❸❉❋▼③❑✹❖❭❈✧◗✻❖ ❖❭▼P❉ ❯ ❚ ❙ ❙ ❚ ❙➎❙ ❯ ❚ ❙➎✍❙ ✌ ❯ u = − Li☞ ❙ ▼✩❴③❑■❴③▼✱◗❡❖✱❑❧❉❪❈❁◗❥❉❜▼✩❉ ❴ ② q✁❴③❑■▼ ❀❑ ❈ ❑❁❉❥❛◆❖ ❉ ❴✫❫❵❉❜▼✩❖①q✁❏✁❈■❉❢②✟❉◆❛❜❖✣❈ ◗➑▼✱❖✩❛❥❖①❉ ❥❉ ▼✩❴✫◗❡❖✈❉ ❞❧❉❥♦ ❑✹❴①②✄⑦✟✝✈▼P❖❭❈✧◗✯▼✩❉❢❑✹▼ ❉♣◗ ❴✫❑■❖✣❈■❏✭❛ ◗✻❉◆❖ P❖ ❏✭❴✫❑■❖✩▼✱❖ ❙ ❙ ❯ ❚ ❙✭❚ ❚✁❯ ❚ ❙ ❙ ❚✭❙ ❙ ❯ q✁❛❥❖✩▼P❴ ❉ ▲❏ ❑❁❉❥▼③❖➎❈❁◗◆❴✱❑✹❖❸❉ ❉❢◗❪❉❜❴✩❛❜❴➎❈❀❖s❴ ☛ ♦✭❖❸❉ ◗◆☛❑ ✎ q ▼✫◗ ❖❾❉❢❫✣q✟❴❭❈✆✏✒✑✄❈❀❴ ✏✔✓ ❉ ▼✩❴✱❑✹❖➎❈❀❏✭❛ ◗❡❉❢❴ ❫❵❴P❉ ❚✟❯ ❯ ❚ ❚ ❙❹❚ ❚ ✭❙ ❚ ❙✭❚ ❯ ❙ ❯ ❚ ❙ ✭❚ ❙ q✁❏✭❴③◗❪❖ ❖③❺✟❏✺❛ ❴❭✜ ⑦ ✝✈▼❭❖P❈✧◗❼▼❭❉⑩❑■▼ ❉♣◗ ❴✫❑❧❖ ❖✩▼ ❴✩◗❥❉❜❖ ❖ ❈❁◗❥❴✩❑❧❖ ❉ ❞✻❏❹❑✻❫❵❴ ❏▲❑✻❫❵❴✩❛❥⑤ ❖✩❏✭❴③❑❁❖P▼③❖ ❞ ▼③◗❥❉❥❴ ❙ ❙ ❚✭❙ ❙ ❯ ❚ ❚ ❯ ✭❙ ❚ ❖③❊✁❉❡❈✧◗❡❴❭❤ ✩❖ ▼✩❉❇❖③▼ ❴③◗❥❉❥❴❸❈ ②⑥❞✻❏▲❑✧❫❵❴ ✕ ■❈ ❖❳q✟❏✭❴✫◗❡❖❸❈■▼✫❑✹❉❥❖❹⑦❭➒ ◗❥☛❑ ✎✧❴ ❖✱❺✟❴✫❑ g −1 ( i ) ❯ ❙ ❙ ❚✭❙ ❚ ❯ i=− L ❴P▼❭❴s❈❁❖❘❉ ❛❥❏✺▼ ❉❜❖❹❈❧◗❪❖✶②✟❏▲②✟❉ ❴❘▼ ❏➅❈ ❑➄❈➄❴ ❖s▼ ❑✹❖ ✗ ◗ ✖③❫ ❛t◗◆❉tq✟❏▲❑✧◗ ❛✁❑✹❖✱❷❭❉❡❈❧◗❪❉❢❺✙✘ ❴✫✹❑ ❖ ❯ ❚ ❙ ❚ ❙ ❙ ❯ ❙ ❚ ❙ ❙ ❚✺❙ g −1 (unde u = g −1 (i )) ❚✭❙ ❈❀❏✺❛ ❥◗ ❉❥❖ ❙ ❙✭❚ ❉❜▼✩❴ q✁❖ ◆◗ ❑ ❚ ❙ ❏▲❑❁❉❜▼❭❖ iL ❯ ❖✩❏✭❴✱❑✹❖P▼✩❖➓❑❁❖✫❷❭❉❪❈✧◗❪❏▲❑ ❛ ✩❖ ❛❜❉ ❉❥❴✫❑ ❙ ❚ ❚ ❚✭❙ ❖✺❈✻◗❪❖❅▼✩❏ ❢◗ ❑✹❏✭❛❥❴✫◗✕❉ ❚ ❚ ▼ ✧❑ ❑❁❖ ◗✕▼ ❙ ❚ ❙ ⑦ i ∈ (−∞,+∞) ✉✟❖s▼✩❏ ❀❈ ❉ ❖③❑❁❴ ▼✩❉t❑✹▼ ❉⑩◗ ✞⑨➋❍⑧☛❉ ▼✩❴✩❑✹❖❦q✁❴✩❑✻◗❪❖✩❴❦❑■❖✫❷❹❉❥❈✧◗❪❉❢❺✁❴s❞✻❏▲❑✧❫❵❖P❴③❷❭❴ ❫ ❛t◗❡❉tq✟❏▲❑❧◗ ❛❥❴❦q✟❏❭❑✻◗✻❉◆❛❥❖s▼③❴✱❑ ❉❥❴ ❚ ❯ ❙✭❚ ❙ ❚ ❙✭❚ ❙ ❙ ❈ ◗✌▼P❏ ❖P▼✫◗❡❴✫◗❡❖❘❖✩❛❥❖✫❫❬❖ ◗❪❖✱❛◆❖ ❉ ❴✫❫❵❉❜▼✩❖✺⑦ ❙✺❚ ❚ ❚ ❯ ❚ ➏❸❴✩▼P❴❘❉ ◗❜❑✯✎ ▼③❉❢❑■▼ ❉t◗ ✞r➋❍⑧☛❈ ◗✌❉ ❖③q✁❛❥❉ ❉❢◗❪❖⑨▼P❏ ❉❢◗❥❉❥❉❢❛❥❖✺➂ ✚✜✛✣✢✥✤✦✛★✧✪✩✬✫✭✫✮✫ ❚ ❙✭❚ ❙ ❙✭❚ ❚✁❯ ❚ ❚✁❯ ❖③❊✁❉❡❈❁◗❡❴➎② ▼❭❛❢❖➅❣❁❈❀❖✩▼➁◗❪❉ ❉❢❝✆❞✧❏▲❑✻❫❵❴✫◗✻❖ ❉ ▼✩❏ ❖ ❈❀❴✫◗✻❏✭❴✫❑■❖✣❣❥②✟❏▲②✁❉ ❖③❝❳❈■✄❉ ✲✧❈■❴ ❈ ❑■❈❀❖✣❉ ❖✫q✟❖ ❖ ◗✻❖ ✰✱ ❚✭❙ ❙ ❙✭❚ ❯ ❚ ❚✟❯ ❚ ❚ ❙ ❙ ❚✟❯ ❚✟❯ ❚ ❖⑨◗❪❖ ❈■❉ ❖❳❣❧▼ ❑❁❖ ◗❥❝ ❯ ❚ ❙✭❚ ❙ ❚ ◗◆❑✹❏✭❛◆❴③◗❇❉ ◗✻❖ ❈■❉ ❖⑨▼P❴✫❑■❖ ❖❭❈✧◗✻❖⑨▼✩❏ ◗◆❑■❏✭❛◆❴➁◗ ❈❁❉❋❉ ▼ ❑■❖ ◗✄❖❭❈❁◗❡❖⑨▼✩❏ ❖✩▼✩◗❥❴③◗❇❉ ✰✳✰ ✱ ❏▲❑✹❉❥▼③❖❦❑✹❖✱❷❭❉❡❈❧◗❪❏▲❑✵▼✩❏ ❚ ❚ ❚ ❙✭❚ ❚✭❙ ❚ ❚ ❙ ❚ ❚ ❚ q✁❴③❑■❴✩❛◆❖✱❛❋▼ ▼❭❏ ❖ ❈➄❴✫◗❪❏▲❑ ❙①❙✭❚ ❚✁❯ ❚ ❏❭❑■❉❥▼③❖⑥❑✹❖③❷✺❉❡❈✻◗✻❏▲❑❳▼✩❏ ◗◆❑■❏✭❛◆❴③◗r❉ ▼ ❑❁❖ ◗✷▼❭❴✫❑■❖ ❖▲❈❁◗❥❖♥▼✩❏ ◗◆❑■❏✭❛◆❴➁◗❘❈■❉✯❉ ◗❪❖ ❈❁❉ ❖❵❖▲❈❁◗❪❖❬▼❭❏ ❖✱▼✫◗❪❴➁◗⑨❉ ✰✳✰✴✰ ✱ ❚ ❚ ❙ ❚ ❚✭❙ ❚ ❚ ❚ ❙✭❚ ❚ ❚ ❈❀❖✫❑■❉❢❖❳▼ ❏s②✟❏▲②✟❉ ❴ ❙ ❚ 19 ✰✁ ✱ ❏▲❑■❉◆▼✩❖✽❑■❖✫❷❭❉❡❈✹◗❪❏▲❑❵▼✩❴③❑■❖ ❚✭❙ ❈❀❴③◗❪❉❪❈■❞✧❴③▼P❖✙▼✱❏ ❚✁❯ ❉❢◗❪❉❜❉◆❛❜❖ ✰✴✰ ✱ ❈➄❉ ✰✴✰ ✰ ✱ ❖❹❈✹◗❥❖❶❈❁◗◆❑✹❉❜▼✫◗✈▼✱❑✹❖✺❈✹▼✩❴③◗◆❏❭❑➊❈❀❉❾❴③❑■❖ q✟❴ ✻◗ ❴ ❚ ▼✩❴✱❑✹❴P▼✫◗❪❖③❑✹❉◆❈❁◗❡❉◆▼P❉❢❉❱❉ ✩▼ ❴ ❑❁❴➁◗❪❴❦❴❭❈❁◗❡❞✧❖③❛ du ❚ ❯ 0 < µ 1< < µ 2< ∞ di ❖ ❈■❴✫◗✻❏❭❑✵❴✫❑✹❖❳❏①▼✱❴✫❑■❴✩▼➁◗❪❖✩❑✹❉❡❈✧◗❡❉❢▼▲❴⑨▼✩❏ ◗◆❑■❏✭❛❜❴✱◗❥❴❳❉ ❈✹❴✫❑■▼✱❉ ❴ ✮✱ ❏▲❑■❉◆▼✩❖❦▼✩❏ ❚✟❯ ❚ ❚ ❚ ❚ ❴❘❴✫❑✹❖❳❏①▼✱❴✱❑✹❴✩▼③◗❪❖③❑✹❉◆❈❁◗❡❉◆▼P❴⑨▼✩❏ ◗❥❑❁❏✭❛❥❴✫◗❪❴❦❉ ❞✧❛ ❊ ✰ ✱ ❏▲❑■❉❜▼✩❖✆②✟❏▲②✟❉ ❚ ❚ ❚ ❙ ❴✫◗ ▼❭❉✁❖✩❊✁❉◆❈✹◗❪❴⑨❖P▼ ❴➆◗✻❉❢❉❜❛◆❖ ❖❾❈✻◗❪❴③❑■❖⑨❴✱❛◆❖❘▼P❉t❑✹▼ ❉⑩◗ ❛ ❉✣❉❥❴✫❑➀▼✩❉❢❑✹▼ ❉t◗ ❛✟❴✫❑✹❖❳▼③❖P❛✁q ◗❡❉ ❏➅❈■❏✭❛ ◗❪❉❥❖❹⑦ ❙✭❚ ❙ ❯ ❙ ❙ ❙ ❙ ❙ ❙ ❚ ❙ ✂☎✄✝✆ ✛ ✤ ➒ ✩✟✞ ✰✄✰ ❖✱❊✁❖✫❫✣q✟❛ ❇ ❛ q✭❑✹❖❭▼✩❖ ❖ ◗ ❴P▼✩❴➅❈✹❖❸❉ ◗❥❑✹❏ ▼✩❖ ▼✩❏ ❖ ❀❈ ❴✫◗✻❏❭❑✶❉ q✟❴✫❑■❴✱❛◆❖✱❛✌▼ ❑■❖➁❷▲❉❡❈✧◗✻❏❭❑ ❛ ✩❖ ❛❜❉ ❉❢❴✩❑■❤✟❴➆◗ ▼▲❉ ❙ ❯ ❚ ❯ ❚ ❯▲❙ ✭❙ ❚ ❚✟❯ ❚ ❚ ❙ ❙ ❚ ❚ ❙✭❚ ✖✫❫ ❛t◗❡❉tq✟❏▲❑✧◗ ❛✭❑■❖✱❷❭❉❥❈❁◗❥❉❢✙ ❺ ❦ ✘ ❴✫❑■❖❘❏➅❈■❏✭❛ ◗✻❉◆❖❘❈❀❉ ❫❵❴✩❉ ✆ ❴ q✟❖ ◗❥❑ ▲❏ ❑✹❉◆▼✱❖ ❈➄❉ ⑦ ❙ ❙ ❙ ❚✭❙ ❙✭❚ ❚ ❙ i L uC ✰✱ ❚ ✣❈■❖③❫❵❴ ➁❴ ◗❪❏▲❑r❈■❖❦◗❥❑■❴➆◗✻❖✩❴③❷▲❴s▼✩❴✫❷ ❛ ❉✄▼P❉t❑■▼ ❉❢◗✥❞✻❏▲❑✧❫❵❴➁◗ ❉ ◆◗ ❑☛✎ ✩▼ ❏ ❖ ❈❀❴✫◗✻❏❭❑✶❉ q✁❴✩❑✹❴✩❛◆❛❜❖✩❛✄▼ ❑■❖❭❈■❉❪❈✧◗❪❏▲❑ ❚ ❙ ✭❙ ❚✺❙ ❙ ❯ ❚ ✭❙ ❚ ❚ ❯ ❚ ✟ ❚ ❙✣❙✭❚ ❖P❛◆❉ ❉❜❴✫❑✵▼✱❴✫❑■❖ ❖▲❈✧◗❪❖❦▼✩❏ ◗◆❑■❏✭❛◆❴③◗❇❉ ◗✻❖ ■❈ ❉ ❖✺⑦ ❚ ❚ ❚✭❙ ❚ ❚ ❚ ✺❙ ❚ ✰✳✰ ✱ ➒ ✝ ❴P▼✩❖▲❈❁◗✥▼▲❴➁❷⑥❈■❖➎❴ ❴ ♦✭❴s②✟❏▲②✟❉ ① ❴ ❉ ❈❀❖✫❑■❉❥❖①▼ ❑✹❖③❷❹❉❪❈✧◗❪❏▲❑ ❛ ❈❀❉✞✖✱❫ ❛♣◗✻❉tq✟❏▲❑❧◗ ❛❋❑■❖✱❷✩❉❪❈❁◗❥❉❢✍❺ ✘①❴③❑✹❖①❏♥❈❀❏✭❛ ◗✻❉❢❖➅❈❀❉ ❫⑥❴▲❉ ❯ ❙ ❚ ❚ ❙ ❙ ❙ ❙ ❙ ❚✺❙ ❴✆q✟❖ ◗❥❑ ❏▲❑■❉◆▼P❖ ❈❀❉ ⑦ ✺❙ ❚ ❚ ❙ iL uC ❚ 20 ✂✁☎✄✝✆✟✞✡✠☞☛✟✞✌☛✎✍✏☛✌✆✒✑✓✍✕✔✖✠✗✠✗✘✗✙✚✞✜✛✢☛✤✣✥✙✂✦✝☛✤✠★✧✩✙✚✦✪✍✕✘✫☛ ✉✟▼✫❑■❉❥❖✫❑■❖③❴✕❖✩▼ ❴ ❉❜❉❢❛❥❏❭❑ ✧ ✛ ✴✢✭✬✍✛ ✰✯✮ ✢✰✬ ✩✟✱✳✛✢✲ ✧ ✛ ✳✢✭✬ ✩✏✳ ✢ ✳✛ ✮✵✴ ✰ ✩✟✱ ✛✵✱✳✢✥✤ ✮ ✢✭✬✟✶✥✤ ✰ ✱✴✢✥✤ ✱ q✟❖ ◗❥❑ ▼P❉t❑✹▼ ❉⑩◗ ❙ ④ ❚ ❙❼❙✭❚ ❙ ❖P❛◆❉ ❉❜❴✫❑✵▼✩❏ ❈❁◗❡⑤ ✄ ➂ ❚ ❚ ❚ ❚ ✎❁❈■▼✫❑■❉❥❖✫❑■❖③❴⑥◗✻❖✩❏▲❑■❖➆❫❵❖P❉✵➒❘❴♥❛ ❉✝✷s❉❢❑■▼✫➌✭➌✟❏✺❞✻❞ ✄ ✎✹✸ ❏ ❑■❉✶▼ ❖③❊✁▼❭❖✫q ❉◆❴ ❏ ❛ ❉ ❖⑥❑✹❖❭❞✻❖③❑❁❉ ⑤⑥q✟❖ ◗◆❑ ❙ ❚ ❚ ❚ ❯▲❙ ❙ ④ ❚ ❯❭❙ ❙ ❯ ❚✭④ ❚ ❙ ▼✩❴✱❑✹✻ ❖ ✺✽✼ ❩✁❤♥❴✫❺✟❻ ❑■❖✫q✭◗ ❖✩▼ ❏✁❈■▼ ◗✻❖❘q✟❏▲◗❡❖ ❉❜❴❭❛❢❖P❛❢❖ ❏ ❑✹❉❥❛◆❏▲❑r✇■❉❃▼ ❑■❖ ❉◆❉ ❉ ❛❥❴✫◗ ❑■❉❜❛◆❖❸▼ ❖✩❛❜❖✫❫❵❖ ◗❡❖❸❴✩❛❜❖ ❚✟❯ ❯ ❚ ❙✺❚ ❙ ❚✭④ ❚ ❯▲❙ ❙ ❚✺④ ❯ ❚ ❙ ❙ ❚ ➇ ▼✩⑤✱❑✹❏▲❑✵▼✩❴✫❑■❴✱▼✫◗❡❖✱❑✹❉❡❈✹◗❪❉❜▼✩❉ ❈ ◗✄▼✩❏ ◗◆❑■❏✭❛◆❴③◗❥☎ ❖ ✄ ◗❪❖ ❈❁❉ ❖ ❚✭❙ ❙✭❚ ❚ ❚ ❚ ❙✭❚ ✎❁❈■▼✫❑■❉❥❖✫❑■❖③❴❳❖❭▼ ❴ ❉❜❉◆❛◆❏▲❑ ❖❘❛◆❖▲♦✺⑤③◗ ❑✹⑤ ❉ ◗❥❑❁❖⑨◗❥❖ ❈❀❉ ❖❾✇■❉❋▼ ❑■❖ ◗❋q✟❖ ◗❥❑ ❞✻❉❥❖✩▼❭❴③❑❁❖❳❖③❛❥❖✫❫⑥❖ ◗✌▼✩❴✩❑✹❖ ❴✱❑✹❖❳❏ ❙ ④ ❯ ❙ ❯ ❚ ❚ ❙✭❚ ❙ ❚ ❚ ❙ ❚ ❚✭❙ ▼✩❴✱❑✹❴P▼✫◗❪❖③❑✹❉◆❈❁◗❡❉◆▼P⑤➐▼▲❏ ◗❥❑✹❏✺❛❜❴✫◗✻⑤ ✄ ◗❪❖ ❈❀❉ ✤❖ ✾✵❞❧❉❥❖✩▼P❴✫❑✹❖ ❖P❛❢❖③❫❵❖ ◗ ❖➊❴✩▼P❖❭❈✧◗✆◗❡❉tq✝❉ ◗◆❑✹❏ ▼✩❖✎▼✩❖✩❛rq ❉ ❏✎❺❋❴✫❑✹❉❥❴✫②✟❉◆❛❜⑤ ❚ ❚ ❚ ❙✭❚ ❚ ❯ ❚ ❯▲❙ ❙✺④ ❚ ❈ q✟❛❥❉♣❫❵❖ ◗❡❴✱❑❧⑤❳❣ ▼ ❑■❖ ◗❥❝➑✇✹❉❇❏①❖✱▼ ❴ ❉◆❖❳❈ q✟❛◆❉t❫♥❖ ◗❪❴③❑✹⑤✺➂ ❙ ❚ ❙✭❚ ❙ ❚ ❙ ④ ❙ ❚ ✎❁❈ ❑❀❈■❴✠❉ ❖❭❴P❛◆⑤ ❖ ◗❪❖ ❈■❉ ❖✠❉ ◗❥❑■❏ ▼✩❖ ❖❭▼ ❏✁❈■▼ ◗❡❴✠❈ q✁❛❥❉t❫❵❖ ◗❥❴③❑❁⑤ ❉❬❣✧▼ ❑■❖ ◗ ❛✣q✭❑■❉ ❙ ❯ ❯ ❚ ❙✭❚ ❚ ❯▲❙ ❚ ❙✭❚ ❙ ❙ ❚ ❙ ❚ ❙ ❚ ❈ ❑➄❈➄⑤✩❝❍✇❀❉❋❖③▼ ❴ ❉◆❴❘❈ q✟❛◆❉⑩❫⑥❖ ◗❡❴✱❑✹⑤ ❙ ❙ ④ ❙ ❚ V j − Vk = e(t ) ✎❪❑■❖③❷▲❉❡❈✧◗✻❏❭❑ ❛❬▼✩❏ ◗◆❑✹❏✭❛❥❴✱◗ ✄ ▼ ❑■❖ ◗➐❉ ◗◆❑■❏ ▼✩❖✕❺✟❴✫❑■❉◆❴③②✟❉❢❛❥❴➔❈ q✟❛❜❉t❫❵❖ ◗❥❴③❑❁⑤✒❀❉ ✿✙q✭❑■❉ ❖✩▼ ❴ ❉❥❴ ❙ ❚ ❚ ❙ ❚ ❚ ❯▲❙ ❙ ❚ ❚ ❙ ④ ✞ ✞ ✞ u R = uˆ R (iR ) ✎✌②✟❏❭②✟❉ ❳ ❴ ❛◆❉ ❉❜❴✫❑■⑤⑨❉ ◗❢❑■❏ ▼P❖ ✄ q✟❛ ❈❍❺✟❴✫❑✹❉◆❴➁②✟❉◆❛❥❴⑨❉❂❁✥❺✟❴✱❑✹❉◆❴③②✟❉❢❛❥❴❘✇❀❉✟❖P▼ ❴ ❉❜❴ ❚ ❚ ❚ ❯▲❙ ❚ ❙ ❙ ④ u L = Li❃ L ✎❪②✟❏▲②✁❉ ❴ ❖✩❛◆❉ ❉◆❴✫❑■⑤ ▼✩❏ ◗◆❑■❏✭❛◆❴➁◗❪⑤ ✄ ❞✧❛ ❊ ❉ ◗❥❑✹❏ ▼❭❖ ❏ ⑤ ❺✟❴✫❑■❉❥❴✫②✟❉◆❛❥❖ ❈ q✟❛❢❉⑩❫❵❖ ◗✻❴✱❑✹❖ ❚ ❚ ❚ ❚ ❚ ❙ ❚ ❯❭❙ ❯ ❙ ❙ ❚ q✭❑■❉ ❖✩▼ ❴ ❉t❉◆❛❥❖ ⑦ ❚ ❙ ④ i L si φ iL = iˆL (φ ) si u L = φ❄ ✎✧▼✩❏ ❖ ❈❀❴✫◗✻❏▲❑ ❛✥▼✩❏ ◗◆❑■❏✭❛◆❴③◗ ✄ ❈■❴✫❑■▼✱❉ ⑤①❉ ◗◆❑✹❏ ▼▲❖s❺✟❴✱❑✹❉❥❴✫②✟❉❢❛❥❖③❛❥❖✈❈ q✟❛◆❉❢❫➅❖ ◗✻❴✫❑✹❖➎❉❀❅❸✇➄❉❇❆✭❅❦q✭❑■❉ ❚✟❯ ❚ ❙ ❚ ❚ ❚ ❚ ❯▲❙ ❙ ❚ ❚ ❖✩▼ ❴ ❉❥❉❢❛❥❖ ❈ ❉ ❙ ④ ic = q si q c = q c ( uc ) ❿✕❊✡❋❍●✚■✝❏▲❑ V4 = 0, V1 = 2 cos t , V2 = 10−3 i▼ 2 , (V2 − V1◆ ) / 2◆ + (V2 − V3 ) / 1 + i2 = 0, 10−6 (V1 − V3 ) + 1(V2 − V3 ) = i3, V3 = 3i32 − i3 1 −6 (V❖1 − V❖ 3 ) = i1, 2 P✝❖➁◗❪❏ ❴ ❏ ✩❴ ❛◆⑤ ❴✱❑✹❖❶❫❬❴❭❉❸q ❉ ❖☛❖✩▼ ❴ ❥❉ ❉ ❖✩▼✱❻✱◗➅❫❵❖➆◗✻❏ ❴✙◗❪❴③②✁❛❥❏ ❛ ❉✈▼✱❖✩❖✩❴✕▼③❖ ❖❭❈❁◗❪❖ ❴③❺✟❴ ◗❥❴ ☛✎q✟❖ ◆◗ ❑ ❯ ❚ ❯ ❙✭④ ❚ ❙ ④ ❯ ❯ ❙ ❙ ❙✭❚ ❚ ❚ ❙ (V1 − V2 ) − 10 21 ✞ ✄ ▼✩❴❭❛❜▼ ❛ ❛✥❫❬❴ ❴✩❛❧⑦ ❦❖③❷▲❏✺❛❢❺✟❴✫❑■❖③❴❬❖✱▼ ❴ ❉❥❉❢❛❥❏❭❑ ❉◆❞✧❖✫❑■❖ ❉❢❴P❛◆❖ ❖✣▼P⑤✫◗❥❑✹❖ ❏▲q✟❖✫❑■❴✫◗✻❏❭❑ ❫❵❴ ❖❭❈❁◗❡❖ ❈■⑤ ❖P❞✻❉◆▼✩❉❜❖ ◗✻⑤ ❙ ❙ ❚✭❙ ❙ ④ ❯ ❚✺④ ❯ ❙✭❚ ❙ ❚ ❚ ❚ ❫❵❴P❉❃q✟❖ ◗❥❑ ▼▲❉t❑✹▼ ❉❢◗❥❖P❛❢❖ ❖➅❏▲❑ ❉ ❛❃➒➄❤ ❖✈❖✱❊✼❉❪❈✧◗❪⑤➅❏➊❈■❉ ♦ ❑❁⑤➅❖✩▼ ❴ ❉❥❖❹⑦ ▼③❖③q✁❻ ▼ ❏▲❑ ❉ ❛ ➒✻➒❦❤❋❞✧❉◆❖⑥❈■❖ ❚✺❙ ❚ ❙ ❙ ❯ ❯ ❚✺❙ ❙✭❚✟❯ ❚ ❙ ❙ ④ ❚ ❚✟❯ ❙ ❯ ❚✭❙ ❖③◗❪❖③❑❧❫♥❉ ⑤➐❣❪❴❭▼✩❏✭❛◆❏ ❖➊❖③❊✁❉❡❈✧◗✻⑤③❝❸❏➍❈❀❏✭❛ ❉◆❖♥❴ ❴P❛❢❉❢◗❥❉❥▼✩⑤ ◗❥❉❥❛❢❉t❷❭❻ ❫♥❴ ❉tq ❛◆❴③◗❪❏▲❑①❈❀❉⑩❫✣②✟❏✺❛❥❉◆▼➊❣ ❖➐❖✱❊✼❖✩❫✈q✟❛ ❯ ❚ ❙✺❚✟❯ ❙✺④ ❚ ❙ ❚✁❯ ❙✺❚ ❚ ❙ ❯ ❙ ✄➋✥❿ ❝❍❈❀❴ ❤✭▼✫➌✟❉◆❴③❑❍✇➄❉✁q✟❖ ◗❢❑ ▼✩❉❢❑❁▼ ❉t◗❡❖✩❛❜❖❘❛❜❉ ❉◆❴③❑❁❖P❤✁❈✹❖s❞✻❏✭❛◆❏✁❈❀❖❭✇❧◗❪❖❘❏❾❫⑥❖✩◗❥❏ ⑤ ❫❬❖✱❑❧❉❥▼✩⑤❹⑦ ❙ ❚ ❙ ❙ ❚ ❯ ❚✭❙ ✆ ✜✝ ✡ P ✂✁☎✄✝✆✟✞✡✠☞☛✍✌✂✎✑✏✒✆☎✎✔✓✖✕✘✗✙✆✚✏✜✛✣✢✚✤ ✉✟❖ ✄ ❉ ■❈ ❉❡❈✧◗❡❖✫❫ ❖♠❖✩▼ ❴ ❉◆❉ ❖ ❈✻◗❪❴③❑✹❖ ❈❀▼✫❑■❉❡❈❁❖ ❞✻❏❭❑✧❫❵❴ ❏▲❑✻❫❵❴✩❛❥⑤ ❙✭❚✭❙ ❯ ❙ ④ ❯ ❚ ❚ ⑦⑨⑧✷➌✁❉❥❴✫❑ ✇❀❉❾q✟❖ ◗◆❑ ▼✱❉t❑✹▼ ❉⑩◗❥❖❭❛t❖ ❛◆❉ ❉❜❴✱❑✹❖ ❴③❛❥❖✙▼P⑤✱❑✹❏▲❑➐❖✩▼ ❴ ❉❜❉①❴ ❈■❏✭❛ ❉❜❖✙❴ ❴✱❛◆❉❢◗◆❉◆▼✩⑤▲❤❘❈■❖✙q✺❑✹❖▲❞✻❖✱❑✹⑤ ✥ ❚ ❙ ❙ ❚ ❙ ④ ❙ ❙✭④ ❚ x = f ( x ,t ) ◗✻❉◆❛❜❉❢❷▲❴✫❑✹❖✩❴⑨❫⑥❖✫◗✻❏ ❖P❛❢❏▲❑ ❫♥❖✫❑■❉◆▼P❖❭⑦ ✈▼✩❖P❴❭❈✻◗❡⑤❦q✭❑✹❖✩❞✧❖✫❑■❉ ⑤❘❖▲❈✧◗❪❖r❫❵❏▲◗❪❉⑩❺✼❴✱◗❪⑤❦❴❭❈✻◗✻❞✧❖✩❛✻➂ ❙ ❯ ❚✭❙ ❚✺④ ❉❢❝✷▼✫➌✟❉◆❴③❑➀q✟❖ ◗◆❑ ▼P❉t❑✹▼ ❉❢◗ ❖❸❏▲❑ ❉ ❛ ❏✺❉❃▼P❴③❛❥▼ ❛◆❖✱❛◆❖s❴ ❴P❛❢❉❢◗❥❉❜▼✩❖➎❈ ◗ ❖▲❈❁◗ ❛ ❖❸▼✩❏▲❫✣q✟❛❜❉◆▼✩❴③◗❪❖❳q✟❖ ◗◆❑ ❚ ❙✣❙✭❚ ❙ ❯ ❯ ❚✭❙ ❯ ❙ ❚ ❙✭❚ ❯ ❙ ❯ ❚ ❙ ❴❘❞✧❉✟❖✩❞✧❖P▼➆◗ ❴③◗❥❖❳❖✩❞✻❉◆▼P❉❢❖ ◗ ❖ ❏▲q✟❖✫❑■❴✫◗✻❏▲❑ ❫♥❴ ⑦ ❙ ❚ ❯ ❙✭❚ ❙ ❚ ❉◆❉❜❝ ❖➆◗✻❖✫❑✧❫⑥❉ ❴✫❑■❖✩❴➊❈■❏✭❛ ❉◆❉❥❛◆❏▲❑❳❴ ❴P❛❢❉❢◗❥❉❥▼③❖➊▼ ❴ ◗✻❏❭❑ ❛ ❏▲❑❘q✭❑■❏✭♦▲❑✹❴✫❫❵❖ ❖✩▼✩❖▲❈■❉t◗❡⑤ ❖✩❞✧❏▲❑✧◗ ❖♥▼✩❴✱❛◆▼ ❛ ❯ ❚ ❙✭④ ❚ ❙ ❙ ❙ ❙✭❚ ❚ ❙✺❚ ❯ ❙ ▼✩❏ ❈❀❉ ❖✫❑■❴✫②✟❉❥❛◆❤✭▼③❴P❛❢▼ ❛◆❴➁◗❪❏✭❴✫❑■❖✩❛❜❖❘❞✧❉❜❉ q✭❑■❏✭❉❢❖▲▼✫◗❡❴✫◗❡❖✆q✟❖ ◗◆❑ ❴⑨❏▲q✟❖✫❑■❴❘▼ ▼❭❉❜❞❪❑■❖❘✇❀❉ ▼ ❈■❉❢❫✣②✟❏✺❛ ❑❁❉❪⑦ ❚ ❯ ❙ ❚✟❯ ❚ ❙ ❙ ❚✭❙ ❙ ❙ ✦❦❑■❉◆▼P❖✠❫⑥❖③◗❪❏ ⑤ ❫♥❖✫❑✹❉◆▼P⑤ q✟❏▲❑ ❖▲✇✧◗❪❖ ❖ ❛❢❴ ▼✩❏ ❉ ❉❥❴❼❉ ❉ ❉❢❴P❛◆⑤ ✇■❉ ❖➁◗❪❖③❑❧❫♥❉ ⑤ q✟❖✠❑✹❻ ❯ ❚✭❙ ❚ ❯ ❚✟❯ ④ ❚ ④ ❯ ❚ ❚✁❯ x( t 0 ) ❖⑨➌➅❖▲❈❁◗❪❖✶q✟❴▲❈ ❛ ❖✆◗❥❉⑩❫✣q✄⑦ ❙ ❯ x( t 0 + h ), x( t 0 + 2 h ),..., ❙✭❚✟❯ ✧✈❏▲◗❪⑤③❫ ⑦✣⑧⑨❖✱❛◆❖✠❫❬❴✩❉✽❈❀❉♣❫✣q✟❛❥❖ ❫⑥❖✩◗❥❏ ❖ ❖ ❉ ◗✻❖✩♦▲❑■❴✫❑■❖ ❫❵❖✫❑■❉❜▼✩⑤ ❈ ◗ ❖P❞❧❉ ❉♣◗❡❖ ❖ ❯ ❯ ❚ ❚✭❙ ❙✺❚ ❯ ❚ ❯ x( t ) = x k k ❑❁❖P❛◆❴ ❉❥❉❢❛❥❖▲➂ ④ ❇❫❵❖✱◗❡❏ ❴⑨❿ ❛◆❖③❑❍❖✱❊❹q✟❛❥❉❢▼P❉♣◗✻⑤❘❣✻➒❧❝ ❯ ❙ x = x + h⋅ f (x ) k +1 k k ❇❫❵❖③◗❪❏ ❴r❿ ❛❥❖✫❑✵❉t❫➅q✁❛❜❉◆▼P❉♣◗❡⑤❘❣✻➒✧➒✧❝ ❯ ❙ x = x + h⋅ f (x ) k +1 k k +1 ❙ ❑✻❫❵⑤✫❑■❖❭✇❁◗❡❖❅❑■❖③❷▲❏✺❛❢❺✼❴✫❑■❖✩❴ ✝ ☛ ✎ ✎ x ✌❫⑥❖✱◗❡❏ ❴✆◗◆❑✹❴③q✁❖✱❷ ❛ ❉❋❣❡➒✧➒✧➒✻❝ h ❯ ❙ ❙ = x + [ f (x ) + f (x )] k k +1 k +1 k 2 ✎ ★✒✩✫✪✟✬✮✭✰✯✲✱✴✳✵✭✶✱✷✯✜✳✸✬✹✩✺✱✫✯✼✻✾✽❀✿✵❁❂✭❃✳✾❄✟✭✶❁❅❄✟✭❆✳✮✳☎✱✴✽❇✭✶✩❉❈❀✳❋❊✍✩✴✭❆✽✘●✮✪❂✬✮✭❆✩❍✩■✿✮✩❏✱✫✯✲✱✫❑✫✬✍✩❇✻✜✽❍❁❅✭✶❈▲✳✵✪▼✩✘❈▲✩✘◆P❖❇✭❆✳✮◆P✩❘◗✡❙❅❚✰✬✮✳❋❊✰❊✰❯✫❱❲◆❳✩✫✬❋❁✟❈▲✩❅✿✵✩ ✪✟✯✟◆❨✩❩✭❃✳❬✱❀✩❘❄✟✭❆✩✫❭■✩❩✪✟✬❬✽✫✬✹✩✖◆P✽■✳✟●❬✪✜✽✴✳✵✪▲✬✍✩❘✪✟✯✲❈▲✽✴✯❪✭✶✩✴❭❩✯✜✿❫✬✍✽✫✬❬✩❍✱■❁❅✭✶✩✫✱✫✬✹✩✟❴❩❵❋✪P✽✴✱■✩❀❚❆✬✚✱■✽✴❭❪❚❛✩✺❊✍❁▼✿✵❁✚❚❆✩❅❚✶✱❘◆P✩✴✬✹❁▼❈▲✩❉❚✰❄✜✩■✱✫✳❬✽■✿✸✩❉❈▲✩ ✬❬✳✮❄✲❜✺✩■✽❇✭❝❴❩❞❘✩❀✿❬✽❢❡✹✳❬✳✹✿✵✩❏✱❩✽✫✭❆✩❉❈✟✩❩❊✍✳✸✪✜✩❅❚❆✱✘◆❣✩✫✬✍❁✟❈▲✩❩✿✵✩❉❜✺✩✴✽✴✭☞❈✟✩❉❁❅✭✶❈▲✳✵✪✟✯▼✿✾❤✡✐❦❥❧❚❆✯✟✪✟✬✶♠ ❪❫❵❖✫◗✻❏ ❴♦s ♥ ❖③❴✱❑ ❖❘❏▲❑ ❉ ❛q♣ ✎ ❯ ❯ ❯ ❚✭❙ 2 4 1 ❣✻➒ ❾❝ x k +1 = x k − x k −1 + h[ f ( x k +1 , t k +1 )] ❪❫❵❖✫◗✻❏ ❴♦3♥s❖③❴✱❑ ❖❘3❏▲❑ ❉ ❛✒r 3 ❯ ❯ ❯ ❚✭❙ 18 9 2 6 xk +1 = xk − xk −1 + xk −2 + h[ f ( xk +1 , t k +1 )] 11 ❪❫❵❖✫◗✻❏ ♦ ❴ 11 ♥s❖③❴✱❑ ❖❘11❏▲❑ ❉ ❛❳s11 ❯ ❯ ❯ ❚✭❙ 12 48 36 16 3 x k +1 = xk − x k −1 + xk − 2 − xk − 3 + h[ f ( x k +1 , t k +1 )] 25 25 25 25 25 ❪❫❵❖✫◗✻❏ ♦ ❴ ♥s❖③❴✱❑ ❖❘❏▲❑ ❉ ✡ ❛ t ❯ ❯ ❯ ❚✭❙ 60 300 300 200 75 12 x k +1 = xk − xk −1 + xk − 2 − xk −3 + xk − 4 + h[ f ( xk +1 , t k +1 )] 137 137 137 137 137 137 ✺ ✎ ✎ ✎ 22 ❪❫❵❖✫◗✻❏ ♦ ❴ ♥s❖③❴✱❑ ❖❘❏▲❑ ❉ ❛✁ ❯ ❯ ❯ ❚✭❙ 60 360 450 400 225 72 10 x k +1 = xk − x k −1 + xk − 2 − xk − 3 + xk − 4 − x k − 5 + h[ f ( x k +1 , t k +1 )] 147 1147 147 147 147 147 147 ✎ ⑧⑨❏ ■❈ ❉ ❖✫❑■❻ ▼✩⑤ ✂❳❫❵❖③◗❪❏ ❖✩❛❜❖ ❉◆❞✧❖✫❑■⑤✶q✭❑■❉ ❫❵❏ ❛ ❖❘▼✩❴✱❛◆▼ ❛✟❴✩❛ ❖✫❑■❉t❺✟❴➁◗❪❖✱❉❧➂ ❚ ❯ ❚✁❯ ❯ ❯ ❚ ❯▲❙ ❯ ❙ ❯ x = x + h⋅ x k +1 k ) = x ( t ) + hx✄ ( t ) ( I ) k k ) ( II ) x( t ) = x ( t ) + hx✄ ( t k +1 k k +1 h ) = x ( t ) + ( x✄ ( t ) + x✄ ( t )) ( III ) x( t k k +1 k +1 k 2 x( t k +1 2 ⑦❸❣❡➒✗✺❾❝ x (t k −1 ) + h x☎ (t k +1 ) 3 3 4 ✉✟❖➐❏▲②❋❈■❖✫❑✧❺✟⑤ ✇➄❏❭❑❸▼❭⑤⑥❫❵❖✫◗✻❏ ❴♥➒➄❤ ✄ ▼P❴✫❑■❖ ❈■❖ ❖③◗❪❖③❑❧❫❬❉ ⑤ ✄ ❞ ▼ ❉◆❖ ❖ ❙ ❯ ❚ ❯ ❚ ❚ ❙✭❚ ④ ❯ x( t ) k +1 ❖③❊✺q✟❛◆❉◆▼✱❉t◗✻⑤✩✣❤ ✄ ◗❡❉t❫✣q ▼③❖⑨❫⑥❖✫◗✻❏ ❖▲❛❢❖⑨➒✻➒➄❤❹➒✧➒✻➒✵✇■❉✁➒ ✺❽❈ ◗✟❫❬❖✱◗❡❏ ❖⑨❉❢❫✈q❋❛❢❉❜▼✩❉⑩◗❪❖✺⑦ ❚ ❯ ❙✭❚ ❯ ✉✟❏✭❛ ❉❥❉t❛◆❖s❏▲② ❉ ◗✻❖❦q✺❑■❉ ❉ ◗✻❖✩♦▲❑■❴✫❑✹❖ ❫❵❖➁❑✹❉◆▼✩⑤❸❞❡❉❥❉ ❴③q✭❑❁❏❹❊✁❉t❫♥❴➆◗✻❉t❺✟❖s❈■❖s▼P❴③❛❥▼ ❙✭④ ④ ❚✺❙ ❚ ❚ ❚✭❙ ❚✟❯ ❙ ❛◆❴s❞✻❉◆❖P▼③❴③❑■❖❦q✁❴P❈✝✆✟✞✡✠☞☛✍✌✎✠✏✞✑✌✓✒✔☛✎✕✑✌✖✒✘❵ ✗ ✇■✁❉ ✞✡✠✏☛✙✌✎✠✚✞✑✌✜✛✁✒✟☛✙✢✎✌✖✒✘⑥ ✗ ❴❦❫⑥❖③◗❪❏ ❖✩❉❢❝ ⑦❭➏❸❖❘❖③❊❱❖✫❫⑥q✁❛ ❤❭q✟❖ ❯ ❙ x (t k +1 ) = ❴❘▼✩⑤③❑❁❖P❉❋❈■❏✭❛ ❙✺④ 3 x (t k ) − 1 x( t ) k ❤➀❖✺❈✧◗❪❖➊❏♥❫❵❖③◗❪❏ ⑤ ❯ ❛❢❖P❴➁❷✺⑤❘❖✫❑■❏▲❑✹❉◆❛❥❖❘❉ ◆◗ ❑■❏ ❈■❖ ❚ ❯▲❙ ◗❥❑ ❖✩▼ ❴ ❉❜✤ ❴ ✣ ❚ ❙ ❙ ④ x = −λ x ❉❢❖❦❴ ✩❴ ❛❜❉t◗✻❉❢▼P⑤⑨❖❹❈❧◗❪❖ − λ t ❈■❖⑨▼P❴✩❛◆▼ ❛❢❖P❴③❷P⑤✼➂ ❚ ❙ x ( t ) = x ( 0 )e ❖✩❑✹❏✺❴✩❑✹❖✩❴ ❛◆❏✭▼✩❴▲❛❢⑤ ❛❥❴ t = t n+1 ε n = x exact ( t n+1 ) − x aprox ( t n+1 ) = ( x n e ❣ ❖ ❴s❞✻❏✁❈❁◗✌▼③❴P❛❢▼ ❛◆❴➁◗❋q✟❛❢❖P▼✩❻ ❖❦◆❛ ❴ ▼P❴✩❛❜▼ ❛❢❴③◗❇❴✫q✭❑■❏❹❊✁❉♣❫❵❴✩◗❥❉❢❺✭❝ ❙✭❚✟❯ xexact ( t n+1 ) ❙ ❚✟❯➎❯ ❙ x( t n ) ✎ −λ h ) − x n+1 λt ❣ ❖ ❴→❞✧❏✁❈❁◗ t = t n+1 este ε t = ( x 0e n+1 ) − x n+1 ❙✭❚✟❯ x exact ( t n+1 ) ❴✫◗✌❉ ❉ ❉❢❴P❛❢❝ ⑦ ▼✩❴❭❛❜▼ ❛◆❴➁◗❋q✟❛❜❖✩▼✩❻ ❖⑨❛❥❴ ❚ ④ ❙ ❚✟❯①❯ x( 0 ) ❯ ✆ ❞✧❉❢♦ ❑■⑤s❈❀❖✆q✭❑■❖③❷P❉ ◗❪⑤❾❈■❏✭❛ ❉❜❉◆❛❜❖❘❖✱❊✼❴P▼✫◗❡⑤s✇✹❉❱❴✱q✭❑✹❏❹❊✼❉❢❫❵❴✫◗❡❉t❺✟⑤✆q✟❖ ◗❥❑ ✱❖ ▼ ❴ ❉❢❴ ✥ ❤ ❖ ⑦ ❚ ❙ ❚ ❙✭④ ❚ ❙ ❙ ④ x = −λ x ❙✺❚✟❯ λ > 0 ☛❖✱❑✹❏✭❴③❑❁❖P❴ ◗❪❏▲◗❪❴✱❛◆⑤→❛◆❴ ✎ ✦❅❫❵❖③◗❪❏ ❖➊❉ ✻◗ ❖❭♦▲❑✹❴✫❑■❖♥q✟❖ ◗❥❑ ▼✩❴✫❑■❖✎❖✫❑■❏✭❴✫❑■❖③❴❵◗❪❏▲◗❡❴✩❛❥⑤ ❖❭❈■▼③❑■❖❭✇✻◗✻❖ ❏ ❴✫◗✻⑤➐▼ ▼✫❑■❖❭✇✧◗❡❖✫❑■❖P❴ ❚ ❚ ❙ ❯ ❯ ❙ ◗❪❉t❫⑥q ❛ ❉➑❖❭❈❁◗❡❖❵❏✧✦✜★✑✩✟✪✍✫✭✬✯✮✰✩✘✱✎✲✖✳✔✴✵❋ ✬ ⑦ ✦ ❫❬❖✫◗✻❏ ⑤♥▼✱❴✱❑✹❖ ❴✫❑✹❖❬❴✩▼P❖③❴▲❈❁◗❪⑤➎q✭❑■❏▲q✭❑❁❉❥❖✫◗✻❴➆◗✻❖❬❖❭❈✧◗❡❖ ❫ ❖③❑■❉❢✶ ❵ ▼ ✳✵✷✍✮✏✩✘✱✎✲✖✳✔✴✸✬✟❤ ❙ ❙ ❯ ❚✺❙ ❚✺❙ ▼✫➌✟❉◆❴③❑ ❴✱▼❭⑤⑨❖③❑❁❏✭❴③❑❁❖P❴⑨❛❥❏✺▼✱❴✩❛❥⑤❘❖❭❈❧◗❪❖r❫❵❉◆▼✱⑤s✇■❉ ❖▲❈❀▼✫❑✹❖❹✇✹◗❥❖☎✄ ◗❪❉⑩❫✣q✄⑦ ❯ ❯ ❚ ✆ ❖③❊✁❖③❫✣q✟❛ ❛ ❖✶❫❵❴P❉❋❈ ❈✥q✟❖ ◗◆❑ ❫⑥❖③◗❪❏ ❴r❿ ❛❥❖✫❑➀❖✩❊✺q✟❛◆❉❜▼✩❉⑩◗❥⑤ ✄ ▼P❴✫❑✹❖ ❚ ❙ ❯ ❙ ❚ ❙ ❯ ❙ ❚ ❯ ⑤ ❚✭❙ ❫❵❖✫❑■❉❥▼✩⑤ ❯ 23 x1 = ( 1 − λ h ) x 0 x 2 = ( 1 − λ h ) x1 = ( 1 − λ h ) 2 x 0 ❖❭❈❁◗❪❖✽▼③❛❥❴✱❑➅▼P⑤ ❯ xn = ( 1 − λ h ) n x0 ❴③◗ ▼❭❉ q✁❖ ◆◗ ❑ xn → ∞ ❚ ❙ 1 − λ h > 1 ❙✭❚ ❴✩▼P⑤ n→∞ ✇❀❉ ❯ ❖P▼✩❉✆q✟❖ ❜◗ ❑ q✟❴❹✇■❉⑨❫❵❴✫❑■❉ ❖✎◗❪❉⑩❫✣q ❚ ❙ ❯ ❫❵❖✫◗✻❏ ❦ ❴ ❖❹❈✧◗❪❖❦❉ ❈✧◗❡❴✱②✟❉❢❛❜⑤✭✑⑦ ✡✄❖ ❜◗ ❑ ▼✱❴❦❫⑥❖③◗❪❏ ❴s❈■⑤s❞✻❉❜❖s❈✧◗✻❴✫②✟❉❢❛❥⑤✶◗❥❑✹❖✱② ❉❥❖⑨▼✱❴ ➄✇ ❉ ❖③▼❭❉ 2⑦ ❯ ❚ ❚ ❙ ❯ ❙ 1− λ h < 1 ❯ h< λ ✡❃❖ ❚ ◗❥❑ ❙ ❫❵❖③◗❪❏ ⑨ ❴ ❿ ❛❥❖✫❑➀❉♣❫⑥q✁❛❜❉◆▼P❉♣◗✻⑤⑨▼ ❯ ❙ ❙ xn = 1 x (1 + λ h) n 0 hλ 1− 2 xn = hλ 1+ 2 n ✁ ☎ ✄ ✝ ✄ ✝ ✄ ✝ x ✄ ✂ ✇❀❉✭❫❵❖③◗❪❏ ✶ ❴ ◗❥❑❁❴③q✟❖③❷ ❛ ❋ ❉ ▼ ❯ ❙ ❙ ❙ ✝ 0 ✆ ❈❀❖s❺✟❖ ➎ ❖ ▼✩⑤P❤ ❴✩▼❭⑤ ❤♥q✟❖ ◗◆❑ ❴✫❺✟❖③❫ ✇❀❉ ❖✩▼▲❉❋❫❬❖✫◗✻❏ ❖❭❛t❖➅❈ ◗✯❈✧◗✻❴✫②✟❉❢❛❥❖✩❤❋❉ ❥❉ ❞✻❖✱❑✹❖ ◗ ❖ ❯ ❯ ❚ ❙ n→∞ ❯ ❙✭❚ ❚✟❯ ❚ ❯ λ >0 xn → 0 ❯ ❫❵⑤✫❑■❉t❫♥❖③❴✆➌✈❴✆q✟❴❹❈ ❛ ❉ ❖✶◗✻❉t❫✣q✄⑦✌➋➑❴①❏➅❫❬❖✱◗❪❏ ⑤✈❈✹◗❪❴✫②✟❉❥❛◆⑤❾❫❵⑤✫❑✹❉❢❫⑥❖❭❴❾q✟❴❭❈ ❛ ❉ ❖s◗✻❉t❫➅q➐❖❹❈✧◗❡❖①❛❥❉t❫⑥❉⑩◗❡❴✫◗❪⑤ ❫❵❴❭❉ ❖ ❙ ❙ ❯ ❯ ❙ ❙ ❯ ✭❚ ❙ ❯ ❖✫❑■❏✭❴✫❑■❖✩❴❦❛◆❏✭▼✩❴P❛◆⑤⑨❉❢❫✣q ❈■⑤⑨▼P❴✱❑✹❖ ❖✫q✟❉ ❖ ❖✶q✭❑■❏▲②✟❛◆❖✫❫♥❴s❈❁◗ ❉❜❴✱◗◆⑤✭⑦ ❙ ❯ ❚✟❯ ❯ ❙✟❯ ➋✵❴ ❫❵❖③◗❪❏ ❴ ❿ ❛◆❖✫❑→❖③❊✺q✟❛◆❉◆▼P❉♣◗✻⑤ ❣ q✟❖ ◗❥❑ ❫❵❖✫◗✻❏ ❴♠❿ ❛◆❖③❑ ❉t➅ ❫ q✟❛◆❉◆▼✱❉t◗❡⑤❭❤ ❯ ❙ ❚ ❙ ❯ ❙ ε e = k1h 2 ε e = k 2 h 2 ε e = k 3h 3 q✟❖ ◆◗ ❑ ❫❵❖③◗❪❏ ❴➃◗◆❑■❴③q✁❖✱❷ ❛ ❉❜❝■❤❦q✟❴❭❈ ❛➅❞✧❉◆❉ ❛◆❉❢❫⑥❉❢◗❥❴③◗❵✇❀❉ ❖ ❚ ❙ ❯ ❙ ❙ ❙ ❚✟❯ ❯ 2 ❈■❖❶q✟❏✭❴✫◗✻❖ ✄ ✻◗ ❻✫❫✣q✼❛❜❴✝▼P❴ ❚ λ max ❛ ✫▼ ❑■❻ ▼ ✟ ❺ ❴✩❛❥❏▲❑❁❉✥❞✧❏✭❴✫❑✧◗❥❖①❫❵❉❜▼✩❉❃❴❭❛t❖✈❛ ❉✌➌ ❫♥⑤✫❑ ❛✄❫❵❴③❑■❖ ❖❸q✟❴❭✇➄❉ ❖✩▼P❖❭❈■❴③❑✹❉❃q✟❖ ◗❥❑ ❖✩◗❥❖✱❑✻❫♥❉ ❴✫❑■❖✱❴✣❈❀❏✭❛ ❉◆❖P❉ ❙ ❚ ❯ ❙ ✟ ❙ ❚✭❙ ❙ ❯ ❚ ❚ ❙➐❯ ❚ ❙✭④ ❈❀⑤ P▼ ⑤⑨❛❥❴ ✻◗ ❉t❫✣q ❖⑨▼P❴✩❛◆▼ ❛ ❖ ☛ ❈❧◗❪❉❥❞❧❉❜▼✱❴✫◗ ❖✶❫♥❴✫❑■❖❭⑦ ❯▲❙ ❙✭❚ ❯ ❙ ❚ ❙ ❯ ✆ ▼❭❴✱❷ ❛ ❖P❉❍❫❵❖✫◗✻❏ ❖♥❖✱❊❹q✟❛❜❉◆▼✱❉t◗✻❖✩❤ q✟⑤♥❴✩❛◆❖P♦✭❖✫❑■❖③❴⑥q✟❴❭❈ ❛ ❉ ❖➅◗❡❉t❫✣q❋❤✵❈➄❖✣q✟❏▲❑ ❖❭✇✧◗❡❖ ❖♥❛❥❴❬▼❭❏ ❉ ❉❥❴ ❚ ❙ ❙✁❚ ❯ ❯▲❙ ❙ ❙ ❯ ❚ ❯ ❚✁❯ ④ ❉ ❉ ❉◆❴P❛❢⑤ ✇■❉♥❈✹❖✒▼P❴③❛❥▼ ❛◆❖✱❴✩❷P⑤ ❴✩▼P❏▲q✁❖③❑■❉ ◗✻❏▲◗➐❉ ◗✻❖✱❑✻❺✟❴✩❛ ❛ ❖❶◗✻❉t❫✣q ▼✩❴③❑■❖ ❚ ④ ❙ ❚✼❯ ❚ ❙ ❯ x( t 0 ) x( t 0 + h ), x( t 0 + 2h ), ... q✺❑■❖✱❷❭❉ ◗✻⑤✈❉ ◗❡❖✫❑■❖❭❈➁⑦ ✆ ▼❭❴✫❷ ❛ ❖✩❉✌❫❵❖✫◗❡❏ ❖✣❉t❫✣q✟❛❜❉◆▼✱❉t◗❡❖✣❛❜❴✣❞✻❉❢❖P▼✩❴③❑❁❖①q✟❴❹❈❘❈■❖✣❞✻❴P▼❸❫❵❴P❉✄❫ ❛❢◗❥❖➅❉t◗❡❖✫❑■❴ ❉◆❉✷⑦✟➋✵❴➎q✭❑✹❉t❫❬❴ ❚ ❚ ❚ ❙ ❙✭❚ ❯ ❙ ④ ⑦✌✺①❴P❛◆❏✭❴✫❑■❖✱❴ ❉t◗❪❖③❑✹❴ ❉❢❖❬❈❁❖✣❞✧❏✭❛◆❏✁❈■❖❭✇✹◗❪❖✈❏⑥❫❵❖➁◗❪❏ ⑤✈❖✱❊✺q❱❛❜❉◆▼✱❉t◗❡⑤❬✇❀❉❍❈❀❖✈❏▲② ❉ ❖①q✭❑❁❖ ❉◆▼➁◗❪❏▲❑ ❛ ④ ❯ ④ ❚ ❯ ❙ x ( 0) = x + hf ( x ) k +1 k k ❏❭② ❉ ◗❪⑤❳q✟❖ ◗◆❑ q✭❑✹❖ ❉❥▼✱◗❥❏▲❑r❈➄❖❾❉ ◗◆❑■❏ ▼P✘ ❖ ✄ ❫❵❖③❫✣②✭❑ ❛ ❑■❖✫q✭◗➀❴✩❛✄❖P▼ ❴ ❉t❖❭❉❱❫⑥❖✱◗❡❏ ❖❭❉❋❉t❫⑥q✁❛❥❉❢▼P❉♣◗✻❖❸❏❭② ❉ ❻ ✹❈➄❖ ④ ❚✭❙ ❚ ❙ ❯ ❚ ❯▲❙ ❚ ❙ ❯ ❙ ④ ❯ ④ ❚ ❚✟❯▲❙ ❏ ❏ ⑤❳❺✟❴✩❛◆❏✭❴③❑❁❖❸❴❸❛ ❉ ✬❣ ✄ ❫♥❖✫❫✣②✭❑ ❛❃❈❁◗❪❴ ♦❹❝✵▼▲❴✫❑✹❖❸❛◆❴❸❉t◗❡❖✫❑■❴ ❉❥❴ ❑✧❫❵⑤③◗❪❏✭❴✫❑✹❖➎❈■❖❸❉ ◗❥❑❁❏ ▼✩❖ ❉ ❏ ✄ ❚ ❙ ❙ x (1) ❚ ❙ ❚ ④ ❙ ❚ ❯▲❙ ❯ ❚⑥❚ ❙ ❚ k+1 ❫❵❖✫❫✣②✭❑ ❛ ❑✹❖✱q✭◗ ✇■❉❃❈■❖❘❏▲② ❉ ❖ ❈✫⑦ ❴❹⑦ ❫ ⑦ ⑦✩q✟❻ ⑤s▼✱❖ ( N −1) − x ( N ) < ε ❉❢❫✣q ❙ ❈✫⑦ ✺①❴✱❛◆❏▲❑✹❉◆❛❜❖ ❙ ❯ ④ ❚ x (2) ❯ ❚ x k +1 k +1 k+1 ✎ x (1) , x ( 2) ... ❈➄❖ ❚✭❙ ❫♥❖❹❈✹▼❘▼P❏▲❑❁❖✱▼✱◗❡❏▲❑❁❉❧⑦ k +1 k +1 24 ✂✁ ✄✆☎✞✝✠✟☛✡☞✟✌✡✍✟✏✎✑☎✓✒✕✔✗✖✙✘✛✚☞☎✓✘ P ❖➆◗✻❏ P❖ ❛◆❖ ❫❵❖✫❑■❉❜▼✩❖ ❖❭❈➄▼✱❑❧❉◆❖ ✄ q✟❴③❑❁❴P♦▲❑✹❴✩❞ ❛❘❴ ◗❪❖③❑■❉◆❏▲❑ ❖P▼③❖▲❈❀❉♣◗✻⑤➍❈❀▼✫❑✹❉❜❖✱❑✹❖✩❴✽❖✩▼ ❴ ❉❜❉⑩❛❥❏▲❑ ❖➃❈✧◗✻❴✫❑■❖ ✄ ❯ ✭❚ ❙ ❯ ❚ ❙ ❚ ❚ ❙ ④ ❯ ❚ ❞❧❏▲❑✧❫⑥❴ ❏▲❑✧❫❵❴✩❛❥⑤ ✜ ⑦ ✉✟▼✫❑❧❉❥❖✱❑✹❖✱❴➍❖✩▼ ❴ ❉t❉◆❛❥❏❭❑✈▼P❉t❑■▼ ❉⑩◗ ❛ ❉ ✄ ❴▲▼✩❖✱❴❭❈❁◗❡❴ ❞✧❏❭❑✻❫♥⑤ ❖P▼③❖✺❈■❉♣◗❡⑤ ❖❭❛t❉❢❫❵❉ ❴✫❑✹❖P❴ ❚ ❙ ④ ❙ ❙ ❙ ❚ ❚ ❚ x = f ( x, t ) ❏▲❑❘❺✟❴✱❑✹❉❥❴✫②✟❉◆❛❥❖ ✇❁❉r❖✱❊❹q✟❛❜❉◆▼✱❉t◗✻❴✫❑✹❖✩❴♥❴P❛♣◗✻❏❭❑✹❴❵q✼❛◆❖P▼③❻ ❖➊❛❜❴❵❖P▼ ❴ ❉◆❉❜❛◆❖❬▼▲❉t❑✹▼ ❉❢◗ ❛ ❉❪⑦ ✝✣▼③❖▲❈❁◗❥❖♥❏▲q✟❖✫❑■❴ ❉ ❉✶❉t❫✣q✟❛❜❉◆▼✱⑤ ❙✺❚ ❚✁❯✽❯ ❙ ④ ❙ ❙ ❙ ④ ❙✭❚ ❑❁❖✱❷❭❏✭❛t❺✟❴③❑❁❖P❴ ❫❵❖③❑✹❉❢▼▲⑤ ❴ ❏▲❑❵❈➄❉❡❈✧◗✻❖✫❫❵❖ ❖➍❖✩▼ ❴ ❉❥❉❘❴✩❛❥♦✭❖✫②✭❑■❉❢▼P❖➍❛❢❉ ❉❥❴✫❑✹❖❶❈■❴ ❖❭❛❜❉ ❉◆❴✩❑✹❖❭❤⑨❏▲q✟❖✫❑■❴ ❉❜❖➍❴✩❛❘▼❭⑤③❑ ❉ ❚✺❙ ❙✺❚ ❯ ❙ ④ ❚ ❙✝❚ ❚ ④ ❙ ❑❁❖✱❷ ❛t◗❡❴✫◗✄q✟❏✭❴✫◗✻❖s❞✧❉✄❴✩❞✧❖✩▼➁◗❪❴➁◗ ❖s❖③❑■❏❭❑■❉❃❈■❖③❫ ❉❢❞✧❉❜▼❭❴➁◗❪❉t❺✟❖❸✇■❉✄❉❢❫✣q✟❛❢❉❜▼✩⑤ ❴ ❫❵❉⑩◗ ❖✩❞✻❏▲❑✧◗ ❖s▼✩❴✩❛❜▼ ❛❧⑦✑✝✈▼❭❖▲❈✻◗❃q✭❑❁❏✭▼P❖ ❖ ❙ ❯ ❚ ❙✭❚ ❚✺❙ ❯ ❙ ❯ ❙ ❖➃❴➍❉ ◗✻❖✩♦▲❑■❴➍❖✩▼ ❴ ❉❥❉❢❛❜❖➃▼③❉❢❑❁▼ ❉t◗ ❛ ❉❸❈■❖❶❈■❉t❫✣q✟❛❥❉◆❞❡❉❥▼✩⑤✽q✭❑✹❉ ✄ ❛◆❏✭▼ ❉♣❑■❖P❴ ❖✱❛◆❖③❫⑥❖ ◗✻❖③❛❥❏P❑ ❉ ❴③❫❵❉◆▼P❖ ▼ ❫❵❏ ❖✩❛❥❖ ❯ ❚ ❙ ④ ❙ ❙ ❙ ❚ ❚ ❙ ❚ ❯ ❚ ❙ ❯ ❑❁❖✱❷❭❉❪❈✧◗❪❉⑩❺✟❖ ❫❵❉t◗✻✣ ❖ ✢✥✤✧✦✩★✫✪✬★✮✭✯✤✧✢✱✰✳✲✌✴✩✵✶✤✧❇ ✴ ⑦ ✡❇❑■❏✺▼P❖ ❻ ❴❭❈❁◗✻❞✻❖✩❛❍❈■❖ ❖➁◗❪❖③❑❧❫❬❉ ⑤①❑❁⑤▲❈❁q ❈ ❛ ❑❁❖✱❷❭❏✭❛t❺✼❻ ▼✩❉❢❑✹▼ ❉♣◗ ❚✭❙ ❯ ❚✟❯ ❯ ❚ ❙✺❚ ❙ ❚✟❯✣❙✭❚ ❙ ❑❁❖✱❷❭❉❪❈✧◗❪❉⑩❺✈❛◆❉ ❉❜❴✫❑✯❈■❴ ❖③❛❥❉ ❉❢❴③❑➀❛❢❴❘❞✧❉❜❖✩▼P❴✫❑✹❖❦q✟❴✩❈ ❖✆◗❡❉t❫⑥q❇⑦ ❚ ❙➎❚ ❚ ❯ ✷ ✤✸✦✩★✯✪✶★✫✪✬★✹✭✫✤✧✢✞✰✳✲✑✴✩✵✺✤✸✴✻✰✛★✼✴✾✽❀✿❂❁❃❁✧✴❄✭✫✤✧✴✩✦✑★✼✴☛❅❆✲✩✽✶✤✧✿✻❅❆✲✩❁❇✤❃❈✩✤✌❈✑✵✺✴✌❉❇✪✶✵❀✴✩✵❊✲✌✿❋❉ ❖✫❑■❉♣❺✟⑤ ❉ ❫♥❖✫◗❪❏ ❴ ❯ ❯ ❚ ❯ ❫❵❖③❑■❉◆▼✩⑤ ❖✝❉ ◗✻❖✩♦▲❑✹❴✱❑❧❖ ◗❪❉❜❛❢❉❜❷✩❴✱◗✻⑤✜✄ q✟❴✫❑■❴✩♦▲❑■❴✩❞ ❛➎❴ ◗❪❖③❑❁❉❜❏▲❑➆✩ ⑦ P✝❖③◗❪❏ ❴ ❿ ❛◆❖✫❑➐❉❢❫①q❋❛❜❉◆▼✩❉⑩◗❪⑤ ❞✻❏✭❛◆❏✁❈❀❖❭✇❧◗❪❖ ❑❁❖P❛❢❴ ❉❥❴ ❚✺❙ ❯ ❚ ❙ ❚ ❙ ❚ ❯ ❙ ④ ❖❵❊➍❖✺❈✧◗❥❖❵❺✟❴✫❑✹❉❜❴✱②✟❉❢❛❜❴ ❖➊❈✧◗❡❴✫❑■❖✎⑦✌➏❸❖P▼③❉➑q✟❖ ◗❢❑ ▼P❏ ❖ ❈❀❴③◗❪❏▲❑①✇✹❉✷②✟❏▲②✁❉ ⑤➐❈➄❖❬q✟❏P◗❘❈■▼✱❑❧❉❥❖ ❯ ❚ ❙ ❚✟❯ ❚ ❚ xn +1 = xn + x● n +1 ❙✭❚✟❯ ❑❁❖P❛◆❴ ❉❥❉❢❛❥❖▲➂ ④ ⑧✆❏ ❍⑨❏▲②✟❉ ⑤ ❖ ❈■❴③◗❪❏▲❑ ❚✁❯ ❚ ❚ u■ n +1 = un +1 = un + hu❑ n +1 in +1 = C h un +1 − C h un 1 C i❏ n +1 = in +1 in +1 = in +1 = h L 1 L un +1 un +1 + in h un +1 + in L ✝✈▼❭❖❭❈✧◗✻❏▲❑✈❑■❖✩❛❜❴ ❉◆❉ ❛❢❖➍▼▲❏❭❑✹❖✺❈✧q ❈❀▼✫➌✟❖✫❫❵❖P❛◆❖➍❖P▼✫➌✟❉t❺✟❴✱❛◆❖ ◗❡❖➃▼✩❴✫❑■❖✎❑■❖✫q✭❑■❖✫❷❭❉ ◗❡⑤✽❫⑥❏ ❭❖ ❛ ❛s▼✩❏▲❫✣q✟❴ ❥❉ ❏ ❴✩❛ ❉ ④ ❙✺❚✟❯ ❚ ❚ ❯ ❙ ❚ ❚ ❙✁❚✭❙ ✩▼ ❏ ❖ ❈➄❴✫◗❪❏▲❑✯✇■❉✟❴P❛ ❖✩❉✭②✟❏▲②✟❉ ❖❘❴❹❈➄❏✺▼✱❉◆❴③◗✼❫♥❖✱◗❥❏ ❖❭❉✁❿ ❛◆❖③❑❍❉❢❫✣q✟❛❢❉❜▼✩❉❢◗◆❖✭⑦ ❚ ❯ ❚ ✟ ✭❙ ❚ ❚ ❯ ❙ P ❖➆◗✻❏ ✆ ❴ ◗❥❑✹❴✱q✟❖➁❷ ❛ ❉❋❞✧❏✭❛❢❏✁❈■❖▲✇❁◗❥❖✆❑✹❖P❛❢❴ ❉❥❴ h ❯ ❙ ❙ ④ xn +1 = xn + [x▲ n + x▲ n +1 ] 2 25 ⑧⑨❏ ✝ un = 1 C in , ❖ ❈■❴✫◗✻❏❭❑ ❚✟❯ ❚ ✝ un +1 = 1 ✝ ✌ in = in +1 C h h un +1 = un + un + un +1 2 2 2C 2C in +1 = un +1 − un + in h h ✝ ❘❏▲②✟❉ ⑤ ❚ ❍ ✄☎✆ 1 L ✌ in +1 = in + ✂ ✁ in +1 = h 2L 1 in +1 = un , h 2 ✌ in + un +1 ✡☛☞ + ✌ L h 2 un +1 in +1 h 2L un + in ✠✞ ✟ ✏✍ ✎✒✑✔✓✖✕✘✗✚✙✛✎✜✗✣✢✤✕✘✎✥✓✦✢✧✕✘★✪✩✘✫✭✬✮✕✘✓✘★✰✯✧✓✘✬✱✓✳✲✴✎✖✵✶✑✷✎✸✢✴✎✜✬✱✗✺✹✻★✻✗ x = 3 x − 1 x + h 2 x✼ ✽✘✾ ✑✷★✻✬✿★❁❀✜❂✺✩✭✕ n n +1 n −1 n +1 3 4 3 ✗✜❃❄✎❅✗❄✲❆✑✱❇❈✢✧✎✜✬❁✗❅✹❉★✿✎❈❊✭✎✺✩✖✑✻✢✔✫✚❃✦✓❄✩✭✕✘✎✣✩❋✲✴✗✺✑✷✓✦✢●✵✴★✳❍■✓✦❍✘❍✳★✿✩✭❇✤❏❁❑▲❊✘✢✴✎✣✫✘✩✭❇▼❃❅✫▲✎✜❃✺✫✭✗✣✹❉★✱★❁✬✱✎▼✕✘✎▼✯✷✫✘✩✭❃✣✹❉★✱✓✦✩✭✗✣✢✴✎▼✬✱✗◆❑❖✓✦❑❖✎✣✩✘✑✔✎✺✬✻✎▼✕✘✎ ✑✷★✪❑P❊◗✩✭❘✒❙❅❚❄✩❯✵❱★✘✩■❲❳❙❨✢✧✎✺❀❩✫■✬✿✑✻❇✘❬ ❡❢❣ 4 1 ❞❝ ❜ ❴❵❛ 2C C ❫❪ ❭ ✵✴★ 3C 2h u − u u i − i = u − = + i i n +1 2h n +1 h n 2h n −1 n +1 3L n +1 ✍❤✓✘✕✘✎✜✬✻✎✺✬✻✎✤❃✜✓✦❑▲❊✭✗✺✩✳★✻✓✦✩✚✗✦✲❱✓✖❃❅★✱✗✐✑✷✎❈❃❅✫❥❑P✎✣✑❉✓✘✕✘✗▼✙✛✎✜✗✺✢❦✕✘✎▼✓✦✢✴✕✘★✪✩✘✫✭✬❋✕✘✓✖★❧✲✶✫✘✩✖✑✧❬ ♠❳✓✦✩✭✕■✎✺✩♥✲♦✗✣✑✷✓✦✢ 3 n 3 n −1 ♣❈✓❄❍✭★✿✩✳❇ ✮q ✓✘✗✺✑✷✎✚❑❖✓✘✕✘✎✜✬✻✎✜✬❁✎❯❃✜✓✦❑▲❊✭✗✣✩✭★✱✓✦✩r❊■✢✧✎❅❀s✎✣✩✘✑❉✗✣✑❉✎❯❃✜✓✦✩✘✹✷★✿✩t❃❩❂✣✑✷✎P✓✉✢✴✎✐❀✦★❉✲✶✑❉✎❅✩✘✹❉❇▲❃❄✓✦✩❋✲✶✑✷✗✺✩s✑✷❇✚❊■✎✣✩✘✑✻✢✈✫r✫✘✩✇❊✭✗❄✲✛✕✘✎ ✑✷★✪❑P❊②①③✗✺✬✻✎❄✲✴❚④❏❁✩⑤❊■✗✣✢✴✗✜✬❁✎✜✬⑥❃✣✫⑦❃✺❂❅✑❉✎⑧✓⑨✲❆✫✘✢♦✲♦❇⑩✕✖✎❶❃✣✫✘✢✴✎✣✩✘✑❷❃✜✓✦❑❖✗✺✩✭✕✖✗✐✑✷❇⑩✕✘✎❸✑✻✎✺✩♥✲❱★✿✫✘✩✳✎❩✗⑩✲✴✗✺✫⑨❃❅✫✘✢✧✎❅✩✘✑❁✫✭✬ ❃✜✓✦✩✭✕✖✎✜✩♥✲♦✗✣✑✷✓✦✢✶✫✭✬✪✫✭★❹✲✴✗✺✫◗❍✭✓✦❍✭★✪✩✭✎✜★♥✬✱✗◆❑❖✓✦❑❖✎✒✩✘✑✻✫✳✬❺✕✖✎◆✑✷★✿❑▲❊✥✗✣✩✘✑✷✎✺✢❆★✻✓✦✢ ✽ ❻❼✎✣✩✘✑✱✢✶✫✥✫✘✩✚❊✳✗✦✲●✕✖✎❈✑✔★✪❑▲❊✚①P✕✘✗✣✑✶❚s❊✭✬✱✎❩❃✺❂✺✩✭✕✥✕✘✎✛✬✱✗❽✲❆✑❉✗✣✢✴✎✜✗❾★✪✩✭★✪✹❉★✱✗❅✬✱❇✛✕✘✗✐✑✷❇✛✗▼✎❩✬✱✎✣❑❯✎❅✩✘✑✻✎❩✬✱✓✦✢❿✕✖★✿✩✭✗✣❑❖★✻❃✺✎❥✲✴✎✛❃❅✗❄✬✪❃❅✫✭✬✱✎❩✗✐❀✦❇ ✢❆✗✦✲❆❊✘✫✖✩❋✲✶✫✭✬❦❃✜★✪✢✧❃✜✫✳★➀✑✱✫✭✬✿✫✳★❦✬✿✗◗❑❖✓✦❑P✎❅✩✘✑✻✫✳✬ t0+h ✫✘✑✻★✻✬✱★✪❀❄❂❅✩✭✕❯✓P❑❖✎✺✑✷✓✘✕✘❇✥✕✘✎✥★✿✩✖✑✔✎✜➁✦✢✴✗✣✢✴✎✸✩✖✫✘❑❖✎✜✢✧★✿❃❩❇✥✕✘✎✥✓✦✢✴✕✘★✪✩✘✫✭✬❧➂ ✽✭➃ ✩ ✗✜❃❄✎✦✲✔✑❧✯✶✎❅✬❋✗✣✩■✗❩✬✿★❁❀❄✗❳✫✘✩✘✫✭★❋❃✜★✿✢✈❃❅✫✭★➄✑❼✕s★✪✩✭✗✣❑❖★❁❃▼✬❁★✪✩✭★✱✗✺✢➅❊✭✓✘✗✣✑❉✎▼✯✶★✭✯✔❇✦❃✣✫✘✑✻❇✤❊✖✢✴★✿✩◗✢✧✎❅❀❄✓✘✬✪➆✭✗✣✢✴✎❅✗❳✫✘✩✘✫✳★❺❃✜★✪✢✧❃✺✫✭★✪✑❋✢✈✎❩❀✜★✷✲✶✑✻★✿➆✚✬❁★✪✩✭★✻✗✣✢ 26 ✱✬ ✗ ✯✶★✱✎❅❃✜✗✺✢❆✎✏❊✭✗s✲ ✕✘✎✏✑✷★✿❑▲❊ ✽ ✁✥❃❩✎✜✲❆✑❖❃✦★➄✢✧❃✺✫✳★✿✑ ❃✜✓✦✩✘✹✷★✿✩✳✎✏❑✉✓✖✕✘✎❩✬✿✎❄✬✪✎ ❃✜✓✦❑▲❊✭✗✺✩✳★✻✓✦✩ ✗✜✬✻✎ ✎✜✬✻✎✣❑❖✎✺✩✘✑✻✎❄✬❁✓✦✢ ✕✘★✪✩✭✗✺❑❖★✿❃❩✎❄❚ ✢❆✎❅❀❄★✷✲✶✑✷✓✘✗✣✢✧✎❾✬✱★➀✩✳★❁✗❅✢✧✎▼✕✘★✿❊✳✓✘✬❁✗✣✢✴✎✛✵✴★✭❑✥✫✭✬➄✑✔★✪❊✭✓✘✬✿✗✐✢✴✎✛✵❱★❹✲✶✫✘✢❱✲✴✎❈★✪✩✭✕✘✎✣❊✭✎❅✩✭✕✘✎✣✩✘✑✷✎ ✽ ➃ ✩✭❃✜✎✣❊✭❂❅✩✭✕✚❃❅✫▲❑P✓✦❑❖✎✺✩■✑❁✫✭✬ t0+2h ✲✴✎❾❊✭✓✖✗✺✑✷✎❈✫✘✑✷★❁✬❁★➀❀s✗❽✓❥❑▲✎❅✑✔✓✖✕✘❇❽✕✘✎✛✓✦✢✧✕✖★✿✩✘✫✳✬❹➂✶✄➂ ✂✶❃❩✗❈❑❖✎✣✑✔✓✖✕✘✗❳✑✻✢✴✗✣❊✭✎✺❀❩✫✭✬➀✫✭★ ✲❱✗✣✫❥❑❖✎✣✑✔✓✖✕✘✗▼✙✛✎❩✗✣✢❿✕✘✎▼✓✦✢❆✕✘★✿✩✖✫✭✬✭➂✔✆➂ ☎ ✞✽ ✝ ✗✜❃✜❇✤❊✳✗✦✲✶✫✭✬❋✕✘✎❳✑✷★✿❑▲❊❥✩✖✫❯✲✴✎❈❑❯✓✖✕✘★✻✯✈★✻❃✜❇❩❚✦❏✱✩✚❃❅★✪✢✴❃✺✫✳★✿✑✱✫✭✬✳✢✧✎❅❀❄★❉✲✈✑✷★✿➆P✲✴✎❈❑❯✓✘✕■★❁✯✶★✿❃❩❇ ✩✖✫✘❑❖✗❩★ ❊✭✗✣✢✧✗✜❑❖✎✒✑✻✢✧★✿★❾✲✶✫✘✢❱✲✴✎✜✬✻✓❄✢❽★✪✩✭✕✘✎❅❊✭✎✣✩✭✕✘✎✣✩✘✑✷✎✦❚✮➆✭✗❅✬✱✓✦✢✧★✱✬✻✎❯✢✧✎❩❀✜★✷✲✶✑✻✎✺✩✘✹✷✎❅✬✱✓✦✢✛✢✴❇✣❑❖❂✺✩✳❂✜✩✳✕ ✗✜❃❄✎❅✬✱✎❩✗✜✵❱★ ✟✽ ✝ ✗✜❃❩❇❯❊✭✗❄✲❆✫✭✬❳✕s✎ ✑✷★✪❑P❊P✲❆✎❳❑❖✓✘✕✘★✱✯✶★✿❃❩❇❳➆✭✗✜✬❁✓✦✢❆★✻✬✿✎✤✢❆✎✣❀❄★✷✲✶✑❉✎❅✩✘✹✻✎❩✬✱✓✦✢❹✑✱✢✴✎✣❍✘✫✭★✱✎◆✢✴✎✜❃❅✗❄✬✿❃✜✫■✬✱✗✣✑❉✎ ✽ ✠☛✡✌☞✎✍✑✏✒✍✓✏✔✍✖✕✑✡✌✗✙✘✄✚✜✛✎✢✒✡✣✛✤✘✥✍✓✛✎✦★✧✪✩✫✍✌✏✒✍✬✗✭✍✮✛✎✦★✍✑✏✒✍✯☞✎✢✰✛✱✚✜✗✭✢✔✕✬✍✲✛✎✍✞✏★✢✳✛✎✢★✚✜✧✴✍ ✲❱✎ ❃❩✓❄✩❋✲✶✑✻✢✶✫✳★✻✎❄✲✴❃r✗❄✲❱✎✺❑❖❇✣✩✭❇✐✑✷✓✦✢◗❃✣✫ ❑❖✓✘✕✖✎❩✬✱✎❩✬✿✎ ✎❩✬✿✎✺❑❖✎✣✩✘✑❉✎✜✬✱✓✦✢ ✬✿★➀✩✭★✱✗❅✢✧✎ ✽▼➃ ✩ ❃✜✓✦✩✘✑✷★✿✩✘✫✳✗✺✢✴✎❄❚❾➆✳✓✦❑ ✕✘✎✣✑✔✎✣✢✶❑❖★✿✩■✗ ❊✭✗✣✢✴✗✣❑❖✎✐✑✱✢✴★✻★✥✗✜❃❩✎❄✲✶✑✔✓✦✢❖❑✉✓✖✕✘✎❄✬❁✎✏❊✭✎✣✩✘✑✱✢✔✫ ❃✜✓✦✩✭✕✖✎✜✩♥✲♦✗✣✑✷✓✦✢✶✫✭✬❾✩✳✎❩✬✱★✿✩■★✱✗✣✢r✵✧★❾❍♥✓✦❍✭★➀✩✳✗ ✩✭✎✺✬✻★➄✩✭★✻✗✣✢✴❇ ❏❁✩❷❃❩✗✺❀✜✫✳✬❾✫s✑✔★✱✬❁★➀❀✦❇✣✢✴★✿★❾❑❖✎✐✑✷✓✘✕✘✎✜✶★ ✵✮✫✭✬✿✎❅✢✉★➄❑▲❊✭✬❁★✱❃❅★✪✑❉✎ ✽ ✍ ✓✘✕✘✎✜✬❁✎✜✬❁✎ ❃✜✓✦✢✴✎❄✲❆❊✘✫✘✩✘❀✦❇✣✑✔✓✖✗✺✢✴✎✤❃✜✎❩✬✱✓✦✢✧✬✱✗❩✬➄✑✔✎ ❑❖✎✐✑✷✓✘✕✘✎✛✕✘✎✤★✿✩✖✑✔✎✜➁✦✢✧✗❅✢✧✎❳✩✳✫✘❑P✎❅✢✶★✱❃✜❇✸✲✴✎✤✕✘✎❅✑✱✎❅✢✔❑❖★✿✩✭❇◆❏❁✩❥❑P✓✘✕❥✗❄✲❱✎✺❑❖❇✣✩✭❇✐✑✷✓✦✢ ✽ ♠◆✓✦✩✭✕■✎✺✩❋✲✴✗✣✑✔✓❄✢ ♣▼✓✦❍✭★✪✩✭❇ ✼ u = uˆ ( q ) in +1 = qn +1 ✼ un +1 = un + hun +1 ✼ un +1 = ✼ duˆ dq n +1 un +1 = un + h ✻ qn +1 = duˆ dq n +1 i = iˆ(ϕ ) duˆ dq n +1 ⋅ in +1 ✷✺ ✷ f (✹ in+✷ 1 ) ✷ ❈ q = qˆ (u ) ✸ in +1 = qn +1 ❈ qn +1 = qn + hqn +1 1 1 in +1 = q (un +1 ) − qˆ (un ) h ❇ h ❂❆❅ ❂❄❃ ( n+1 ) ⋅ in +1 ❁ u n +1 = ϕ n +1 ❁ in +1 = in + hin +1 ❁ ❁ diˆ diˆ in +1 = ⋅ in +1 ϕ n +1 = dϕ n +1 dϕ n +1 diˆ in +1 = in + h ⋅ un +1 ❀ dϕ n +1 ✾ ✽ ✽ f (✿u n+✽ 1 ) ✽ ■ ϕ = ϕˆ (i ) un +1 = ϕn +1 ■ ϕ n +1 = ϕ n + hϕ n +1 ϕ ϕ 1 1 un +1 = n +1 − n = ϕˆ (in +1 ) − ϕˆ (in ) h h h ❍ h❉❆● ❉❋❊ ( n+1 ) f i f u ♠✤★➀✢✴❃✣✫✭★➀✑✱✫✭✬♥✢✴✎✺❀❩★❉✲✶✑❉★✪➆❯❃✜✗✣✢✴✎❽✲❱✎❈✢✧✎❅❀❄✓✘✬✪➆✭❇✛✬✱✗✛✯✶★✪✎❄❃❩✗✣✢✴✎❈❊✭✗❄✲●✕■✎❈✑❉★➀❑▲❊P✎✦✲❆✑✻✎▼❏✱✩P✗✜❃❩✎❄✲✶✑➅❃❩✗✐❀❾✩✭✎❩✬✿★✿✩✳★✻✗✣✢❱❚■❃✺①✭★✿✗✺✢ ✕✘✗✜❃❩❇ 27 ✢❆✎❅❀❄★✷✲✶✑✷✓✘✗✣✢✧✎✦✬✿✎✉✕✘★✪✩ ❃❩★✪✢✈❃❅✫✭★➀✑ ✲❆✫✘✩✖✑◆✬✻★✪✩✭★❁✗✣✢✴✎ ✽ ✁✥❃❩✎❄✲✶✑✤❃❩★✪✢✴❃✣✫✭★➀✑❈✲❱✎P✢✧✎❅❀❄✓✘✬✪➆✭❇❖❊✖✢✴★✿✩✖✑❁✢✴❘✶✓r❑❖✎✣✑✱✓✘✕✘❇ ★✿✑✷✎✺✢❆✗✐✑✷★✿➆✭❇✜❚❿✕✘✎✉✓✦❍✳★✻❃✜✎❩★ ❑❖✎✣✑✔✓✖✕✘✗✁✥✎✄✂▼✑✷✓✦✩✭✆❘ ☎▼✗✺❊✖①❋✲✴✓✦✩ ✽ ✝✟✞✡✠☞☛✍✌✏✎✒✑✔✓✖✕✗✕ ★★☎ ★✱★✰☎ ✏✍ ✓✘✕✖✎❩✬✱✎❅✬✱✎✉❃✜✓✦❑▲❊✭✗✺✩✳★✻✓✦✩ ✗✜✬❁✎✉✎✺✬✻✎❅❑✚✎✜✩✖✑✔✎✺✬✻✓✦✢✸✕✘★✪✩✭✗✣❑❖★❁❃✜✎✉✬❁★✪✩✭★❁✗❅✢✧✎P❊✳✓✦✑✤✯✶★ ❃✜✓✦✩❋✲✶✑✱✢✶✫✭★✪✑❉✎r✵✴★●❃✣✫❤✲✶✫✘✢❱✲✴✎r✕✘✎ ✑✔✎✣✩❋✲❱★✪✫✘✩✭✎❆✂✶✲❱✎◗❊✳✓✘✗✺✑✻✎◗✑✱✢✧✗✣✩❋✲❱✯✶✓✦✢✔❑❖✗▲✲❆✫✘✢✴✲❱✗◗✢✧✎✜✗❄✬❁❇✥✕✘✎✥❃✺✫✘✢✧✎❅✩✘✑✰❏✱✩t✲✧✫✖✢❱✲✴❇✚✕✘✎❽✑❉✎✣✩❋✲❱★✪✫✘✩✭✬✎ ☎ ✽✙✘ ❘✶✗✣✫✉❊✘✢✴✎✜✯✶✎✣✢✧✗✜✑ ❃❩✎❄✬✿✎✚❃❅✫ ✲❆✫✘✢✴✲❱❇▲✕✘✎▲❃✣✫✘✢✴✎✣✩s✑●❃❄✗✣✢✈✎❖✲❆✫✖✩✘✑●✗❄✕✘✎✺❃✺➆✳✗✺✑✷✎◗❑❖✎✣✑✔✓✖✕✘✎❅★✰❊✭✓✦✑✷✎✣✩✘✹❉★✱✗❩✬✿✎❩✬✱✓✦✢ ✩✭✓✘✕✦✫✘✢❆★✻✬✱✓✦✢ ✽ ✍ ✓✘✕✘✎✜✬❁✎✜✬✻✎ ❃❩✓❄❑P❊✳✗✜✩✳★❁✓✦✩t✗❩✬✱✎▲✎❅✬✱✎✺❑P✎❅✩✘✑❉✎✜✬✻✓❄✢ ✩✭✎✜✬✻★✪✩✭★❁✗✣✢✴✎✥✗✺✫t✯✔✓✳✲❆✑ ❃❩✓✦✩♥✲❆✑✻✢✔✫✭★✪✑✷✎P✗❄✲✶✑✔✯✔✎✜✬➅❏✱✩✭❃✜❂✐✑ ✲✶✫✘✢♦✲♦✗❯✲✴❇❽✢✴✎✺❊✖✢✴✎❅❀❄★✪✩✘✑❉✎ ✍✎ ✚■❃✦✬➄✫❋✲❱★✪➆ ★➀✩✭✯✶✬✪✫✭✎✣✩✘✹❉✗✛➆✭✗✜✬✻✓✦✢❆★✻✬✱✓✦✢❨✕✘✎❥✬✱✗✸❑❖✓❩❑❖✎✣✩✘✑✻✫✳✬✰✗✣✩✘✑❉✎✣✢✴★✱✓✦✢ ✎✺➆✭✎✣✩✘✑✱✫✭✗✜✬❁❇✸✑✱✢✴✗✣✩❋✲❱✯✔✓✦✢✶❑❖✗✣✢✴✎❥✗✥✲✶✫✘✢❱✲✴✎❄★ ✜✽ ✛ ✗✺✢ ✯✶✗✜❃❩✎✜❘✶✓ ✲❆❇❷✕✘✎✺➆✳★✿✩✭❇ ✕✖★✿✩ ✲✶✫✘✢❱✲✴❇ ★➀✩✭✕✖✎✺❊✭✎✣✩✭✕✘✎✣✩✘✑✷❇ ✓ ✲❆✫✖✢✴✲❱❇ ❃❩✓✦❑❖✗✣✩✭✕✘✗✣✑❉❇⑥✩✳✎❅✬✱★✿✩✭★➀✗✺✢t✬✿✫✭❃✣✢✶✫ ❃❩✗❅✢✧✎ ❃❩✓❄❑P❊✳✬✻★✱❃❅❇▼✗❅✩✭✗❅✬✱★✪❀❄✗✤❃❩★✪✢✧❃✺✫✳★✿✑✱✫✭✬✪✫■★✭✢✧✎✜❀✜★✷✲✶✑❉★✪➆ ✽ ➂✈✩✘✑✷✎❩➁❄✢✴✗✺✢❆✎❩✗❯✎❄❃✣✫✭✗✐✹✷★❁★✱✬✻✓❄✢❽❃✺★✿✢❆❃✺✫✭★➄✑✻✫✳✬✿✫✭★◆✲❆✎ ✯✶✗❅❃✜✎r✕✘✎▲✢✧✎✦➁❄✫✭✬✻❇❯❃✣✫ ❊✭✗❄✲❆✫✭✬❦① ➆✭✗✣✢✴★❁✗❅❍■★✱✬ ✽ ❻❼✎✚❑❖❇s✲✧✫✖✢✴❇✉❃✜✎P① ✲❱❃❩✗✜✕✘✎❄❚✳✢✴✎✣❀❄★❉✲✧✑✻✎❅✩✘✹❉✗ RC ✕✘★✪✩❯❑❖✓✘✕✘✎✜✬✿✫✭✬➅❃❩✓❄✩✭✕✘✎✣✩❹✲❱✗✣✑✔✓✦✢✔✫✳✬✿✫✭★✮✲✴❃✦✗✜✕✘✎◗★❁✗✣✢ ✢❆✎❅❀❄★✷✲✶✑✷✎✺✩✖✹❉✗ RL ✕✘★✪✩❯❑✉✓✖✕✳✎✜✬✿✫✭✬ ❍✭✓✦❍✭★✪✩✭✎✜★❳❃✣✢✴✎❄✵✧✑✷✎ ✽ ✂ RC h2 = RL LC ✬✱✗❖❑❖✎✣✑✔✓✖✕✘✲✗ ✵❼✫✭✬✻✎✣✢✸★✿❑▲❊✭✬✱★✱❃❩★➄✑✔❇✬☎ ✽✣✢ ★✻✎❯✫✘✩ ❃❩★✪✢✴❃✣✫✭★✿✁✑ ✤ ♠ ✕✖✎✺✢✴★✪➆✭✗✣✹✔★✿✎✉❃✣✫ ✴✵ ★ C = 1 pF ❚●✗✣➆✭❂✣✩✭✕ ✯✷✢✧✎❄❃✣➆✭✎✺✩✖✹✔✗✇✕✘✎❖✢❆✎❅❀❄✓✦✩✭✗❅✩✘✹✻❇t✕✘✎ 1,6 ⋅10 Hz ❃✜✓✦✢✧✎s✲✶❊✘✫✳✩■❀❄✗✣✑✔✓✘✗❅✢✧✎ ✫✘✩✭✎✜★❷❊♥✎✺✢✧★✱✓✘✗✜✕✘✎⑨✕s✎ 6,3ns ✽ ♠❳✓❄✩❋✲❱★✻✕✖✎✺✢✧❂❅✩✭✕⑦✓③✯✶✓✦✢✔❑❖❇ ✕✘✎ ✫✖✩✭✕✘❇ ❃✺✫ ✲✴❃❅①■★✿❑✥❍✭❇❅✢✧★ ✗✣❍✘✢✔✫✘❊✘✑✷✎ ❊✘✢✴✎❄✲❆✫✘❊✘✫✘✩✳✎✺❑②❃✜❇❯★✿✩✳★✿✹✻★✻✗✺✬ ✲♦✎▲★✱✗ h0 = T = 6,3 ⋅10−11 s ★✻✗✣✢▼✗✜❃✜✎❅✗s✲✈✑✷❇✚➆✭✗✺✬✻✓✘✗✣✢✴✎❯✗✣✢❳❊✘✫✖✑✔✎✺✗✉✲♦❃✜❇✜✕✘✎❩✗ L = 1µH 8 100 ✘✕ ✎ ❙✦✥✧✥ ✕✘✎ ✓✦✢✧★✛✬✿✗ h = 6,3 ⋅10−13 s ✽❿➃ ✩ ✫✭✬✪✑✔★✪❑✥✫✭✬▼❃❅✗✺❀ RC = 4 ⋅10−7 ✕✘✎❄❃❩★✤❃❩★✪✢✧❃✣✫✭★✿✑✱✫✭✬✤✗✺✢❆✎r❏❁✩ 1 RL ❊✭✗✣✢✴✗❩✬✱✎❅✬✮✕✘✓✦✫✳❇◗✢✴✎✐❀✦★❉✲✶✑❉✓✘✗❅✢✧✎◗✗❩✬✿✎❥❃❄❇✺✢✧✓❄✢ ✢✴✎✺❀❩★❉✲✈✑✷✎✺✩✖✹✔✎✥✲❆✫✘✩✘✑❿✕✖★✻✯✔✎✣✢✴★✪✑❉✎❥❃✣✫ ✵✴✗✣❊✘✑❉✎✥✓✦✢✴✕✘★✪✩✭✎✥✕✘✎❾❑❖❇✣✢✴★➀❑❖✎ ✽✘✾ ✩ ✗✦✲❆✑✷✯✔✎✜✬❯✕✘✎ ❃✺★✿✢✧❃❅✫✭★ ✑❖✩✘✫ ❊✭✓✘✗✣✑✔✎ ✯✈★P✢✧✎❅❀❄✓✘✬✪➆■✗✺✑r❃❩✓✦✢❆✎❅❃✣✑ ❃❩✗✜✬❁❃❅✫✭✬✿❂✜✩✳✕ ❃✣✫ ❊✘✢✧✎❄❃❅★✿❀❩★✱✎ ✲✧★✿❑✥❊✭✬✱❇ ✂✖★ ❃❩★✱✯✷✢✧✎ ✲❱✎✺❑▲✩✭★✱✯✔★✱❃❅✗❅✑❉★➄➆✭✓✎ ☎ ✽ ➃ ✩ ✗✺✩✖✫✘❑❖★✿✑✷✎ ❃✺✗❅❀❅✫✘✢✴★ ❃✣①✭★✱✗✺✢②❃✜✗❅✬✱❃✣✫✭✬❁✎✜✬✻✎ ❏✱✩ ❊✘✢❆✎❩❃✜★❁❀✜★✻✎ ✕✦✫✘❍✭✬✱❇ ✂✐❙✍✩ ❃❩★✱✯✷✢✧✎ ✲❱✎✺❑▲✩✭★✱✯✔★✱❃❅✗❅✑❉★➄➆✭✓✎ ☎❧❊✭✓✦✑➅✲✴❇✏✩✘✫▲❃✜✓✦✩✭✕❄✫✭❃❩❇▼✬✿✗❈✢✶✎✺❀❩✫✭✬✪✑✱✗❅✑❉✎▼❃❩✓❄✢✴✎❅❃✣✑❉✎ ✽ ✪✬✫ ✭✯✮✖✰✲✱✴✳✁✮✆✵✍✶✷✱✜✸✙✰✹✶✻✺✏✳✁✼✁✱✾✽✿✮❁❀❂✼✁✶✜✸✹❃✣❄❆❅❇✸❉❈✔✶✴❊❋✼✁✸●❅❇✶❍❊■✮✖✱✧✮ ✾ ✩⑧✎✜✬✱✎✺❑❖✎✣✩✘✑ ✩✭✎❅✬✱★✿✩✳★❁✗✣✢ ✕✘✎ ❃❄★✿✢✧❃✣✫✭★➀✑ ✯✷✫✖✩✭❃✺✹✷★✱✓✦✩✭✎✜✗✣❀❄❇ ✬✱✗ ✲♦✎❅❑▲✩✭✗✺✬✻✎ ❑P★❁❃✜★t✗✣✑✻✫✖✩✭❃❄★r❃❅❂❅✩✭✕❑✦❏ ✎☞✳✚ ❃✣✫✘✢❱✲✴★✱✔✗ ▲ ❊✖✫✘✩✭❃✣✑✻✫✭✬✪✫✭★❋✕✘✎▼✯✷✫✘✩✳❃✺✹✷★✻✓❄✩✭✗✺✢❆✎✤❊✳✎❾❃✺✗❅✢✧✗✦❃✒✑✔✎✣✢✴★❁✲❆✑✻★✻❃✜✗❳✩✭✎❅✬✱★✪✩✭★❁✗✣✢✴❇✤✎✦✲❆✑✷✎❈✎✜❃✣①✭★✿➆■✗✜✬✱✎✺✩✘✑✻❇✛❃✺✫✚✕✘✎✣❊✭✬❁✗✜✲❱✗✺✢✴✎✜✗✤✬✿✫✭★✳❊✭✎❳✑✻✗✺✩✭➁✖✎✺✩✘✑✷✗✤✬✻✗ ✗✜❃❄✎❅✗❄✲❆✑✱❇❥❃❄✗✣✢✧✗❩❃✣✑✷✎✺✢✧★❉✲✶✑❉★✱❃❅❇❽❏❁✩❯❊✘✫✘✩✳❃✺✑✱✫✭✬❦✲✶✑✷✗✺✑✷★❁❃✸✕✘✎❥✯✷✫■✩✭❃✣✹✔★✱✓✦✩✭✗✣✢✧✎ ✽✖✾ ✩✉✎❄✬✿✎✜❑P✎✣✩✘✑❼✩✭✎❄✬✱★➀✩■★✱✗✜✢❨❃✣✫ ❃❩✗✣✢✴✗✜❃✺✑✻✎❅✢✧★✷✲✶✑✷★✪❃❄❇❥✬❁★➄✩✭★✱✗✜✢✧❇ ❊✳✎P❊✭✓✦✢✶✹✷★➀✫✖✩✭★✷❚✮✯✷✫✖✩✭❃✺✹✷★✱✓✦✩✭✎✜✗✣❀❄❇❯✬✻✗r✲✧✎✣❑P✩✳✗❩✬✿✎✚❑✉★✱❃✜★●✗✐✑✱✫✘✩✭❃✜★●❃❩❂✣✩✭✕✉❊✘✫✘✩❋❃✣✑✱✫✭✬●✕✘✎✉✯✻✫✘✩♥❃✺✹✷★❁✓❄✩■✗✜✢✧✎ ✲❱✎❯✕s✎✜❊✳✬❁✗s✲❆✎✜✗✜❀❩❇P❊✳✎❯✓ ✲❱★➄✩✭➁✖✫✖✢✧✗❾❊✭✓✦✢✶✹✷★✪✫✘✩✭✎❽✬✱★✿✩✳★❁✗❅✢✈❇ ✌✽ ✝ ✗✺❃❩❇❈✑✷✓✘✗✣✑❉✎❽✎✺✬❁✎❅❑❯✎❅✩✘✑✷✎❅✬✱✎❈✫✘✩✘✫✳★❼❃❩★✪✢✧❃✣✫✭★➀✑❼✩✳✎❅✬✱★✿✩✭★✿✗✺✢ ❏❁✩✭✕✘✎✣❊✭✬✱★➀✩✭✎❄✲✴❃❽✗❩❃✜✎✜✗✦✲✶✑✷❇✛❃❩✓✦✩✳✕✘★✿✹✷★❁✎◗✲♦✎ ✲❆❊✖✫✘✩✭✎▼❃❩❇▼❃✜★➀✢✴❃✣✫✭★➄✑✻✫✭✬❋✯✷✫✖✩✭❃✺✹✻★✻✓❄✩✭✎❩✗✺❀❩❇✛✬➀✗❽✲❆✎✣❑✚✩✳✗❩✬✱✎❈❑▲★✱❃❩★ ✽ 28 ★✱✎❈✎✜❃✺✫♥✗✐✹✷★❁★✿✬✻✎ ❑❖✎✺✑✻✓✳✕✘✎✜★■✑✻✗❅❍✭✬✻✓❄✫✭✬➀✫✭★✘❊✭✎❅✩✘✑✿✢✶✫❽✫✘✩✚❃❩★✪✢✴❃✣✫✭★✿✑✭✩✭✎✜✬❁★✪✩✭★✱✗✺✢ ❬ ✢ Ai = 0 U = A TV h( u , u , i , i ,φ ,φ , q , q , t ) = 0 ❺❻ ✢✴✎❄✲✶✫✘❊✘✫✘✩✭✎❅❑ ❃❩❇◗❃✦★✪✢✧❃✣✫✭★➀✑✱✫✭✬✰❃✜✓✦✩✘✹✷★✪✩✭✎✚✲✶✫✘✢❱✲✧✎❥❃❅✫❯❊✭✗✣✢✧✗❅❑❯✎❅✑✱✢✈★✰❃✜✓✦✩❋✲❆✑✷✗✺✩✘✑✿✹❉★❧❏✱✩❯✑✻★✿❑▲❊ ✂✶✕✘✎◗❃✺✫✘✢❆✎✺✩✘✑❦❃❩✓❄✩✘✑✔★✪✩✘✫✘✜✫ ☎ ✵❱★●✲❆✫✘✢✴✲❱✎▲❃❅✫r❊✳✗✜✢✧✗✣❑❖✎❅✑❁✢✴★❦➆✭✗✣✢✧★✱✗✺❍✭★✿✬❁★❿❏❁✩r✑✷★✿❑✥❊ ✽❺➃ ✩ ✗✜❃✜✎✦✗❄✲✶✑✔❇✉✲✴★➄✑✻✫✭✗✣✹❉★✱✎✉✲✴✓✘✬✪✫✘✹❉★✱✗❯✗❅❃✜✎✦✲✶✑✱✫✭★●✲❱★✷✲✶✑✷✎✐❑⑦✲✴✎✚❊✭✓✖✗✺✑✷✎✉✲✴❃✜✢✧★✿✎❯❃❩✗ ✲❆✫✖❑✉✗▼✗▼✕✘✓✘★✘✑✷✎✺✢✶❑P✎✣✩✭★✶❬ ~ i = I q + i (t ), u = U q + u~ (t ), v = Vq + v~ (t ), q = Qq + q~ (t ), φ = Φ q + ϕ~ (t ) ❻❺✢❆★✿❑✥✫✭✬❾✑✻✎✜✢✔❑❖✎✣✩ ( I q ,U q ,Vq ,Qq , Φ q) ✗✺✢❆✎ ✓❤➆✳✗✦✬✿✓✳✗✣✢✧✎④❃✜✓✦✩❋✲❆✑✱✗❅✩✘✑✷❇ ✵✧★❽❃❩✓❄✢✧✎✖✲✶❊✘✫✖✩✭✕✘✎ ✁✄✂✆☎✞✝✠✟✡✂☞☛✌✂☞✍✏✎✑✟✓✒✞✟✔✍✌✝✖✕✞✗ ✘ ✂✆☎✞✝✚✙✛✍✡✜✆☎✢✒☞✣✤✗ ✂✻❊ ✲ ✯ ☎●❃❅✗✣✢✴✎❽✎❄✲✧✑✷✎❥✲✴✓✘✬✪✫✘✹❉★✱✗❥✲✧★✻✲❆✑✷✎✺❑▲✫✭✬✪✫✭★❹✕✘✎✛✎❄❃✣✫✭✗✐✹✷★❁★❧✗❩✬❹❃✜★✿✢✈❃❅✫✭★➄✑✻✫✭✬✪✫✭★❋❏❁✩❖❃❩✗✒✢✴✎❥✲✶✫✘✢❱✲❱✎✜✬✻✎✸❃✺✫▲❊✭✗✣✢✴✗✣❑❯✎✣✑✻✢✧★ ✽✽✽ ➆✳✗✺✢✴★✱✗✣❍✭★✻✬✿★❼❏✪✩❯✑✔★➄❑P❊✇✲✶✫✘✩✖✑✰❊✭✗❄✲❱★➀➆✭★✿❀❄✗✺✑✷✎❩❚✳❍✭✓❄❍✭★✿✩✳✎❩✬✱✎✚✲✶✫✘✩✘✑✰❏✱✩✳✬✻✓✘❃✣✫✭★➄✑✔✎◗❃✺✫❯✢✧✎✺❀✦★❉✲✶✑❉✎✣✩✘✹✷✎✸✩✘✫✭✬✱✎❥✵✴★✰❃✜✓✦✩✭✕✖✎✜✩♥✲♦✗✣✑✷✓✘✗✺✢❆✎❅✬✱✎✚✲❆✫✘✩✘✑ ❏❁✩✭✬✱✓✘❃✣✫✭★✿✑✻✎ ❃✺✫④✢❆✎❅❀❄★✷✲✶✑✻✎✜✩✖✹✔✎✇★✿✩✳✯✶★✿✩✳★ ✑✔✎ ✽ ✁✥✬✤✕✘✓✘★✱✬✻✎✜✗r✑❉✎✣✢✶❑❖✎✺✩ ( i~, u~, v~, q~, φ~ ) ✎✦✲✶✑✷✎r➆✭✗✺✢❆★✪✗❅❍✭★✻✬◆❏❁✩④✑✔★✪❑▲❊ ✵✧★✤➆✳✎✺✢✴★✿✯✶★✱❃❩❇ ✎✜❃❅✫✭✗✣✹❉★✱★❁✬❁✎ Ai~ = 0 si U~ = A V~; ✁✥❃✜✎❄✲❆✑❉✎✉❑✉❇✣✢✧★✪❑✉★ ✢✴✎✣❊✘✢✴✎✐❀✦★✪✩✘✑❉❇ ✕✘✎ ✯✶✗✣❊✘✑✛✕✖✎✺➆✭★✱✗✣✹❉★✱★❸❑❖★❁❃✜★◆❏✱✦✩ ✥✧✫✘✢✶✫✭✬ ❊ ✽ ✲ ✽ ✯ ✽ T I q ,U q ,Vq , Qq , Φ q . ✎✜❀✜➆✳✓✘✬✿✑✻❂❅✩✭✕▲❏✱✩ ✲✴✎✜✢✧★✿✎▲q❦✗✤✧✭✬✱✓✦✢◆❏❁✩★✥♦✫✖✢✔✫✭✬✰❊ ✽ ✲ ✽ ✯ ✽ ✎✺❃✺✫✳✗✺✹✻★✻✗✥❃❩✓❄✩❋✲❆✑✔★➄✑❁✫✘✑✷★✪➆✭❇✚✗✥✯✶★✱✎❅❃✜❇✺✢✈✫✭★❦✎✜✬❁✎✒❑❖✎✺✩✘✑❿✩✭✎✺✬✻★✪✩✭★✿✗✺✢▼✵❱★ ✢❆✎✺✹✷★✿✩✳❂✺✩✭✕ ✩✘✫✘❑❖✗✜★✥✑✷✎✣✢✶❑❖✎✺✩✘✫✳✬P✕✖✎⑥✓❄✢✴✕✘★✪✩✘✫✭✬✥➂❖✢✴✎✺❀✜✫✳✬✿✑✷❇ ❃❩❇✪i✩ , u✩ , v✩ ,q✩ ,φ✩ ➆✳✎✺✢✴★✱✯✔★✱❃❩❇✏✩✭★✱✵❆✑❉✎ ✎✜❃✺✫✭✗✣✹✷★✪★✚✬✱★✪✩✭★✻✗✣✢✴✎ ❃❅✗❅✢✧✎ ❏❁✩✭✬✱✓✘❃✣✫✭★✻✎❄✲✴❃▼✎❅❃✣✫✭✗❅✹❉★❁✗ h( ⋅ ) = 0. ✵❿❃✣✫✭✗✺✹✷★✱✗❈❃✜✓✦✩❋✲❆✑✷★➀✑✿✫✘✑✔★✪➆✭❇ ✗❳✫✘✩✘✫✭★✳✢✴✎✺❀❩★❉✲❆✑✷✓✦✢✮❃✜✓✦✩✘✑❁✢❆✓✘✬✻✗✣✑♥❏✱✩◗✑❉✎✣✩❋✲✴★✿✫✘✩✳✎✛✕✘✎✺➆✳★✿✩✭✎✦❬ i = f (u) ✝ i ( t ) + I q = f (U q ) + f ' (U ) U u~ +... q ✥✓❄✑✔❂✣✩✭✕ ❃❩✓❄✩✭✕✦✫✭❃✣✑✔✗✣✩✘✹✷✗④✕✘★➄✩✭✗❅❑❯★✿❃✦❇✇❏❁✩✏❊ ✽ ✲ ✽ ✯ ✽ ❃✺✫ Gdq = f '( u) U ✵❱★✛✕✘✎✜✓✘✗✣✢✴✎❅❃✜✎ q I q = f (U q ) ✢✴✎✺❀✜✫✳✬✿✑✻❇ ✓ ✖✕ ✎✺❊✭✎❅✩✭✕✖✎✺✩✘✹✷❇ ✬✻★✪✩✭★✿✗✺✢✴❇✇❏❁✩✘✑✱✢✧✎ ~i (t ) si u~(t ): i~(t ) = G u~(t ). ❻❹✎❅✩✘✑✻✢✔✫❤✫✘✩❤✢✴✎✺❀✦★❉✲✈✑✷✓✦✢✚❃❩✓✦✩✖✑✻✢✴✓✖✬✻✗✣✑❾❏✱✩✏❃✜✫✖✢✧✎✜✩✖✑ dq ✢❆✎❅❀❅✫✭✬✪✑❉❇ u~(t ) = R ⋅ ~i (t ) ✫✘✩✭✕✖✎ R ✎❄✲❆✑❉✎❾✢✈✎✺❀s★✻✲✶✑✔✎✣✩s✹✷✗✛✕✘★✿✩✳✗✺❑P★✻❃✺❇✤❏✱✩✥❊ ✽ ✲ ✽ ✯ ✽ ❻❺✢✴★✪✩✘✑❁✢❆❘✷✫✘✩✥✢✧✗✜✹✷★❁✓❄✩✭✗❅❑✉✎✣✩s✑✮✲✴★✿❑P★✱✬❁✗✣✢ dq dq ✢❆✎❅❀❅✫✭✬✪✑❉❇▼✎❩❃✣✫✭✗✣✹❉★✱★❁✬✿✎❾✬✱★➀✩✳★❁✗❅✢✧✎❳❊✭✎✺✩✖✑✻✢✔✫❧❬ ❘ ✲❆✫✘✢✴✲❱✎❩✬✱✎✤❃❩✓✦❑❖✗✣✩✭✕✘✗❅✑❉✎❳✩✭✎✜✬❁★✪✩✭★✱✗✺✢ i1 = f ( u2 ) → i~1 = G u~2 dq ~ i1 = f (i2 ) → i1 = β dq i~2 u1 = f ( u2 ) → u~1 = α dq u~2 u1 = f (i2 ) → u~1 = Rdq i~1 29 ❘❧❍✳✓✦❍✭★✪✩✭✗▼❃❩✓✦✩✖✑✻✢✴✓✖✬✻✗✣✑✔❇◆❏❁✩✥❃✺✫✖✢✴✎✺✩✖✑ df ~ φ = f (i ) → Φ = Ldq i~ cu Ldq = q di ~ u~ = φ si ❘❧❍✳✓✦❍✭★✪✩✭✗▼❃❩✓✦✩✖✑✻✢✴✓✖✬✻✗✣✑✔❇◆❏❁✩▲✯✔✬➀✧ ✫ ✚ ~ ~ i = f (φ ) → i = Γdq φ df cu Γdq = q dφ ❘✰❃✜✓✦✩✭✕✖✎✺✩❋✲❱✗✣✑✷✓✦✢✶✫✭✬✭❃❄✓✦✩✘✑❁✢❆✓✘✬✻✗✒✑❺❏✱✩❽✑✷✎✣✩❋✲❱★➀✫✘✩♥✎ df q = f ( u) → q~ = Cdq u~ cu Cdq = q du si ✁ q~ = Cdq u~ ❘✰❃✜✓✦✩✭✕✖✎✺✩❋✲❱✗✣✑✷✓✦✢✶✫✭✬✭❃❄✓✦✩✘✑❁✢❆✓✘✬✻✗✒✑❺❏✱✩P✲♦✗✣✢✧❃✜★✿✩✭❇ u = f (q ) → u~ = Sdq q~ cu ❻❺✢❆★✿✩ ❏❁✩✭✬✻✓✖❃✺✫✭★✪✢✴✎✜✗✉✯✔★✱✎❩❃✜❇✺✢✔✫✳★❳✎❅✬✱✎✣❑❯✎✣✩✘✑❳✩✳✎❩✬✿★➀✩✭★✱✗✺✢✸✕✘✎ S dq = df q dq ❃❩★✪✢✈❃❅✫✭★✿✑◆❃✣✫ ✎✜✬✻✎✒❑✉✎✣✩✘✑✻✎❩✬✱✎r✕s★✪✩✭✗✣❑✉★✿❃❩✎ ❃✜★✪✢✴❃✺✫✳★➀✑✱✫✭✬♥✎✜❃✣①✭★✿➆✳✗❅✬✱✎✺✩■✑❋✬✱✗✸✲✴✎✣❑▲✩✭✗❅✬✱✎◆❑❖★✿❃❩★✭❃❩✗✣✢✧✎✛✎❄✲✈✑✷✎◆✫✖✩▲❃❄★➀✢✴❃✣✫■★✪✑❹✬✱★✿✩✳★❁✗✣✢ ✕✘✎P❑❖✗✜★❳✲✶✫❋✲✚✲❱✎❖✓✦❍✘✹✷★✿✩■✎ ✽ ✁✥✬✱➁✘✓✦✢✴★✪✑✻❑▲✫✭✬✭✕✘✎▼✗✣✩✭✗❩✬✿★❁❀✜❇▼✗❩✬✘✫✘✩✘✫✭★❋❃✜★➀✢✴❃✣✫✭★➄✑❹❃✺✗✜✢✧✎▼✯✷✫✖✩✭❃✺✹✷★✱✓✦✩✭✎❅✗✜❀❄❇❈✬✿✗✛✲❱✎✐❑▲✩✭✗✜✬❁✎❳❑❖★❁❃✜★♥✎❄✲✶✑❉✎✖❬ ❙✪☎ ✎❾✕✖✎✺✑✷✎✺✢✈❑✉★✪✩✭❇❳❊✘✫✘✩✳❃✺✑✱✫✭✬❹✲✈✑✷✗✣✑❉★✱❃❾✕✖✎❾✯❉✫✘✩✭❃✐✹✷★✱✓✦✩✭✗✣✢✴✎❳❊✘✢✴★✪✩▲✗✣✩✭✗❩✬✿★❁❀✜✗▼❃✦★➄✢✧❃✺✫✳★✿✑✱✫✭✬✪✫■★♥❏❁✩▲❃✜✗✣✢✧✎✤✑✷✓✘✗✣✑✔✎✛✲❆✫✘✢✴✲❱✎✺✬✻✎✤❃✣✫ ✘ ❊✳✗✺✢✴✗✣❑❖✎✺✑✱✢✧★⑥➆✭✗✣✢✴★✿✗✺❍✭★✱✬❁★ ❏❁✩⑦✑❉★✪❑▲❊ ✲❆✫✖✩✘✑ ❊✭✗❄✲♦★✪➆✭★✿❀❩✗❅✑❉✎❄❚④❍✭✓✦❍✭★✪✩✭✎✜✬❁✎⑤✲❆✫✖✩✘✑❷❏✱✩✳✬✻✓✘❃✣✫✭★✪✑❉✎⑩❃❅✫⑦✢✧✎✜❀❩★❉✲✧✑✻✎✺✩✖✹✔✎❶✩✖✫✭✬✻✎②✵♦★ ❃✜✓✦✩✭✕✖✎✜✩♥✲♦✗✣✑✷✓✘✗✺✢❆✎❩✬✿✎✸✲❆✫✖✩✘✑❋❏❁✩✭✬✿✓✘❃❅✫✭★✿✑❁✎❾❃✣✫❥✢❆✎❅❀✜★✻✲✶✑✔✎✣✩✳✹✻✎▼★✿✩✭✯✔★✪✩✭★➄✑✔✎ ✂✜☎ ✘ ✎ ❃✜✗❩✬✱❃✺✫■✬✱✎✜✗❅❀❄❇ ❊❋✗✣✢✈✗✜❑❖✎✒✑✻✢✧★✱★ ( Rdq , Gdq , Ldq , Γdq , Cdq , S dq , α dq , β dq ) ✄✣☎ ✎ ✘ ✯✶✗❩❃✜✎ ✗✺✩✳✗❩✬✱★✪❀❄✗ ❃❩★✪✢✧❃❅✫✭★ ✑✱✫✭✬✿✫✳★ ✽ ❃❩★✪✢✧❃❅✫✭★ ✑✱✫✭✬✪✫✭★ ✎❅❃✣①✭★✪➆✭✗✜✬❁✎✣✩✘✑ ❊✳✎✺✩✘✑✱✢✔✫▲✯✈★✻✎✜❃❅✗✣✢✴✎✤✎✦✬✿✎✺❑❖✎✣✩✘✑✭✩✭✎❩✬✿★✿✩✳★✻✗✣✢ ✎✜❃✣①✭★➀➆✭✗✜✬✱✎✺✩✘✑ ✕✖✎ ✲✴✎❅❑✥✩✭✗❩✬r❑P★✻❃ ✕✘✎ ✲❱✎✣❑P✩✳✗❩✬ ❑P★✻❃ ❃✜✓✦✩✘✹✷★✪✩✭✎ ✩✘✫✘❑❖✗❩★ ✲❆✫✘✢♦✲♦✎✜✬❁✎ ✽ ❃✜✗✺✢❆✎ ★✪✩✭✕✘✎✣❊✭✎❅✩✭✕✘✎✣✩✘✑✷✎❳➆✭✗✺✢❆★❁✗✣❍✭★✻✬✿✎❳❏❁✩❥✑✷★ ❑▲❊ ✂✧✲✧✫■✢✴✲❱✎❩✬✿✎▼✕✘✎❈❃✣✫✘✢✴✎✣✩✘✑❧❃✜✓✦✩✘✑❉★✪✩✘✫✘✫✚✯✶★✱★✿✩✭✕✛❊✭✗❄✲♦★✪➆✭★➀❀✦✗✣✑❉✎✓☎ ☎✜☎ ❃✜✓✦✢✴✎✜❃✣✑❉❇ ✎✇✯✔✗❄❃❩✎❯➆❋✎✣✢✧★✻✯✶★✱❃✜✗✺✢✴✎✺✗ ✘ ❊✭✎✣✩✘✑✱✢✶✫⑥✯✶★❁✎✜❃❩✗✣✢✴✎ ✢✧✎✜❀❅✫✭✬➀✑✷✗✣✑❉✎✜✬✻✓✦✢✛✑✷✎❩✲✶✑✷❂❅✩✭✕ ❏✱✩ ✽ ❃❄✎❯❑✉❇❄✲❆✫✘✢✧❇✇✗✺❊✖✢✴✓✡✚✳★✪❑❖✗✺✢✴✎✜✗r✕■✎t✲❱✎✣❑▲✩✭✗❩✬●❑❖★❁❃✇✎✦✲✶✑❉✎ ✎❄✬✪✎❅❑❖✎✺✩✘✑✛✩✳✎❅✬✱★✿✩■★✿✗✺✢P✕✘✎④❃✜★✿✢✈❃❅✫✭★➄✑❆✂✔✕✘✗❅❃❄❇ ❏✜✎✔■ ✚ ❃✺✫✘✢✴✲❱★✱✗✍▲ ❊✘✫✘✩✭❃✣✑❁✫✭✬✪✫✭★✛✕✘✎ ✯❉✫✘✩✭❃✣✹✔★✱✓✦✩✭✗✣✢✴✎t❊✭✎ ❃✜✗❅✢✧✗❩❃✣✑✷✎✺✢✧★✱✲❆✑❉★✱❃❩✗ ✩✭✎✜✬❁★✪✩✭★✻✗✣✢✧❇❳❊✭✓✘✗✣✑✔✎✤✯✶★✭✗✺❊✖✢✴✓✡✚✳★✪❑❖✗✺✑✷❇❈❃✣✫ ❏✜✎✄✚✳❃✺✫✖✢❱✲✴★✻☞ ✗ ▲✤❊✳✎✤✑❁✗✜✩✳➁✘✎✺✩✖✑✔✗◆❏✱✩◗❊ ✲ ✯ ☎ ✽ ✽ ✽ ✽ ✆✞✝ ✟✡✠☞☛✍✌✞✎✑✏✓✒✔✌✕☛✖✠☞✗✑✘✚✙✛✏✢✜✛✠☞✜✣✌✕✤✥✗✔✌✦✎★✧✩✤✪✙✫✜✣✌✬✤✭✒✮✠✯✗✰☛✍✤✢✧✲✱✯✙✣✤✢✱✯✧✔✙✳☛✖✠✭✎✔✠✴✱✯✧✰✤✵✠✯✗✔☛✶✏✢✜✳☛✷✠✸✤✪✗✮✏✹☛✍✙✻✺ ✲❱❘❉✗❯✗✺✢✧❇✣✑✔✗✣✑❦❏✱✩✇❊✭✗✣✢✧✗✦➁❄✢✧✗✦✯✻✫✳★ ✬ ✂ ✰ ❚ ✲✴✎✚❊✘✢✧✎❄✯✔✎❅✢✧❇▲❃❩✓✦❑❖✗✣✩✭✕✘✗✥❏❁✩t✑✻✎✺✩❋✲✴★✪✫✘✩✭✎❯✕✘✎✜✓✘✗❅✢✈✎❩❃❄✎❥❑P❇✜✢✧★➄❑❖✎❩✗▲✕✘✎ ✽ ❃✜✓✦❑❖✗✣✩❋✕✘❇✚❊✭✓✖✗✜✑✻✎✉✯✔★ ✲❱❃✣✢✧★✷✲✴❇❯❃✜✗✉✓t✕✖★✻✯✶✎✣✢✴✎✣✩s✹❉❇✉✕✖✎P❊✭✓✦✑✻✎✜✩✖✹❉★❁✗✜✬✱✎ ♠❈✗❥✫✘✢✶❑❖✗✣✢✴✎❄❚✰❃✜★✿✢✧❃❅✫✭★➄✑❁✫✭✬●✎✜❃✣①✭★✿➆✳✗❩✬✿✎✜✩✖✑ ❃✣✫ ✲✶✫✘✢❱✲✴✎✉✕✘✎ ✽ ✑✷✎✣✩❋✲❱★✿✫✖✩✭✎ ❃✜✓✦❑❖✗✺✩✭✕✖✗✺✑✻✎ ❏✱✩ ❃✣✫✘✢✴✎✣✩✘✑✇✗✜✬P❍✭✓✦❍✳★✿✩✭✎✜✬✱✓✦✢ ❃❅✫✘❊✳✬✻✗✣✑✔✎ ✩✘✫ ✎✦✲✶✑✷✎⑥❊✭✓✦✑✱✢✴★✪➆✭★➀✑❯❊✭✎❅✩✘✑✱✢✔✫❸✲❱❃✺✢✧★✱✎✣✢✴✎❩✗❷✎❩❃✣✫✭✗✣✹❉★✱★❁✬✱✓✦✢ ✁▲✲❱✗❯❃✣✫✘❑ ❑❖✎✣✑✔✓✖✕✘✎❩★ ✩✭✓✘✕✖✗❩✬✿✎ ✾ ❃✜✓✦❑❖✗❅✩✭✕✘✗✣✑✔✎ ✤ ➂ ✼✾✽ ω ✼ ✼✾✿ ✥ ✥ ❻❼✎✣✩✳✑✿✢✶✫❤✗❅❃✜✎❄✲✧✑✷✎❖❍✳✓✦❍✭★✪✩✭✎ ✲✴✎✉❊✭✓✘✗✣✑❉✎✇❃❩✓❄✩❋✲✶✑✻✢✶✫✳★❨✫✘✩❤❃✜★✪✢✶❃❅✫✭★✪✑✤✎✜❃✺①✳★✿➆✭✗✺✬✻✎❅✩✘✑❈❃✺✫❤✲❆✫✖✢❱✲✴✎t✕✘✎❖❃✜✫✖✢✧✎✜✩✖✑ ✽ ❏❁✩ ✑✷✎✺✩❋✲❱★✪✫✘✩✭✎ ✩ ✗✜❃✦✎❄✲✔✑✥✲♦❃✜✓❩❊❷✲♦✎t✢✧✎✺❀s✓✘✬✪➆✭❇ ✎❩❃✣✫❋✗✒✹✔★✿★❁✬❁✎ ✕✖✎ ✯❉✫✘✩✭❃✣✹✔★✱✓✦✩✭✗✣✢✴✎ ✗❩✬✱✎r❍✭✓✦❍✭★✪✩✭✎✜✬✻✓❄✢▲❃✣✫✘❊✭✬✱✗✺✑✻✎✳❬ ✽❿➃ ✍ ➂ ❀ ❚ ❀ ✤✞❀✿➂ ❀ ✍④➂ ❏✱✩◗✢✧✗❅❊✭✓✦✢✔✑❧❃✺✫◗✩✭✎✜❃✺✫✘✩✳✓✳✲✴❃✜✫✖✑❉✎❄✬❁✎✤➂ ✵♦★■➂ ❀ ω ✾ ✽ ω ✿ ω ✼ ✼ ✽ ✥ ✥ 30 ✂✁☎✄✝✆✂✞✠✟☛✡✌☞ ✢ U 1 = 2 jI 1 + jI 2 ★✱✎ ❍✭✓✦❍✳★✿✩✭✎✜✬✱✎ si ❃✺✫✖❊✭✬✻✗✣✑❉✎ ❃✺✫ ✎✜❃✺✫✳✗✺✹✻★❁★✱✬✻✎ ✕✖✎ ✯✻✫✘✩✭❃❅✹✻★✻✓✦✩✳✗✜✢✧✎ U 2 = jI 1 + 2 jI 2 ☎▼✎✺❀✦✓✘✬✪➆■❂✣✩✭✕❥✗✜❃❩✎❄✲✶✑➅✲♦★✱✲❆✑❉✎✣❑❸✕✘✎✤✎❅❃✣✫❋✗✣✹❉★✱★✘❏❁✩❥✢✧✗✣❊✭✓✦✢✶✑❧❃✣✫✥➂ ✵♦★✭➂ ✢✴✎✣❀❅✫✭✬✪✑❉❇✖❬ ❀ ✼ I1 = ✗✜✕✘★✱❃❩❇▼✎✺❃✺✫✭✗✣✹❉★✱★✱✬❁✎ ✍ ✞✎✟✑✏☛✒✔✓✖✕✗✏✘✄✙☞ ✘ U1 j + U2 j3 3 2 si U2 j + U 3 3 1 j 2 I2 = ✫✘✢✶❑❖❇✣✑✔✓✦✢✈✫✭✬➀✫✭★✭❃❄★➀✢✴❃✣✫✭★➄✑❹✎✜❃✺①✳★➀➆✭✗❄✬✱✎✺✩✘✑✶❬ ❇✛✲✴✎❾✲❱❃✣✢✧★✱✎❈✎✜❃✺✫✳✗✺✹✻★❁★✱✬✻✎ ❊✭✓✦✑✷✎✺✩✖✹❉★❁✗✜✬✱✎❩✬✿✓✦✢t✩✳✓✘✕✦✫✘✢✴★✱✬✱✓✦✢❼❊✭✎✣✩✘✑✱✢✶✫▲❃✜★✿✢❆❃✺✫✭★➄✑✻✫✳✬ e ( t ) = 2 cos 2t π i ( t ) = 2 sin( 2t + ) s 4 ✎❄✓✘✗✣✢✧✎❄❃❩✎✸❃✺✫✘❊✳✬✻✗ ♦✫✭✬❹✩✘✫✇✲✴✎✸❊✭✓✖✗✺✑✻✎✚✲❆❊✭✗✣✢✴➁✘✎❽❃✜✓✦✩❋✲✶✑✱✢✶✫✭★✪❑⑧❃✜★✿✢✧❃✣✫✭★✪✑✻✫■✬➅✎❅❃✣①✭★✪➆✭✗❩✬✿✎❅✩✘✑✮❃✣✫✇✲❆✫✘✢✧✲❱✎❥✕✘✎◗❃✺✫✘✢❆✎✺✩✘✑✰❃❩✓✦❑❖✗✣✩✭✕✘✗✣✑❉✎ ✥ ✝ ❏❁✩✥✑✔✎✣✩❋✲✴★✿✫✖✩✭✎❽✗❅✬❹❃✺✎❩✬✱✓✦✢ ✕✘✓✦✫✭❇❈❍✳✓✦❍✭★✪✩✭✎✛❃✺✫✖❊✭✬✻✗✣✑✷✎ ✕✖✓✦✫✭❇✸❍✳✓✦❍✭★✪✩✭✎✚✲❆✫✘✩✖✑✴❬ U1 = 2 jI 1 + jI 2 ❏❁✩▲✎☞✚✳✎✣❑▲❊✭✬✿✫✳✬■❊✘✢❆✎❩❃✜✎❩✕✖✎✜✩s✑ ✘ ✎▼✓✦❍♥✲❱✎✺✢✶➆✳❇❾❃✜❇▼✗✺➆✳✎✺❑ ❃✜✗❅✢✧✎ ✽ ✘ ✽ ♠❨✫✥✩✭✓✦✑✷✗✺✹✻★✻★✱✬❁✎✛✕✘★✪✩P✯✶★❁➁✦✫✖✢✴❇✛✎❩❃❅✫✭✗✒✹✔★✿★✻✬✿✎✛✕✘✎▼✯✷✫✘✩✭❃✣✹✷★✻✓✦✩✳✗✺✢✧✎❽✗✜✬❁✎✛✗✜❃❅✎❄✲❆✑✷✓✦✢ ✕✘✎❄❃✜★❹❃✦★✪✢✈❃✺✫✳★✿✑✱✫✭✬❼✎✺❃✺①✭★➄➆✭✗❄✬❁✎❅✩✘✑➅✎s✲✶✑❉✎◗❃❅✎✜✬❺❊✘✢✴✎✺❀✦✎✣✩✘✑✻✗✜✑ , U 2 = 2 jI 2 + jI 1 ❃❅①✭✎✣❑✉✗✤✎✜❃✺①■★✪➆✭✗✜✬✻✎❅✩✘✑✻❇❳❏❁✩▲❃✜✓✦❑▲❊✭✬✱✍ ✎ ✚❥✎❄✲✶✑❉✎■❬ ✫✘✩✚❃❩★➄✢✴❃✣✫✭★✿✑❋✢✧✎✺❀✦✓❄✩✭✗✺✩✖✑✙☎ ✤❦♠ ✲✴✎✺✢❆★❁✎▼❃✺✛ ✫ ✚ ✜ ✥ ✽ ✩✖✫❤✲❱✎❖❊✭✓s✑❳✑✱✢✧✗✜✩♥✲♦✯✈✓✦✢✔❑❖✗r❏❁✩❤✲❆✫✘✢✴✲❱✎r✕✖✎t❃✣✫✘✢✴✎✣✩✘✑ ✽ ✕✖★✿✩✘✑✱✢✴✎▼✗❅❃✜✎❄✲✧✑✷✎✛✲✶✫✘✢❱✲❱✎✤★❁✗✣✢✰❊✭✎✣✩✘✑✱✢✔✫▲❃✜✎❩✗✜✬❁✗✜✬➀✑✷❇❈★➄✩✘✑✱✢✴✓✘✕✦✫✭❃✜✎✣❑ ▲✬❁✎✜➁✘✎✣❑③❃✜✗ ✁ ❘ ❱❲ ✥ ✚✵❱★❋✕✘✓❄✫✭❇❾✲✶✫✘✢♦✲♦✎✛★✱✕✘✎✜✗❩✬✿✎▼✕✘✎❳✑❉✎✣✩❋✲❱★✿✫✖✩✭✎ ✄ ✄ ✽ ❊✳✓✦✑❉✎✣✩✘✹✷★❁✗✜✬❈✕✘✎❯✢✴✎✜✯✔✎✺✢✴★✪✩✘✹✻❇t✓ ✩✭✎❩❃✣✫✘✩✭✓■✲❱❃✺✫✘✑✷✗✛✲❆✫✘❊✭✬✱★ ❑❖✎❅✩✘✑✱✗❅✢✧❇✤➂ 31 ✢ ✽ ❍✳✓✦✢✔✩✭❇✇✗ ✫✘✩✳✎❩★✱✗ ☎❈✎✺❀✜✫✳✬✿✑✻❇❾✎✺❃✺✫✳✗✺✹✻★✻★✿✬✻✎✦❬ V5 = 0 V1 = j ✄ j 1 2 1☎✝✆ 2 + V 2 ✁✂ − V1 − V 3 = I 4 + U 2 3j 3 3j 3 3 U2 = −V3 V 2 − V 4 = 3I 3 I3 = V1 − V 3 1− j ✄ V V 1 1 2 j V3 + + ☎ ✆ − 1 − 2 = 1 + j + U1 1 − j 3 3j 1 − j 3 3 ✂✁ U1 = V1 − V 2 ✄ 1 V 4 ✁✂ ☎ ✆ = −1 − j − I 4 2 ✗✜✕✘★✱❃❩❇❳✫✘✩P✲✴★✷✲✶✑❉✎❅❑ ✕✘✎✟✞❥✎✺❃✺✫✳✗✺✹✻★✻★✭❃✺✫◗✩✭✎✜❃✜✫✖✩■✓✳✲✴❃✺✫✘✑✷✎❄✬✪✎ V 1 ,..., V 5 , U1 , U 2 , I 3 , I 4 . ✎❄❃✜★♥✗✜✬✱➁✘✓✦✢✧★✪✑❁❑▲✫✭✬❋✕✘✎▼✲❱❃✺✢✴★➀✎✜✢✧✎▼✗▼✎❅❃✣✫✭✗✣✹❉★✱★❁✬✱✓✦✢✰❊✳✓✦✑✔✎✣✩✘✹✷★❁✗✜✬✻✎✺✬✻✓❄✢r✩✭✓✘✕❄✫✘✢✴★✱✬✻✓✦✢❦✎✦✲✶✑✻✎✘❬ ✝ • • • ✲❱✎❽✯✈✗❩❃❾✑✷✓✘✗✣✑❉✎❾✑✱✢✴✗✣✩❋✲✴✯✶✓✦✢✶❑P✗❅✢✧★❁✬✱✎❾❊✭✓✳✲❱★➄❍❋★✿✬❁✎❽✗❄✬✪✎✥✲✧✫✖✢❱✲✴✎❩✬✱✓✦✢❨✕✘✎❾✑✷✎✺✩♥✲♦★✪✫✘✩✭✎❾❏✱✩ ✲✶✫✘✢❱✲❱✎❽✕✖✎❽❃✺✫✖✢❱✎✣✩✘✑✮✵❱★❼✗✜✬❁✎❽❃✜✓✦❑❖✎✺✩✘❀❩★✱✬✻✓✦✢ ❏❁✩▲❃❅✫✘✢✧✎✣✩✘✑♥❏✱✩✚❃❩✓✦❑❖✎❅✩■❀✜★✳❏❁✩❥✑✱✎❅✩❋✲✴★✪✫✘✩✭✎❄❚ ✲❱✎▲✗✜✬✻✎✜➁s✎❥❊✭✓✦✑✷✎✺✩✘✹✷★❁✗✜✬✪✫✭✬❿✕✘✎◗✢❆✎❩✯✶✎✣✢✴★✪✩✘✹❉❇▲✗❄✲✈✑✷✯✶✎✺✬❦❏✪✩✭❃❄❂✐✑❷❃✜❂✜✑✮❑P✗❩★✰❑▲✫✭✬➀✑✻✎❥❊✭✓✦✑✷✎✺✩✘✹✷★❁✗✜✬❁✎▲✗✜✬❁✎❥✩✭✓✘✕❄✫✳✢✴★✱✬✻✓❄✢▼✲✴❇✥❊✳✓✘✗❅✑❉❇▲✯✶★ ✎✄✚✖❊✘✢✧★✪❑❖✗✺✑✷✎✤❃❩✗✛✲✶✫✘❑❖✎❈✕✖✎❈✑❉✎✣✩❋✲✴★✿✫✘✩✳★♥✎✜✬✻✎✺❃❅✑❁✢✴✓❩❑❖✓✦✑✷✓✘✗✣✢✴✎✦❚ ❃❄✓✦✩❋✲✴★✱✕✘✎✣✢✴❂✺✩✳✕r✵✴★➅✩✭✎✜❃✣✫✘✩✭✓✳✲❱❃✣✫✘✑✷✎❅✬✱✎P✲✶✫✘❊✭✬✱★✪❑❯✎✒✩✘✑✔✗✣✢✴❆ ✎ ✶✂ ❃✣✫✘✢✴✎✣✩✘✹✔★✿★❧✫✘✩✳✓✦✢▼✲✶✫✘✢❱✲✴✎✥✕✘✎◗✑✷✎✣✩❋✲❱★✿✫✖✩✭✎❥❃✜✓✦✩✭✎✜❃✺✑✷✗✐✑✷✎❽❏❁✩✘✑✱✢✴✎❥✗✜✬➀✑✷✎ ✩✭✓■✕✦✫✘✢✴★❋✕✘✎✜❃✜❂✺✑❧❃❅✎✜✬❁✎▼✕✘✎✤✬✱✗◆❊✘✫✖✩❋❃✣✑✻✫✭✬✘❊✘✢✴✎✜❃✜✎❩✕✘✎✣✩✘✑➅✵✴★♥❃✣✫✘✢✴✎✣✩✘✹✻★✻★❋✕s✎▼❃❩✓❄❑P❑P✗✣✩✭✕✘✓ ❇ ☎❦✲❱✎✛✲❆❃❅✢✧★✿✎❾✲❱★❉✲✶✑✔✎✒❑P✫✳✬❺✕✖✎❈✎❄❃✣✫✭✗✐✹✷★❁★✈❬ Vj ✠ k ∈j Yk − ✠ k ∈i , j ViYk = ✵✴★✭✎❩❃✣✫✭✗✐✹✷★❁★✱✬✱✎❾✲✶✫✘❊✭✬✱★✪❑❯✎✣✩✘✑✷✗✺✢✴✎ ✠ k ∈j I sk ✽ 32 CAPITOLUL 2. UTILIZAREA PROGRAMULUI PSPICE ✂✁☎✄✝✆✟✞✡✠☞☛✍✌✏✎✏✑☞✒☞✠☞✒ ✓✕✔✗✖✙✘✟✚✜✛✣✢✥✤✧✦✩★✫✪✭✬✝✮✯✤✱✰✳✲✵✴✷✶✙✰✹✸✥✶✺✬✫✦✼✻✽✤✣✮✧✾❀✿❁✲✝✮✭❂✺✸✥✶✺✬✝✮✭❂❄❃❆❅❇✤✧✶✙❈✹★❉✤✯✮✍❊✷✦☞❋☎✾✫✬❍●❄✤■●❑❏▼▲❖◆◗P❘▲❚❙✕❯❲❱✫❳❩❨✫❬✡❳❩❭❫❪❵❴✫▲❖◆◗P❘❛❜❯❝❭❫P ❁◆ ❛❜❪❞❙❝❡✯❢❫❳❩❛✯❛❤❣✹❛❜❳◗❣❫❙✝❛❜P❘▲✹❡✯❨✡❳✷▲✹❡✯▲✹❣❫P✣❳◗❛✯❣✹▲❍✐❥✔✂✓❉✔❦✖✙✘✽✚❧❳◗▲❫❱✕❳◗▲❄♠❖❛❜❯✕P❘❢♦♥❝❭❫❳❩❛✯❭❫❯✫P♣❭rq✫s✥t◗✉❇✈❚❱❝▲❫❯✫P✣❳✇❙①❣✹❭✹❡✯❣❫❙❝❡✯❭❫P♣❨✫❭❫❳◗▲✟❱❝▲❫❳❁◆②❨✳❯❝❭✹❡✯▲✏③❁❛ ❭✽④⑤❨❉◆◗P⑥❴✕▲❄♠⑦♥❝❨✫❡❜P❘❭❫P⑧❴✫▲♦⑨❲⑩❘❶✹❷✹❸❉✓✕⑩❘❹❻❺✯❯❝❣✹▲❫❱✝❼❫❯❝❴❽❣❫❙❚❾❫❿✝➀➂➁❤✐ ➃r❡❜P❘▲➄♥❝▲❫❳❁◆❁❛❜❙✫❯❝❛➅❭✹❡✯▲➄❱✫❳❩❨✫❬✡❳❩❭❫❪➆❙❝❡✭❙❝❛♦q✫s✥t◗✉❇✈➇◆➈❙✕❯✫P✩➉➋➊➆q❍s⑥t➈✉❇✈➌❱✫❳❩❨✫❴✡❙⑥◆✩❴✕▲①➍❀▲❫P❘❭❖➎➈q❉❨✫④❘P✯➏✩❭❫❳❩▲❖➐☞t❩➎◗q❍s⑥t◗✉❇✈ ❱✕❳◗❨✫❴✳❙✥◆➆❴✫▲➒➑➓✉✽q⑥q→➔✗❛❜❪①▲❖◆◗➣❝❭❫❳◗❛✭❯❝❬✵◆②❛✍t◗↔✏➎◗q❍s⑥t◗✉❇✈↕❱✫❳❩❨✫❴✡❙✥◆❽❴✫▲➒➃▼✐ ➙✩✐⑧➃❽◆✺◆②❨✕❣✹❛✯❭❫P❘▲❖◆❞➐❇q✫s✥✈☎✉✍➔☞➛✍✈ ❱✫❳❩❨✫❴✡❙⑥◆➓❴✫▲ ✉✍❭✹❴✕▲❫❯❝❣✹▲➝➜✩▲❖◆❁❛✯❬✡❯✥➐❉q✫s⑥t➈✉❇✈✂➎➞s⑧❡❜❙⑥◆➟❱✫❳◗❨✕❴✡❙✥◆➠❴✕▲✍➡r❭✹❡✯❛✯❴r➢☎❨✫❬✫❛✯❣❍✐ ➤⑥➥⑤➤✥➥⑧➦r➧❉➨❉➧❉➩➋➫➯➭❫➨❉➩➳➲✂➧➸➵❉➺➋➻➼➩⑧➽➅➲✷➾ ❤➚ ▲❫❳❩▲✹❭❖◆➈P✣❳◗❭➪❴✫▲➪❣✹❨✡❪①❭❫❯❝❴✫❢➟◆❁▲➄❱❝❨✫❭❫P❘▲➸❴✕▲❖◆②❣❫➣✝❛✯❴✫▲➸❴✫❛❜❯❧q✳P❘❭❫❳⑤P✇➎➞s✝❳❩❨✫❬✡❳❩❭❫❪➸◆➓◆❁❭❫❙➶❣❫❙❀❣✹❡✯❛✯❣❫➹➳❱❝▲➄❙✕❯➶④✇❛❘◆②❛✣▲❫❳➘❴✫▲ ❛✭❯✫P✯❳❩❭❫❳◗▲➘✛❘❴✕▲✍P❘❛✭❱➷➴✕✐➬❣✹❛❜❳☎◆②❭❫❙➆s❦q✳❱✝❛✯❣✹▲➮✉✍❛❜❳❩❣❫❙❝❛❜P✥➚❤❛✯❡✯▲❄❏✗◆❁❭❫❙❆❱✝▲✍❙✫❯▼④⑤❛➞◆❁❛✯▲❫❳❦❴✫▲➝❳❩▲❄♠⑦❙❝❡❜P❘❭❫P❘▲➘✛♣❴✫▲✍P♣❛❜❱➪➴✫✐➬❴✕❭❫P❤◆❁❭❫❙ s➱◆◗❱❝❛✯❣✹▲➪q❉❛❜❪❞❙❝❡✯❭❫P❘❛✣❨✡❯❚➛♦▲❖◆◗❙❝❡❜P⑤◆◗❏❑✐✥s✝❳❩❨✫❬✡❳❩❭❫❪❞❙❝❡❇◆❁▲❞❪①❭✹❛✗❱❝❨✕❭❫P❘▲①❡✯❭❫❯✥◆❁❭➸③❁❛✂❴✫❛❜❯✃④⑤▲❫❳❩▲✹❭❖◆➈P✣❳◗❭➄❈✹✰✳✦❐✦➘✬✫✲✫❃❚❣❫❙➷✴☎✢✳✴☞✿❫❅☞❊ ✲❝★✳✦✏❂❫❒➈❮❍✤Ï❰✹✤✭❂❄✶r❃❉❂➸✤✧✲✥✮✧✶✺✬✫✶✙❂➸◆❁❭❫❙✃❴✫❛❜❯✕P✯❳②➎✣❯Ð❱✕❳◗❨✫❬✳❳◗❭❫❪Ñ◆❁❣❫❳◗❛❘◆✽❺✯❯✃✉♦➐➋➚❤Ò✟➛♦➔☞➛➅➃✟➑Ó◆❁❭❫❙❚❭✹❡❜P❇❡✯❛❜❪❞Ô❝❭➈Õ✩❴✫▲❽❱✕❳◗❨✫❬✳❳◗❭❫❪①❭❫❳◗▲ ④⑤❨✫❡✯❨❝◆②❛✭❯❝❴❽❛❜❯✥◆◗P✯❳⑤❙❝❣❫P❘❛❜❙✫❯✝❛❝❴✕▲✍P❘❛✭❱➓●❘Ö❝●❄✮✭❂❄✦✩× Ø ❯▼④✇▲❫❳❩▲✹❭❖◆◗P✯❳❩❭✟❴✫▲✽❣✹❨✡❪①❭❫❯❝❴✫❢✽▲❄Ù❉❛➞◆◗P❘❢➝❙✫❳⑤❪➄❭❫P❘❨✕❭❫❳◗▲✹❡✯▲➝❪①▲❫❯✫❙✥➎✣❙✫❳◗❛◗➉ Ú✗⑩✯Û❘Ü ➎Ð❣✹❭➌❺✯❯❻P❘❨✫❭❫P❘▲↕❭❫❱❝❡✯❛✯❣✹❭❫Ý♣❛✯❛✯❡✯▲ßÞ❲t❘➑✩➜✩Ò✟Þßq✝➐❽❱❝❨✕❭❫P❘▲à❴✕▲❖◆②❣❫➣✝❛✯❴✫▲➌❙✫❯á④✇❛➞③❁❛✯▲❫❳❁➐❽❺❘❡➒❱❝❨✕❭❫P❘▲➼P♣❛❜❱❝❢❫❳❩❛➞➐ ❪①▲❫❪①❨✡❳◗▲✹❭❄♠❖❢➶❙❝❡❜P♣❛❜❪➄▲✹❡✣▲➶④⑤❛➞③❁❛✯▲❫❳❩▲❀❭✹❣✹❣✹▲❖◆②❭❫P♣▲→③❁❛➘❙❝❡✭P❘❛❜❪①▲✹❡✯▲✵◆②❛✭❪➆❙❝❡✣❢❫❳◗❛r④⑤❢✹❣❫❙✫P❘▲❖➐➝❱❝❨✫❭❫P❘▲✵◆❁❭✹❡❜♥❝❭✵◆❁❭❫❙➼❱❝❨✫❭❫P❘▲ ❺✣❯❝❣❫➣❝❛✯❴✫▲➝❙✫❯▼④⑤❛➞③❁❛✯▲❫❳➠③❁❛✕❱❝❨✫❭❫P❘▲➝❺✯❯❝❣❫➣❝❛✣❴✫▲✟④⑤▲❫❳❩▲✹❭❖◆➈P✣❳◗❭✽❴✫▲✽❣✹❨✡❪➄❭❫❯✝❴✫❢❍✐ ✚♦â✝⑩✯ã❤ä❽❱❝▲❫❳⑤❪①❛❜P❘▲➝❪➄❨✫❴✕❛✯④✇❛✣❣✹❢❫❳◗❛✯❡✣▲✍❙✕♠⑦❙✝❭✹❡✯▲✍❺✣❯❞④⑤❛➞③❁❛✯▲❫❳⑤❙❝❡✝❴✫▲❖◆❁❣❫➣❝❛➞◆❦❺✯❯▼④✇▲❫❳❩▲✹❭❖◆◗P✯❳◗❭✽❴✫▲✽❣✹❨✡❪①❭❫❯❝❴✫❢❍✐ å➘⑩❘Ü❫æÑä➼❱❝▲❫❳⑤❪➄❛❜P♣▲➳♥✝❛✭♠⑦❙❝❭✹❡✯❛✭♠❖❭❫❳❩▲✹❭❀④✇❛❘③②❛✣▲❫❳◗▲✹❡✯❨✳❳➒❴✫▲❀❛❜❯✫P✣❳◗❭❫❳❩▲✵✛❩➴✫✐ç❣✹❛❜❳➈❏➸③❁❛✏❴✫▲è❛✣▲❖③②❛✭❳◗▲❀✛②➴✫✐ç❨✡❙✫P✣❏②➐➝❭è④⑤▲❫❳❩▲❖◆➈P✣❳◗▲✹❛ ❬✡❳❩❭✹④✇❛✣❣✹▲r❺✯❯Ð❣✹❭❫❳❩▲❞◆②▲✩❱✫❳❩▲❄♠❖❛❜❯✫P♣❢✏❳❩▲❄♠⑦❙❝❡❜P♣❭❫P❘▲✹❡✯▲❞◆❁❛❜❪➆❙✝❡✯❢❫❳◗❛✣❛➞➐✥❭❽④⑤▲❫❳◗▲❖◆◗P✯❳❩▲✹❛❤❺✯❯é❣✹❭❫❳❩▲➆◆❁▲❽❣✹❭❫❳❩❭✹❣❫P❘▲❫❳◗❛✱♠❖▲✹❭❄♠❖❭➓❬✕❡✯❨✡Ô❝❭✹❡ ◆❁❛❜❪❞❙❝❡✯❭❫❳❩▲✹❭r✛❘❴✕❭✹❣✹❢r◆◗❙✫❯✫P➱▲❫❳❩❨✡❳❩❛✝❺✯❯➸④✇❛❘③②❛✣▲❫❳✇❙❝❡❤❴✫▲✏❛❜❯✕P✯❳◗❭❫❳❩▲❖➐❉❴✫❭✹❣✹❢✽❙✫❯➸❭❫❯✫❙✫❪①▲✽P❘❛❜❱①❴✫▲➘❭❫❯❝❭✹❡✯❛✭♠❖❢✟❯✫❙➄❣✹❨✳❯✫♥❝▲❫❳❩❬✫▲❖➐ ❴✫❭✹❣✹❢➘◆❁❛❜❪➆❙✝❡✯❭❫❳◗▲✹❭➘◆✺➎❘❭➝❺✯❯❝❣❫➣✝▲✹❛✯❭❫P✣❏❑✐ ✓❉⑩➞❹➪ê❝Û♣ë✡ã✇⑩➞❸✕ì❞ä❝❣✹❨✡❯✫Ý♣❛❜❯❝▲✽❣✹❨✡❪➄▲❫❯❉♠❖❛✯❡✯▲✽❴✫▲✍❺✣❯❝❣✹▲❫❱❝▲❫❳❩▲✟❭➘◆❁❛❜❪➆❙✝❡✯❢❫❳◗❛✣❛➞➐❖❱❝❭❫❙✕♠❖❢➘③❁❛✝❨✡❱✫❳❩❛❜❳◗▲✽❭➘◆②❛✭❪➆❙❝❡✣❢❫❳◗❛✯❛➈✐ í✍❷❖ë✳❶✹Ü➓äÐ❣✹❨✡❯✫Ý❘❛❜❯✝▲➆❣✹❨✡❪①▲❫❯✕♠❖❛✯❡✣▲➆❴✫▲❞❭✹❴✫❢❫❙❝❬✕❭❫❳◗▲➄③❁❛➠③◗P❘▲❫❳❩❬✫▲❫❳❩▲➆❭➓❙✫❯❝▲✹❛➱❳❩▲❫❱✫❳❩▲❄♠❖▲❫❯✫P❘❢❫❳❩❛❦❬✳❳◗❭✹④⑤❛✯❣✹▲➄③②❛✗❣✹❨✡❪①▲❫❯✕♠❖❛ ❱✫❳❩❛❜♥❝❛❜P♣❨✫❭❫❳◗▲✽❡✯❭✽❣❫❙✫❳❁◆②❨✳❳❑✐ ✔✗Û➞❸✳ã②➎✟❣✹❨✡❯✕Ý❘❛❜❯❝▲➪❣✹❨✡❪①▲❫❯✕♠❖❛✂❱✫❳❩❛❜♥❝❛❜P♣❨✫❭❫❳◗▲➪❡✯❭➪④⑤❨✡❳⑤❪➄❭❫P✯❙✝❡✍❴✫▲➪❭✹④✇❛➞③❁❭❫❳❩▲➪❣❫❙✫❪î◆◗❙✫❯✫P➠➉➱❪①❨✫❴✫❛✯④⑤❛✯❣✹❭❫❳❩▲✹❭➟◆②❣✹❢❫❳❩❛✯❛✂❱❝▲ ❭❄Ù❉▲✹❡✯▲➒ïÓ◆②❛✷ð➓➐➱❪➄❨✫❴✕❛✯④✇❛✣❣✹❭❫❳◗▲✹❭➸❛❜❯✕P❘▲❫❳✇♥✝❭✹❡❜❙❝❡❜❙❝❛➅❴✫▲➸❭✹④✇❛❘③②❭❫❳❩▲❖➐☎◆②❣❫➣✝❛❜❪➆Ô✝❭❫❳◗▲✹❭➄❪①❢❫❳◗❛✭❪➄❛✯❛➅❭❖◆②❨✕❣✹❛✯❭❫P❘▲➄❙✫❯❝▲✹❛➅❭❄Ù❉▲❖➐ ❛❜❯✫P✣❳◗❨✫❴✳❙❝❣✹▲❫❳◗▲✹❭➘③❁❛⑥③➈P♣▲❫❳◗❬✫▲❫❳❩▲✹❭➝❙✫❯❝▲✹❛✝❭❄Ù❉▲➝ð❆◆➈❙✕❱❝❡✯❛❜❪①▲❫❯✫P❘❭❫❳❩▲❍✐ í✍❸✫❸✳Û❘ñ❑➎✽❱✝▲❫❳✇❪①❛❜P❘▲❚❛❜❯✫P✯❳❩❨✫❴✡❙❝❣✹▲❫❳❩▲✹❭➳◆❁❭❫❙➮▲✹❡✯❛❜❪①❛❜❯❝❭❫❳❩▲✹❭➪❙✫❯✝❨✡❳rÔ❝❭❫❳❩▲Ð❣❫❙➮❛✯❣✹❨✫❭❫❯❝▲❚❣✹❭❫❳◗▲➷❳◗▲❫❱✕❳◗▲❄♠❖❛❜❯✕P❘❢é❴✕❛❜♥❝▲❫❳❁◆②▲ ❣✹❨✡❪①▲❫❯✕♠❖❛⑤✐ ò❆⑩➞ì✥â⑥❸✡æ❽➎✯❙✫❯❽❪①▲❫❯❝❛❜❙▼❣✹❡✯❭❖◆❁❛✯❣✽❣✹❭❫❳◗▲➝❱❝▲❫❳⑤❪➄❛✭P❘▲✟❡✭❙❝❣❫❳✇❙✝❡❝❣❫❙▼④⑤▲❫❳◗▲❖◆◗P✯❳❩▲✹❡✯▲✽❴✫❛❜❯➆s❦q❍s⑥t➈✉❇✈➝✐ ó➄Ü❄Û✯ô➋➎⑥❙✕❯➓❪①▲❫❯❝❛❜❙▼❣✹❡✯❭❖◆❁❛✯❣✽❣✹❭❫❳❩▲✍❱❝▲❫❳⑤❪①❛❜P❘▲✽❣✹❨✡❯✥◆◗❙❝❡❜P❘❭❫❳❩▲✹❭✽❴✫❨✫❣❫❙✫❪①▲❫❯✫P❘❭❫Ý❘❛✣▲✹❛✥③❁❛✝❭✹❣✹❣✹▲❖◆◗❙❝❡✕❱❝▲✽❛❜❯✫P❘▲❫❳⑤❯❝▲❫P⑧❡✯❭➘◆②❛✭P❘▲❖➎ ❙✫❳❩❛✯❡✯▲➝❱✫❳❩❨✫❴✡❙❝❣✹❢❫P❘❨✳❳✇❙❝❡✭❙❝❛⑤✐ 33 ➤✥➥✁☞➥⑧➦✄✂✆☎✝✂②➧❉➨❉✟➧ ②✞ ➧➸➲✷➧✠✂◗➽➠➭❫➨❉➩➋➨❉➧✡☎✝✂➅➲✷➧✠✂②➧☛☎✝✂❩➨❉➧ ☞✍✌✏✎✑✌✓✒✕✔✗✖✙✘✙✚✛✒✜✌✣✢✥✤✓✔✧✦★✔✧✒ ➚❤❛➞③❁❛✯▲❫❳⑤❙❝❡❇❴✫▲➸❛✭❯✫P✯❳❩❭❫❳◗▲➄P✯❳❩▲❫Ô✫❙❝❛✣▲➪◆②❢➸❭✹❛✭Ô❝❢➸▲❄Ù❍P❘▲❫❯✥◆❁❛✯❭❚✐ç✉❇t⑤➛ ✐✥➃➘❳◗▲➸❨è◆◗P✯❳⑤❙❝❣❫P✯❙✕❳◗❢➸❡✯❛✭Ô❝▲❫❳◗❢❖➐➱❺✯❯→◆❁▲❫❯✥◆◗❙❝❡✂❣✹❢❞❯✫❙ ❄▲ Ù❝❛➞◆◗P❘❢✏❨①◆◗❙❝❣✹❣✹▲❖◆❁❛❜❙✕❯❝▲r◆◗P✯❳❩❛✯❣❫P❘❢✏❭✏❡✣❛❜❯❝❛✯❛✯❡✣❨✡❳②➐❉❣❫❙➸▲❄Ù❉❣✹▲❫❱✫Ý❘❛✯❭❐❱✫❳◗❛✭❪➄▲✹❛❤❡✯❛❜❯❝❛✯❛❤❣✹❭❫❳◗▲❐❳◗▲❫❱✫❳❩▲❄♠❖❛❜❯✫P♣❢✽P❘❛❜P❘❡✭❙❝❡⑧❣✹❛❜❳❩❣❫❙❝❛❜P✣❙❝❡❜❙❝❛➱◆②❛❤❭❫❳◗▲ ♥✝❭✹❡✯❨✫❭❫❳❩▲✹❭①❙✕❯✫❙❝❛✍❣✹❨✡❪①▲❫❯✫P❘❭❫❳❩❛❜❙❲✛✣❱✫❳◗❛✭❪➆❙❝❡✍❣✹❭❫❳❩❭✹❣❫P❘▲❫❳r▲❖◆➈P♣▲➳➴✹❏r③❁❛✍❭➒❙❝❡❜P❘❛❜❪①▲✹❛✍❡✯❛❜❯✝❛✯❛✍✛②✐ ✈➱➑✩➜➘❏❐❣✹❭❫❳❩▲➸❛❜❯❝❴✕❛✯❣✹❢➪◆❁④✇❼❫❳❁③❁❛❜P✯❙❝❡ ④⑤❛➞③❁❛✯▲❫❳⑤❙❝❡❜❙❝❛✝❴✫▲✽❛❜❯✫P✣❳◗❭❫❳❩▲❍✐➂✉✍❨✡❯✫P♣❛❜❯✫❙❝❭❫❳❩▲✹❭✍❙✕❯❝▲✹❛✝❡✯❛❜❯❝❛✯❛⑥◆❁▲✽④✇❭✹❣✹▲➝P❘❭❖◆◗P❘❼❫❯❝❴✪✩ ❺✯❯❽❱✫❳◗❛✭❪➄❭✽❣✹❨✫❡✯❨✫❭❫❯✝❢✟❭✽❡✯❛❜❯❝❛✣▲✹❛✫❙✕❳✇❪①❢❫P❘❨✫❭❫❳❩▲❍✐ s❦q✫s✥t◗✉❇✈➟❯✫❙▼④⑤❭✹❣✹▲✟❴✕❛✯④✇▲❫❳❩▲❫❯✫Ý❘❭➝❺✯❯✫P✣❳◗▲✽❡✯❛❜P❘▲❫❳❩▲➝❪➄❭❫❳❩❛⑥③②❛✝❡✯❛✭P❘▲❫❳◗▲➝❪①❛✯❣✹❛⑤✐ ✫➘❯❽❯✫❙✫❪①▲✍P✯❳❩▲❫Ô✫❙❝❛✣▲❐◆❁❢➝❺✯❯❝❣✹▲✹❭❫❱❝❢✽❣❫❙▼❨➓❡✣❛❜P❘▲❫❳❩❢❐③❁❛✝❣✹❨✡❯✫Ý❘❛✭❯❝▲✍❪①❭❄Ù❉❛❜❪ ➀➘❣✹❭❫❳❩❭✹❣❫P❘▲❫❳❩▲❍✐ ✫➘❯▼❯✫❙✕❪➄❢❫❳✗❱❝❨✫❭❫P♣▲✏④✇❛✥❺✯❯✫P✯❳❩▲✹❬①◆❁❭❫❙➄❙✫❯▼❯✫❙✕❪➄❭❫❳☞❺✯❯❞♥❝❛❜❳❩❬✡❙❝❡✣❢♦❪①❨✡Ô❝❛✯❡✯❢✩③❁❛❉❱❝❨✕❭❫P❘▲❐④⑤❛❉❺✣❯✥◆❁❨✡Ý❘❛❜P➋❴✫▲♦❙✕❯➄▲❄Ù❍❱✝❨✡❯❝▲❫❯✫P ❺✣❯✫P✯❳❩▲✹❬⑥✐⑦✈✗Ù❉❛➞◆◗P❘❢➝❙✫❳⑤❪➄❢❫P❘❨✳❳◗❛✯❛✝④⑤❭✹❣❫P❘❨✡❳❩❛✝❴✫▲❐◆❁❣✹❭✹❡✯❢✫➉ ➔ ✬✍❾❑✈➘✗❾ ✮✰➓ ✭ ✯ ✜ ↔ ✬✍❾❑✈☎✱ ❿ ✯✩➍è✈☎✜ ↔ ✬✍❾❑✈✳✲✴✯✶✵✄✬✍❾❑✳✈ ✷✴✯ ➍ ✬✍❾❑✈✂✹➎ ✷✴✯✺✫✻✬✍❾❑✈✂✹➎ ✲✴✯❐✶ ✸ ➑ ✬➝❾❑✈✂➎❘❿✱✳✯ s✥✬✍❾❑✈✂➎✙✗❾ ✮✰✯✡✥➚ ✬✍❾✙✈✷➎✙✽❾ ✼➄✐ ➢☎❛❜P♣▲❫❳◗▲✹❡✯▲✽❣✹❭❫❳❩▲❐◆❁▲➘◆➈❙✝❣✹❣✹▲✹❴➓❛✭❪➄▲✹❴✫❛✣❭❫P⑥❴✳❙✫❱❝❢➝❙✫❯❽❯✫❙✫❪①❢❫❳➠③❁❛✫❯✕❙➄◆◗❙✫❯✕P⑥④⑤❭✹❣❫P❘❨✡❳❩❛✝❴✫▲➘◆②❣✹❭✹❡✣❢❐◆◗❙✫❯✫P⑧❛✯❬✳❯❝❨✡❳❩❭❫P❘▲❍✐ ➜✩▲❖◆❁❣❫❳◗❛✯▲❫❳❩▲✹❭➪❣✹❛❜❳❩❣❫❙❝❛❜P✯❙❝❡✭❙❝❛✍✛✣P❘❨✡❱❝❨✫❡✣❨✫❬✫❛✯❭➪③❁❛❇❴✫▲❖◆❁❣❫❳❩❛✯▲❫❳❩▲✹❭➸▲✹❡✯▲❫❪➄▲❫❯✕P❘▲✹❡✯❨✡❳✏❴✫▲➸❣✹❛❜❳❩❣❫❙❝❛❜P✣❏✩◆❁▲➸④✇❭✹❣✹▲➸❣❫❙❀❭➈Õ②❙✫P❘❨✡❳⑤❙❝❡ ❡✣❛❜❯❝❛✯❛✯❡✣❨✡❳❦▲✹❡✯▲❫❪①▲❫❯✫P❘▲✹❡✯❨✳❳❦❴✫▲✽❣✹❛❜❳◗❣❫❙✝❛❜P❩✐ Ø ❯▼❡✯▲✹❬✫❢❫P✣❙✫❳◗❢✽❣❫❙▼❭✹❣✹▲❖◆◗P❘▲✹❭✍P✣❳◗▲❫Ô✫❙✝❛✯▲✟④⑤❢✹❣❫❙✫P❘▲✽❣✹❼❫P❘▲❫♥❝❭➝❱✫❳❩▲✹❣✹❛✭♠❖❢❫❳❩❛✥③❁❛✝❭❫❯✫❙✫❪①▲✫➉ ➴ ◆❁▲➇❙✫P❘❛✯❡✯❛✱♠❖▲✹❭❄♠❖❢↕❳◗▲✹❬✳❙❝❡✯❭ ❴✫▲ ❡✯❭↕❳❩▲✹❣✹▲❫❱✫P❘❨✫❭❫❳❩▲↕❱❝▲❫❯✫P✣❳✇❙Ó◆②▲❫❯⑥◆➈❙✕❳◗❛✯❡✣▲ ❴✫▲↕❳❩▲✹④✇▲❫❳❩❛❜❯✫Ý♣❢ ❭✹❡✯▲↕P❘▲❫❯⑥◆②❛✭❙✫❯❝❛✯❡✯❨✳❳❧③❁❛ ❣❫❙✕❳◗▲❫❯✫Ý♣❛✯❡✯❨✡❳✙✐ ➴ ❯❝❨✫❴✳❙❝❡✿✾➓❳❩▲❫❱✫❳◗▲❄♠❖❛✭❯✫P❘❢♦❪①❭❖◆❁❭✏③❁❛⑥▲❖◆➈P♣▲❐❨✡Ô❝❡✣❛✯❬✫❭❫P❘❨✡❳❩❛❜❙✥➐✫❣✹▲✹❡✣▲✹❡✯❭✹❡❜P❘▲✟❯❝❨✫❴✡❙✫❳❩❛✝④✇❛✣❛❜❯❝❴✩❯✫❙✫❪①▲❫❳◗❨✳P❘❭❫P❘▲➘◆➈❙✝❣✹❣✹▲❖◆②❛✭♥✥➐❖❺✯❯ ✽ ❨✳❳◗❴✫❛✭❯❝▲❐❣❫❳❩▲❖◆❁❣✹❢❫P❘❨✫❭❫❳❩▲❖➐✡❺✯❯❝❣✹▲❫❱❝❼❫❯❝❴❞❣❫❙è❾❖✐➂➚⑧❛✣▲✹❣✹❭❫❳◗▲✟❯❝❨✫❴✩P✯❳❩▲❫Ô✫❙❝❛✯▲➘◆❁❢✟❭✹❛❜Ô✝❢✟❨❽❣✹❭✹❡✯▲✽❣✹❢❫P✯❳❩▲✍❪①❭❖◆❁❢✍❺✯❯❽❳❩▲✹❬✫❛❜❪ ❴✕▲✟❣❫❙✫❳❩▲❫❯✫P ❣✹❨✳❯✫P❘❛❜❯✫❙✕❙➇❴✫▲✹❨✕❭❫❳◗▲✹❣✹▲→❺✯❯❆❬✫▲❫❯❝▲❫❳❩❭✹❡➞➐❐❣❫➣❝❛✯❭❫❳➪❴✫❭✹❣✹❢❲◆❁▲✵❣✹▲❫❳◗▲✵❨ß❭❫❯❝❭✹❡✯❛✭♠❖❢➶❺✯❯❆❣❫❙✫❳❩▲❫❯✫P▼❭✹❡❜P❘▲❫❳⑤❯❝❭❫P❘❛❜♥ ◆❁❭❫❙ß❺✯❯ß❳◗▲✹❬✕❛❜❪ P✣❳◗❭❫❯✕♠❖❛✭P❘❨✡❳❩❛❜❙✥➐❥s❦q❍s⑥t◗✉❇✈➳④⑤❭✹❣✹▲✟❡✯❭➝❺✯❯❝❣✹▲❫❱✕❙✫P✝❙✫❯▼❣✹❭✹❡✯❣❫❙❝❡✝❭✹❡✕❱✫❙✫❯❝❣❫P✯❙✝❡❜❙❝❛⑥◆◗P❘❭❫P❘❛✯❣✽❴✫▲✽④➞❙✫❯✝❣❫Ý❘❛✯❨✡❯❝❭❫❳❩▲❍✐ ➴①❣✹❛❜❳❩❣❫❙❝❛❜P✯❙✝❡➋❯✕❙➷P✣❳◗▲❫Ô✫❙✝❛✯▲▼◆❁❢❞❣✹❨✡❯✫Ý❘❛✭❯❝❢rÔ✕❙❝❣✹❡✯▲➄✛✇◆❁▲✹❣❫Ý❘❛❜❙✕❯❝❛✭❏✍④⑤❨✡❳⑤❪➄❭❫P❘▲❞▲❄Ù❉❣✹❡❜❙⑥◆②❛✭♥➟❴✕❛❜❯ ❬✫▲❫❯✝▲❫❳◗❭❫P❘❨✕❭❫❳◗▲❞❛✯❴✫▲✹❭✹❡✯▲ ❴✕▲✍P❘▲❫❯⑥◆②❛✭❙✫❯❝▲➘✛❘❣❫❙✫❳❩▲❫❯✫P✣❏❑✐ ➃➘❯❝❭✹❡✯❛✭♠❖❭r❣✹❛✭❳◗❣❫❙❝❛✭P✯❙❝❡❜❙❝❛❦✛✯P❘❛✭❱✫❙❝❡➋❭❫❯❝❭✹❡✣❛✭♠❖▲✹❛☞③❁❛⑧❴✫❛✭♥❝▲❫❳②③❁❛✯❛✥❱❝❭❫❳❩❭❫❪➄▲❫P✯❳❩❛✭❏☎▲❖◆◗P❘▲✏❴✫▲❖◆❁❣❫❳❩❛➞◆❁❢✏❴✫▲✏❡✣❛❜❯❝❛✯❛✯❡✣▲✏❴✫▲✏❣✹❨✡❪①❭❫❯❝❴✫❢ ✐⑦➃✩❣✹▲❖◆◗P❘▲✹❭✽❭❫❙➓❙✕❯➓❱✕❙✫❯❝❣❫P✝❺✯❯❽❱✫❳❩❛❜❪➄❭✽❣✹❨✫❡✯❨✕❭❫❯❝❢❍✐ ☞✍✌✏✎✑✌✓✒✕✔✗✖✙✘✙✚✛✒✜✌✁✒❀✎✝✌✁✔❁✒ ❤➚ ❛➞③❁❛✯▲❫❳⑤❙❝❡✝❴✫▲✽❛✯▲❖③❁❛❜❳◗▲✽❭❫❳❩▲✟▲❄Ù❍P♣▲❫❯✥◆❁❛✯❭r✐çÒ❂✫r➔➳③❁❛✝▲❖◆◗P❘▲✟❬✕▲❫❯❝▲❫❳◗❭❫P⑧❭❫❙✫P♣❨✡❪➄❭❫P✗✐ ➃r❣✹▲❖◆◗P❘❭r❣✹❨✡❯✫Ý❘❛✭❯❝▲r❴✫❛❜♥❝▲❫❳❁◆❁▲r❛❜❯❝④⑤❨✡❳⑤❪➄❭❫Ý❘❛✣❛➋❣✹❭①➉❉❡✯❛➞◆◗P❘❭❫❳❩▲✹❭❽◆❁❭❫❙①❯✫❙➷❭r④⑤❛➞③❁❛✯▲❫❳⑤❙❝❡❜❙❝❛➱❴✫▲✏❛❜❯✫P✣❳◗❭❫❳❩▲❖➐❉❡✯❛➞◆◗P❘❭❫❳❩▲✹❭r◆❁❭❫❙▼❯✫❙ ❭Ð❪➄❨✫❴✕▲✹❡✯▲✹❡✯❨✡❳❁➐✍❭➳❴✫❛❜♥❝▲❫❳❁③❁❛✯❡✯❨✡❳❽❱❝❭❫❳❩❭❫❪①▲❫P✯❳◗❛✩✛✯❺✣❯ ④❘❙✫❯❝❣❫Ý❘❛✯▲❚❴✫▲Ð❨✳❱✫Ý❘❛❜❙✫❯✝❛✯❡✯▲Ð❴✕❛❜❯❧❡✣❛❜❯❝❛✯❭❚❴✫▲Ð❣✹❨✡❪①❭❫❯❝❴✫❢→✐➬Ò✽s⑧➔☞t◗Ò✍➑➄q❍❏②➐ ❡✣❛➞◆◗P❘❭❫❳◗▲✹❭➝♥❝❭✹❡✯❨✳❳◗❛✯❡✣❨✡❳➋♥❝❭❫❳❩❛✯❭❫Ô❝❛✯❡✣▲✹❡✯❨✡❳❦❴✫▲✽❛✯▲❖③❁❛❜❳❩▲❐✛♣❛❜❯❝❴✕❛✯❣✹❭❫P❘▲➝❺✯❯▼❡✯❛❜❯❝❛✯❭✽❴✫▲✽❣✹❨✡❪①❭❫❯❝❴✫❢➓✐ s⑥➛✍t➞➑❽➔☞❏❩➐✳▲❫P♣❣❍✐ ➤⑥➥✏❃✗➥❅❄❆②✞ ➧❉➻➼➧✫➽➠➭❫➧✟②✞ ➧➸➲✷➧➸➵✟②✂ ➨❉➵✛❇❈✂⑤➭ s❦q✫s✥t◗✉❇✈ ◆❁❛❜❪❞❙❝❡✯▲✹❭❄♠❖❢➟❣✹❛❜❳❩❣❫❙❝❛❜P❘▲➟▲✹❡✯▲✹❣❫P✯❳❩❛✯❣✹▲➟❣✹❭❫❳◗▲➪❣✹❨✡❯✫Ý♣❛❜❯è❙✫❳⑤❪➄❢❫P❘❨✕❭❫❳◗▲✹❡✯▲➪▲✹❡✯▲❫❪①▲❫❯✫P❘▲➪❴✫▲➪❣✹❛❜❳❩❣❫❙❝❛❜P⑤➐➠❡✯❛❜❯✝❛✯❭❫❳◗▲ ◆❁❭❫❙❽❯❝▲✹❡✯❛✭❯❝❛✯❭❫❳❩▲✫➉ ➎✥❳◗▲❄♠❖❛➞◆◗P❘❨✳❳ ❉❋❊★❊✛❊ ➎❤❣✹❨✡❯❝❴✫▲❫❯✥◆❁❭❫P❘❨✳❳ ✘✪❊✛❊★❊ ➎✥Ô❝❨✡Ô❝❛❜❯✝❢ ●✄❊✛❊✛❊ ➎✥Ô❝❨✡Ô❝❛❜❯✝❢✟❣❫❙✫❱✝❡✯❭❫P❘❢ ❍■❊✛❊✛❊ ➎❤❡✯❛❜❯❝❛✯▲✽❴✫▲➝P✯❳❩❭❫❯✥◆◗❪➄❛➞◆❁❛✯▲✽í✜❊✛❊✛❊ ➎➱◆➈❙✕❳②◆❁▲✽❛❜❯❝❴✫▲❫❱❝▲❫❯✝❴✫▲❫❯✫P❘▲ 34 ➎⑧❴✫▲➝P❘▲❫❯✥◆❁❛❜❙✫❯❝▲ å✴❊✛❊✛❊ ➎⑧❴✫▲✽❣❫❙✫❳◗▲❫❯✕P ✖ ❊✛❊✛❊ ➚❤❨✡❳⑤❪➄▲✹❡✣▲✟❴✫▲➝❙✫❯❝❴✕❢✍❺✯❯❽❳❩▲✹❬✫❛❜❪ P✣❳◗❭❫❯✕♠❖❛✭P❘❨✡❳❩❛❜❙❽❱❝❨✡P⑧④✇❛✝❴✫▲➝P❘❛❜❱✕❙❝❡➈➉ ➎➱◆②❛✭❯✫❙✥◆❁❨✫❛✯❴✫❭✹❡ ✓✕✂✖ ✁ ➎❤▲❄Ù❍❱❝❨✡❯❝▲❫❯✕Ý❘❛✯❭✹❡ ✚ ➘ ☎ ✄ ✔ ➎❤❛❜❪➆❱✕❙❝❡➞◆ ✔ ✆✄●✽✓✕✚ ✝ ➎❤❡✯❛❜❯❝❛✯❭❫❳➱❱❝▲✍❱✝❨✡❳✇Ý♣❛❜❙✫❯❝❛ ✔❦ò ● ➎➱◆②❛✭❯✫❙✥◆❁❨✫❛✯❴✫❭✹❡✕❪①❨✫❴✡❙❝❡✣❭❫P❝❺✣❯❞④❘❳❩▲✹❣❫♥❝▲❫❯✫Ý❘❢➼✓❉Ú✗Ú✗⑨ ➎➱◆➈❙✕❳②◆❁▲✽❣✹❨✡❪➄❭❫❯✝❴✫❭❫P❘▲ ➎➋◆◗❙✫❳❁◆❁❭✟❴✕▲✍P❘▲❫❯⑥◆②❛✭❙✫❯❝▲✽❣✹❨✡❪➄❭❫❯✝❴✫❭❫P❘❢➝❺✯❯❽P❘▲❫❯✥◆❁❛❜❙✫❯❝▲♦✚✄❊✛❊✛❊ ➎➋◆◗❙✫❳❁◆❁❭✟❴✕▲✍P❘▲❫❯⑥◆②❛✭❙✫❯❝▲✽❣✹❨✡❪➄❭❫❯✝❴✫❭❫P❘❢➝❺✯❯▼❣❫❙✫❳◗▲❫❯✕P ó■❊✛❊✛❊ ➎➋◆◗❙✫❳❁◆❁❭✟❴✕▲✟❣❫❙✫❳❩▲❫❯✫P⑧❣✹❨✡❪①❭❫❯❝❴✫❭❫P❘❢➝❺✯❯❽P❘▲❫❯✥◆❁❛❜❙✕❯❝▲ ✞✱❊✛❊✛❊ ➎➋◆◗❙✫❳❁◆❁❭✟❴✕▲✟❣❫❙✫❳❩▲❫❯✫P⑧❣✹❨✡❪①❭❫❯❝❴✫❭❫P❘❢➝❺✯❯▼❣❫❙✫❳❩▲❫❯✫P Ú ❊✛❊★❊ ✉✍❨✳❪➄❭❫❯❝❴✕❭✍❱❝❨✕❭❫P❘▲✟④⑤❛✝❴✫▲➝P❘❛❜❱✫❙❝❡◗➉ ➎❤❡✯❛❜❯❝❛✯❭❫❳❩❢ ➎✥❱❝❨✫❡✯❛❜❯✝❨✡❪➄❛✣❭✹❡✯❢ ✔✠✟ ●☎✡ ➎❤❨➓④❘❙✫❯❝❣❫Ý♣❛✯▲✟❭❫❯✝❭✹❡✯❛❜P❘❛✯❣✹❢ å ✻ ☞ ☛ ●✌✆❐✚ ➎❤❴✫❭❫P❘❢➝❱✫❳◗❛✭❯➓❱✕❙✫❯❝❣❫P❘▲ í ☛✏✎ ●✍✚ ✍ ➎❤❨➓▲❄Ù✫❱✫❳◗▲❖◆❁❛✯▲➝❺✯❯▼❴✫❨✡❪①▲❫❯❝❛❜❙✝❡✕➢☎❭❫❱❝❡✣❭✹❣✹▲ ●☎☛➘✔ ●☎☛✩✘✽✚ ➎⑧❴✕❛➞◆◗❱❝❨✳♠❖❛❜P❘❛✭♥❝▲✹❡✯▲➘◆②▲❫❪①❛✯❣✹❨✡❯✝❴✡❙❝❣❫P❘❨✫❭❫❳❩▲➝❙✕♠⑦❙❝❭✹❡✯▲✽❭✹❣✹❣✹▲❫❱✫P❘❭❫P❘▲➘◆◗❙✫❯✫P➠➉ ➎⑧❴✫❛✣❨✫❴✫❭ ✑❋❊✛❊✛❊ ➎✝P✯❳❩❭❫❯✕♠❖❛➞◆◗P❘❨✳❳✇❙❝❡✕Ô❝❛✭❱❝❨✫❡✯❭❫❳ ✒ ❊★❊✛❊ ➎✝P✯❳❩❭❫❯✕♠❖❛➞◆◗P❘❨✳❳✇❙❝❡✝❣❫❙▼▲✹④⑤▲✹❣❫P⑥❴✕▲✟❣✹❼❫❪❞❱✔✓ ➚⑥✈☎➔ ✕❅❊✛❊★❊ ➎✝P✯❳❩❭❫❯✕♠❖❛➞◆◗P❘❨✳❳✇❙❝❡❞➍❀Òrq✫➚✥✈☎➔ ⑨ ❊✛❊✛❊ ➎✝P✯❳❩❭❫❯✕♠❖❛➞◆◗P❘❨✳❳✇❙❝❡✝↔➘❭❫➃➓◆❩➚⑧▲❫P ✎✄❊✛❊✛❊ ◆◗❙✥◆❩❏❑✐ ➎➘◆➈❙✕Ô❝❣✹❛❜❳❩❣❫❙❝❛❜P✯❙❝❡ ✄❆❊✛❊★❊ ✛❘❣✹❭❫❳❩▲➸▲❖◆◗P❘▲➸❴✫▲✹④⑤❛❜❯❝❛❜P➝❣❫❙❀❭➈Õ②❙✫P❘❨✡❳⑤❙❝❡❇▲✹❡✣▲❫❪➄▲❫❯✫P♣▲✹❡✯❨✡❳✏❴✫▲➸❣✹❛❜❳❩❣❫❙❝❛❜P❇❱✫❳❩▲❄♠❖▲❫❯✫P❘❭❫P♣▲➄❪➄❭✹❛ ➜ ▲❖◆❁❣❫❳◗❛✯▲❫❳❩▲✹❭➟❺✯❯➼❨✡❳❩❴✫❛❜❯❝▲➳❭✹❡✣④✇❭❫Ô❝▲❫P♣❛✯❣✹❢➳❭➳▲✹❡✯▲❫❪①▲❫❯✫P❘▲✹❡✣❨✡❳▼❴✫▲Ð❣✹❛❜❳❩❣❫❙❝❛❜P✏▲❖◆◗P❘▲Ð❴✕❭❫P❘❢➪❺✯❯✵❱❝❭❫❳❩❭✹❬✡❳❩❭✹④➞❙❝❂ ✩ ❡ ✮✝➐❇❛✯❭❫❳➆❭ ❪①❨✫❴✫▲✹❡✣▲✹❡✯❨✡❳❦❭❖◆❁❨✫❣✹❛✯❭❫P♣▲✟❡✯❨✳❳➷❺✣❯➓❱✝❭❫❳◗❭✹❬✡❳❩❭✹④❘❙❝✥❡ ✷✫➐✳❡✯❭✽❡✯❛❜❯❝❛✣❭✟❴✫▲✽❣✹❨✡❪①❭❫❯❝❴✫❢➓✐ ➍❀Ò✽➜❐✈✗➢➝✐ ➤⑥➥✗✖☞➥✙✘✶✂✛✚ ➅ ❇ ➨✟✂ ✞②➧➸➲✷➧➸➩❤➽❇✿ ➩ ✞✹✂✢✜⑥➾ ✣✶✢❅✦✛✘✁✌✥✤✑✧ ✦ ✦✣✢✩★✕✖ ✭ ✔❁✒ ✢❅✤✪★✬✫✛✢✥✤✁✌✁✢ ✖✥✖ ✏✘ß➎❤❣✹❭✹❡✯❣❫❙❝❡✯▲✹❭❄♠❖❢➝❱✫❙✫❯❝❣❫P✣❙❝❡⑥◆➈P♣❭❫P❘❛✯❣✽❴✫▲➮④❘❙✫❯❝❣❫Ý❘❛✯❨✳❯❝❭❫❳◗▲✽❭✹❡✝❣✹❛❜❳❩❣❫❙❝❛❜P✯❙❝❡✭❙❝❛⑥✛✯❺✯❯▼❣❫❙✫❳❩▲❫❯✫P⑧❣✹❨✡❯✫P♣❛❜❯✫❙✫❙✕❏❑✐ ✑ ✟❽✔↕➎➅❴✕▲❫P❘▲❫❳✇❪①❛❜❯❝❢➆❱✫❙✫❯❝❣❫P✣❙❝❡✷◆◗P❘❭❫P♣❛✯❣▼❴✫▲➄④➞❙✕❯❝❣❫Ý❘❛✯❨✡❯✝❭❫❳◗▲❍✐❉➃r❣✹▲✹❭❖◆➈P♣❢▼❭❫❯❝❭✹❡✯❛✱♠❖❢▼▲❖◆◗P❘▲▼❭❫❙✕P❘❨✡❪①❭❫P✂▲✹④⑤▲✹❣❫P✯❙❝❭❫P❘❢➆❺✯❯❝❭✹❛❜❯✕P❘▲▼❴✫▲ ❫❭✭ ❯✝❭✹❡✯❛✭♠❖❭➶❺✯❯ß❳◗▲✹❬✫❛✭❪ P✯❳❩❭❫❯✕♠❖❛❜P❘❨✡❳❩❛❜❙✥➐✽❴✫❭✹❣✹❢✃❯✫❙❆◆◗❙✫❯✫P▼◆◗❱❝▲✹❣✹❛✯④⑤❛✯❣✹❭❫P❘▲❀❣✹❨✡❯✝❴✫❛❜Ý❘❛✯❛✣❡✯▲❀❛❜❯❝❛❜Ý❘❛✣❭✹❡✯▲❖➐✍❱✫❳❩▲✹❣❫❙✫❪ ③②❛➘❺✣❯❝❭✹❛❜❯✫P❘▲❀❴✫▲ ❭❫❯✝❭✹❡✯❛✭♠❖❭❀❺✣❯à❳❩▲✹❬✫❛❜❪ ◆❁❛❜❯✫❙⑥◆②❨✕❛✯❴✫❭✹❡➞➐✽❱❝▲❫❯✕P✯❳✇❙ ❴✫▲❫P❘▲❫❳⑤❪①❛❜❯❝❭❫❳❩▲✹❭❀❪①❨✫❴✫▲✹❡✯▲✹❡✯❨✳❳é❴✫▲❲◆❁▲❫❪❞❯❝❭✹❡✩❪➄❛✣❣→❭✹❡✯▲✵❴✫❛➞◆◗❱❝❨❍♠❖❛❜P❘❛❜♥❝▲✹❡✣❨✡❳ ❯✝▲✹❡✯❛❜❯❝❛✯❭❫❳❩▲➮❴✫▲✽❣✹❛❜❳❩❣❫❙❝❛❜P❩✐ í❇Ú➇➎➝❴✕▲❫P❘▲❫❳✇❪①❛❜❯❝❢➸❨✳❳◗❛✯❣✹▲➸④❘❙✫❯❝❣❫Ý♣❛✯▲①❴✫▲①❣✹❛✭❳◗❣❫❙❝❛✭P✷❺✯❯è❣❫❙✫❳◗▲❫❯✕P➅❣✹❨✡❯✫P♣❛❜❯✫❙✫❙è❣✹❨✡❯✥◆❁❛✯❴✫▲❫❳❩❼❫❯❝❴➪❪①❨✫❴✫▲✹❡✭❙❝❡✷❴✫▲➷◆②▲❫❪❞❯❝❭✹❡➠❪➄❛✯❣ ✹❣✭ ❨✳❳◗▲❖◆◗❱✫❙✫❯❉♠❖❢❫P❘❨✡❳➋❱✫❙✕❯❝❣❫P✯❙❝❡❜❙✝❛✥◆◗P❘❭❫P♣❛✯❣✟❴✕▲➮④➞❙✕❯❝❣❫Ý❘❛✯❨✡❯✝❭❫❳◗▲❍✐ 35 ✓❉✚✌✁r✓Ó➎è❴✫▲❫P❘▲❫❳⑤❪➄❛❜❯✝❢ ◆❁▲❫❯✕♠❖❛❜P❘❛❜♥✝❛❜P❘❢❫Ý❘❛✯❡✣▲↕❙✫❯❝❨✡❳❀P❘▲❫❯⑥◆②❛✭❙✫❯❝❛➳③❁❛ ⑤◆②❭❫❙ ❣❫❙✫❳❩▲❫❯✫Ý❘❛➪❺✣❯ ④❘❙✫❯❝❣❫Ý❘❛✣▲❆❴✫▲àP❘❨✡Ý❘❛➷❱❝❭❫❳❩❭❫❪➄▲❫P✯❳❩❛✯❛ ❪ ✭ ❨✫❴✫▲✹❡✭❙❝❡❜❙❝❛✝❴✫▲➘◆❁▲❫❪➆❯❝❭✹❡✕❪①❛✯❣✽❣✹❨✡❳◗▲❖◆◗❱✫❙✕❯✕♠❖❢❫P❘❨✡❳➋❱✕❙✫❯❝❣❫P✯❙❝❡✭❙❝❛⑥◆➈P♣❭❫P❘❛✯❣✽❴✫▲➮④❘❙✫❯❝❣❫Ý❘❛✯❨✳❯❝❭❫❳◗▲❍✐ ① ✁ ✣✶✢❅✦✛✘✁✌✥✑✤ ✦✧✦✣✢✡✔✧✒✄✂ ✌✆☎✞✗ ✝ ✌✣✢ ✫ ✖✟✝ ✛✌✁✚★✦✛✘✙✚✛✒✠✝✗✒✡☎✴✢✥✦★✘✟☎✠✌ ★ ❡✕❺✯❯❽❳◗▲✹❬✫❛✭❪ ◆❁❛❜❯✫❙✥◆❁❨✫❛✯❴✕❭✹❡➞➐❖❺✯❯✫P✣❳②➎♣❨✩Ô❝❭❫❯❝❴✫❢✽❴✫❭❫P❘❢✽❴✫▲✽④➞❳❩▲✹❣❫♥❝▲❫❯✫Ý❘▲❖➐❍④✇❨✫❡✣❨❉◆❁❛❜❯❝❴ ✭ ☛✩✘✩➎⑧❪➄❭❫❨✫❯❝❴✕❭✹▲✹❡✯❛✱❡❜♠❖❙❝▲✹❡✝❭❄❡✯♠❖❛❜❯❝❢✟❛✣❭❫❣✹❳❦❛❜❳❩❴✫❣❫❙❝▲➘❛❜◆②P✣❙❝▲❫❪❞ ❯❝❭✹❡✕❪①❛✯❣✟❣✹❨✳❳◗▲❖◆◗❱✫❙✫❯❉♠❖❢❫P❘❨✡❳➋❱✫❙✕❯❝❣❫P✯❙❝❡❜❙✝❛✥◆◗P❘❭❫P♣❛✯❣✟❴✕▲➮④➞❙✕❯❝❣❫Ý❘❛✯❨✡❯✝❭❫❳◗▲❍✐ ✁ ✟❽✖❫✓✕✚ ➎⑧❣✹❭✹❡✯❣❫❙✝❡✯▲✹❭❄♠❖❢➝♠❖❬✕❨✡❪➄❨✳P✯❙❝❡✕❱✫❳❩❨✫❴✡❙✥◆☎❡✯❭✟❨❽❭❫❯✫❙✕❪➄❛❜P♣❢✟❛✯▲❖③◗❳❩▲❐✛✣P❘▲❫❯✥◆❁❛❜❙✕❯❝▲❐◆❁❭❫❙▼❣❫❙✫❳❩▲❫❯✫P✣❏❤❺✣❯✫P✯❳❁➎❘❨✩Ô❝❭❫❯❝❴✕❢✟❴✫▲ ❘④✭ ❳❩▲✹❣❫♥❝▲❫❯✫Ý❘▲➘◆◗❱❝▲✹❣✹❛✯④⑤❛✯❣✹❭❫P❘❢➝❺✯❯▼❣✹❨✡❪➄❭❫❯✝❴✫❭✟❭❖◆❁❨✫❣✹❛✣❭❫P❘❢r✐ ➃✩✉➘✐ ✣✶✢❅✦✛✘✁✌✥✤✑✧ ✦ ✦✣✢✡✔✧✒✄✂ ✤ ✗✫ ✌✆☎ ✤✁✔❁✦ ✢ ✝✌✁✤ ✛✔❁✌ ✖ í ❉ ☛ ✁↕ä➷❛✭❯✫P❘▲✹❬✡❳❩▲✹❭❄♠❖❢r▲✹❣❫❙❝❭❫Ý❘❛✯❛✣❡✯▲➘❪➄❨✕❴✫▲✹❡❜❙❝❡❜❙✝❛➋❴✫▲❽◆❁▲❫❪➆❯✝❭✹❡⑥❪➄❭❫❳❩▲r❭✹❡⑧❣✹❛✭❳◗❣❫❙❝❛✭P✯❙❝❡❜❙❝❛✥❱❝▲✽❙✫❯➸❛❜❯✫P❘▲❫❳⑤♥❝❭✹❡❤❴✫▲✽P❘❛❜❪❞❱➒❴✕❭❫P❩✐ ✉✭ ❨✳❯❝❴✫❛❜Ý❘❛✣❛✯❡✯▲➳❛❜❯✝❛❜Ý❘❛✯❭✹❡✯▲➶◆➈❙✕❯✫P✩❣✹❭✹❡✯❣❫❙✝❡✯❭❫P❘▲➟❱✫❳❩❛❜❯✫P✣❳②➎♣❨❲❭❫❯❝❭✹❡✣❛✭♠❖❭➟❺✯❯➼❣❫❙✫❳◗▲❫❯✕P✩❣✹❨✡❯✫P♣❛❜❯✫❙✫❙❧❺✯❯ ❣✹❭❫❳❩▲è◆❁▲Ð❣✹❨✡❯⑥◆②❛✣❴✫▲❫❳◗❢éP❘❨✫❭❫P❘▲ ✍ ◆◗❙✫❳❁◆❁▲✹❡✯▲✽❛❜❯❝❴✫▲❫❱❝▲❫❯✝❴✫▲❫❯✫P❘▲✽❣❫❙❽♥❝❭✹❡✯❨✡❳❩❛✯❡✯▲✽❴✫▲✽❡✯❭✍❪①❨✡❪①▲❫❯✫P✯❙❝❡✝❛❜❯✝❛❜Ý❘❛✯❭✹❡⑥◆❁❭❫❙①◆➈❙✕❯✫P⑧❴✫❭❫P❘▲✽❴✫▲✍❙✕P❘❛✯❡✯❛✭♠❖❭❫P♣❨✡❳❑✐ Ú✠✟ ✄ ✆ ❉ ä❝④⑤❭✹❣✹▲✟❭❫❯❝❭✹❡✣❛✭♠❖❭✟❭❫❳⑤❪①❨✡❯❝❛✯❣✹❢✽❭✍❙✕❯❝▲✹❛✕P❘▲❫❯✥◆❁❛❜❙✫❯❝❛⑥◆❁❭❫❙▼❭✍❙✕❯✫❙❝❛✝❣❫❙✫❳❩▲❫❯✫P⑧❨✡Ô✫Ý❘❛❜❯✕❙✫P⑧❣❫❙❞❣✹❨✳❪➄❭❫❯❝❴✕❭r✐ç➔☞➛➅➃✟➑ ✝❱ ✭ ▲❫❯✫P✯❳⑤❙❞❨❽④❘❳◗▲✹❣❫♥✝▲❫❯✫Ý❘❢✽④➞❙✫❯✝❴✫❭❫❪➄▲❫❯✕P❘❭✹❡✯❢✽❴✫❭❫P❘❢➘③②❛✕❙✫❯❽❯✫❙✕❪➄❢❫❳➠◆◗❱❝▲✹❣✹❛✣④✇❛✯❣✹❭❫P⑧❴✫▲✽❣✹❨✡❪❞❱❝❨✡❯❝▲❫❯✕P❘▲✟❭❫❳⑤❪①❨✡❯❝❛✯❣✹▲❍✐ ➃r❣✹▲❖◆◗P❘▲✟❣✹❨✳❪➄▲❫❯✕♠❖❛⑥◆◗❙✫❯✫P⑧❴✫▲❖◆❁❣❫❳❩❛➞◆❁▲✍❺✣❯❞❴✕▲❫P❘❭✹❡✯❛❜❙❽❺✯❯❽❱❝❭❫❳❩❭✹❬✡❳❩❭✹④➞❙❝❡✥✷✝✐ ➤⑥➥ ☞➥ r➵✛❇➅➩ ☞☛ ✍✌ ❩➧➸✟➵ ✂②➨❉✛➵ ❇❈✂⑤➭✑❇❈✞ ❇❈✂ ✏✎❁✂ ✂✹✞ Ø ❯➶❣❫❙✫❳❩▲❫❯✫P➝❣✹❨✳❯✫P❘❛❜❯✫❙✕❙→③❁❛☎❺✯❯➶❣❫❙✫❳❩▲❫❯✫P➝❭✹❡✭P❘▲❫❳✇❯✝❭❫P❘❛❜♥→◆❁▲➄❙✫P❘❛✣❡✯❛✭♠❖▲✹❭❄♠❖❢➸▲✹❣❫❙❝❭❫Ý❘❛✣❛✯❡✯▲➄❪①▲❫P❘❨✫❴✫▲✹❛➠❱❝❨✡P❘▲❫❯✫Ý♣❛✯❭✹❡✯▲✹❡✯❨✡❳♦❯❝❨✕❴✡❙✫❳❩❛✯❡✯❨✡❳✙✐ ➃➘❯❝❭✹❡✯❛✭♠❖❭ ❺✯❯ ❳❩▲✹❬✫❛❜❪ P✣❳◗❭❫❯✕♠❖❛✭P❘❨✡❳❩❛❜❙ ④⑤❨✫❡✯❨❝◆②▲❖③◗P❘▲ ❪➄❨✕❴✫▲✹❡✯▲✹❡✯▲ ❣✹❨✡❪➆❱✝❭❫❯❝❛✯❨✡❯ ❭✹❡✯▲ ▲✹❡✣▲❫❪➄▲❫❯✫P♣▲✹❡✯❨✡❳ ❴✫❛❜❯✝❭❫❪➄❛✯❣✹▲ ❣✹❨✳❳◗▲❖◆◗❱✫❙✫❯❉♠❖❢❫P❘❨✫❭❫❳❩▲✽❙✫❯❝▲✹❛❤❭❫❯✫❙✫❪①❛❜P❘▲✽❪①▲❫P❘❨✫❴✫▲❐❯✫❙✫❪①▲❫❳◗❛✯❣✹▲ ✯✕❱❝▲❫❯✫P✯❳⑤❙➒❣✹❛✭❳◗❣❫❙❝❛✭P✯❙❝❡✝❳❩▲❄♠❖❛➞◆◗P❘❛❜♥➸❣✹❭❫❳◗▲❐❳◗▲❄♠⑦❙❝❡✭P❘❢✏◆❁▲♦❙✕P❘❛✯❡✯❛✭♠❖▲✹❭❄♠❖❢ ▲✹❣❫❙✝❭❫Ý❘❛✯❛✯❡✯▲➝❪①▲❫P❘❨✫❴✫▲✹❛✕❱❝❨✡P♣▲❫❯✫Ý❘❛✯❭✹❡✯▲✹❡✣❨✡❳➋❯❝❨✫❴✳❙✫❳◗❛✣❡✯❨✡❳✙✐ ➤⑥➥ ➋➥ Ó➧✳➭❫➺➋➲✷➧✟✞②➧①➽ ➅ ❇ ➻➼➧❉➨✟✂❩➵❉➧ ✒✑ ✍✓ ✣✕✔✗✖ ✒✕✤✗✛✫ ✚✛✒✜✚✛✒✜✌✣✢✥✤✓✒✄✂❅✔❁✦✛✔❁✒❂✢ ✖✟☎✠✒✕✔❁✌ ★✙✘ ➍❀▲❫P♣❨✫❴✫▲✹❡✯▲✽❴✫▲✽❛❜❯✫P❘▲✹❬✡❳❩❭❫❳❩▲✍❯✫❙✕❪➄▲❫❳❩❛✯❣✹❢✽④✇❨✫❡✣❨❉◆❁❛❜P❘▲✽❴✫▲➝s❦q✫s✥t◗✉❇✈è◆◗❙✫❯✫P❁➉ ➎✥❪➄▲❫P❘❨✕❴✫❭✍P✣❳◗❭❫❱❝▲❄♠⑦❙✝❡❜❙❝❛➞➐ ➎✥❪➄▲❫P❘❨✕❴✫❭✟↔➘▲✹❭❫❳❦❴✫▲✽❨✡❳❩❴✫❛❜❆ ❯ ➓ ✮ ❱❝❼❫❯❝❢✽❡✯✍ ❭ ✲✥✐ ✫➘❯❝▲✹❡✯▲➆♥❝❭❫❳◗❛✣❭❫❯✫P❘▲▼❴✕▲➆s❦q❍s⑥t➈✉❇✈➮❱❝▲❫❳✇❪①❛❜P☎❙✫P❘❡✯❛✱♠❖❭❫P❘❨✡❳⑤❙❝❡❜❙❝❛✂◆❁❢▼❭✹❡✯▲✹❭✹❬✫❢➓❺✯❯✫P✣❳◗▲➓❪➄▲❫P♣❨✫❴✫❭➓P✯❳◗❭❫❱✝▲❄♠⑦❙❝❡❜❙❝❛☎③❁❛➱❪➄▲❫P❘❨✕❴✫❭➆↔➘▲✹❭❫❳ ❴✕▲✍❙✫❯▼❭❫❯✫❙✕❪➄❛❜P⑧❨✡❳❩❴✫❛❜❯❽❪①❭❄Ù❉❛❜❪➟✐⑦➢☎❭✟❭✹❡✭P❘▲✍♥✝❭❫❳◗❛✯❭❫❯✕P❘▲✵❪①▲❫P❘❨✫❴✫❭✽❴✫▲✽❛❜❯✫P❘▲✹❬✡❳❩❭❫❳❩▲✍❯✫❙✕❪➄▲❫❳❩❛✯❣✹❢✽▲❖◆➈P♣▲✟❭✹❡✯▲✹❭❖◆❁❢✽❴✫▲➝❱✫❳◗❨✕❬✡❳◗❭❫❪➟✐ ✒✕✗✤ ★ ✫ ✚✛✒ ✚★✒ ✔❁✒ ✤ ✛✫ ✘ ✽✦✛✔❁✒✜✦ ✽★ ✌✁✔ ✕★ ✖ ✌✁✤✓✒✕✘✗✛✫ ✔ ✔❁✒ ✝✤ ✌ ✗✤✁✌ ✽✒ ✢❅✒✕✘✁✌✣✢❅✌✁✦✛✔❁✒ s❦q✫s✥t◗✉❇✈ ❙✫P❘❛✣❡✯❛✭♠❖▲✹❭❄♠❖❢ ❺✯❯ ❭✹❣✹▲❖◆➈P ◆❁❣✹❨✡❱ ❪①▲❫P❘❨✫❴✕❭❞➑➓▲❫➏❐P♣❨✡❯✥➎➞➛♦❭❫❱✫➣✥◆❁❨✡❯➶❭❫P❘❼❫P➅❱❝▲❫❯✫P✯❳⑤❙➶❭❫❯❝❭✹❡✯❛✱♠❖❭➄❺✯❯➶❣❫❙✫❳❩▲❫❯✫P➝❣✹❨✡❯✕P❘❛❜❯✫❙✫❙❀❣✹❼❫P✟③❁❛☎❱❝▲❫❯✫P✯❳⑤❙➶❭❫❯❝❭✹❡✣❛✭♠❖❭①❣✹❛❜❳❩❣❫❙❝❛❜P❘▲✹❡✣❨✡❳ ❣✹❭❫❳❩▲✽❣✹❨✡❯✫Ý❘❛❜❯❽❪①❨✫❴✫▲✹❡✯▲✽❣✹❨✡❪❞❱❝❭❫❯❝❛✯❨✳❯➋✐ ✚✛✔✜✖ ✒✢ ✆✝ ✒✢ ✟✣ ➤⑥➥ ☞➥ ➪➨✟✂✇➭❫➧❉✟➨ ✂✹✂❇➲✂➧➸➵❉➺⑧➽ ➠➧❉➨ ➱➧✫➽ ❫➾➳➩ ⑤✂ ➭❫➧❉➨❉➩ ☞✤ ✦✥ ★✧ ✟✩ ✪✎ ❩➺➱➨ Ð➧ é➭❫➺⑧➽ ❚➩✪✚ ❇➫✹➺⑧➽ ✏❁ ✎ ✂ ✂✹✞ ✬✫ ✮✭ ✰✯✲✱ ✴✳ t✇P❘▲❫❳❩❭❫Ý❘❛✯❛✯❡✣▲❇➑❽▲❫➏❐P❘❨✡❯⑥➎✣➛♦❭❫❱✫➣⑥◆②❨✳❯➄◆❁▲✽❨✡❱✫❳❩▲❖◆❁❣✟❴✫❭✹❣✹❢➘◆◗❙✫❯✫P✝❺✯❯❝❴✕▲❫❱❝❡✯❛❜❯❝❛✭P❘▲✍P♣❨✫❭❫P❘▲✽❣✹❨✡❯❝❴✫❛❜Ý♣❛✯❛✯❡✯▲➝❙✫❳⑤❪➄❢❫P❘❨✕❭❫❳◗▲✫➉ 36 ❴✕▲✹❣✹❼❫P ➎♣▲❫❳❩❨✫❭❫❳◗▲✹❭✽❭❫Ô✥◆❁❨✫❡❜❙✕P❘❢✍❺✣❯✫P✯❳❩▲✍♥❝❭✹❡✣❨✡❳◗❛✣❡✯▲✟❨✳❳◗❛✯❣✹❢❫❳⑤❙❝❛✝❣❫❙✫❳❩▲❫❯✫P⑧❡✯❭➝❙❝❡❜P❘❛❜❪①▲✹❡✯▲✽❴✫❨✡❙❝❢✽❛❜P❘▲❫❳❩❭❫Ý❘❛✯❛✝▲❖◆◗P❘▲➝❪➄❭✹❛✕❪①❛✯❣✹❢✽❴✫▲✹❣✹❼❫P ♥❝❭✹❡✯❨✫❭❫❳❩▲✹❭➝❱❝❭❫❳◗❭❫❪①▲❫P✯❳⑤❙❝❡❜❙❝❛✕➃✏➙➘q✕➔✗Ò✟➢✂✛♣❴✫❭✹❣✹❢➝❯✫❙①◆②▲➘◆◗❱❝▲✹❣✹❛✯④⑤❛✯❣✹❢✽❭✹❡❜P❘④⑤▲✹❡✫➃✏➙➘q✕➔✗Ò✽➢ ✬➝❾✺❱✫➃✏❏ ➎♣▲❫❳❩❨✫❭❫❳◗▲✹❭✃❭❫Ô✥◆❁❨✫❡❜❙✫P❘❢Ð❺✯❯✫P✯❳❩▲➟♥❝❭✹❡✯❨✳❳◗❛✯❡✣▲➳❨✡❳❩❛✯❣✹❢❫❳◗▲✹❛✟P❘▲❫❯✥◆❁❛❜❙✫❯✝❛❐❡✯❭➟❙✝❡❜P❘❛❜❪①▲✹❡✯▲➳❴✫❨✳❙❝❢➳❛❜P❘▲❫❳❩❭❫Ý❘❛✯❛✽▲❖◆◗P❘▲➪❪①❭✹❛➝❪➄❛✯❣✹❢ ❝♥ ❭✹❡✯❨✫❭❫❳❩▲✹❭➝❱❝❭❫❳◗❭❫❪①▲❫P✯❳⑤❙❝❡❜❙❝❛✕➡✽➑➓➔✗Ò✟➢✂✛♣❴✫❭✹❣✹❢✍❯✕❙➄◆❁▲➘◆◗❱❝▲✹❣✹❛✯④⑤❛✯❣✹❢✟❭✹❡✭P❘④✇▲✹❡✕➡✽➑➓➔✗Ò✟➢ ✬✍❾✁❥➡✏❏ ➎♣▲❫❳❩❨✫❭❫❳◗▲✹❭➝❳❩▲✹❡✯❭❫P❘❛❜♥❝❢➝❺✯❯✫P✣❳◗▲➝♥❝❭✹❡✯❨✡❳❩❛✯❡✯▲✽❨✡❳❩❛✯❣✹❢❫❳⑤❙❝❛✝❣❫❙✫❳◗▲❫❯✕P❤◆❁❭❫❙▼❨✡❳❩❛✯❣✹❢❫❳❩▲✹❛✫P♣▲❫❯✥◆❁❛❜❙✫❯❝❛✝❡✯❭➝❙❝❡❜P❘❛✭❪➄▲✹❡✯▲✽❴✫❨✡❙✝❢✟❛❜P♣▲❫❳◗❭❫Ý❘❛✣❛ ▲❖◆◗P❘▲✍❪①❭✹❛✕❪➄❛✣❣✹❢✟❴✫▲✹❣✹❼❫P✝♥❝❭✹❡✣❨✫❭❫❳◗▲✹❭➝❱❝❭❫❳❩❭❫❪➄▲❫P✣❳✇❙❝❡✭❙❝❛❉➛✍✈❦➢☎➔✗Ò✟➢✂✛♣❴✫❭✹❣✹❢➝❯✫❙①◆②▲➘◆◗❱❝▲✹❣✹❛✯④⑤❛✯❣✹❢✽❭✹❡❜P❘④⑤▲✹❡➆➛✍✈✗➢☎➔❦Ò✽➢ ✬ ✾⑥✐➞❾✄✂➸❏ ➃➘P❘❼❫P☎♥❝❭✹❡✯❨✡❳❩❛✯❡✯▲➆❱❝❭❫❳◗❭❫❪①▲❫P✯❳❩❛✯❡✯❨✡❳✍➃✏➙➘q✕➔✗Ò✽➢✷➐⑥➡✽➑➓➔✗Ò✟➢✂➐⑧➛✍✈✗➢☎➔✗Ò✟➢➼❣✹❼❫P✍③❁❛☞❯✫❙✕❪➄❢❫❳⑤❙❝❡➱❪➄❭❄Ù❝❛❜❪ ❴✫▲❞❛❜P❘▲❫❳❩❭❫Ý❘❛✯❛ ➑❽▲❫➏❐P❘❨✳❯✥➎✣➛♦❭❫❱✕➣✥◆❁❨✡❯❽❱❝❨✡P⑧④✇❛⑥◆❁❣❫➣❝❛❜❪❞Ô❝❭❫P❘▲➝❺✯❯▼❡✯❛❜❯✝❛✯❭✟❴✕▲✟❣✹❨✡❪①❭❫❯❝❴✫❢ ✭ ✟➓✔☎í❇✖ ✟ ✁➓✓➋✐ ➜✩❭✹❣✹❢✽❱✫❳❩❨✫❣✹▲✹❴✫▲❫❙❝❡❤❛❜P❘▲❫❳❩❭❫P❘❛❜♥➄❯✫❙➒▲❖◆◗P❘▲✏❣✹❨✳❯✫♥❝▲❫❳❩❬✫▲❫❯✫P❦◆❁▲✽P♣❛❜❱❝❢❫❳❩▲❖◆②❣❐❱❝❨✡P❘▲❫❯✫Ý♣❛✯❭✹❡✯▲✹❡✯▲✽❯✝❨✫❴✡❙✫❳❩❛✯❡✯❨✡❳✗❱❝▲❫❯✕P✯❳✇❙❞❙❝❡❜P♣❛❜❪➄❭ ❛✭P❘▲❫❳◗❭❫Ý♣❛✯▲❍✐ ➤⑥➥✆☞☎ ➥✞✝ ➅ ❇ ➨❉➫✹➧➸➲✷➧➸➧❉➨❉➺➋✟➨ ✂ ✗✈ Ù❉❛❘◆➈P♣▲❫❯✫Ý❘❭➘❙✫❯❝❨✳❳✂▲❫❳❩❨✡❳❩❛⑥❺✯❯➷❴✫▲❖◆❁❣❫❳❩❛✯▲❫❳❩▲✹❭r❣✹❛❜❳◗❣❫❙✝❛❜P✯❙❝❡❜❙✝❛➞➐❝❣✹❭❽③❁❛➋❭➘❙✕❯❝❨✡❳❇▲❫❳◗❨✡❳❩❛⑧❴✕▲✏❣✹❭✹❡✯❣❫❙❝❡❤▲❖◆➈P♣▲r◆❁▲❫❪➆❯❝❭✹❡✣❭❫P❘❢✽❺✯❯ ④⑤▲❫❳❩▲✹❭❖◆➈P✣❳◗❭➝❺✯❯▼❣✹❭❫❳◗▲➘◆❁▲✟❣✹❭❫❳❩❭✹❣❫P❘▲❫❳❩❛✭♠❖▲✹❭❄♠❖❭✽❬✫❡✯❨✡Ô❝❭✹❡⑥◆❁❛❜❪❞❙❝❡✯❭❫❳❩▲✹❭❍✐✙➜✩▲❖◆❁❣❫❳◗❛✯▲❫❳❩▲✹❭①❴✫▲❫P♣❭✹❡✯❛✯❭❫P❘❢①❭➒❭✹❣✹▲❖◆➈P♣❨✡❳❐▲❫❳◗❨✡❳❩❛❇◆❁▲①❬✕❢❖◆②▲❖③◗P❘▲ ❺✣❯❞④⑤❛➞③❁❛✯▲❫❳⑤❙❝❡✝❴✫▲✽❛✯▲❖③❁❛❜❳◗▲❍✐ ➜✩❭✹❣✹❢é❣✹❛✭❳◗❣❫❙❝❛✭P✯❙❝❡♦❭❫❳◗▲➷❙✫❯❀❯❝❨✫❴✃❱❝▲❫❯✕P✯❳✇❙→❣✹❭❫❳❩▲➒❯✫❙→▲❄Ù❉❛➞◆◗P❘❢➪❨è❣✹❭✹❡✯▲➪❴✫▲➒❳❩▲❄♠❖❛➞◆◗P❘▲❫❯✫Ý❘❢➪④⑤❛❜❯❝❛❜P♣❢é◆◗❱✫❳❩▲➒❪➄❭❖◆❁❢➒❺✯❯ ❣❫❙✕❳◗▲❫❯✫Pé❣✹❨✳❯✫P❘❛❜❯✫❙✕❙ ❱✕❙✫❯❝❣❫P✯❙❝❡➪◆◗P❘❭❫P♣❛✯❣➼❴✫▲➼④❘❙✫❯❝❣❫Ý❘❛✯❨✳❯❝❭❫❳◗▲❧❯✕❙ ◆❁▲❧❱❝❨✫❭❫P❘▲➼❣✹❭✹❡✯❣❫❙✝❡✯❭ß③❁❛❞❱✫❳❩❨✫❬✡❳❩❭❫❪➆❙❝❡➷◆②▲➼❨✳❱✫❳◗▲❖③◗P❘▲ ◆❁▲❫❪❞❯❝❭✹❡✯❼❫❯❝❴➟❭✹❣✹▲✹❭❖◆◗P❘❢❞▲❫❳◗❨✫❭❫❳❩▲❍✐✕➜✩❭✹❣✹❢➆❣✹▲✹❡✯▲❞❴✫❨✡❙✝❢➆❛✯▲❖③❁❛❜❳❩❛✗❭✹❡✯▲❞❡✯❛❜❯❝❛✯▲✹❛✗❴✫▲➓P✯❳❩❭❫❯✥◆◗❪➄❛❘◆②❛✣▲r❯✫❙Ð❭❫❙➪❺✯❯✫P✣❳◗▲❞▲✹❡✯▲❞❨➷❣✹❭✹❡✣▲➆❴✫▲ ❳❩▲❄♠❖❛➞◆◗P❘▲❫❯✫Ý♣❢é④⑤❛❜❯❝❛❜P♣❢➸❺✯❯❲❣❫❙✫❳❩▲❫❯✫P❐❣✹❨✡❯✫P♣❛❜❯✫❙✫❙➮◆❁❭❫❙❲❴✫❭✹❣✹❢➒❯❝❨✫❴✡❙✫❳❩❛✯❡✯▲➪❴✫▲➪❣✹❨✡❪①❭❫❯❝❴✫❢➪❭✹❡✯▲➒❙✫❯✝▲✹❛♦◆➈❙✕❳②◆❁▲➪❣✹❨✡❪①❭❫❯❝❴✫❭❫P❘▲➒❺✯❯ P♣▲❫❯✥◆❁❛❜❙✫❯❝▲✽❯✕❙➒❭❫❙▼❺✣❯✫P✯❳❩▲✏▲✹❡✯▲✏❨➄❣✹❭✹❡✯▲✏❴✫▲✽❳❩▲❄♠❖❛➞◆◗P❘▲❫❯✫Ý♣❢✏④✇❛✭❯❝❛❜P❘❢✟❺✯❯➄❣❫❙✕❳◗▲❫❯✫P➋❣✹❨✡❯✫P♣❛❜❯✫❙✫❙①▲✹④✇▲✹❣❫P✣❙❝❡✥▲❖◆◗P❘▲➘❭✹❣✹▲✹❡✯❭❖③❁❛⑤✐✹➃✩❣✹▲❖◆◗P⑥P❘❛❜❱ ❴✕▲✟▲❫❳❩❨✡❳❩❛✥◆❁▲✽▲✹❡✯❛❜❪①❛❜❯❝❢✽❣✹❨✡❯❝▲✹❣❫P❘❼❫❯✝❴➓❣✹❨✳❳◗▲❖◆◗❱✫❙✫❯❉♠❖❢❫P❘❨✡❳➪❙✫❯❽❳◗▲❄♠❖❛❘◆➈P♣❨✡❳❦❣❫❙❽❳◗▲❄♠❖❛❘◆➈P♣▲❫❯✫Ý❘❭✽④✇❨✫❭❫❳⑤P❘▲➝❪➄❭❫❳❩▲❍✐ s❤❨✡P✏▲❄Ù❉❛➞◆◗P❘❭❚▲❫❳◗❨✡❳❩❛♦❴✫❭❫P❘❨✡❳❩❭❫P❘▲➟▲❄Ù❉❛➞◆◗P❘▲❫❯✫Ý♣▲✹❛❇❙✕❯❝❨✡❳✏Ô✫❙✝❣✹❡✯▲é④⑤❨✡❳⑤❪➄❭❫P♣▲➸❯✫❙✕❪➄❭✹❛♦❴✫❛❜❯➮◆◗❙✫❳❁◆❁▲é❴✫▲➷P❘▲❫❯✥◆❁❛❜❙✕❯❝▲Ð◆❁❭❫❙ ❙✕❯❝❨✡❳✟◆❁▲✹❣❫Ý❘❛❜❙✕❯❝❛✗④✇❨✡❳⑤❪①❭❫P❘▲r❯✕❙✫❪➄❭✹❛☞❴✕❛❜❯Ð◆◗❙✫❳②◆❁▲❽❴✫▲❽❣❫❙✫❳❩▲❫❯✫P❩✐❍➃✩❣✹▲❖◆◗P❘▲❽▲❫❳◗❨✳❳◗❛✗◆❁▲❽❣✹❨✡❳❩▲✹❣❫P❘▲✹❭❄♠❖❢❽❛❜❯✫P✯❳❩❨✫❴✡❙❝❣✹❼❫❯✝❴➄❺✯❯➸Ô✫❙❝❣✹❡✯❭ ❳❩▲❖◆◗❱❝▲✹❣❫P❘❛❜♥✝❢✽❙✫❯▼❳❩▲❄♠❖❛➞◆◗P❘❨✡❳✷❣❫❙➄❳◗▲❄♠❖❛➞◆◗P❘▲❫❯✕Ý❘❭✏④⑤❨✫❭❫❳✇P♣▲✽❪➄❛✣❣✹❢r◆❁❭❫❙➒❛❜❯✫P✣❳◗❨✫❴✳❙❝❣✹❼❫❯❝❴➓❺✯❯➒◆❁▲✹❣❫Ý❘❛✭❙✫❯❝▲✹❭♦❳❩▲❖◆◗❱❝▲✹❣❫P❘❛❜♥✝❢♦❙✫❯❞❳◗▲❄♠❖❛❘◆➈P♣❨✡❳ ❣❫❙❽❳❩▲❄♠❖❛➞◆◗P❘▲❫❯✫Ý❘❭✽④⑤❨✫❭❫❳✇P♣▲✍❪①❭❫❳◗▲❍✐ s❤❨✡P✷❭❫❱❝❢❫❳❩▲✹❭r❱✕❳◗❨✡Ô✝❡✯▲❫❪➄▲❞❴✫▲❞❣✹❨✡❯✫♥❝▲❫❳❩❬✫▲❫❯✫Ý♣❢➆❭❞❛❜P❘▲❫❳❩❭❫Ý❘❛✯❛✯❡✣❨✡❳✷➑➓▲❫➏➘P❘❨✡❯✥➎➞➛♦❭❫❱✫➣✥◆❁❨✡❯➪❺✯❯Ð❣❫❙✫❳❩▲❫❯✫P➠❣✹❨✡❯✕P❘❛❜❯✫❙✫❙Ð◆❁❭❫❙ ❺✣❯➮❳◗▲✹❬✕❛❜❪ P✣❳◗❭❫❯✕♠❖❛✭P❘❨✡❳❩❛❜❙➋✐ Ø ❯➌❣❫❙✕❳◗▲❫❯✫P❽❣✹❨✡❯✫P❘❛❜❯✕❙✫❙à◆❁▲❚❳◗▲✹❣✹❨✳❪➄❭❫❯❝❴✕❢➟❙✫P❘❛✯❡✣❛✭♠❖❭❫❳◗▲✹❭✃❣✹❨✡❪➄▲❫❯❉♠❖❛✯❛ ✭ ✁ ✟✧❐ ✑ ✚✽✓❉✚♦í ❱❝▲❫❯✫P✣❳✇❙ ❭✹❡✣▲✹❬✫▲❫❳◗▲✹❭ ❙✫❯✝▲✹❛➸❭✹❡❜P❘▲ß❭❫❱✫❳❩❨✳Ù❉❛❜❪①❭❫Ý❘❛✯❛➸❛❜❯✝❛❜Ý❘❛✯❭✹❡✯▲❍✐r➔✗❨✡P➸❺✯❯ ❭✹❣✹▲❖◆◗Pé❣✹❭❄♠➇◆②▲ ❱❝❨✕❭❫P❘▲ ❺✯❯❝❣✹▲❫❳❩❣✹❭➮❙✕P❘❛✯❡✯❛✭♠❖❭❫❳❩▲✹❭ß❨✡❱✫Ý❘❛✭❙✫❯❝❛✯❛ ✓✝í❇✚✍✍✔ ✞✩⑨❧✂✖ ✁✼❣✹❭❫❳❩▲➳❬✫▲❫❯❝▲❫❳❩▲✹❭❄♠❖❢➟❙✫❯❧❱✫❳❩❨✫❣✹▲❖◆①❛❜P❘▲❫❳❩❭❫P❘❛❜♥➼❴✫▲➳❣✹❨✡❳❩▲✹❣❫P❘❭❫❳❩▲➳❭➳❣✹❨✡❯❝❴✳❙❝❣❫P❘❭❫Ý❘▲✹❛✟❪➄❛❜❯✝❛❜❪➄▲❚❴✫❛❜❯➮❣✹❛❜❳❩❣❫❙❝❛❜P ✯ ❭✹❣✹▲✹❭❖◆◗P❘❢➪❨✡❱✕Ý❘❛❜❙✫❯❝▲➒❱❝❨✕❭❫P❘▲➪④✇❛✍❭✹❣❫P❘❛✭♥❝❭❫P❘❢➒❱✫❳❩❛❜❯→❣✹❨✡❪①❭❫❯❝❴✫❭ ✭ ✟❽✔☎í❇✖ ✟ ✁➓✓➋✐ Ø ❯è❳❩▲✹❬✫❛❜❪ P✯❳◗❭❫❯❉♠❖❛❜P❘❨✡❳❩❛❜❙→❣✹❭❫❙✕♠❖❭➒❱❝❨✫❭❫P♣▲➸④✇❛ ❙✕P❘❛✯❡✯❛✭♠❖❭❫❳❩▲✹❭➌❙✫❯❝❨✡❳➳❪①❨✫❴✫▲✹❡✣▲➼❯❝▲❫❳❩▲✹❭✹❡✯❛➞◆◗P❘▲❖➐➄④✇❢❫❳❩❢à❣✹❭❫❱✝❭✹❣✹❛❜P❘❢❫Ý❘❛➟③❁❛➒❳◗▲❄♠❖❛➞◆◗P❘▲❫❯✕Ý❘▲➼❱✝❭❫❳◗❭❄♠❖❛❜P♣▲❆✛♣❴✫▲ß▲❄Ù❉▲❫❪❞❱❝❡❜❙ ❣✹❛❜❳❩❣❫❙❝❛❜P❘▲ ❣✹❨✳❯✫Ý❘❛❜❯❝❼❫❯✝❴➷❴✕❛✯❨✫❴✫▲➄③❁❛➋Ô✝❨✡Ô❝❛❜❯❝▲❖➐❤④✇❢❫❳❩❢➆❣✹❭❫❱✝❭✹❣✹❛❜P❘❢❫Ý❘❛☎③❁❛➱❳◗▲❄♠❖❛➞◆◗P❘▲❫❯✕Ý❘▲r❱✝❭❫❳◗❭❄♠❖❛❜P♣▲❄❏❩➐➋◆❁❭❫❙➳◆❁▲❫P❘❭❫❳❩▲✹❭r❯❝▲✹❣✹❨✳❳◗▲❖◆◗❱✫❙✫❯❉♠❖❢❫P❘❨✫❭❫❳❩▲➓❭ ❴✕❛✯④✇▲❫❳❩❛❜P❘▲✹❡✣❨✡❳❦▲❫❳❩❨✡❳◗❛✕❙✫P♣❛✯❡✯❛✭♠❖❭❫P❘▲✽❴✡❳❩▲❫❱✫P⑧❣❫❳◗❛✭P❘▲❫❳◗❛✣❛❝❴✕▲✟❣✹❨✡❯✕♥❝▲❫❳◗❬✕▲❫❯✫Ý❘❢✽❭✹❡✯▲✟❛✭P❘▲❫❳◗❭❫Ý♣❛✯❛✯❡✯❨✡❳❤➑➓▲❫➏❐P♣❨✡❯✥➎➞➛♦❭❫❱✫➣✥◆❁❨✡❯➋✐ 37 ✽✁✂✁ß✒☎✄❉✑☞✠✝✆✺✒☞✠☞✒✟✞ß✝✒ ✠✺✒☎✡ ✒☞✆✟✞✡✒✝✠✺☛✍✠✵✌✏✒→✝✑ ✆✺✠☞✑☞✎☛✆❩✞ ➱s ◆◗❱❝❛✯❣✹▲✵❙✕P❘❛✯❡✯❛✭♠❖▲✹❭❄♠❖❢➝❙✫❳⑤❪➄❢❫P♣❨✫❭❫❳◗▲✹❡✣▲✟▲✹❡✯▲❫❪①▲❫❯✫P❘▲✽❴✫▲✽❣✹❛❜❳❩❣❫❙❝❛❜P✇➐❖❺✣❯❞❨✳❳◗❴✫❛✭❯❝▲✟❭✹❡✣④✇❭❫Ô❝▲❫P♣❛✯❣✹❢✫➉ ✎✄❊★❊✛❊✌☞☞ í✍❷❖ë❍ì✎✍➂⑩➞ñ❁ã◗❸✫❷✏✞➆ë ☛rñ❁Ú❦✚♦í ✭ ➚❤❨✡❳⑤❪➄❭✽❬✫▲❫❯❝▲❫❳❩❭✹❡✯❢✫➉ B<name> ❴✒✑☞➐ <d> ❬✓✑➟<g> ➐ ◆✔✑✖<s> ✕➒❯❝❨✕<model> ❴✡❙✫❳❩❛✯❡✯▲✽❣✹❨✡[<area>] ❳◗▲❖◆◗❱✫❙✕❯✕♠❖❢❫P❘❨✫❭❫❳❩▲✟❴✳❳◗▲❫❯❝▲✹❛❘➐✳❬✡❳❩❛✯❡✯▲✹❛⑥③❁❛✥◆◗❙✫❳❁◆❁▲✹❛ ✏ ✏ ✏ ✏ ❪①❨✫❴✫▲✹❡✗✑ ✏ ❭❫❳❩▲✹❭✘✑ ✬➒❯✫❙✫❪①▲✹❡✯▲➝❪➄❨✕❴✫▲✹❡❜❙❝❡❜❙✝❛ ✬➪④⑤❭✹❣❫P❘❨✡❳❦❴✕▲✟❭❫❳❩❛✯▲ ✈✗Ù❉▲❫❪❞❱❝❡✯▲✫➉ ➙✍t❘➑ ✝❾ ✾✛✾ ❾ ✾❆↔✽➚✝➃❞q✕➔ ➙❞✽❾ ✷ ✮✛✮❚❾❑➁❋✮✙✷à↔➝➑➓Ò✟➍ ✮❤✐ ✾ ✘ ❊✛❊✛❊✙☞☞✘➘❸✫ì✥â⑥Ü✹ì⑥ñ❑ë✡ã➈❸✫❷ ➚❤❨✡❳⑤❪➄❭✽❬✫▲❫❯❝▲❫❳❩❭✹❡✯❢✫➉ C<name> ✩⑧❯❝❨✫❴✫<+node> ✘▲ ✑➟➐ ➎✣❯❝❨✫❴✫<-node> ▲✘✑✡✬➒❯✝❨✫❴✡[<model>]<value>[IC=<initial>] ❙✫❳❩❛✯❡✯▲➝❱❝❨✳♠❖❛❜P♣❛❜♥①③②❛✕❯❝▲✹❬✕❭❫P❘❛❜♥ ✏ ✏ ✏ ❝♥ ❭✹❡✭❙❝▲✘✑ t➈✉ ✬ ✏ ❛❜❯✝❛❜P❘❛✯❭✹❡✚✑ ✬➒♥❝❭✹❡✯❨✕❭❫❳◗▲✹❭✽❣✹❭❫❱❝❭✹❣✹❛❜P❘❢❫Ý♣❛✯❛✕❺✯❯➆➚ ✬➒P♣▲❫❯✥◆❁❛❜❙✫❯❝▲✹❭✽❛❜❯❝❛❜Ý♣❛✯❭✹❡✯❭❖➐❖❺✯❯❽➡❽➐❖❱❝▲✽❣✹❨✡❯❝❴✫▲❫❯✥◆❁❭❫P❘❨✳❳ ✛✯❺✣❯❞❭✹❣✹▲❖◆◗P⑧❣✹❭❄♠✩◆②▲✽④⑤❨✫❡✯❨❉◆❁▲❖③◗P❘▲✽❣✹❨✡❪➄❭❫❯✝❴✫❭r✐ç➔☞➛➅➃✟➑❆❣❫❙▼❨✡❱✫Ý♣❛❜❙✫❯❝▲✹❭✩q ✵➘t✇s⑥➙✍s⑥❏ ✈✗Ù❉▲❫❪❞❱❝❡✯▲✫➉ ✉❇➢☎Ò✟➃➘➜ ✽❾ ✼ ✾ ✮☛✾✡❱✕➚ ✉❇➚⑥➜❐➙ ✵ ✷ ✷★✷à❇ ✉ ➍❀Ò✟➜ ❾❀✾✡❱✕➚①t◗✉ ✬✍❾❖✐ ✼❖♥ ✑❋❊★❊✛❊✙☞ ✑❐⑩➞❸✕â✜✛ ➚❤❨✡❳✇❪①❭✽❬✫▲❫❯❝▲❫❳❩❭✹❡✯❢✫➉ D<name> ✩⑥❯✝❨✫❴✫▲✘✑➟<+node> ➐ ➎✣❯❝❨✫❴✫▲✘✑✡<-node> ✬➒❯❝❨✕❴✡❙✫❳❩❛✯<model>[<area>] ❡✯▲➝❱❝❨✳♠❖❛❜P❘❛✭♥➄③❁❛✕❯❝▲✹❬✫❭❫P♣❛❜♥ ✏ ✏ ❪✏ ①❨✫❴✫▲✹❡✚✑ ✬ ❯✫❙✫❪①▲✹❡✯▲➝❪➄❨✫❴✕▲✹❡❜❙❝❡❜❙❝❛ ➒ ✬ ④✇❭✹❣❫P❘❨✳❳❦❴✫▲✽❭❫❳◗❛✯▲ ➪ ✏ ❭❫❳❩▲✹❭✘✑ ✈✗Ù❉▲❫❪❞❱❝❡✯▲✫➉ ➜✏✉❇➢❦➃✏➍ès✼❾✙➁ ✾à➜➘➍❀Ò✟➜ ➜➸❾ ✷ ✽❾ ✼➪❾✣✢ q✫Þ❲t◗➔✗✉❇➊ ❾❖✐ ✼ ✚✄❊✛❊✛❊✙☞❩✓✝ê⑥❷✹ñ✤r ✛ â⑥Ü➝ã◗Ü✹ì⑥ñ②⑩❘ê✥ì⑥Ü✟❶✹❸✕❹éë❍ì✥â❤ë❥ã✥✛✧✦◗ì▼ã➈Ü✹ì⑥ñ❁⑩➞ê⑥ì✥Ü UC 38 ✂✁☎✄✝✆✟✞✡✠☞☛✍✌✎☛✍✄✏✞✒✑✔✓☞✕ E<name> <+node> <-node> <+control> <-control> <gain> E<name> <+node> <-node> POLY(<value>) < <+control> <-control> >* + < <coeff> >* E<name> <+node> <-node> VALUE={<exp>} E<name> <+node> <-node> TABLE {<exp>} < (inval), (outval) >* E<name> <+node> <-node> LAPLACE {<exp>} {<sexp>} E<name> ✌✘✁☞✙☞<+node> ☛✛✚✢✜ ✌✘✁☞✙✥<-node> ☛✛✚ ✦✧✌✘✁☞FREQ ✙✩★☞✄✏✪✫✑✔☛✭✬✎{<exp>} ✁✩✮✯✪✱✰✲✪✱✳✟✴✵✪✥<✌✘(freq, ☛✒✠☞✞✍✰✲✪✶✳✷magdb, ✞✒✑✫☛✭✬✘✁✩✄✹✸✲✪✫phasedeg) ✪✘✺✒✁✩✆✻✞✍✌✘✙✥✞✍✰✲☛>* ✖✂✗ ✖✤✣ ✖ ✗ ✒✺ ✁☎✌☞✰✔✄✼✁✩✚✽✜ ✖✽✣ ✺✒✁☎✌✥✰✔✄✏✁✥✑✶✚✾✦✧✌✘✁☞✙☎★✥✄✏✪✔✑✫☛✭✬✘✁✿✮✯✪✱✰✲✪✱✳✟✴✵✪✥✌✘☛✒✠☞✞✍✰✲✪✱✳❀✞✒✑✔☛❁✬✘✁☎✄✝✸✲✪✔✪✎✙☞☛❂✺✒✁☎✆✟✞✍✌✘✙☞✓ ✂ ✦✧★✥❃❅❄✲☛❆✜✩✙✥✞✒✺✒✓❈❇✏★☞✄✵❇✵✞✡☛✯❇✏✰✲☛❂✺✒✁☎✆✟✞✒✙☞✞✍✰✲✓❂✑✔✪✱✌✘✪✔✞✍✄ ✖ ✠✥✞✒✪✱✌✥✚ ❉❋❊✡●✽❍❏■ ✖ ✳✘✞✒✑✶★✘☛✛✚✂❑ ✖✾✖ ✺✒✁✥☛✒▲✹▲▼✚✾✚❖◆ ✣ ❇✏★☞✄P❇✵✞❂☛✯❇✏✰✲☛✡✺✒✁✩✆✻✞✍✌✘✙✥✞✍✰✲✓✭✬✎✁☞✑✔✪✱✌✘✁✩✆✻✪✔✞✒✑ ✦◗✙☞✪✱✆✟☛✍✌❘❇✵✪✱★☞✌✘☛✒✞❁✬✘✁☞✑✫✪✱✌✘✁☎✆✷★✘✑✱★✘✪❙■❯❚✒❱❳❲❙❱❩❨❬❑ ✖ ✳✘✞✒✑✱★✎☛✛✚ ✖ ✺✒✁☞☛✒▲✹▲▼✚✾✦◗✺✒✁☞☛✒▲✝✪✔✺✒✪✔☛✍✌☞✸▼✪✔✪✥✬✘✁☞✑✔✪✱✌✎✁☎✆❭★✎✑✱★✘✪ ❪❴❫❵☛✍✆✷✬✘✑✔☛☞✕ ✣ ✬✎✁☞✑✔✪✱✌✘✁✩✆❛✙☞☛❂✙☞✪✱✆✟☛✍✌❘❇✵✪✱★☞✌✘❢ ☛✻❚ ☛❜✦◗✞✛❝ ✗ ✞☎❞❡★✥❃ ✗ ✞✛❢✫★✥❃ ✗❖❣✱❣✱❣ ✣ ✬✎✁☞✑✔✪✱✌✘✁✩✆❛✙☞☛❤✙☞✪✶✆✻☛✍✌❘❇✵✪✱★✥✌✘☛❁❲ ❢ ❢ ❥ ☛❜✦◗✞✛❝ ✗ ✞☎❞❡★✥❢ ❃✐❞ ✗ ✞✛❢✫★✥❃✱❢ ✗ ❢✞✛❥✫★✥❃✲❞ ✗ ❥✞✍❦✐★❵❃✐❞❡★✥❃✱❢ ✗ ✞✒❧❩★✥❃✱❢ ✗ ✞✛♠✫★❵❃✐❞ ✗ ✗ ✞✛♥✫★✥❃✲❞ ★✥❃✱❢ ✗ ✞✒♦❩★✥❃✐❞❡★❵❃❅❢ ✗ ✞✛♣✫★✥❃✱❢ ✗q❣❅❣✱❣ ✣ ✬✘✁☞✑✔✪✶✌✘✁☎✆❛✙☞☛❂✙☞✪✱✆✟☛✍✌❘❇✵✪✱★☞✌✎☛❈❨ ❢ ☛❜✦◗✞✛❝ ✗ ✞☎❢ ❞❡★✥❃✐❞ ✗ ✞✛❢✫★✥❃✱❢ ✗ ✞✛❥✫★❵❃❅❢ ❥ ✗ ✞✍❦✐★✥❃✲❞ ❥ ✗ ✞✒❧❩★✥❃✐❞❡★❵❢ ❃❅❢ ✗ ✞✛♠✫★✥❃✲❞❡★✥❃✱❥ ✗ ✗ ✞ ♥ ★ ❃✱❢ ❢ ✗ ✞ ♦ ★ ❃✱❢ ★ ❃✱❥ ✗ ✞ ♣ ❢ ★ ❃✱❥ ✗ ✞ ❞❅❝ ★ ❃✲❞ ✗ ✞ ❞✐❞ ★ ❃✐❞ ★ ❃✱❢ ✗ ❢ ✗ ✞☎❞❅❢✫★✥❃✲❞ ❥ ★✥❃❅❥ ✗ ✞☎❞❅❥✫★❵❢ ❃✐❞❡★✥❃✱❢ ✗ ✞☎❞❡❦✐★✥❃✐❞❡★❵❢ ❃❅❢✫★✥❃✱❥ ✗ ❚✛r✯❥ ★❵❃✐❞❡★✥❃✱❥ ✗ ✗ ✞☎❞❅♠✫★✥❃✱❢ ✗ ✞☎❞❅♥✫★✥❃✱❢ ★✥❃✱❥ ✗ ✞☎❞✶♦❩★❵❃❅❢✫★✥❃✱❥ ✗ ✞☎❞❅♣✫★✥❃✱❥ ✗q❣❅❣✱❣ s✉t✉●✽✈✉❪❴✦ ✣ ❇✇★✥✄P❇✵✞❂☛✯❇✇✰▼☛✡✁①☛✛❫✿✬☞✄✼☛✯❇✵✪✔☛❂✞✍✌✘✞✒✑✔✪✱✰✲✪✫✺✒✓✡✙☞☛❁✳✘✞✍✄✼✪✔✞✍②✘✪✔✑✫☛✒✑✔☛❁❫☞③ {<exp>}■✫✰✲☛✍✌❘❇✵✪✱★☞✌✎✪✐✜✩✺✍★☞✄✼☛✍✌☞✸✲✪✥✬☞✄✼✪✱✌✟❇✏★☞✄P❇✵☛❂✪✔✙☞☛✒✞✒✑✔☛❂✙☞☛❁✰✲☛✍✌❘❇✵✪✱★☞✌✎☛✯✜✯✰✲✪✱✆✷✬✥❑ ☛✛✦④▲✏■✫❫ ③ ❑❴✜✯⑤✥✦❁❚✛✜⑥❲❘✜ ❣✱❣ ✎★✥✌✘✺✍✸✲✪✔✪✥★☞✰✲✪✫✑✔✪✶✮✯✞✍②✘✪✔✑✫☛✭⑦✔✌❀☛✛❫✿✬✥✄✏☛✯❇✵✪✔✪✇✕ ⑧⑩⑨❘❶❙❷✛❸✝❹▼❺ ❻✘❼✒❽◗❶✎❹✐❾▼❹✐❷✯❺❬❸✝❹▼❺ ABS(x) SQRT(x) EXP(x) LOG(x) LOG10(x) PWR(x,y) PWRS(x,y) SIN(x) COS(x) TAN(x) ATAN(x) |x| x1/2 ex ln x log x |x|y +|x|y (x>0), -|x|y (x<0) ✵❇ ✪✱✌❭❫❀✜❿❫❏⑦✔✌①✄✼✞✒✙☞✪✔✞✍✌✘✪ ✺✒✁❵❇✽❫✻✜❬❫❏⑦✔✌①✄✼✙☞✪✔✞✍✌✘✪ ✰✲✞✍✌❭❫✎✜➀❫❏⑦✔✌①✄✼✞✒✙☞✪✔✞✍✌✘✪ ✞✍✄✏✺✍✰▼✞✍✌①❫❘✜✯⑦✔✌①✄✏✞✒✙✥✪✔✞✍✌✘✪ ✞✍✄✏✺✒✺✍✰▼✞✍✌①❫❘✜✯⑦✔✌①✄✏✞✒✙✥✪✔✞✍✌✘✪ t✉➁❜➂✭➃➄t❂➅❭■✐❫☞❑ 39 ➃➄t✁✭●✽❪ ✚q◆ {<exp>} < (inval), (outval) ✣ ✆✻✓✍✄✼✪✱✆✟☛✒✞✡✙☞☛❂✺✒✁☎✆✟✞✍✌✘✙☞✓❂☛✯❇✏✰✲☛✡☛✛❫☞✬✄✂❀✁①☛✛❫✿✬☞✄✼☛✯❇✵✪✔☛❂✺✒✞✍✄✏☛❂✙☞☛✍✬✘✪✱✌✎✙☞☛✡✙✥☛✡✁ ✰✲☛✍✌❘❇✵✪✱★✥✌✘☛✡✙✥✪✱✌❀✺✒✪✱✄✏✺✍★✎✪✱✰④✜ ✣ ✙☞☛✍✬✘☛✍✌✘✙☞☛✍✌✥✸✲✞✡☛✯❇✏✰✲☛❂☛✛❫✿✬☞✄✼✪✱✆✟✞✍✰✲✓ ✬☞✄✼✪✱✌①✬☞★☞✌✎✺✍✰✲☛✡✙✥☛✡✺✒✁☞✁✩✄✏✙☞✁✩✌✘✞✍✰✲☛ ✖ ■▼✪✱✌☞✳✘✞✒✑✶❑P✜☞■▼✁☎★☞✰✔✳✎✞✒✑✶❑④✚⑩◆❈✞✍✄✼✞✍✆✌ ☎ ✞✍✰✲☛❁⑦✔✌ ✁☎✄✼✙☞✪✱✌✎☛✒✞✡✺✍✄✼☛✯❇✵✺✒✓✍✰✲✁☞✞✍✄✼☛✡✞❁✳✘✞✒✑✔✁☎✄✼✪✔✑✔✁✩✄ ✖ ✪✱✌✥✳✘✞✒✑✶✚✤✜❬★☞✌✘✙☞☛❁✳✘✞✒✑✔✁☎✄✼✪✔✑✔☛ ✖ ✪✶✌☞✳✘✞✒✑✶✚ ❇✏★☞✌☞✰✂✞✒✑✔☛❂✑✱★✘✪ ✖ ☛✛❫✿✬❵✚◗✪✔✞✍✄ ✳✘✞✒✑✫✁☎✄✏✪✫✑✔☛ ✖ ✁☎★☞✰✔✳✎✞✒✑✶✚✢❇✏★☞✌☞✰✂✞✒✑✔☛❁✰✲☛✍✌❘❇✵✪✱★☞✌✘✪✫✪❘❇✏★☞✄✵❇✵☛✒✪❙✜ ✣ ✺✒✁☎✌☞✰✲✪✱✌✥★✘✪✱✰✲✞✍✰✲☛✒✞❂✞✒✺✒☛✯❇✏✰✲☛✒✪✎✙☞☛✍✬✘☛✍✌✘✙✥☛✍✌☞✸✲☛❂☛✯❇✇✰▼☛✡✞✯❇✵✪✔✠☎★✥✄✏✞✍✰✲✓❁✬☞✄✼✪✱✌❀✪✱✌☞✰✲☛✍✄✝✬✘✁☞✑✫✞✍✄✏☛❂✑✔✪✱✌✘✪✫✞✍✄✏✓✯✜ ✪✔✞✍✄④⑦✫✌✷✞✒▲✝✞✍✄✼✞ ✪✱✌☞✰✲☛✍✄✝✳✘✞✒✑✱★✎✑✱★✘✪✥✳✘✞✒✑✔✁☎✄✼✪✔✑✔✁✩✄ ✖ ✪✱✌☞✳✘✞✒✑✶✚✧✰▼☛✍✌❘❇✵✪✱★☞✌✘☛✒✞✉❇✏★☞✄P❇✵☛✒✪✎✪✔✞❁✳✘✞✒✑✔✁☎✄✼✪✎✺✒✁☎✌❘❇✏✰✲✞✍✌☞✰▼☛✡✺✒✁☎✄✼☛✯❇✏✬☞★☞✌✥✮✯✓✍✰▼✁☞✞✍✄✏☛ ✺✒✞✍✬✘☛✍✰▼☛✒✑✔✁☎✄✽✪✱✌✥✰✲☛✍✄✹✳✎✞✒✑✱★✘✑✱★✘✪ ❣ ●✽t✉❉❙●✽t ➂ ❪ {<exp>} ☛{<sexp>} ✺✒✁☎✆✟✞✍✌✘✙☞✓❂☛✯❇✇✰▼☛✡☛✛❫✿✬✝✂❀✁①☛✛❫✿✬☞✄✼☛✯❇✵✪✔☛✡✺✒✞✍✄✼☛❂✙☞☛✍✬✘✪✱✌✘✙✥☛✡✙☞☛❂✁ ✰✲☛✍✌❘❇✵✪✱★☞✌✎☛✡✙☞✪✶✌✷✺✒✪✶✄✏✺✍★✘✪✶✰❋✜ ✣ ✆✟✓✍✄✼✪✱✆✻☛✒✞❂✙☞❂ ✝ ▲ ✒ ✞ ✍ ✺ ✲ ✰ ☎ ✁ ✝ ✄ ✘ ★ ✎ ✑ ☞ ✙ ❁ ☛ ✰ ✔ ✼ ✄ ✞✍✌❘❇✵▲✹☛✍✄✽☛✯❇✏✰✲☛❂✙☞✞✍✰✂✺✒✞❂▲✐★☞✌✎✺✍✸✲✪✔✞ ✖ ❇✵☛✛❫✿✬✥✚◗✙☞☛❁✳✘✞✍✄✼✪✔✞✍②✘✪✔✑✫✞✡✺✒✁☎✆✷✬✘✑✔☛✛❫✘✓❈❇ ❣ ✣ ❘➁✭❪✟✞ ✖ ■✲▲✲✄✼☛✡✠❵✜✯✆✟✞✒✠☞✙☎②❘✜✯✬☞☛✘✞✯❇✵☛✒✙☞☛✒✠✩❑④✚q◆ {<exp>} ✣ ✆✻✓✍✄✼✪✱✆✟☛✒✞✡✙☞☛❂✺✒✁☎✆✟✞✍✌✘✙☞✓❂☛✯❇✏✰✲☛✡☛✛❫☞✬✄✂❀✁①☛✛❫✿✬☞✄✼☛✯❇✵✪✔☛❂✺✒✞✍✄✏☛❂✙☞☛✍✬✘✪✱✌✎✙☞☛✡✙✥☛✡✁ ✲✰ ☛✍✌❘❇✵✪✱★✥✌✘☛✡✙✥✪✱✌❀✺✒✪✱✄✏✺✍★✎✪✱✰④✜ ✣ ▲✹✞✒✺✍✰✲✁✩✄✹★✘✑✎✙☞☛❁✰✔✄✼✞✍✌❘❇✵▲✹☛✍✄❴☛✯❇✇✰▼☛✡✙☞✞✍✰✎✬✘☛✍✌✥✰✔✄✹★❀✞✍✌☞★✥✆✻✪✱✰▼☛✡▲✲✄✏☛✒✺✍✳✎☛✍✌☞✸✲☛ ✬☞✄✼✪✱✌①✳✘✞✒✑✔✁✥✞✍✄✏☛✒✞❁✆✻✁✥✙☎★✘✑✱★✘✑✶★✘✪❙■✔⑦✔✌❀✙✌✭❑ ✴P✪ ▲✝✞✛✮✯☛✒✪❙■✔⑦✔✌❀✠☎✄✼✞✒✙☞☛✛❑ ❣ s ✞✒✑✫✁☎✄✏✪✫✑✔☛✭✬✎☛✍✌☞✰✔✄✝★✷▲✲✄✼☛✒✺✍✳✘☛✍✌☞✸✲☛✒✑✔☛❂✪✱✌☞✰▼☛✍✄✹✆✟☛✒✙☞✪✔✞✍✄✼☛❈❇✵☛❂✁☎②☞✸✲✪✶✌❏✬✥✄✏✪✱✌❀✪✱✌✥✰✲☛✍✄✹✬✎✁☞✑✔✞✍✄✼☛✡✑✔✪✶✌✘✪✔✞✍✄✼✓ ■✫✬✘☛✍✌☞✰✫✄✹★❀▲✝✞✛✮✯✓✛❑❴❇✵✞✍★ ✑✔✁☞✠✥✞✍✄✏✪✱✰✫✆✻✪✔✺✒✓✉■✫✬✘☛✍✌☞✰✔✄✝★①✆✻✁☞✙✩★✘✑✶❑ ❣ ❪❴❫❵☛✍✆✷✬✘✑✔☛☞✕ EBUFF 1 2 10 11 1.0 EAMP 13 0 POLY(1) 26 0 500 ENLIN 100 101 POLY(2) 3 0 4 0 0.0 13.6 0.2 0.005 ESQRT 10 0 VALUE = {SQRT(V(5))} ETAB 20 5 TABLE {10*V(2)}(-5v,5v) (0v,0v) (5v,-5v) E1POLE 10 0 LAPLACE {V(1)} {1 / (1 + s)} EATTEN 20 0 FREQ {V(100)} (0,0,0 10,-2,-5 20,-6,-10) ⑧✎✍✏✍☞✍✒✑ ❻✘✔⑨ ✓✡✕✗✖✙✘❙❼❂❷✒⑨✚✓✒❼✒✜❶ ✛✤❷✡✢☞❽✢❺✩❶✔✘✂❺✣✛✤✖✦✇✥ ❶✟❷✒⑨✔✒✓ ❼✒❶✜✛ ❋✁☎✄✝✆✻✞❂✠☞☛✍✌✘☛✍✄✼✞✒✑✔✓☞✕ ⑧★✧➄❶✂❺✩❽◗✪❼ ✩✫✧✭✬④✔❶ ✢✏✘❘❼✪✩✫✧✭✑✇❶✔✢✏✘❙❼✪✩✫✧✔✮✘❶❋❺✩❽◗❼✪✩✫✧✰✯✘❺❬❹✐❶✜✩ ⑧★✧➄❶✂❺✩❽◗✪❼ ✩✫✧✭✬④✔❶ ✢✏✘❘✪❼ ✩✫✧✭✑✇✔❶ ✢✏✘❙❼✪✩✫✱✟✲✄✳✎✴✶✵✷✧✸❘✮ ✺❺ ✹✐⑨❙✪❼ ✩✸✻✰✧✫✧✸✮✘❶❋❺✩❽◗❼✪✩✫✩★✍✼✧✫✧④❷✡✢☞❼✒❾✝❾✽✩✫✩★✍ ✖✂✗ ✌✘✁☞✙☞☛✛✚✽✜ ✖✤✣ ✌✘✁☞✙✥☛✛✚✾✦✧✌✘✁☞✙☎★☞✄✼✪✔✑✔☛❁✬✘✁✩✮✯✪✶✰✲✪✱✳✟✴✵✪☞✌✎☛✒✠☞✞✍✰✲✪✱✳ ✦➀❇✇★✥✄P❇✵✞✾✙☞☛✻✰✲☛✍✌❙❇P✪✶★☞✌✘☛✾✞✒✑ ✺✒✓✍✄✼☛✒✪ ✺✍★☞✄✼☛✍✌☞✰ ✄✼☛✍✬☞✄✼☛✛✮✯✪✱✌☞✰✲✓✻✆✟✓✍✄✏✪✶✆✻☛✒✞✾✙☞☛ ✖ ✳✥✌✘✞✍✆✻☛✛✚ ✺✒✁✩✆✻✞✍✌✘✙✥✓ ✦◗✪✐❇✏❄✲✪✔✺✉✜✩✺✡✾✍✌✘✙✷❇✏★☞✄✵❇✵✞✡☛✯❇✏✰✲☛❂✺✒✁☎✆✟✞✍✌✘✙☞✞✍✰✲✓❂✑✔✪✱✌✘✪✫✞✍✄ ✖ ✠✥✞✒✪✱✌✥✚ ❉✂❊❂●✤❍ ■ ✖ ✳✎✞✒✑✱★✘☛✛✚✂❑ ✣ ❇✏★☞✄P❇✵✞❂☛✯❇✏✰✲☛✡✺✒✁✩✆✻✞✍✌✘✙✥✞✍✰✲✓✭✬✎✁☞✑✔✪✱✌✘✁✩✆✻✪✔✞✒✑❙■✫✳✘☛✛✮✯✪✎☛✛❫✿✬✘✑✔✪✫✺✒✞✍✸✲✪✔✪✔✑✔☛❁✬✘☛✍✌☞✰✫✄✹★ ❇✏★☞✄P❇✵✞❂✙☞☛❁✰✲☛✍✌❘❇✵✪✱★☞✌✘☛❂✺✒✁☎✆✟✞✍✌✘✙☞✞✍✰✲✓❁⑦✔✌①✰✲☛✍✌❘❇✵✪✱★✥✌✘☛❁❪❁◆☞◆✥◆ ❑ ❪❴❫❵☛✍✆✷✬✘✑✔☛☞✕ ❀✿☞❪④➅❁✿✿❪ ❚ ❲ s❂✿☞❪➄➅❃✿☞❪ ❚❅❄ ❣ ❄ ❘t❇❆ ❉ ❚✛❨ ❄ ❉❋❊✡●✽❍①■ ❚ ❑④s❇❈✲➅ ✌r ❄✏❄ ✘➅ ★ ● ❈✲➅ ❅❚ ❄✏❄✢❚❅❄✤❚ ❉✂❊❂●✽❍❏■✐❲☞❑❋s ❚ ✥✳ ❲ ❄ ❣✝ ❄ ❄ ❣❊❉ ❄ ❣ ❋ ❲ ❄ ❣ ❄✏❵❄ r 40 ❁✍✏✍✏✍✼✑❴❻✎✚⑨ ✓✡✕✗✖✶✘❙❼✡❷✒⑨✔✓✒❼✒❶ ✛✽❷✡☞✢ ❽✢❺✩❶✚✘❋❺✺✛ ✖✦✇✥ ❶❃✛✏❼✒❶✚✵✕ ❹✐⑨❙❶❘❼ ❋✁☎✄✝✆✻✞❂✠☞☛✍✌✘☛✍✄✼✞✒✑✔✓☞✕ G<name> <+node> <-node> <+control> <-control> <gain> G<name> <+node> <-node> POLY(<value>) < <+control> <-control> >* + < <coeff> >* G<name> <+node> <-node> VALUE={<exp>} G<name> <+node> <-node> TABLE {<exp>}= < (inval), (outval) >* G<name> <+node> <-node> LAPLACE {<exp>} <sexp>} <+node> phasedeg) ❪❴❫✿✬✎✑✔✪✔✺✒✞✍✸✲G<name> ✪✔✪✫✑✔☛❈❇✏★☞✌✥✰❋❇✵✪✱✆✟ ✪✔✑✔✞✍✄✼☛❂✺✍★✷✺✒<-node> ☛✒✑✫☛✡✙☞☛❂✑✔✞✉FREQ ❇✏★☞✄P❇✵✞❜❪❁{<exp>}< ◆☞◆☞◆❿■✹❇✏★☞✄✵❇✵✓✡(freq, ✙✥☛✭✰✲☛✍✌❙magdb, ❇P✪✶★☞✌✘☛❂✺✒✁☎✆✻ ✞✍✌✎✙☞✞✍✰✲✓❁⑦✔✌①>* ✰✲☛✍✌❘❇✵✪✱★☞✌✘☛✛❑ ❪❴❫❵☛✍✆✷✬✘✑✔☛☞✕ GBUFF 1 2 10 11 1.0 GAMP 13 0 POLY(1) 26 0 500 GNLIN 100 101 POLY(2) 3 0 4 0 0.0 13.6 0.2 0.005 GSQRT 10 0 VALUE = {SQRT(V(5))} GTAB 20 5 TABLE{V(2)} = (-5v,5v) (0v,0v) (5v,-5v) G1POLE 10 0 LAPLACE {V(1)} {1 / (1 + s)} GATTEN 20 0 FREQ {V(100)}(0,0,0 10,-2,-5 20,-6,-10) ✍✏✍✏✍✼✑❴❻✎⑨✔✓✡✕ ✖✶✘❙❼ ✛✏❼✒❶✔✕P❹✲⑨❘❶❙❼✡❷✡✢✥❽ ❺✿❶✚✘❋❺✺✛ ✖ ✥✏❶✟❷✒⑨✚✓✒❼✒❶✜✛ ✁ ❋✁☎✄✹✆✟✞❂✠☞☛✍✌✘☛✍✄✼✞✒✑✔✓☞✕ ✁❋✧④❶❋❺✩❽◗✪❼ ✩✫✧✸✬④✔ ❶ ✢✏✘❙❼✪✩✫✧✸✑✏❶✚✢☞✘❘❼✪✩✫✧✸✮✘❶✂❺✿❽ ❼✪✩✫✧ ✯❵❺❬❹✐✜❶ ✩ ✁❋✧④❶❋❺✩❽◗✪❼ ✩✫✧✸④ ✬ ✔❶ ✢✏✘❙✪❼ ✩✫✧✸✑✏❶✚✢☞✘❘✪❼ ✩✫✱✟✲✝✳✎✴✙✵✽✧✔✮❘❺✺✹✲⑨❘❼✪✩✭✻ ✧✫✧✸✮✘❶✂❺✿❽ ❼✪✩✫✩ ✍✼✧✫✧④❷✡✢☞❼✒❾✝❾✷✩✫✩ ✍ ❪❴❫✿✬✎✑✔✪✔✺✒✞✍✸✲✪✔✪✫✑✔☛❈❇✏★☞✌✥✰❋❇✵✪✱✆✟✪✔✑✔✞✍✄✼☛❂✺✍★✷✺✒☛✒✑✫☛✭✬✘☛✍✌✥✰✔✄✹★✟❇✏★☞✄✵❇✵✞❁ ◆☞◆☞◆❿■✝❇✇★✥✄P❇✵✞❂✙☞☛✡✺✍★✥✄✏☛✍✌☞✰ ✒✺ ✁✩✆✻✞✍✌✘✙✥✞✍✰✲✓✭⑦✫✌✷✺✍★✥✄✏☛✍✌☞✰✐❑ ❣ ❪❴❫❵☛✍✆✷✬✘✑✔☛☞✕ ✂❂✿✿❪➄➅❁✿✿❪ ❚ ❲ ❂ s ☞✿ ❪④➅❁✿✿ ❪ ✆❚ ❄ ❣ ❄ ✂❈❇ t ❆ ❉ ❚✛❨ ❄ ❉❋❊✡●✽❍❏■❯❚ ❑❋✁ s ✐❈ ➅ r ❄✏❄ ✂✡➅ ★ ● ❈✲➅ ❅❚ ❄✏❄✢❅❚ ❄✤❚ ❉✂❊❂●✤❍①■✫❲✥❑❋s ❚ ✳✥❲ ❄ ❣ ❄✝❄ ❊❣ ❉ ❄ ❣ ❂ ❲ ❄ ❣ ❄✏✘❄ r ✄ ✍✏✍✏✍✼✑❴❻✎⑨✚✓✡✕✗✖❿❹✐❶✔✘❘❼✆☎❋❼✒❶✔✘❘❼✒❶✜✛✤✖✶✘❙❼✡❷✒⑨✔✓✒❼✒❶✜✛ 41 ❣ ❋✁☎✄✝✆✻✞❂✠☞☛✍✌✘☛✍✄✼✞✒✑✔✓☞✕ I<name> <+node> <-node> [[DC] <value>] [AC <mag> [<phase>]] + [ <transient> ] ✌✘✁☞✙☎★✥✄✏✪✔✑✫☛✭✬✘✁✿✮✯✪✱✰✲✪✱✳✟✴✵✪✥✌✘☛✒✠☞✞✍✰✲✪✱✳ <+node>, <-node> = [DC] <value>] = semnal de curent continuu de tip i = <value> AC <mag> [<phase>] = semnal de curent alternativ de tipul ▲✐✄✼☛✒✺✍✳✘☛✍✌☞✸✲✞ i =❇✇<mag> * sin(2 *3.14*f*t), unde f este ✬✎☛✒✺✒✪✔▲✹✪✫✺✒✞✍✰✲✓✭⑦✫✌✷✑✫✪✱✌✘✪✔✞❂✙☞☛❂✺✒✁☎✆✻✞✍✌✎✙☞✓ t ➂ ❣ <transient> = semnal tranzitoriu ✿❵☛✍✆❭✌✘✞✒✑✫☛✒✑✔☛✭✰✫✄✏✞✍✌✥✮✯✪✶✰✲✁☎✄✼✪✔✪✥✬✘✁☎✰✂▲✹✪✎✙☞☛❁✰✲✪✱✬✥★✘✑✤✕ ✣ ☛✛❫☞✬✘✁☎✌✘☛✍✌☞✸▼✪✔✞✒✑ ✂✁✁✱✟✲✵ ❹☎✄◗✝❹ ✆✟✞ ✓✡✘❙✪❼ ✹▼✡❺ ✠ ✓✪✛✇❷ ❾ ❙✘ ✪❼ ✹▼✡❺ ✠ ✷❾ ✏✛ ❷✪✻ i= ❆ ✓✍✄✏✪✶✆✻☛ ✪✵❚ ✪✱✬☞⑤ ✄✏✙✥☛✒✑✔☞✞ ☛ ✄✹✰▼✺ ▲✹✙✥☛✒✑✔☞✞ ☛ ▲✐✰▼✺ i1, 0 < t < rdelay i1 + (ipk-i1){1-exp[-(t-rdelay)/rtc]}, rdelay < t < fdelay ipk + (i1 - ipk){1-exp{-(t-fdelay)/ftc]}, fdelay < t < TSTOP ✿❵☛✍✆✷✌✘✪✔▲✝✪✔✺✒✞✍✸✲✪✔☛ s ✞✒✑✔✁☞✞✍✄✼☛❂✪✱✌✘✪✱✸✲✪✫✞✒✑✔✓ s ✞✒✑✔✁☞✞✍✄✼☛❂✙☞☛✭✜✳ ✾✍✄✏▲ ➃✽✪✶✆❭✬❀✙☞☛❁⑦✔✌☞✷✰ ✾✍✄✇✮✯✪✔☛✍✄✼☛✭✬✎☛✍✌☞✰✔✄✝★✷▲✲✄✼✁☎✌☞✰✔★✘✑✎✺✍✄✼☛✯❇✵✺✒✓✍✰✲✁☎✄ ➂✭✁☎✌❘❇✏✰✲✞✍✌☞✰✲✞❂✙☞☛❁✰✲✪✱✆✷✬①✬✘☛✍✌☞✰✔✄✝★❀▲✐✄✼✁☎✌☞✰✔★✎✑✘✺✍✄✼☛✯❇✵✺✒✓✍✰✲✁☎✄ ➃✽✪✶✆❭✬ ✙✥☛ ⑦✔✌☞✷✰ ✾✍✄✇✮✯✪✔☛✍✄✼☛ ✬✘☛✍✌☞✰✫✄✹★ ▲✐✄✼✁☎✌☞✰✔★✎✑ ✙☞☛✯❇✵✺✍✄✏☛✯❇✵✺✒✓✍✰✲✁☎✄ ➂✭✁☎✌❘❇✏✰✲✞✍✌☞✰✲✞❂✙☞☛❁✰✲✪✱✆✷✬①✬✘☛✍✌☞✰✔✄✝★❀▲✐✄✼✁☎✌☞✰✔★✎✑✘✙✥☛✯❇P✺✍✄✼☛✯❇✵✺✒✓✍✰✲✁☎✄ ✣ ✬✥★✘✑✐❇ ✱✍✼✌ ✳❂❻✎✦✵▼❹☎✄✘❹✑✏✚✛✓✒✰✛✓✔✖✕ ✗✙✘✏✓✛ ✚✜✛✣✢ ✢ ✥✤ ☎ 42 ❋❼✡✓✪✻ ☎ s ✞✒✑✔✁☞✞✍✄✼☛ ☞✬ ✄✼☛✒✙☞☛✒▲✹✪✶✌✘✪✱✰✲✓ ✣ ✣ ❄ ❣❄ ➃ ✿✥➃✤❪❴❉ ✄✼✙☞☛✒✑✔✞☞☛ ✗ ➃ ✿❵➃✤❪❴❉ ➃ ❵✿ ➃✤❪✽❉ ✈❈✌✘✪✱✰▼✞✍✰✲☛✡✙✥☛ ✆✻✓✯❇✏★☞✄✼✓ t t ❇ ❇ ❇ ❇ ❆ ✓✍✄✏✪✱✆✟☛ ✪❞ ✪❢ ✰✁ ✰ ✂☎✄ ✆✞✝ ✰✁✟ ✠☛✡ ✡ ✬✌☞ ✘✬ ☛✍✄ ✿❵☛✍✆❭✌✘✪✫▲✹✪✔✺✒✞✍✸▼✪✔☛ s ✞✒✑✔✁☞✞✍✄✼☛ ✬☞✄✼☛✒✙☞☛✒▲✝✪✱✌✘✪✱✰✲✓ ✣ ✣ ❄ ❣❄ ➃ ✿✥➃✤❪❴❉ ➃ ✿✥➃✤❪❴❉ ➃ ✿✥➃❴❊❂❉ ➃ ✿✥➃❴❊❂❉ s ✞✒✑✔✁☞✞✍✄✼☛✡✪✱✌✎✪✱✸✲✪✔✞✒✑✔✓ s ✞✒✑✔✁☞✞✍✄✼☛✡✙☞☛❁✳ ✾✍✄✼▲ ➃❴✪✱✆✷✬❀✙☞☛✭⑦✫✌☞✷✰ ✾✍✄✇✮✯✪✔☛✍✄✏☛ ➃❴✪✱✆✷✬❀✙☞☛✡✺✍✄✼☛✯✴✏✰✲☛✍✄✼☛ ➃❴✪✱✆✷✬❀✙☞☛✡✙✥☛✯❇P✺✍✄✼☛✯✴✏✰✲☛✍✄✼☛ ✍❈★☞✄✏✞✍✰▼✞✡✪✱✆✷✬☞★✘✑✲❇✇★✎✑✱★✘✪ ❉❋☛✍✄✏✪✫✁☞✞✒✙☞✞ ✣ ✑✫✪✱✌✘✪✔✞✍✄④✬✎☛✭✬✘✁✩✄✹✸✲✪✶★☞✌✘✪ ✱✏✎ ✈✉✌✘✪✱✰✲✞✍✰▼☛✡✙☞☛ ✆✟✓✯❇✏★☞✄✼✓ t t ❇ ❇ ❇ ❇ ❇ ✳✦✵✽✟✛ ✎ ✄ ❹☎✄✜✛ ❙ ✏ ❹✣✏✒✑✓✑☛✺ ✑ ✛✕✔✤❾✖✔ ✻ s ✞✒✑✔✁☎✄✼✪✔✑✔☛❂✪✱✌☞✰▼☛✍✄✹✆✟☛✒✙☞✪✔✞✍✄✼☛❈❇✏★☞✌✥✰❙✁✩②☞✸✲✪✱✌☞★✥✰✲☛✭✬✥✄✏✪✱✌❀✪✱✌✥✰✲☛✍✄✹✬✎✁☞✑✔✞✍✄✼☛✡✑✔✪✶✌✘✪✔✞✍✄✼✓ ❣ ✘ ❾✝❽✫✻ ✣ ❇P✪✶✌☞★❘❇✵✁☞✪✔✙☞✞✒✑✥✆✟✁☞✙☎★✘✑✫✞✍✰✘⑦✫✌✷▲✲✄✼☛✒✺✍✳✘☛✍✌☞✸✲✓✉❻✥⑧❴⑧✘✗ ✵▼❹ ✢☞❾✝❾✂❹▼❺✿❽ ✔✹❙❾✹❷❂❽ ✢✏✟ ☎ i(t) = ioff + iampl*sin[(2*3.14*fc*t)+mod*sin(2*3.14*fm*t)] 43 ❆ ✿❵☛✍✆❭✌✘✪✫▲✹✪✔✺✒✞✍✸▼✪✔☛ ✓✍✄✏✪✱✆✟☛ ✪✔✁☞▲✝▲ ✪✔✞✍✆❭✬✎✑ ▲✹✺ ✆✻✁☞✙ ▲✐✆ s ✒✞ ✑✔✁☞✞✍✄✼☛✡✙☞☛❂✁☞▲✝▲✇❇✵☛✍✰ t✉✆❭✬✘✑✫✪✱✰✔★✘✙☞✪✶✌✘☛✒✞ ❘✄✏☛✒✺✍✳✎☛✍✌☞✸✲✞❁✬☞★☞✄✝✰✲✓✍✰✲✁☞✞✍✄✼☛ ❈✹✌✘✙☞✪✔✺✒☛❂✙☞☛❁✆✻✁✥✙☎★✘✑✔✞✍✄✼☛ ❘✄✏☛✒✺✍✳✎☛✍✌☞✸✲✞✉❇P☛✍✆✷✌✘✞✒✑✱★✎✑✱★✘✪ ❇✵✪✱✌✥★❘❇✵✁☞✪✔✙☞✞✒✑✧❻ ✣ ❈④✦ ❆ ✁✵▼❹ ✢☞❾✝❾✂❹▼❺✿❽ ✔✹❙❾ ✓✒❼✄✂❁✛ ✘ ✘❙❾ ✁ ✄ ☎ ✪✔✁☞▲✝▲✏✜ ❄ ✆☎✂❺ ☎ ✂ ✂ ✕ ❼✪✻ ioff + iampl*exp[-(t-td)*df]*sin[2*3.14*freq*(t-td) + + 3.14*Phase/180] , td < t < TSTOP ✿❵☛✍✆❭✌✘✪✫▲✹✪✔✺✒✞✍✸▼✪✔☛ s ✞✒✑✔✁☞✞✍✄✼☛ ✬☞✄✼☛✒✙☞☛✒▲✝✪✱✌✘✪✱✰✲✓ ✣ ✣ ❚ ❄✲➃ ✿❵➃❴❊✡❉ ❄ ❣❄ ❄ ❣❄ ❄ ❣❄ s ✒✞ ✑✔✁☞✞✍✄✼☛✡✙☞☛❂✁☞▲✝▲✇❇✵☛✍✰ t✉✆❭✬✘✑✫✪✱✰✔★✘✙☞✪✶✌✘☛✒✞ ❘✄✏☛✒✺✍✳✎☛✍✌☞✸✲✞ ➃❴✪✱✆✷✬❀✙☞☛✭⑦✫✌☞✷✰ ✾✍✄✇✮✯✪✔☛✍✄✏☛ ❋✞✒✺✍✰✲✁☎✄✽✙✥☛✡✞✍✆✟✁☎✄✝✰✲✪✶✮✯✞✍✄✼☛ ❋✞✛✮✯✞ ❪❴❫❵☛✍✆✷✬✘✑✔☛☞✕ ❈ ❈✹❂ t ✿ ❚✛❨ ❄ ❲ ❣ ❨✯✆❭t ❈✹t ➂ ❲ ❨❆t ➂ ❣ ❄✏❄✤❚ ❈✹t ➂ ❉ ✝✿①❲ ❨❆t ➂ ❣ ❄✏❄✤❚ ❉ ❄ ❈✝❉❘✈✉✎● ✿✿❪➀❚ ❄ ❉✎✈❿● ✿☞❪ ■ ✣ ❚ ✆✷t ❚ ✆✷t➀❲✿✌❘❇✽❲☞✌❘❇✽❲✿✌❙❇q✌r ❄☎✌❙❇❈❚❅❄✏❄✩✌❘❇✼❑ ❈P❨ ✠❲ ✟☛✡☞✡ ❿➂ ❣ ❄☞❄✩❲❏t ➂ ❚ ✿✏✐❈ ➅✷■ ❣ ❄✏❄✩❲ ❣ ❄✏❄✩❲ ❚ ❣ r✺❆ ❪✍✡✌ ❑ ✂ ✍ J ✈✉✌✘✪✱✰✲✞✍✰▼☛✡✙☞☛ ✆✟✓✯❇✏★☞✄✼✓ t t ❈✮ ✣ ❈✮ ✖ ✰ ✖ ✰✲✙ ✓✍✄✏✪✱✆✟☛ ✪✔✁☞▲✝▲ ✪✔✞✍✆❭✬✎✑ ▲✐✄✼✡☛ ✠ ✰✲✙ ✙☞▲ ✜❉ ☛✘✞✯❇✵☛ s ✞✒✑✔✁☞✞✍✄✼☛ ☞✬ ✄✼☛✒✙☞☛✒▲✝✪✱✌✘✪✱✰✲✓ ✣ ✣ ❚ ❄▼➃ ✿❵➃❴❊✡❉ ✣ ❚ ❄✂➃ ✿❵➃✽❊❂❉ ✑✏✎ ✓✯❺✩❶✒✑ ✷❹ ✕ ✛ ✢✏✓❂❷✒⑨✟❼✒❾✝❼✒❷✪✛ ✘❘❼❂❷✔✓✩❽ ✖✕✥✍ ⑧ ✗✎ ☎ 44 ✈✉✌✘✪✱✰✲✞✍✰▼☛✡✙☞☛ ✆✟✓✯❇✏★☞✄✼✓ t t ❈✮ ❇ ❞ ❇✞✝ ✠☎✄✼✞✒✙☞☛ ✂ ❋✁☎✄✝✆✻✞❂✠☞☛✍✌✘☛✍✄✼✞✒✑✔✓☞✕ J<name> <d> <g> <s> <model> [<area> >] <d> <g> <s> = nodurile corespunzatoare drenei, grilei si sursei <model> = numele modelului <area> = factor de arie Exemple: JIN 100 1 0 JFAST J13 22 14 23 JNOM + 2.0 K*** - bobine cuplate Forma generala: K<name> L<name> < L<name> >* <coupling> K<name> < L<name> >* <coupling> <model> [<size>] L<name>* = numele bobinelor cuplate M <coupling> = factor de cuplaj 0 ≤ K = ∠1 L1 + L2 <model> = numele modelului. Se poate defini bobina, neliniara, cu miez feromagnetic, cu histerezis. <size> = factor de scala pentru aria transversala a bobinei neliniare. Exemple: KTUNED L3OUT L4IN .8 KXFR1 LPRIM LSEC .99 KXFR2 L1 L2 L3 L4 .98 KPOT_3C8 L*** bobina Forma generala: L<name> <+node> <-node> [model] <value> [IC=<initial>] <+node>, <-node> = nodurile pozitiv si negativ 45 [model] = numele modelului <value> = valoare in H ✪✱✌✎✪✱✸✲✪✔✞✒✑✶✚✾✦✧✳✎✞✒✑✔✁☞✞✍✄✼☛✡✪✱✌✎✪✱✸✲✪✔✞✒✑✔✓❁✬✘☛✍✌☞✰✫✄✹★❀✺✍★☞✄✼☛✍✌☞✰✔★✘✑✎✙☞✪✶✌❏②✎✁☎②✘✪✱✌✘✞✉■④❇✵☛❂▲✹✁☞✑✫✁❵❇✵☛✯❇✇✰▼☛✡✺✡✾✍✌✘✙ IC=< ⑦✔✌❀✑✔✪✱✌✘✪✔✞❂✙☞☛❂✺✒✁☎✆✟✞✍✌✘✙☞✞❂➃➄➁ t❂➅ ❇✵☛❈❇✏✬✘☛✒✺✒✪✔▲✝✪✔✺✒✞✙✿✁❇❈✹❉✔✭❉❙❑ ❣ Exemple: LLOAD 15 0 20mH L2 1 2 .2e-6 LCHOKE 3 42 LMOD .03 LSENSE 5 12 2uH IC=2mA M ***- tranzistor MOSFET. Forma generala: M<name> <d> <g> <s> <sub> <mdl> [L=<value>] [W=<value>] +[AD=<value>] [AS=<value>] [PD=<value>] [PS=<value>] +[NRD=<value>] +[NRS=<value>] [NRG=<value>] [NRB=<value>] ✺✒✁☎✄✼☛✯❇✏✬☞★☞✌✥✮✯✓✍✰▼✁☞✞✍✄✏☛❂✙☎✄✼☛✍✌✘☛✒✪✐✜✩✠✩✄✏✪✔✑✫☛✒✪ <d>, <g> <s>, <sub>= nodurile , ✴ sursei i substratului ✖ ✆✻✙✥✑✶✚ ✦✧✌✥★☞✆✻☛✒✑✫☛✭✆✟✁☞✙☞☛✒✑✱★✎✑✱★✘✪ ● ✜✄✂ ✦◗✑✱★☞✌✘✠✥✪✱✆✻☛✒✞✉✴✵✪✎✑✔✓✍✸✲✪✱✆✟☛✒✞✡✺✒✞✍✌✎✞✒✑✱★✘✑✱★✘✪ t ✜⑥t❂❏ ✿ ✦◗✞✍✄✼✪✔✪✔✑✫☛✡✙☞☛❂✙☞✪✔▲✲★✥✮✯✞✍✄✼☛✡✞✒✑✫☛✡✙☎✄✼☛✍✌✘☛✒✪❙✴✵✪❙❇✇★✥✄P❇✵☛✒✪ ❉ ✜❀ ❉ ❏ ✿ ✦✧✬✘✞✍✄✼✞✍✆✻☛✍✰✔✄✼✪✔✪ ✪ ☎ ✁☎✌✎✺✍✸✲✪✱★☞✌✘✪✫✪ ➅ ➁ ✜➅ ❇ ➁ ✿✎✜ ➅ ✗ ➁ ✌❿✜ ➅ ➁ ✦✧✌☞★☞✆✟☛✍✄✏☛✒✑✫☛✡✙☞☛❁✬✘✓✍✰✔✄✼✞✍✰✲☛❂☛✒✆✺ ☛✘✪✱✳✘✞✒✑✫☛✍✌☞✰✲☛❂✞✒✑✔☛✡✙✥✪✔▲✐★❵✮✯✪✔✪✔✑✔✁☎✄❴✙☎✄✏☛✍✌✎☛✒✪✐✜ ❇✏★☞✄✵❇✵☛✒✪✐✜✩✠☎✄✼✪✔✑✔☛✒✪❙✴✵✪❙❇✏★☞②❘❇✏✰✔✄✼✞✍✰✔★✘✑✱★✘✪ ❣ ✍ ✍ ✍ Exemple: M1 14 2 13 0 PNOM L=25u W=12u M13 15 3 0 0 PSTRONG M2A 0 2 100 100 PWEAK L=33u w=12u AD=288p AS=288p PD=60u PS=60u + NRD=14 NRS=24 NRG=10 Q*** - tranzistor bipolar Q< < name> > <c> > <b> > <e> > [< <subs> >] <model> > [< <area> >] <c> <b> <e> = nodurile corespunzatoare colectorului si bazei <subs>= nod al substratului ( optional ) <model> = numele modelului <area> = factor de arie Exemple: 46 Q1 14 2 13 PNPNOM Q13 15 3 0 1 NPNSTRONG 1.5 Q7 VC 5 12 [SUB] LATPNP R*** - rezistor Forma generala: R<name> <+node> <-node> [<model>] <value> <+node>, <-node> = nodurile pozitiv si negativ <model> = numele modelului <value> = valoarea rezistentei in Ω Exemple: RLOAD 15 0 2k R2 1 2 2.4e4 R*** N1 N 2 <VALUE> <MNAME> <L= LENGTH> <W = WIDTH> S*** - intrerupator controlat in tensiune Forma generala: S<name> <+node> <-node> ✌✎✁☞<+control> <-control> <model> ✙☎★☞✄✼✪✔✑✔☛❁✬✘✁✩✮✯✪✱✰▼✪✱✳✟✴P✪✥✌✘☛✒✠✥✞✍✰✲✪✱✳❀✞✒✑✔☛❂✪✱✌☞✰✔✄✼☛✍✄✹★✥✬✘✞✍✰✲✁☎✄✝★✘✑✱★✘✪ <+node>, <-node> = ✖❋✗ ✺✒✁☎✌✥✰✔✄✏✁✥✑✶✚✤✜ ✖✽✣ ✺✒✁✩✌☞✰✔✄✼✁☞✑✶✚✾✦✧✌✘✁☞✙✩★☞✄✏✪✫✑✔☛✭✬✎✁✩✮✯✪✱✰✲✪✱✳✟✴✵✪✥✌✘☛✒✠☞✞✍✰✲✪✶✳✷✞✒✑✫☛✭✰✲☛✍✌❙❇P✪✶★☞✌✘✪✔✪✎✙☞☛ ✓ command Uc <model> = numele modelului Exemple: S12 13 17 2 0 SMOD SRESET 5 0 15 3 RELAY T*** - linie de transmisie ✓ Forma general : T<name><A+><A-><B+><B->Z∅ ∅=<value>[TD=<val> | F=<val>[NL=<val>]] 47 ✌✘✁☞✙✩★☞✄✏✪✫✑✔☛✭✬✎✁☎✄✹✰▼✪✔✑✔✁☎✄✡❚❜✴✵✪❵❲ <A+>, <A-> = ✦◗✪✱✆❭✬✎☛✒✙☞✞✍✌☞✸✲✞❂✺✆☛✘✞✍✄✼✞✒✺✍✰✲☛✍✄✏✪✲❇✇✰▼✪✔✺✒✓ ∅ ➃✏✍❆✦✧⑦✔✌✥✰✷✾✍✄✝✮✯✪✫☛✍✄✏☛✒✞❁✰✔✄✼✞✍✌❘❇✏✆✻✪✐❇✵✪✔☛✒✪ ✧✦◗▲✐✄✼☛✒✺✍✳✘☛✍✌☞✸✲✞ ➅ ● ✦ ✑✶★☞✌✘✠☞✪✱✆✟☛✒✞❿☛✒✑✔☛✒✺✍✰✫✄✏✪✔✺✒✓ ✞❿✑✔✪✱✌✘✪✫☛✒✪✐✜☎★☞✌✘✙☞☛✉✑✱★☞✌✘✠✥✪✱✆✻☛✒✞✉▲✹✪❩✮✯✪✔✺✒✓✡●❴✦✎➅ ●❁◆ ✴✵✪ ☛✯❇✇✰▼☛❈✑✱★✘✠✥✪✱✆✻☛✒✞ λ λ ✙✥☛✭★☞✌✎✙☞✓✡✞✉❇✵☛✍✆❭✌✎✞✒✑✱★✘✑✱★✘✪✎✙☞☛❂▲✲✄✏☛✒✺✍✳✘☛✍✌✥✸✲✓❁ ■✎✳✘✞✒✑✔✁☞✞✍✄✼☛✒✞❁✬☞✄✏☛✒✙✥☛✒▲✹✪✱✌✎✪✱✰✲✓❜✦ ❄ ❣ ❲❘r❬❑ ❪❴❫❵☛✍✆✷✬✘✑✔☛☞✕ T1 1 2 3 4 Z0=220 TD=115ns T2 1 2 3 4 Z0=50 F=5MEG NL=0.5 ✖ V*** - sursa independent de tensiune ✓ Forma general : V<name> <+node> <-node> [[DC] <value>] [AC <mag> [<phase>]] + [ <transient> ] ❪❴❫✿✬✎✑✔✪✔✺✒✞✍✸✲✪✔✪✫✑✔☛❈❇✏★☞✌✥✰❋❇✵✪✱✆✟✪✔✑✔✞✍✄✼☛❂✺✍★✷✺✒☛✒✑✫☛✡✙☞☛❂✑✔✞✉❇✏★☞✄P❇✵✞❂✪✱✌✘✙☞☛✍✬✎☛✍✌✘✙☞☛✍✌☞✰✲✓❂✙☞☛❂✺✍★☞✄✼☛✍✌☞✰ I***. Exemple: VBIAS 13 0 2.3mV VAC 2 3 AC .001 VACPHS 2 3 AC .001 90 VPULSE 1 0 + PULSE(-1mV 1mV 2ns 2ns 2ns 50ns 100ns) V3 26 77 DC .002 AC 1 + SIN(.002 .002 1.5MEG) W*** - ❹✐❶✜✛ ✓✒❼✡✓✒⑨ ✰✖✣✛ ✢✏✓❂❷✡✢☞❶✜✛ ✓✡✢ ✹▼✺❺ ✛✜✥✇❶✧❷✒⑨✚✓✡✓✒❼✒❶✜✛ ☎ ✓ Forma general : W<name>✌✘✁✥<+node> ✙☞☛✛✚✤✜ ✖✽✣ ✌✎✁☞<-node> ✙☞☛✛✚✾✦✧✌✘✁☞✙✩<vname> ★☞✄✏✪✫✑✔☛✭✬✎✁✩✮✯✪✱✰✲<model> ✪✱✳✟✴✵✪✥✌✘☛✒✠☞✞✍✰✲✪✶✳✷✞✒✑✫☛✭⑦✔✌✥✰✔✄✏☛✍✄✝★☞✬✘✓✍✰▼✁☎✄✹★✎✑✱★✘✪ ✖❋✗ ✖ ✳☞✌✘✞✍✆✟☛✛✚ ✦✧✌✥★☞✆✻☛✒✑✫☛❈❇✏★☞✄✵❇✵☛✒✪✘✙✥☛✭✰✲☛✍✌❙❇P✪✶★☞✌✘☛❂✞✒✑✘✺✒✓✍✄✼☛✒✪✎✺✍★☞✄✼☛✍✌☞✰✎✄✏☛✍✬☞✄✼☛✛✮✯✪✱✌☞✰▼✓✭✆✟✓✍✄✏✪✶✆✻☛✒✞❂✙☞☛ ✺✒✁☎✆❭✆✟✞✍✌✘✙☞✓ ✖ ✆✟✁☞✙☞☛✒✑✶✚ ✦✧✌✥★☞✆✻☛✒✑✫☛✭✆✟✁☞✙☞☛✒✑✱★✎✑✱★✘✪ ❣ Exemple: W12 13 17 VC WMOD WRESET 5 0 VRESET RELAY X*** - subcircuit 48 Forma generala: X<name> [<node>]* <sname> [PARAMS: <<par>=<val>*>] <node>* = nodurile la care este conectat subcircuitul <sname> = numele subcircuitului. Acesta este definit asfel: SUBCKT <sname> [<node>]* .......... ENDS <sname> Examples: X12 100 101 200 201 DIFFAMP XBUFF 13 15 UNITAMP 3. Liniile de comand ✁✄✂ ☎✑ ✁ ❶❋✺❺ ✹✔❹ ✩✦ ❺ ✥✇❶✟❷✒⑨✔✓✒❼✒✜❶ ✛❴❺✺✹ ✛✏❼✡✓✒❶✂✣❺ ✛✹❹ ✮ ✑ ✓ Forma general : .AC [LIN][OCT][DEC] <points> <start> <end> ❇✵☛✭✳✎✞✍✄✏✪✔✞✛✮✯✓❂▲✲✄✏☛✒✺✍✳✘☛✍✌✥✸✲✞✡✑✫✪✱✌✘✪✔✞✍✄✵✜✯✬✘☛❂✁☞✺✍✰✲✞✍✳✘☛✉❇✵✞✍★①✬✘☛✡✙✥☛✒✺✒✞✒✙☞☛ ❣ [LIN], [OCT], [DEC] ✬✘✁☞✪✱✌✥✰✹❇✼✚✾✦✧✌☞★☞✆✟✞✍✄✹★✘✑✎✙☞☛❁✬☞★☞✌✎✺✍✰✲☛❈■✫✰✲✁☎✰▼✞✒✑☞✬✎☛✍✌☞✰✔✄✝★❏✳✎✞✍✄✏✪✔✞✍✌✥✰✲✞❁★ ● ❈✲➅❭✜✯✬✎☛✡✁☞✺✍✰▼✞✍✳✘✓✭✬✎☛✍✌☞✰✔✄✝★❏✳✎✞✍✄✏✪✔✞✍✌✥✰✲✞ ✖ ❊ ➂✭➃ ✴✵✪✥✬✘☛❂✙☞☛✒✺✒✞✒✙☞✓❁✬✘☛✍✌☞✰✔✄✝★①✳✘✞✍✄✼✪✔✞✍✌☞✰✲✞ ❈❪⑩➂ ❑❋⑦✔✌❀✺✒✞✍✄✼☛❈❇✵☛❂☛✒▲✹☛✒✺✍✰✫★✘☛✒✞✛✮✯✓✡✞✍✌✎✞✒✑✔✪✶✮✯✞❁t ➂ ❈ ✖ ❇✏✰✲✞✍✄✹✰✐✚✾✦◗▲✐✄✼☛✒✺✍✳✘☛✍✌☞✸✲✞❂✪✱✌✘✪✶✸✲✪✔✞✒✑✔✓ ✖ ☛✍✌✘✙✩✚✾✦◗▲✲✄✏☛✒✺✍✳✎☛✍✌☞✸✲✞❂▲✹✪✱✌✎✞✒✑✔✓ ✍ Example: .AC LIN 101 10Hz 200Hz .AC OCT 10 1KHz 16KHz .AC DEC 20 1MEG 100MEG ✆✝✂ ✣ ✁ ❶❋✺❺ ✹✔❹ ✩❺✦✥✇❶✟❷✒⑨✔✓✒❼✒❶✜✛✤❷✡✢☞❶✜✛✹❹✐❶❙⑨❙⑨ ✑ ✓ Forma general : .DC [LIN] <varname> <start> <end> <incr> [<nest>] .DC [OCT][DEC] <varname> <start> <end> <points> [<nest>] .DC <varname> LIST ✌☞★☞✆✟<value>* ☛✒✑✔☛❂✠☞☛✍✌✘☛✍✄✼✞✍[<nest>] ✰✲✁☎✄✝★✘✑✱★✘✪✎✪✱✌✘✙✥☛✍✬✘☛✍✌✘✙☞☛✍✌☞✰✂✙☞☛❁✰✲☛✍✌❘❇✵✪✱★✥✌✘☛❈❇✵✞✍★❀✺✍★☞✄✼☛✍✌☞✰✂✞ <varname> = ✺✒✞✍✄✝★✘✪✥✳✘✞✒✑✔✁☞✞✍✄✼☛❁✳✘✞✍✄✏✪✫✞✛✮✯✓ [LIN], ✠✥☛✍✌✘☛✍✄✏✞✍✰▼[OCT], ✁☎✄✹★✎✑✱★✘✪ [DEC] ✦✧✳✘✞✍✄✏✪✫✞✍✸✲✪✔☛❂✑✔✪✱✌✘✪✔✞✍✄✼✓✯✜✯✬✘☛❂✁☞✺✍✰✲✞✍✳✎☛❈❇✵✞✍★①✬✘☛❂✙☞☛✒✺✒✞✒✙☞☛❂✞✭✳✘✞✒✑✫✁☎✄✏✪✫✪ ✖ ✏❇ ✰✲✞✍✄✹✰✐✚✤✜ ✖ ☛✍✌✘✙✩✚✾✦✧✳✘✞✒✑✔✁✩✄✏✪✔✑✫☛✡✪✱✌✎✪✱✸✲✪✔✞✒✑✔☛✉✴✵✪✎▲✹✪✱✌✎✞✒✑✔☛✭✬✎☛✍✌☞✰✔✄✝★❏✳✎✞✒✑✔✁☞✞✍✄✼☛✒✞✡✠☞☛✍✌✎☛✍✄✏✞✍✰✲✁✩✄✹★✘✑✶★✘✪ ✖ ✬✘✁☞✪✱✌✥✰✹❇✼✚✾✦✧✌☞★☞✆✟✓✍✄✹★✘✑✎✙☞☛❁✬☞★☞✌✎✺✍✰✲☛✭✬✎☛✍✌☞✰✔✄✝★❏✳✎✞✒✑✔✁☎✄✼✪✔✑✔☛❂✠☞☛✍✌✘☛✍✄✼✞✍✰✲✁☎✄✝★✘✑✱★✘✪✥✬✘☛✍✌☞✰✫✄✹★❀✺✒✞✍✄✼☛❈❇✵☛ ▲✝✞✒✺✒☛❂✞✍✌✘✞✒✑✔✪✶✮✯✞ ❿➂ ✖ ✪✱✌✘✺✍✄✇✚✾✦✧✬✘✞✯❇✏★✘✑✎✙☞☛❂✪✱✌✘✺✍✄✼☛✍✆✻☛✍✌☞✰▼✞✍✄✏☛❁✬✘☛✍✌☞✰✔✄✝★①✳✘✞✒✑✔✁☞✞✍✄✼☛✒✞❂✠☞☛✍✌✘☛✍✄✼✞✍✰✲✁☎✄✝★✘✑✱★✘✪ ✖ ✌✘☛✯❇✏✰✫✚✾✦◗✞✒✑✱✰✂✠☞☛✍✌✘☛✍✄✼✞✍✰✲✁☎✄❴✪✱✌✘✙☞☛✍✬✘☛✍✌✎✙☞☛✍✌☞✰✂✞✒✑✎✺✒✞✍✄✹★✘✪✥✬✘✞✍✄✼✞✍✆✟☛✍✰✔✄✹★✟❇✵☛❁✳✘✞✍✄✏✪✫✞✛✮✯✓ ● ❈ ✿✥➃✢✦◗✞✍✌✘✞✒✑✔✪✶✮✯✞ ❿➂ ✬✘☛✍✌☞✰✫✄✹★❀✁①✑✔✪✐❇✏✰✲✓❂✙☞☛❁✳✘✞✒✑✔✁☎✄✼✪ ✖ ✳✘✞✒✑✱★✘☛✛✚◗✞✒✑✔☛❁✳✘✞✒✑✔✁✩✄✏✪✔✪ ★ ✠☞☛✍✌✎☛✍✄✏✞✍✰✲✁✩✄✹★✘✑✶★✘✪ ✖ ✳✘✞✍✄✝✌✘✞✍✆✟☛✛✚ ✍ ✍ Exemple: 49 .DC VIN -.25 .25 .05 .DC LIN I2 5mA -2mA 0.1mA .DC VCE 0v 10v .5v + IB 0mA 1mA 50uA .DC RES RMOD(R) 0.9 1.1 .001 .DC DEC NPN QFAST(IS) + 1e-18 1e-14 5 .DC TEMP LIST 0 20 27 50 80 .DC PARAM RS -1 1 0.1 .DC SRCNAM VSTART VSTOP VINCR < SRC2 START2 STOP2 INCR2 > ❻✘❾✞✓✌✓✁✵❹ ✛✏⑨ ❙✹ ❾▼❹✂✵❹✐❼✡✓✒⑨✜✹✐⑨✘❹✔✘❙❼❁❹✲❶ ✛ ✓✯❺✌✓✒❼✦✱q❻ END ❋✁- ☎✄✝✆✻✞❂✠☞☛✍✌✘☛✍✄✼✞✒✑✔✓☞✕ ✱ ✄ ✂ . END .ENDS - ❻✘❾✞✓✌✁✓ ✵❹ ✛✏⑨ ✹ ✺ ✑ ✢☞❶❙❼✛❹✔✘❘❼✒✘❙❼✒❾✲❹✐❶✎❹✔❸✝❹✐❼✉❺ ⑨❙❶❙⑨✘❹✔✕ ⑨☎✄❋❷✛❹ ✓✒❷✒⑨✎❹ ✛ ❋✁☎✄✝✆✻✞❂✠☞☛✍✌✘☛✍✄✼✞✒✑✔✓☞✕ .ENDS [<name>] <name> = numele subcircuitului Exemplu: .ENDS 741 .FOUR -Analiza Fourier ✓ Forma general : .FOUR <freq> <output ▲✲✄✏☛✒✺✍✳✘☛✍✌✥✸✲✞✡var>* ✺✒✁✩✆❭✬✘✁✩✌✘☛✍✌☞✰✲☛✒✪✎▲✲★☞✌✘✙☞✞✍✆✟☛✍✌☞✰✲✞✒✑✔☛ <freq> = ✖ ✁☎★☞✰✔✬✥★☞✰❴✳✘✞✍✄✝✚q◆❭✦❖✳✘✞✍✄✏✪✫✞✍②✘✪✔✑✔☛✒✑✔☛ ✬✘☛✍✌✥✰✔✄✹★ ✺✒✞✍✄✼☛✷❇✵☛❏✙✥✁☎✄✏☛✯✴✏✰✲☛①✞✍✌✘✞✒✑✫✪✶✮✯✞ ✂✁☎★✥✄✏✪✔☛✍✄ ❱✂✞✒✺✒☛✒✞✯❇✏✰✲✓①✞✍✌✘✞✒✑✔✪✶✮✯✓ ❇✵☛❁✬✘✁☞✞✍✰✲☛❂▲✝✞✒✺✒☛✭✌☞★✥✆✻✞✒✪✥✬✘✑✔☛✒✺✡✾✍✌✎✙❏✙✥☛✡✑✔✞❂✙☞✞✍✰✲☛✒✑✫☛✡✺✒✞✒✑✔✺✍★✎✑✔✞✍✰✲☛❂✺✍★✷✞✍✌✎✞✒✑✔✪✶✮✯✞❁⑦✔✌①✄✏☛✒✠☞✪✶✆ ✰✔✄✼✞✍✌✥✮✯✪✱✰✲✁☎✄✼✪✱★ ❣ Exemplu: .FOUR 10KHz v(5) v(6,7) .IC - Conditii initiale pentru regimul tranzitoriu Forma generala: .IC < <vnode> = <value> >* ✪✱✌✘✪✶✸✲✪✔✞✒✑✔☛ <value> <vnode> = <value> - se atribuie nodurilor < vnode> valorile Exemplu: .IC V(2)=3.4 V(102)=0 .INC - Includerea unui fisier ✓ Forma general : .INC <name> ▲✝✪✐✴✵✪✔☛✍✄✝★✘✑✱★✘✪ , inclusiv calea <name> = numele Exemple: 50 .INC SETUP.CIR .INC C:\\PSLIB\\VCO.CIR .LIB - Utilizarea unei biblioteci de modele si subcircuite ✓ Forma ✌✘✞✍✆✟☛✛✚ general ✦ ✌☞★☞:✆✟.LIB ☛✒✑✔☛ ✑✔[<name>] ✪✱②✥✄✏✞✍✄✼✪✔☛✒✪✐✜ ✪✱✌✘✺✒✑✱★❘❇✵✪✱✳ ✺✒✞✒✑✫☛✒✞ ✍❿✞✒✺✒✓ ✖ ✬✥✄✏✪✱✌✎✺✒✪✱✬✘✞✒✑✔✓ ❣ ✑✔✪✶✬❘❇✵☛✯✴✇✰▼☛✯✜ ❇✵☛ ★☞✰✲✪✫✑✔✪✶✮✯☛✒✞✛✮✯✓ ②✘✪✱②✎✑✔✪✔✁☎✰✲☛✒✺✒✞ NOM.LIB Exemple: .LIB .LIB OPNOM.LIB .LIB C:\\PSLIB\\QNOM.LIB .MC -Analiza Monte Carlo ✓ Forma general : .MC <#runs> [DC][AC][TRAN] <opvar> <func> <option>* <#runs> = numar de simulari Monte Carlo ✓ [DC] [AC] [TRAN] = tipul de analiz ✙☞☛❂✪✔☛✯✴✵✪✱✄✼☛✭✬✘☛✍✌✥✰✔✄✹★❀✺✒✞✍✄✼☛❈❇✵☛❂▲✹✞✒✺✒☛❂✞✍✌✘✞✒✑✔✪❩✮✯✞ ❆ ✁☎✆✷✰✲☛❂➂✭✞✍✄✏✑✫✁ <opvar> ▲✲★☞✌✘✺✛✚✾✦◗=▲✲★☞marimile ✌✘✺✍✸✲✪✔✞❂✞✍✬✘✑✔✪✫✺✒✞✍✰✲✓✭✳✎✞✒✑✔✁☎✄✼✪✔✑✔✁☎✄④✆✟✓✍✄✼✪✱✆✻✪✫✑✔✁☎✄✽✙☞☛❂✪✔☛✯✴✵✪✱✄✼☛✭✬✎☛✍✌☞✰✔✄✝★✷✞❂✁☎②☞✸▼✪✱✌✘☛ ✖ ✁✷❇✵✪✱✌✘✠☎★✥✄✏✓❁✳✘✞✒✑✔✁☞✞✍✄✼☛❂✺✒✞✍✄✏✺✒✞✍✰✲☛✍✄✼✪✐❇✏✰✲✪✔✺✒✓ ❣ t ✺✒☛✯❇✏✰✲☛✒✞❁✬✘✁☎✰✂▲✹✪✤✕ ❇ ❍ ❆❖t✁❏✜ ❪ ✍ ✡❪ ■✫✳✘✞✒✑✱★✘☛✛❑P✜❬❘t✉●❴● ❪✘✍ ✡❪ ✫■ ✳✘✞✒✑✱★✘☛✛❑ ✂ ✖ ✁✩✬☞✰✫✚✾✦◗✙☞✪✱✳✎☛✍✄P❇✵☛❂✁☎✬☞✸✲✪✱★✥✌✘✪✎✺✒✞✍✄✏☛✉✂❇✵☛✭✬✘✁✩✰❋❇✵☛✍✰✲✞ ❆❖t✁❏✜✺❆ ❈✲➅❭✜❬➁ ❈ ✿☞❪ ✌ ✌ : [LIST], .PLOT, .PROBE, ALL, FIRST <n> , EVERY <n> , RUNS <n> . Exemple: .MC 10 TRAN V(5) YMAX .MC 50 DC IC(Q7) MIN LIST .MC 20 AC VP(13,5) RISE_EDGE(1.0) LIST OUTPUT ALL .WCASE -Wort Case Analysis ✓ Forma general : .WCASE <analysis> <opvar> <func> <option>* Exemple: .WCASE DC V(4,5) YMAX .WCASE TRAN V(1) FALL EDGE(3.5v) VARY BOTH BY RELTOL DEVICES RL .MODEL - Model. Forma generala: .MODEL <name> <type> [<param>=<value> [<tol>]]* <name> = numele modelului <type> = tipul dispozitivului <name> C*** L*** R*** D*** Q*** Q*** <type> CAP IND RES D NPN PNP Tip condensator bobina rezistor dioda tr.bipolar NPN tr.bipolar PNP 51 Q*** J*** J*** M*** M*** B*** K*** S*** W*** N*** O*** LPNP NJF PJF NMOS PMOS GASFET CORE VSWITCH ISWITCH DINPUT DOUTPUT tr.lateral PNP JFET canal N JFET canal P MOSFET canal N MOSFET canal P GaAsFet canal N miez neliniar ⑦✔✌✥✰✔✄✏☛✍✄✝★☞✬✘✞✍✰▼✁☎✄✽✺✒✁☎✌☞✰✫✄✏✁☞✑✫✞✍✰✘⑦✫✌❏✰▼☛✍✌❘❇✵✪✱★☞✌✘☛ ⑦✔✌✥✰✔✄✏☛✍✄✝★☞✬✘✞✍✰▼✁☎✄✽✺✒✁☎✌☞✰✫✄✏✁☞✑✫✞✍✰✘⑦✫✌✷✺✍★✥✄✏☛✍✌☞✰ dispozitiv digital de intrare dispozitiv digital de iesire ✚✢✴✵✪ <param> parametri la valorile <value ☛✍✳✘☛✍✌☞✰✔★✘✞✒✑✎=✺✍★①<value> ✰✲✁☞✑✫☛✍✄✏✞✍✌☞✸▼[☛✒✑✔<☛ tol>] ✰✲✁☞✑✶✚✧-⑦✫setarea ✌❏✬✥✄✏✁☞✺✒☛✍✌✥anumitor ✰✲☛ ✖ ✈✉✄✹✆✟☛✒✞✛✮✯✓❂✑✔✪✐❇✏✰✲✞❁✬✘✞✍✄✏✞✍✆✟☛✍✰✔✄✼✪✔✑✔✁☎✄❴✺✡✾✍✰✲✁☎✄✝✳✘✞✭✰▼✪✱✬☞★☞✄✼✪✎✙☞☛✡✙✥✪✐❇✏✬✘✁✩✮✯✪✱✰✲✪✶✳✘☛❈✴✵✪✎✞✍✌☞★☞✆✟☛☞✕ ** pentru rezistor Nume R TC 1 TC 2 TCE R*** ✺✒✁☞☛✒▲✝✪✔✺✒✪✔☛✍Parametru ✌✥✰❙✙✥☛✡✞✍✆✷✬✘✑✔✪✔▲✝✪✔✺✒✞✍✄✼☛ ✬✘☛✍✌☞✰✫✄✹★①✄✼☛✛✮✯✪✐❇✏✰✲☛✍✌☞✸✲✞ ✺✒✁☞☛✒▲✝✪✔✺✒✪✔☛✍✌☞✰✂✑✔✪✱✌✎✪✔✞✍✄✽✙☞☛ ✰✲☛✍✆✷✬✘☛✍✄✼✞✍✰✔★☞✄✼✞ ✺✒✁☞☛✒▲✝✪✔✺✒✪✔☛✍✌☞✰✎✬✘✓✍✰✔✄✼✞✍✰✲✪✔✺❂✙☞☛ ✰✲☛✍✆✷✬✘☛✍✄✼✞✍✰✔★☞✄✼✞ ✺✒✁☞☛✒▲✝✪✔✺✒✪✔☛✍✌✥✰❙☛✛❫☞✬✘✁☎✌✘☛✍✌☞✸▼✪✔✞✒✑✎✙☞☛ ✰✲☛✍✆✷✬✘☛✍✄✼✞✍✰✔★☞✄✼✞ Unitate Val. predefinita 1.0 _ C-1 0 0.0 0 0.0 C-2 0 /0 - 0 C 0.0 <valoare> = RNOM* R * [ 1+ TC1( T - TNOM) + TC2( T - TNOM)2] sau TCE ( T - TNOM ) <valoare> = ✦✧ RNOM ✳✘✞✒✑✔✁☞✞✍✄✼☛✒*✞❁R* ✄✏☛✛✮✯✪✐1❇✏✰✲.01 ☛✍✌✥✸✲☛✒✪✎✑✔✞✭✰▼☛✍✆❭✬✘☛✍✄✼✞✍✰✔★☞✄✼✞❂➃✂➅①❊ ❆ RNOM ➃❋➅①❊ ❆ ✦✧✰✲☛✍✆❭✬✎☛✍✄✏✞✍✰✔★✥✄✏✞❁✌✘✁☎✆✟✪✱✌✘✞✒✑✔✓ ❣ ❉✂✁☞✞✍✰▼☛✡▲✝✪❙❇✇✬✎☛✒✺✒✪✔▲✹✪✫✺✒✞✍✰✲✓✭⑦✫✌✷✑✫✪✱✌✘✪✔✞❂✙☞☛❂✺✒✁☎✆❭✆✟✞✍✌✘✙☞✓ ❣ ✡ ❊ ❉❋➃ ❈✏❊✭➅❁✿ ** pentru condensator C*** Nume C VC1 VC2 TC1 Parametru coeficient de amplificare pentru capacitate coeficient liniar de tensiune coeficient patratic de tensiune ✓ coeficient liniar de temperature 52 Unitate - Val.predefinita 1.0 V-1 V-2 0 -1 C 0.0 0.0 0.0 coeficient patratic de temperature TC2 ✓ 0 C-2 0.0 <valoare> = CNOM.C.(1+VC1.ϑ ϑ✓✍✸✲+VC ϑ 2).[1+TC1(T-TNOM)+TC2(TTNOM)2] ✪✔✪ 2.ϑ CNOM = valoarea capacit ✓ la temperatura TNOM ⑦ ✓ TNOM = valoarea nominal a temperaturii. Se poate specifica n linia de comand .OPTIONS ϑ= tensiunea pe condensator T = temperatura ** pentru bobina L*** Nume L ✺✒✁☞☛✒▲✹✪✫✺✒✪✔☛✍✌☞✰✂✙☞Parametru ☛❂✞✍✆❭✬✘✑✫✪✔▲✹✪✫✺✒✞✍✄✏☛❁✬✘☛✍✌☞✰✔✄✝★ ✪✱✌✘✙☎★✎✺✍✸✲✞✍✌☞✰✲☛ ✺✒✁☞☛✒▲✝✪✔✺✒✪✔☛✍✌☞✰✂✑✔✪✱✌✎✪✔✞✍✄✽✙☞☛❂✺✍★☞✄✼☛✍✌☞✰ ✺✒✁☞☛✒▲✝✪✔✺✒✪✔☛✍✌☞✰✎✬✘✓✍✰✔✄✼✞✍✰✲✪✔✺❂✙☞☛❂✺✍★☞✄✼☛✍✌☞✰ ✺✒✁☞☛✒▲✝✪✔✺✒✪✔☛✍✌✥✰❙✑✫✪✱✌✘✪✔✞✍✄✽✙✥☛✭✰✲☛✍✆✷✬✘☛✍✄✼✞✍✰✔★☞✄✼✞ ✺✒✁☞☛✒▲✝✪✔✺✒✪✔☛✍✌✥✰✘✬✎✓✍✰✔✄✏✞✍✰▼✪✔✺✡✙✥☛✭✰✲☛✍✆✷✬✘☛✍✄✼✞✍✰✔★☞✄✼✞ IL1 IL2 TC1 TC2 Unitate - Val.predefinita 1.0 A-1 A-2 0 -1 C 0 -2 C 0.0 0.0 0.0 0.0 2 2 <valoare> = LNOM*L*(1+IL ).[1+TC (T-TNOM)+TC 2(T-TNOM) ] ✪✱✌✘✙☎★✘✺✍✰▼✞✍1✌☞.i+IL ✸✲☛✒✪✎✑✔✞❁2.i✰✲☛✍✆❭ ✬✎☛✍✄✏✞✍✰✔★✥✄✏1✞❂ ➃❋➅❏❊ ❆ LNOM ➃❋➅①❊ ❆ =✦✧valoarea ✰✲☛✍✆❭✬✎☛✍✄✏✞✍✰✔★✥✄✏✞❁✌✘✁☎✆✟✪✱✌✘✞✒✑✔✓ ✪❵✦◗✺✍★☞✄✼☛✍✌☞✰✔★✎✑☞✬✥✄✏✪✱✌①②✘✁✩②☞②✘✪✱✌✘✓ T = temperatura ** ✬✎☛✍✌☞✰✔✄✝★❇✥✏❶ ✛ ✓✒❼✡✓✒⑨ ➅❿★☞✆✟☛ ☎ ❉ ✞✍✄✼✞✍✆✻☛✍✰✫✄✏✪✔✪ ✂ ➁❜☛✛✮✯✪✐❇✏✰✲☛✍✌✥✸✲✞✂✁✒✁✩✌☎✄ ➁❜☛✛✮✯✪✐❇✏✰✲☛✍✌✥✸✲✞✂✁✒✁✥▲✹▲✆✄ ✰✲☛✍✌❘❇✵✪✱★☞✌✘☛✒✞❂✙☞☛❂✺✒✁☎✌☞✰✔✄✼✁☞✑✥✬✘☛✍✌☞✰✫✄✹★✟❇✏✰✲✞✍✄✼☛✒✞ ✁✒✁✩☎✌ ✄ ✰✲☛✍✌❘❇✵✪✱★☞✌✘☛✒✞❂✙☞☛❂✺✒✁☎✌☞✰✔✄✼✁☞✑✥✬✘☛✍✌☞✰✫✄✹★✟❇✏✰✲✞✍✄✼☛✒✞ ✁✒✁✥▲✹✆▲ ✄ ✙✲ ✁✲①⑧✽⑧ ✝✶✲ ✝✙✲①⑧✽⑧ ◆✥◆❁✬✎☛✍✌☞✰✔✄✝★❇✥✏❶ ✛ ✓✒✡❼ ✓✒⑨ ➅❿★☞✆✟☛ ✙✲ ✁✲①⑧✽⑧ ✗✲ ✗✲①⑧❴⑧ ✄ ✄ ✖✺✛ ✢✏❂✓ ❷✡✢☞❶ ✛ ✓✡✢✌✹▼❺✣✛✜✥✇❶❃✛✏❼✒❶✚✵✕ ❹✐⑨❙❶❙❼ ✭❻ ✍✏✍✏✍ ☎ ✖✺✛ ✢✏✓❂❷✡✢☞❶ ✛ ✓✡✢✌✹▼❺✣✛✜✥✇❶✟❷✒⑨✔✓✒❼✒❶ ✛ ✎ ✈❈✌✘✪✶✰✲✞✍✰✲☛ s s ✒✞ ✑ ❣ ✬☞✄✼☛✒✙☞☛✒▲✝✪✱✌✘✪✱✰✲✞ ❚ ❣❄ ❚❪ ✗ ❚ ❣❄ s ❄ ❣❄ ✈❈✌✘✪✶✰✲✞✍✰✲☛ s ✞✒✑ ❣ ✬☞✄✼☛✒✙☞☛✒▲✝✪✱✌✘✪✱✰✲✞ ❚ ❣❄ ❚❪ ✗ ❚❪ ✣❨ ❄ ❣❄ Ω Ω ✟ ✍☞✍✏✍ ❉ ✞✍✄✼✞✍✆✻☛✍✰✫✄✏✪ ✂ ➁❜☛✛✮✯✪✐❇✏✰✲☛✍✌✥✸✲✞✂✁✒✁✩✌☎✄ ➁❜☛✛✮✯✪✐❇✏✰✲☛✍✌✥✸✲✞✂✁✒✁✥▲✹▲✆✄ ✺✍★✥✄✏☛✍✌☞✰✂✙☞☛❂✺✒✁☎✌☞✰✫✄✏✁☞✑✥✬✘☛✍✌✥✰✔✄✹★✟❇✏✰✲✞✍✄✼☛✒✞✂✁✒✁✩✌☎✄ ✺✍★✥✄✏☛✍✌☞✰✂✙☞☛❂✺✒✁☎✌☞✰✫✄✏✁☞✑✥✬✘☛✍✌✥✰✔✄✹★✟❇✏✰✲✞✍✄✼☛✒✞✂✁✒✁✥▲✹▲✆✄ 53 Ω Ω t t ✟ ◆✥◆❁✬✎☛✍✌☞✰✔✄✝★❋✎✘ ❹ ✢✏❋ ✘❺ ➅❿★☞✆✟☛ ❻ ❻ ✄ ✎ ✎ ✗✕ ✲ ✝ ✕ ✂ ✗ ✁ ✄ ✎ ✂ ☎❭⑧ ❿⑧ ⑧ ✁ ✂ ✆ ✝ ✝✆ ✝ ✄ ❂✍✏✍✏✍ ✆ ❉✂✞✍✄✼✞✍✆✻☛✍✰✫✄✏✪ ✍✺ ★☞✄✼☛✍✌☞✰✂✙☞☛✉❇✵✞✍✰✔★☞✄✼✞✍✸✲✪✔☛ ➁❜☛✛✮✯✪✲❇✇✰▼☛✍✌☞✸✲✞❂✁✣☛☞✆✟✪✔✺✒✞ ✺✒✁☞☛✒▲✹✪✫✺✒✪✔☛✍✌☞✰✂✙☞☛❂☛✍✆✻✪✐❇✵✪✔☛ ✰✲✪✱✆✷✬❀✙☞☛❁✰✔✄✏✞✍✌❵✮✯✪✱✰ ✺✒✞✍✬✎✞✒✺✒✪✱✰✲✞✍✰✲☛✒✞ ☎ ✁☎✌✘✺✍✸▼✪✱★☞✌✘✪✔✪ ✬✎✁☎✰✲☛✍✌☞✰✲✪✫✞✒✑✱★✘✪✑ ☎ ✁☎✌✘✺✍✸▼✪✱★☞✌✘✪✔✪ ✺✒✁☞☛✒▲✹✪✫✺✒✪✔☛✍✌☞✰✔★✘✑✎✙☞☛❂✠☎✄✼✞✒✙☞✞✍✄✼☛ ☛✍✌✘☛✍✄✏✠✥✪✔☛✡✙✥☛✡✞✒✺✍✰✲✪✶✳✘✞✍✄✏☛ ☛✛❫✿✬✎✁☎✌✘☛✍✌☞✰✂✙☞☛❁✰✲☛✍✆✷✬✘☛✍✄✏✞✍✰✫★☞✄✏✞❂✞✒✑✎✺✍★☞✄✼☛✍✌☞✰✔★✘✑✱★✎✪ ✙☞☛❈❇✵✞✍✰✔★✥✄✏✞✍✸✲✪✫☛ ✺✒✁☞☛✒▲✹✪✫✺✒✪✔☛✍✌☞✰✂✙☞☛❜✮✯✠☞✁☎✆✟✁☎✰✂▲✹✑✔✪✫✺✍⑤✘☛✍✄ ☛✛❫✿✬✘✁☎✌✎☛✍✌☞✰✂✙☞☛❁✮✯✠✥✁☎✆✻✁✩✰❙▲✝✑✔✪✔✺✍⑤✎☛✍✄ ✺✒✁☞☛✒▲✝✪✔✺✒✪✔☛✍✌☞✰✎✬✘☛✍✌☞✰✫✄✹★❀▲✝✁☎✄✹✆✷★✘✑✔✞❂✺✒✞✍✬✘✞✒✺✒✪✱✰▼✓✍✸✲✪✔✪ ⑦✔✌①✬✘✁☞✑✔✞✍✄✼✪✶✮✯✞✍✄✼☛✡✙☞✪✶✄✏☛✒✺✍✰✲✓ ✰▼☛✍✌❘❇✵✪✱★☞✌✘☛✒✞❂✙☞☛✉❇✇✰✫✄✏✓✍✬☞★✥✌✘✠☞☛✍✄✼☛ ✺✍★☞✄✼☛✍✌☞✰✔★✘✑✎✑✔✞❁✰✲☛✍✌❘❇✵✪✱★✥✌✘☛✒✞✡✙✥☛❈❇✏✰✔✄✼✓✍✬☞★☞✌✘✠✥☛✍✄✏☛ ✈❈✌✘✪✶✰✲✞✍✰✲☛ t s ✞✒✑ ❣ ✬☞✄✼☛✒✙☞☛✒▲✝✪✱✌✘✪✱✰✲✞ ❚ ❣ ❄✩❪ ✣ ✁❚ ❄ ❣❄ ❚ ❣❄ ❄ ❣❄ ❄ ❣❄ ❚ ❣❄ ❄ ❣r ❚ ❣ ❚☞❚ ❨ ❣❄ Ω ✿ ✣ s ✣ ☛✍s ✣ ❄ ✣ ✣ ❄ ✣ s ❚ ❣r ❚ ❣ ❄✩∞❪ ✣ ❨ t ❉❋✞✍✄✼✞✍✆✻☛✍✰✔✄✼✪✔✪✥✆✟✁☞✙☞☛✒✑✔☛✒✑✔✁✩✄✽✞✒✑✱✰✲✁☎✄❴✙☞✪✐❇✏✬✘✁✩✮✯✪✱✰▼✪✱✳✘☛✉❇✇★✥✌☞✰✎✬☞✄✏☛✛✮✯☛✍✌✥✰✲✞✍✸✲✪✥⑦✔✌❀✙☞☛✍✰✲✞✒✑✔✪✶★❏⑦✫✌❏②✎✪✱②✘✑✔✪✔✁✥✠☎✄✏✞✒▲✝✪✔☛ ❣ Exemple: .MODEL RMAX RES (R=1.5 TC=.02 TC2=.005) .MODEL QDRIV NPN (IS=1e-7 BF=30) .MODEL DLOAD D (IS=1e-9 DEV 5% LOT 10%) ✥✏❽ ❋⑨❙❶✂✺❺ ✛ ✖❬❸✝❹ ✓✒❼✯❺ ❷✡✢☞❶ ✮✘❼✡✓✡✯☞❼✒❶✎❸✇❼✛❹✎❹ ✛✏❼✡✓✯❺❬❸✝❹✔❹ ✹ ✢✏✗ ✓ ❿❼☞✁✤ ✛ ✥✢ ❶ ❋✁☎✄✝✆✻✞❂✠☞☛✍✌✘☛✍✄✼✞✒✑✔✓☞✕ .NODESET ✄ ✑✆ ❺ ✆☎✔✕ ✥✢ ❶ - Setarea valorilor potentialelor nodurilor ✥✇❶ ❷✒⑨✚✒✓ ❼✒❶✜✛ ❷✡✢☞❶ ✝✛ ❹✐❶❙⑨❘⑨ ❋❼✒❶ ✛ ✓✒⑨ ☎ ☎ .NODESET < <node> = <value> >* <node> = <value> - nodurilor <node> li se atribuie valorile <value>.* Exemplu: .NODESET V(2)=3.4 V(3)=-1V .NOISE - Analiza zgomotului ✓ Forma general : .NOISE <opvar> <name> [<ival>] ✺✒☛✒✞✯❇✏✰✲✓✟✞✍✌✘✞✒✑✔✪❩✮✯✓✾❇✵☛✷✬✘✁☞✞✍✰✲☛ pentru zgomotul ; a ★✎✑✱✰✲✪✔✑✔✪❩✮✯✞✭✌☞★✥✆✻✞✒✪✥⑦✔<opvar> ✆✷✬☞✄✼☛✍★☞✌✘✓❂=✺✍★✷variabila ✞✍✌✎✞✒✑✔✪✶✮✯✞❁⑦✔✌❀ ✺✍★☞✄✏☛✍✌✥care ✰❙✞✒✑✶✰✲☛✍se ✄✹✌✎✞✍analizeaza ✰✲✪✱✳ t ➂ ❣ ✖ ✌✘✞✍✆✟☛✛✚✾✦✧✌☞★☞✆✟☛✒✑✔☛✡✠✥☛✍✌✘☛✍✄✏✞✍✰▼✁☎✄✹★✎✑✱★✘✪✎✪✱✌✘✙☞☛✍✬✎☛✍✌✘✙☞☛✍✌☞✰✂✙☞☛❁✰✲☛✍✌❘❇✵✪✱★☞✌✎☛❈❇✵✞✍★❀✺✍★☞✄✼☛✍✌☞✰✂✺✒✞✍✄✏☛ ✺✒✁☎✌❘❇✏✰✲✪✱✰✫★✘✪✔☛✉❇✇★✥✄P❇✵✞❂✙☞☛❁✮✯✠✥✁☎✆✻✁✩✰ ✖ ✪✱✳✘✞✒✑✶✚✾✦ ▲✲✄✏☛✒✺✍✳✎☛✍✌☞✸✲✞❁✬✘☛✍✌☞✰✔✄✝★❀✺✒✞✍✄✏☛✉❇✵☛✭✰✲✪✶✬✘✞✍✄✏☛✯❇✏✰✲☛✉❇✏★☞✬✘✑✔✪✶✆✻☛✍✌☞✰▼✞✍✄✽✺✒✁☎✌☞✰✔✄✼✪✱②☞★✥✸✲✪✔✞❂▲✹✪✔☛✒✺✒✓✍✄✼☛✒✪ ❇✏★☞✄✵❇P☛❂✑✔✞❜✮✯✠☞✁☎✆✟✁☎✰✔★✘✑✥✰✲✁✩✰✲✞✒✑✎✞✒✑✘✺✒✪✶✄✏✺✍★✘✪✶✰✔★✘✑✱★✘✪ ❣ 54 ✱⑩⑨❙❶❘❷✪✛ ✕ ✛P❺✺✛✝❹✐❷ ✘❙❼❂❾✹⑨❙❶❘❷✛❸✝❹ ✢☞❶❋❺✌✒✓ ❼ ✥✇❶✟❷✒⑨✔✓✒❼✒❶ ✛✽❷✡✢☞❶ .OP ✓ Forma general : .OP tinuu .OPTIONS - Optiuni ✓ Forma general : .OPTIONS [<fopt>*] [<vopt>=<value>*] ✸ <fopt>,<vopt> = op iuni care nu cer sau cer atribuirea unor valori <value> ✁✄✂ ✂ ✎ Acestea✑❴sunt: ❇✵☛✭✰▼✪✱✬✘✓✍✄✼☛✯❇P✺❁✰✲✪✱✆✷✬✘✪✎✙☞☛❂✺✒✞✒✑✔✺✍★✘✑ ✂✁✁✱ ✁ ✆ ✣ ✪✱✌✘✙☞✪✔✺✒✓❂☛✒✑✔☛✍✆✟☛✍✌☞✰✲☛✒✑✔☛❂✙☞✪✱✌✟❇✏★☞②✘✺✒✪✶✄✏✺✍★✘✪✶✰✲☛❈✴✵✪✥✬✘✞✍✄✼✞✍✆✻☛✍✰✫✄✏✪✔✪✎✞✒✺✒☛✯❇✏✰✲✁☎✄✼✞ ✳ ✝✆ ✁ ✒✴ ✣ ✑✔✪✐❇✏✰✲☛✒✞✛✮✯✓❂✑✔✪✱✌✎✪✔✪✔✑✔☛❂✙☞✪✱✌❀▲✝✪✐✴✵✪✔☛✍✄✏☛❂✞✒✑✔☛❁②✘✪✱②✘✑✫✪✔✁☎✰✲☛✒✺✒✪✔✑✫✁☎✄ ✳ ✍❻✠✎ ✣ ✑✫✪✐❇✏✰✲☛✒✞✛✮✯✓✡✙✥✞✍✰✲☛✒✑✔☛❂✙☞☛✡✪✶✌☞✰✔✄✼✞✍✄✏☛ ✙✲ ✆ ✣ ✰✲✪✶✬✘✓✍✄✏☛✯✴✏✰✲☛❁✰✲✞✍②✘☛✒✑✶★✘✑✥✌✘✁☞✙☎★☞✄✼✪✔✑✔✁✩✄ ✙✲ ✂ ❃✲ ✣ ❇✏★☞✬☞✄✼✪✱✆✟✓✡✑✫✪✐❇✏✰✲✞✍✄✏☛✒✞ ✙✲ ✲ ✆ ✣ ❇✏★☞✬☞✄✼✪✱✆✟✓✡✑✫✪✐❇✏✰✲✞✍✄✏☛✒✞❁✬✘✞✍✄✼✞✍✆✻☛✍✰✫✄✏✪✔✑✫✁☎✄④✆✟✁☞✙☞☛✒✑✱★✘✑✶★✘✪ ✙✲✄✱ ✁ ✣ ❇✏★☞✬☞✄✼✪✱✆✟✓✭✬✘✞✒✠✥✪✱✌✘✞✍✄✼☛✒✞✡✞✍★☞✰▼✁☎✆✻✞✍✰▼✓ ✲✝✱✍✎❜❻ ✣ ✰▼✪✱✬✘✓✍✄✼☛✯✴✇✰▼☛✭✳✘✞✒✑✫✁☎✄✏✪✎✞✒✑✔☛❂✙☞✪✔▲✝☛✍✄✼✪✱✰✲☛✒✑✔✁☎✄❴✁☎✬☞✸✲✪✱★✥✌✘✪ ✁ ✆❂❻ ✎ ✲✝✳ ✦ ✖ ✳✎✞✒✑✱★✘☛✛✚ ✣ ❇✵✺✆✘☛ ✪✱✆✷②✘✓❁✳✘✞✒✑✔✁☞✞✍✄✼☛✒✞❂☛✍✄✏✁☎✄✼✪✔✪✎✞✍②❘❇✵✁☞✑✱★✥✰✲☛✭✬✎☛✍✌☞✰✔✄✝★✷✺✍★✥✄✏☛✍✌☞✰ ❣ s ✄ ✄ ✁ ✗ ✂ ✂✒✱✍✎ ✆ ✁ ✆ ✄ ✭⑧ ✆ ✗ ✭⑧ ❏ ✁ ❻❭✦ ✆ ✭⑧★✳ ✆ ✭⑧ ✦ ✦ ✎ ✁❆✦ ✗ ✄ ✎ ✳✄✂ ✦ s ✞✒✑✔✁☞✞✍✄✼☛✒✞ ✄ ✎ ✳✆☎①✦ ✖ ✳✎✞✒✑✱★✘☛✛✚ ✎ ✳✆✝①✦ ✖ ✳✎✞✒✑✱★✘☛✛✚ ✄ ✄ ✬☞✄✼☛✒✙☞☛✒▲✝✪✱✌✘✪✱✰▼✓✡☛✯❇✏✰✲☛✻❚ ✬☞t ✖ ✳✘✞✒✑✱★✘☛✛✚ ✣ ❇P✺✆☛✎✪✱✆❭②✎✓✭✳✘✞✒✑✫✁☞✞✍✄✏☛✒✞❂☛✍✄✼✁☎✄✏✪✫✪✘✞✍②❙❇P✁✥✑✱★☞✰✲☛❁✬✘☛✍✌☞✰✔✄✝★✟❇✵✞✍✄✏✺✒✪✱✌✎✓✍❄❙▲✝✑✱★✥❫ ❣ s ✞✒✑✔✁☞✞✍✄✼☛✒✞ ✬☞✄✏☛✒✙✥☛✒▲✹✪✱✌✎✪✱✰✲✞❂☛✯❇✇✰▼☛❀❚ ❣ ❄✩❪ ✣ ✁❚ ✦ ✖ ✳✘✞✒✑✶★✘☛✛✚ ✣ ▲✝✪✶❫❵☛✒✞✛✮✯✓❁✳✘✞✒✑✔✁☞✞✍✄✼☛✒✞✭✆✟✞✛❫❵✪✱✆✟✓✉■✎⑦✔✌✟❇✵☛✒✺✍★☞✌✘✙☞☛✛❑✤✞❂➂ ❉❘✈❆✰✲✪✱✆✟☛ ✦ ✖ ✳✘✞✒✑✱★✎☛✛✚ ✣ ❇✵✆✺ ☛✘✪✶✆❭②✘✓❁✳✘✞✒✑✔✁✥✞✍✄✏☛✒✞❂✞✍✄✏✪✫☛✒✪✘✙✥☛✡✙☞✪✫▲✐★✥✮✯✞✍✄✼☛❂✞✡✙☎✄✼☛✍✌✘☛✒✪✥✬✘☛✍✌☞✰✫✄✹★❀✙☞✪✐❇✏✬✘✁✿✮✯✪✱✰✲✪✱✳✘☛✦❆ ❊✶✿ ❣ s ✞✒✑✔✁☞✞✍✄✼☛✒✞✭✬☞✄✼☛✒✙☞☛✒▲✝✪✱✌✘✪✱✰▼✓✡☛✯❇✏✰✲✒ ☛ ❄ ❣❄ ✘ ✳ ✒ ✞ ✱ ✑ ✘ ★ ✛ ☛ ✚ ✵ ❇ ✺ ✆ ✘ ☛ ✱ ✪ ✷ ✆ ✘ ② ❁ ✓ ✘ ✳ ✒ ✞ ✔ ✑ ☞ ✁ ✍ ✞ ✼ ✄ ✒ ☛ ✡ ✞ ✍ ✞ ✼ ✄ ✔ ✪ ✒ ☛ ✎ ✪ ☞ ✙ ❂ ☛ ☞ ✙ ✔ ✪ ✲ ▲ ✥ ★ ✮✯✞✍✄✼☛✡✞✉❇✏★☞✄P❇✵☛✒✪✥✬✘☛✍✌☞✰✫✄✹★❀✙☞✪✐❇✏✬✘✁✿✮✯✪✱✰✲✪✱✳✘☛✦❆ ✶ ❊ ✿ ❣ ✖ ✣ s ✞✒✑✔✁☞✞✍✄✼☛✒✞❁✬☞✄✏☛✒✙✥☛✒▲✹✪✱✌✎✪✱✰✲✓❂☛✯❇✇✰▼☛ ❄ ❣ ❄ ✳✖ ✘✞✒✑✶★✘☛✛✚ ✣ ❇✵✆✺ ☛✘✪✱✆✷②✘✓✭✳✎✞✒✑✔✁☞✞✍✄✼☛✒✞✡✑✱★✥✌✘✠☞✪✱✆✟✪✔✪✎✺✒✞✍✌✘✞✒✑✱★✘✑✶★✘✪✥✬✘☛✍✌☞✰✔✄✝★❀✙☞✪✐❇✏✬✘✁✩✮✯✪✱✰▼✪✱✳✘☛✦❆ ✶ ❊ ✿ ❣ s ✞✒✑✔✁☞✞✍✄✼☛✒✞ ❉❘✄✏☛✒✙✥☛✒▲✹✪✱✌✎✪✱✰✲✓❂☛✯❇✇✰▼☛❀❅❚ ❄☞❄ ❣ ❄ ✆ ❣ µ ✘ ✳ ✒ ✞ ✱ ✑ ✘ ★ ✛ ☛ ✚ ❊ ✿ ❣ s ✞✒✑✫✁☞✞✍✄✏☛✒✞ ✖ ✣ ❇✵✆✺ ☛✘✪✱✆✷②✘✓✭✳✎✞✒✑✔✁☞✞✍✄✼☛✒✞✡✑✔✞✍✄✼✠☞✪✱✆✟✪✔✪✎✺✒✞✍✌✘✞✒✑✱★✎✑✱★✘✪✥✬✘☛✍✌☞✰✔✄✝★❀✙☞✪✐❇✏✬✘✁✩✮✯✪✶✰✲✪✱✳✘☛✦❆ ✶ ✬☞✄✼☛✒✙☞☛✒▲✝✪✱✌✘✪✱✰▼✓✡☛✯❇✏✰✲☛✻❅❚ ❄✏❄ ❣ ❄ ✆ ❣ µ ✖ ✳✘✞✒✑✱★✘☛✛✚ ✣ ❇✵✆✺ ☛✘✪✱✆✷②✘✓✭✳✎✞✒✑✔✁☞✞✍✄✼☛✒✞✡✑✱★✎✪ ✌✒❆ ❈✐➅✷✜✩✺✒✁☎✌✘✙✩★✘✺✍✰✲✞✍✌☞✸✲✞❁✆✟✪✱✌✘✪✱✆✟✓✡✞✒✺✒✺✒☛✍✬✥✰✲✞✍✰✲✓❂✞✭★☞✌✎☛✒✪ ✑✔✞✍✰✔★✥✄✏✪ ❣ s ✞✒✑✔✁☞✞✍✄✼☛✒✞❁✬☞✄✏☛✒✙✥☛✒▲✹✪✱✌✎✪✱✰✲✓❂☛✯❇✇✰▼☛❀❚ ❣ ❄ ❪ ✣ ❚ ❲ ❣ ✖ ✳✘✞✒✑✶★✘☛✛✚ ✣ ❇✵✆✺ ✘☛ ✪✱✆✷②✘✓✢✌✥★☞✆✻✓✍✄✝★✘✑✡✆✻✞✛❫❵✪✶✆ ✞✒✙✩✆✻✪✐❇✟✙✥☛ ✪✱✰✲☛✍✄✼✞✍✸✲✪✔✪✡✬✘☛✍✌☞✰✔✄✝★ ✙☞☛✍✰▼☛✍✄✹✆✟✪✱✌✘✞✍✄✼☛✒✞✢✬ ❣ ❇ ❣ ▲ ❣ ✲✝✳ ✁ ✎ ✁ ✎ ✳✟✞ ✦ ✖ ✳✘✞✒✑✱★✘☛✛✚ ✰✫✄✏✞✍✌✥✮✯✪✶✰✲✁☎✄✼✪✱★✂❱ ✄ ✳ ✗ ✄ ✗ ✗✎ ✞✒✑✔✁✥✞✍✄✏☛✒✞ ✬☞✄✼☛✒✙☞☛✒▲✝✪✱✌✘✪✱✰✲✓❂☛✯❇✏✰✲☛✻❚✛r✌❄ ❣ ✣ ❇✵✆✺ ☛✘✪✶✆❭②✘✓❁✌☞★☞✆✟✞✍✄✝★✘✑✥✆✻✞✛❫❵✪✶✆❛✞✒✙☎✆✟✪✐❇ ✙☞☛❂✞✍✬☞✄✼✁✩❫❵✪✱✆✟✞✍✰✲✪✔✪✎✪✱✌✎✪✱✸✲✪✔✞✒✑✔☛❁✬✘☛✍✌☞✰✫✄✹★❀✞✍✌✘✞✒✑✔✪❩✮✯✞✭⑦✔✌ ✺✍★☞✄✏☛✍✌✥✰❙✺✒✁✩✌☞✰✲✪✱✌☞★✥★✧❱☎✳✘✞✒✑✫✁☞✞✍✄✏☛✒✞❁✬☞✄✼☛✒✙☞☛✒▲✝✪✱✌✘✪✱✰✲✓❂☛✯❇✏✰✲☛❜❲ ❄ ❣ ✣ ❇✵✆✺ ☛✘✪✶✆❭②✘✓❁✌☞★☞✆✟✞✍✄✝★✘✑✥✆✻✞✛❫❵✪✶✆❛✞✒✙☎✆✟✪✐❇ ✙☞☛❂✪✱✰✲☛✍✄✼✞✍✸✲✪✔✪✥✬✘☛✍✌✥✰✔✄✹★❀▲✝✪✔☛✒✺✒✞✍✄✼☛✭✬☞★✥✌✘✺✍✰✎⑦✔✌❀✺✒✞✒✙☎✄✹★✎✑ ✞✍✌✎✞✒✑✔✪✶✮✯☛✒✪✥✄✏☛✒✠✥✪✱✆❭★✎✑✱★✘✪✥✰✔✄✼✞✍✌✥✮✯✪✱✰✲✁☎✄✼✪✱★✾❱☎✳✘✞✒✑✔✁☞✞✍✄✼☛✒✞❁✬☞✄✏☛✒✙✥☛✒▲✹✪✱✌✎✪✱✰✲✓❂☛✯❇✇✰▼☛❀❅❚ ❄ ❣ ✣ ❇✵✆✺ ✘☛ ✪✱✆✷②✘✓ ✌☞★✥✆✻✞✍✄✝★✘✑◗✆✟✞✛❫❵✪✱✆ ✰✲✁✩✰✲✞✒✑ ✞✒✙☎✆✟✪✐❇❤✙☞☛ ✪✱✰✲☛✍✄✼✞✍✸✲✪✔✪ ⑦✔✌ ✺✒✞✒✙✩✄✹★✘✑ ✄✼☛✒✠☞✪✱✆✷★✘✑✱★✘✪ ✳✘✞✒✑✔✁✥✞✍✄✏☛✒✞❁✬☞✄✼☛✒✙☞☛✒▲✹✪✶✌✘✪✱✰✲✓❂☛✯❇✏✰✲☛❈r ❄✏❄✏❵❄ ✜✯✬✎☛✍✌☞✰✔✄✝✝ ★ ❈✏➃✤● r❬✦✰❄✷❇✵☛✡✁✩✆✻✪✱✰▼☛✡✞✒✺✒☛✒✞✯❇✏✰✲✓❂✑✔✪✱✆✟✪✱✰✲✓ ❣ ✱ ✎❜❻✷✦ ✖ ✳✘✞✒✑✱★✎☛✛✚ ✣ ❇P✆✺ ☛✎✪✱✆❭②✎✓✭✌☞★✥✆✻✞✍✄✝★✘✑✥✆✻✞✛❫✘✪✱✆❛✙☞☛❁✬☞★☞✌✘✺✍✰▼☛✭✬✘☛✍✌✥✰✔✄✹★①★☞✌❀✠☎✄✼✞✒▲✝✪✔✺✡✙✥✪✱✌❀▲✹✪✐✴✵✪✔☛✍✄✝★✘✑➄◆ ❣ ✁☎★☞✰ ❣ ❃✲ ✆❆✦ ✖ ✌✎✞✍✆✻☛✛✚ ✣ ❇✏✰✲✞✍②✎✪✔✑✔☛✯✴✏✰✲☛❁✆✻☛✍✰✲✁✥✙☞✞✡✙✥☛✡✪✱✌✥✰✲☛✒✠☎✄✼✞✍✄✏☛❁✌☞★☞✆✟☛✍✄✏✪✫✺✒✓✯✜✩✺✒☛✭✬✎✁☞✞✍✰✲☛❂▲✹✪✠✌❈☛✒✞✍✄⑩❇P✞✍★①✆✟☛✍✰✲✁☞✙☞✞ ✰✫✄✏✞✍✬✘☛✛✮✯☛✒✑✫✁☎✄ ❣ ✖ ✌✎✞✍✆✻☛✛✚✧✳✘✞❂▲✝✪ ✌✉☛✒✞✍✄ ❇✵✞✍★①✰✔✄✏✞✍✬✎☛✛✮✯✁☞✪✔✙☞✞✒✑ ❣ s ✞✒✑✫✁☞✞✍✄✏☛✒✞❁✬☞✄✼☛✒✙☞☛✒▲✝✪✱✌✘✪✱✰✲✓❂☛✯❇✏✰✲☛ ✰✫✄✏✞✍✬✘☛✛✮✯✁✥✪✔✙☞✞✒✑ ❣ ✁ 55 ✗ ✶✲ ❈✦ ✖ ✳✎✞✒✑✱★✘☛✛✚ ✣ ▲✝✪✶❫❵☛✒✞✛✮✯✓❂✁☎✄✼✙☞✪✱✌☞★✘✑✥✆✟✞✛❫❵✪✱✆ ✬✎☛✍✌☞✰✔✄✝★❏✆✟☛✍✰✲✁☞✙✥✞✡✙☞☛❂✪✱✌☞✰▼☛✒✠☎✄✏✞✍✄✼☛❁✌☞★☞✆✟☛✍✄✏✪✔✺✒✓ ✌❈☛✒✞✍✄ ❣ ❉❋✁☞✞✍✰✲☛❂▲✹✪✥⑦✔✌✥✰✔✄✏☛❜❲✻❇✵✪ ✟ ❣ s ✞✒✑✔✁✥✞✍✄✏☛✒✞❁✬☞✄✼☛✒✙☞☛✒▲✹✪✶✌✘✪✱✰✲✓❂☛✯❇✏✰✲☛✛❲ ❣ ✎ ✦ ✖ ✳✘✞✒✑✱★✘☛✛✚ ✣ ❇✵✆✺ ☛✘✪✱✆✷②✘✓❁✌☞★☞✆✟✞✍✄✹★✘✑✎✙☞☛❂✺✒✪✔▲✲✄✏☛✉❇✵☛✍✆❭✌✎✪✔▲✹✪✫✺✒✞✍✰✲✪✱✳✘☛❂✞✒✑✔☛❁✄✏☛✛✮ ★✘✑✶✰✲✞✍✰✲☛✒✑✔✁☎✄➄✌☞★☞✆✟☛✍✄✏✪✔✺✒☛ ❣ ✖ ✳✘✞✒✑✱★✘☛✛✚✧✰✫✄✏☛✍②☞★✎✪✔☛❈❇✵✓❂▲✹✪✫☛✭⑦✔✌✥✰✔✄✏☛✒❄✷✴✵✪✁ ❣ s ✞✒✑✔✁✥✞✍✄✏☛✒✞❁✬☞✄✼☛✒✙☞☛✒▲✹✪✶✌✘✪✱✰✲✓❂☛✯❇✏✰✲☛ ❣ ✳ ✦ ✖ ✳✘✞✒✑✱★✘☛✛✚ ✣ ✄✏✞✍✬✎✁☎✄✹✰✫★✘✑✥✆✻✞✛❫❵✪✶✆❛✞✒✙☎✆✟✪✐❇✵✪✱②✘✪✔✑✥⑦✔✌☞✰✫✄✏☛❁✳✘✞✒✑✔✁☞✞✍✄✼☛✒✞❁✬✘✪✱✳✘✁☎✰✫★✘✑✱★✘✪✥★☞✌✘☛✒✪✥✆✟✞✍✰✔✄✼✪✔✺✒✪❙✴P✪ ✳✘✞✒✑✔✁☞✞✍✄✼☛✒✞❂✺✒☛✒✑✱★✘✪✥✆✻✞✒✪✥✆✟✞✍✄✏☛❂☛✒✑✔☛✍✆✟☛✍✌☞✰✂✙☞✪✱✌☞✰✫✄ ✣ ✁①✺✒✁☞✑✔✁☞✞✍✌✘✓✉■✫✳✘✞✒✑✔✁☞✞✍✄✼☛✭✬✥✄✏☛✒✙☞☛✒▲✝✪✱✌✘✪✶✰✲✓❀❚ ❣ ❪ ✣ ❨❬❑ ❣ ✎ ✲✝✳✭✦ ✖ ✳✎✞✒✑✱★✘☛✛✚ ✣ ❇✵✆✺ ☛✘✪✱✆✷②✘✓✭✳✎✞✒✑✔✁☞✞✍✄✼☛✒✞✭✆✟✪✱✌✘✪✱✆✟✓❂✞✍②❘❇✵✁☞✑✱★☞✰✲☛✒✓❂✞✒✙☎✆✟✪✐❇✵✓✭✬✎☛✍✌☞✰✔✄✝★❏★✥✌❏✬✎✪✱✳✘✁☎✰✤❱ ✳✎✞✒✑✔✁☞✞✍✄✼☛✒✞✭✬☞✄✼☛✒✙☞☛✒▲✝✪✱✌✘✪✱✰▼✓✡☛✯❇✏✰✲☛✻❚ ❣ ❄ ❪ ✣ ❚✛❨ ❣ ✳✗✎ ✲✄✳ ✦ ✖ ✳✎✞✒✑✱★✘☛✛✚ ✣ ✳✘✞✒✑✫✁☞✞✍✄✏☛✒✞❂☛✍✄✼✁☎✄✏✪✫✪☞✄✼☛✒✑✔✞✍✰✲✪✶✳✘☛✡✞✒✙✩✆✻✪✐❇✵☛✯✜✩✞✍✰✽✾✍✰✘✬✎☛✍✌☞✰✔✄✝★❏✰▼☛✍✌❘❇✵✪✱★☞✌✘☛✯✜✩✡✺ ✾✍✰④✴✵✪✥✬✘☛✍✌☞✰✔✄✝★ ✺✍★☞✄✼☛✍✌☞✰④■✔✳✘✞✒✑✫✁☞✞✍✄✏☛✒✞❁✬☞✄✼☛✒✙☞☛✒▲✝✪✱✌✘✪✱✰✲✒ ✓ ❄ ❣ ❄✏❄✤❚ ❑ ❣ ✙✲ ✦ ✖ ✳✘✞✒✑✱★✘☛✛✚ ✣ ✰✲☛✍✆✷✬✘☛✍✄✼✞✍✰✔★☞✄✼✞✭✌✘✁✩✆✻✪✱✌✎✞✒✑✔✓❈■✫✳✘✞✒✑✔✁✥✞✍✄✏☛✒✞❁✬☞✄✼☛✒✙☞☛✒▲✹✪✶✌✘✪✱✰✲✓✩❲ ✡✄✂▼➂ ❑ ✎ ✲✝✳✭✦ ✖ ✳✘✞✒✑✱★✘☛✛✚ ✣ ☛✍✄✏✁✥✞✍✄✏☛✒✞❂✞✍②❘❇✵✁☞✑✱★☞✰▼✓✭✬✘☛✍✌✥✰✔✄✹★①✰✲☛✍✌❙❇P✪✶★☞✌✘☛✉■✔✳✘✞✒✑✫✁☞✞✍✄✏☛✒✞❁✬☞✄✼☛✒✙☞☛✒▲✝✪✱✌✘✪✱✰✲✓✻❚ µs✉❑ ✎ ✦ ✖ ✳✘✞✒✑✶★✘☛✛✚ ✣ ✑✫✓✍✸✲✪✱✆✟☛✒✞✡▲✝✪✐✴✵✪✔☛✍✄✝★✘✑✱★✘✪✎✙☞☛❂✪✔☛✯✴✵✪✱✄✼☛✯✜✯⑦✔✌①✌☞★☞✆✟✓✍✄✽✙☞☛❂✺✒✞✍✄✏✞✒✺✍✰▼☛✍✄✏☛✉■✔✳✎✞✒✑✔✁☞✞✍✄✼☛✭✬☞✄✼☛✒✙☞☛✒▲✝✪✱✌✘✪✱✰▼☎ ✓ ✺❄✿❑ ✁ ✁ ✌ ✱ ✄ ✱ ✄ ✎ ✗ ✆ ✆ ✝ ✝ ✗ ✝ ✎ ✄ ✆ ✁ ❪❴❫❵☛✍✆✷✬✘✑✔☛☞✕ ❣ ❊✡❉❋➃ ❈✇❊❁➅❃✿❿➅①❊✡❪⑩➂ ❿❊ ➅①❊ ❆ ❊ ❆➁✭❪✽●⑩➃❴❊✡●❴✦ ❣ ❄✤❚ ❪ ❙●❴✦✭❚ ❲☞★ ✉❴ ❪ ✦✆✒★ ❣ ❊✡❉❋➃ ❈✇❊❁➅❃✿ t ➂✭➂✭➃ ✉✽ ✂ ✍ ✍ ✑ ✱ ✁ ✁ ✣ ✱ ✌❺ ✯✓ ❺✩❽◗❼✪✛ ✓✒⑨ ❋✁☎✄✝✆✻✞❂✠☞☛✍✌✘☛✍✄✼✞✒✑✔✓☞✕ ✗ ✍ ✂ ✯✌✹ ✢ ➄❺✺✹ ✄ .PARAM < <name>=<value> >* <<name>={expr}>* <name>=<value> - parametrului <name> i se atribuie valoarea<value>. <name>={expr} - parametrului <name> i se atribuie expresia {expr}. Referirea la valoarea unui parametru global se face utilizand acoladele { } Exemple: .PARAM pi=3.14159265 .PARAM RSHEET=120, VCC=5V .PLOT - Trasarea unui grafic ✓ Forma general : .PLOT [DC][AC][NOISE][TRAN] [ [<opvar>*] [(<lo>,<hi>)] ]* [DC],[AC],[NOISE],[TRAN] - tipul de analiza pentru care se cere trasarea graficului ☛✯✴✵✪✱✄ ✓ ✓ <opvar> = numele variabilei de i e pentru care se traseaz graficul. Num rul maxim al acestor variabile este de 8. (<lo>,<hi>) = punct de origine al axelor pentru graficul respectiv. Exemple: .PLOT DC V(3) V(2,3) V(R1) I(VIN) .PLOT AC VM(2) VP(2) VG(2) .PLOT TRAN V(3) V(2,3) (0,5V) ID(M2) I(VCC) (-50mA,50mA) .PRINT - Tiparirea rezultatelor ✓ Forma general : .PRINT [DC][AC][NOISE][TRAN] [<opvar>*] 56 ✼✄ ☛✛✮ ★✘✑✱✰▼✞✍✰✲[DC],[AC],[NOISE],[TRAN] ☛✒✑✔✁☎✄ ✰▼✪✱✬☞★✘✑✎✙☞☛❂✞✍✌✘✞✒✑✔✪✶✮✯✓❁✬✘☛✍✌☞✰✫✄✹★❀✺✒✞✍✄✼☛❈❇✵☛❂✺✒☛✍✄✏☛❁✰✲✪✱✬✘✓✍✄✼✪✱✄✼☛✒✞ ✖ ✁☎✬☞✳✘✞✍✄✇✚✾✦✧✳✘✞✍✄✼✪✔✞✍②✘✪✔✑✫☛✒✑✔☛✡✺✒✞✍✄✼☛❁★☞✄✹✆✟☛✒✞✛✮✯✓❤✞❂▲✹✪✥✰✲✪✶✬✘✞✍✄✏✪✶✰✲☛ Exemple: .PRINT DC V(3) V(2,3) V(R1) IB(Q13) .PRINT AC VM(2) VP(2) VG(5) II(7) .PRINT NOISE INOISE ONOISE DB(INOISE) .PROBE -Postprocesorul grafic PROBE ✓ Forma general : .PROBE[/CSDF] .PROBE[/CSDF] ✁☎✬☞✳✘✞✍✄✇✚✾✦✧[<opvar>*] ✳✘✞✍✄✼✪✔✞✍②✘✪✔✑✫☛✒✑✔☛✭✬✎☛✍✌☞✰✔✄✝★✷✺✒✞✍✄✼☛✉❇P☛❂✙☞✁☎✄✼☛✯❇✏✰✲☛❂✁❏✞✍✌✎✞✒✑✔✪✶✮✯✓❂✠☎✄✏✞✍☞✬ ☛✘✪✔✺✒✓ ❉❙➁❜❊ ✭❪ <✬✘☛✍✄✝✆✻✪✱✰▼☛✭✳✘✪❩✮ ★✘✞✒✑✔✪✶✮✯✞✍✄✼☛✒✞✭★✥✌✘✁☎✄✽☛✛❫✿✬✥✄✏☛✯❇✵✪✔✪✎✙☞☛❁✳✘✞✍✄✼✪✔✞✍②✘✪✔✑✔☛❂✙☞☛❂✪✔☛✯✴✵✪✱✄✼☛ s ✞✍✄✼✪✔✞✍②✘✪✔✑✫☛✒✑✔☛✡✙✥☛ ❣ ✪✫☛✯✴P✪✶✄✏☛❁✬✘✁❵❇✵✪✱②✘✪✫✑✔☛❈❇✏★☞✌✥✰P✕ ✑④✬✘☛✍✌☞✰✔✄✝★①✄✏☛✒✠☞✪✶✆❭★✘✑✎✙☞☛❂✺✍★☞✄✼☛✍✌☞✰✂✺✒✁☎✌☞✰✲✪✶✌☞★☞★✟❇✵✞✍★❏✰✫✄✏✞✍✌✥✮✯✪✶✰✲✁☎✄✼✪✱★➄✕ ✶✵✷✧➄✚❶ ✢☞✘❘❼✪✩✭✻ ✦✧✬✎✁☎✰✲☛✍✌☞✸✲✪✫✞✒✑✱★✘✑✥✌✘✁☞✙☎★✎✑✱★✘✪ ✖ ✌✘✁✥✙☞☛✛✚ ✶✵✷✧✫✬➄✚❶ ✢☞✘❘✪❼ ✩✁ ✧✭✇✑ ❶✔✢✏✘❙❼✪✩✸✻④✦✧✰✲☛✍✌❙❇P✪✶★☞✌✘☛✒✞❁⑦✔✌☞✰✔✄✼☛✭✌✎✁☞✙☎★✘✑ ✖✂✗ ✌✘✁☞✙☞☛✛✚✢✴✵✪ ✖✤✣ ✌✘✁☞✙✥☛✛✚ ✶✵✷✧➄❶✂❺✿❽ ✪❼ ✩✭✻④✦✧✰✲☛✍✌❘❇✵✪✱★✥✌✘☛✒✞✭✬✎☛✡☛✒✑✔☛✍✆✟☛✍✌☞✰✔★✎✑✘✙✥✪✱✬✘✁☞✑✔✞✍✄ ✖ ✌✘✞✍✆✟☛✛✚ ✄❋ ✂ ✵✷✧➄❶✂❺✩❽◗✪❼ ✩✭✻④✦✧✬✘✁☎✰✲☛✍✌✥✰✲✪✔✞✒✑✱★✘✑✥✌✘✁✥✙☎★✘✑✱★✘✪❵❫✷✞✒✑✥★☞✌☞★✘✪✎☛✒✑✔☛✍✆✟☛✍✌☞✰ ✖ ✌✎✞✍✆✻☛✛✚ ✄✂ ✠ ✵✽④✧ ❶❋❺✩❽◗✪❼ ✩✸✻④✦✧✰▼☛✍✌❘❇✵✪✱★☞✌✘☛✒✞❁⑦✔✌☞✰✔✄✼☛❁✌✘✁☞✙☎★☞✄✼✪✔✑✔☛❜❫✻✴✵✪ ☛❀✞✒✑✔☛❁★☞✌☞★✎✪✘☛✒✑✫☛✍✆✻☛✍✌☞✰✎✆✷★✘✑✱✰✲✪✶✬✘✁☞✑✔✞✍✄ ✗✵✽✧④❶❋❺✩❽◗✪❼ ✩✸✻④✦◗✺✍★✥✄✏☛✍✌☞✰✫★✘✑✥✬☞✄✏✪✶✌✷☛✒✑✫☛✍✆✻☛✍✌☞✰✫★✘✑✎✙☞✪✱✬✘✁☞✑✫✞✍✄ ✖ ✌✘✞✍✆✟☛✛✚ ☎✂ ✵✽✧④❶❋❺✩❽◗✪❼ ✩✸✻➄✦◗✺✍★☞✄✏☛✍✌✥✰✔★✘✑✥✬☞✄✼✪✱✌①②✘✁☎✄✝✌✘✞❁❫✷✞❁★☞✌☞★✘✪✎☛✒✑✔☛✍✆✟☛✍✌☞✰✎✆✷★✘✑✱✰✲✪✱✬✎✁☞✑✔✞✍✄ ✖ ✌✎✞✍✆✻☛✛✚✧✬✘✁✥✞✍✰✲☛✡▲✝✪✥✌☞★☞✆✟☛✒✑✔☛❂✁☎✄✏✪✫✺✒✓✍✄✹★✘✪✎☛✒✑✔☛✍✆✟☛✍✌☞✰✂✙☞☛❁✰✲✪✱✬☞★✎✑✇✕☎➂ ✁❄ ✍❂❄✫❪✽❄✫❘❄ ✌✡❄ ❂❄ ❈✹❄✫●✽❄✫➁ ❄✔s ❫ ☛❏✬✎✁☎✰✂▲✹✪✎✁☎✄✼✪✔✺✒✞✍✄✼☛✡✙☞✪✶✌❏②✎✁☎✄✹✌✎☛✒✑✔☛☞✕ ✣ ✍❂❄ ✌❜❄ ✿ ■ ✭❑ ✝ ✝ ✝ ✝ ✝ ✄ ✄ ✂ - D/G/S (J) - D/G/S/B (M) - C/B/E/S (Q) - pentru regimul sinusoidal: M - amplitudinea DB - amplitudinea in dB P - faza ⑦✫✌☞✰✲✞✍✄✇✮✯✪✔☛✍✄✏☛✒✞❂✙☞☛❂✠☎✄✝★☞✬ G– ✣ ✘✬ ✞✍✄✹✰▼☛✒✞✭✄✼☛✒✞✒✑✔✓ ✣ ✬✘✞✍✄✝✰✲☛✒✞❂✪✱✆✻✞✒✠✥✪✱✌✘✞✍✄✼✓ ✄ ✣ ✬✘☛✍✌☞✰✔✄✝★❀✞✍✌✘✞✒✑✔✪✶✮✯✞❜✮✯✠☞✁☎✆✟✁☎✰✔★✘✑✶★✘✪✇✕ ✙✲ ✍❻✎ ✣ ✮✯✠✥✁☎✆✻✁✩✰✔★✘✑✎✙☞☛❂✪✱✌☞✰✔✄✼✞✍✄✏☛ ✲ ✁✲ ✍❻ ✣ ✮✯✠☞✁✩✆✻✁☎✰✫★✘✑✎✙☞☛✡✪✫☛✯❇P✪✶✄✏☛ ✵ ✁✁✲ ✍❻ ✻ ✣ ✮✯✠☞✁☎✆✟✁☎✰✔★✎✑✘✙✥☛✡✪✱✌✥✰✔✄✏✞✍✄✼☛❁⑦✔✌❀✙✌ ✵ ✲ ✁✲ ❻✎ ✻✰✑④✮✯✠✥✁☎✆✻✁✩✰✔★✘✑✎✙☞☛❂✪✔☛✯✴✵✪✱✄✏☛❁⑦✔✌❀✙✌ ✁ ✄ ✄ ✄ ✆ ✆ ✆ ✆ ✄ ✄ ✄ ❘★☞✌✘✺✍✸▼✪✔✪✔✑✔☛❁★☞✰✲✪✔✑✫✪✶✮✯✞✍②✘✪✔✑✔☛✉❇✏★☞✌☞✰P✕ 57 ❘★☞✌✘✺✍✸✲✪✔☛ t❇❇✿✎■✐❫☞❑ ✿✒✌✭➅❭■✐❫☞❑ ✿ ✞✡➁❜➃⑩■✫❫✥❑ ❪ ✉❉➄■✫❫✥❑ ●⑩❊ ✌ ■✫❫☞❑ ●⑩❊ ✌✧❅❚ ❄❵■✐❫☞❑ ❆ ■✐❫☞❑ ❉➄■✫❫☞❑ ➁❂■✐❫☞❑ ❈ ❆ ✌❿■✐❫☞❑ ✌❿■✐❫☞❑ ❉ ➁❂■✐❫✎✜✜ ☛❵❑ ✿✏❈✐➅✷■✫❫☞❑ ➂✭✶ ❊ ✿✎■✐❫☞❑ ➃➄t✡➅✷■✫❫☞❑ t ➃➄t❂➅❭■✐❫☞❑ t✉➁❜➂✭➃➄t❂➅❭■✐❫☞❑ ✙❵■✐❫☞❑ ❇ ■✫❫✥❑ t❈s ✌❿■✐❫☞❑ ➁ ❆ ✿✎■✫❫☞❑ ❆ ❈✲➅❭■✐❫☞❑ ❆❖t ①■✫❫✥❑ ✂ ✿✥☛✍✆✷✌✘✪✔▲✝✪✔✺✒✞✍✸✲✪✫☛ ■✫❫✥❑ ✗ ❚✛■✐❫ ❄✩❑P✜ ❄❵■✐❫☞✦ ❄✩❑✼✜ ✣ ❚✛■✫❫ ✖ ❄✩❑ ❫✁✄✂ ☎ > ☛✝✆ ✑✱✌❘■✐❫☞❑ ✑✔✁☞✠❵■✐❫☞❑ ✆✻✁✥✙☎★✘✑✱★✘✑✎✑✱★✎✪✥❫ ▲✹✞✛✮✯✞✉■✫⑦✔✌❀✠☎✄✏✞✒✙✥☛✛❑✤✞✡✑✶★✘✪❵❫ ✬✘✞✍✄✝✰✲☛✒✞✭✄✼☛✒✞✒✑✔✓❂✞✡✑✶★✘✪❵❫ ✬✘✞✍✄✝✰✲☛✒✞✡✪✶✆✻✞✒✠☞✪✶✌✘✞✍✄✏✓❂✞❂✑✱★✘✪❵❫ ⑦✔✌☞✰✲✞✍✄✇✮✯✪✔☛✍✄✼☛✒✞✡✙✥☛✡✠☎✄✝★☞✬①✬✘☛✍✌☞✰✫✄✹★❭❫✻■✲✪✶✌✻❇✼❑ ■✫❫✥❑✟✞ ❇✵✪✱✌❭❫✎✜❬❫❏⑦✔✌①✄✏✞✒✙✥✪✔✞✍✌✘✪ ✺✒✁❵❇✽❫❘✜❬❫❏⑦✔✌①✄✏✞✒✙☞✪✫✞✍✌✘✪ ✰✲✞✍✌❭❫✎✜❬❫❏⑦✔✌①✄✼✞✒✙☞✪✔✞✍✌✘✪ ✞✍✄✏✺✍✰▼✞✍✌①❫❘✜✯⑦✔✌①✄✏✞✒✙✥✪✔✞✍✌✘✪ ✞✍✄✏✺✒✺✍✰▼✞✍✌①❫❘✜✯⑦✔✌①✄✏✞✒✙✥✪✔✞✍✌✘✪ ✙☞☛✍✄✼✪✱✳✘✞✍✰✲✞❂✑✱★✘✪❵❫❏⑦✔✌①✄✏✞✍✬✎✁☎✄✹✰✂✺✍★❀✞✍②❘❇✵✺✒✪✐❇✵✞ ✪✱✌☞✰✲☛✒✠✩✄✏✞✒✑✔✞❂✑✱★✘✪❵❫❏⑦✔✌❀✪✱✌☞✰▼☛✍✄✹✳✘✞✒✑✶★✘✑✎✙☞☛✒▲✹✪✶✌✘✪✱✰✎✬✘☛✍✌☞✰✔✄✝★❀✞✍②❘❇✵✺✒✪✐❇✵✞ ✳✘✞✒✑✔✁☞✞✍✄✼☛✒✞❁★☞✌✘✙☞☛✒✪✎✞❂✑✱★✘✪❵❫❏✬✘☛✡✪✶✌☞✰✲☛✍✄✝✳✘✞✒✑✱★✘✑✎✙☞☛✒▲✝✪✱✌✘✪✱✰✎✬✘☛✍✌✥✰✔✄✹★❀✞✍②❘❇✵✺✒✪✐❇✵✞ ✳✘✞✒✑✔✁☞✞✍✄✼☛✒✞❂☛✒▲✹☛✒✺✍✰✲✪✶✳✘✞✡✞❂✑✱★✘✪❵❫✷✪✱✌❀✪✱✌☞✰▼☛✍✄✹✳✘✞✒✑✶★✘✑✎✙☞☛✒▲✹✪✶✌✘✪✱✰✎✬✘☛✍✌☞✰✔✄✝★❀✞✍②❘❇✵✺✒✪✐❇✵✞ ✆✻✪✶✌✘✪✱✆✷★✘✑✥✬✘✞✍✄✹✸▼✪✔✪✥✄✏☛✒✞✒✑✔☛❂✞❂✑✱★✘✪❵❫ ✆✻✞✛❫✘✪✱✆❭★✎✑☞✬✎✞✍✄✹✸✲✪✫✪☞✄✼☛✒✞✒✑✔☛❂✞✡✑✶★✘✪❵❫ Exemple: .PROBE .PROBE v(2) I(R2) VBE(Q13) VDB(5) .PROBE/CSDF ●⑩✞✟✞✒✺✒☛✒✑✫✞✯✴P✪ ✄✏☛✛✮ ★✎✑✱✰✲✞✍✰❁❇✵☛✟✞ ☎✵★☞✌✘✠☞☛❭★☞✰✲✪✔✑✔✪❩✮ ✾✍✌✘✙✾✆✟☛✍✌✘✪✱★✎✑ ➃➄✄✼✞✒✺✒☛✍❄✔t ✙☞✙✢➃➄✄✏✞✒✺✒☛✍❄▼➃④✄✼✞✒✺✒☛❀☛✛❫☞✬☞✄✏☛✯❇ ❇✵✪✔✁☎✌ ❣ ➂✒✿ ✉ ☛✯❇✇✰▼☛①★☞✌ ✍ ▲✝✁☎✄✝✆✻✞✍✰④❇✏✬✘☛✒✺✒✪✔▲✝✪✔✺❁✬✘☛✍✌☞✰✔✄✝★❀✙☞✞✍✰✲☛✒✑✔☛❁✌☞★☞✆✟☛✍✄✼✪✔✺✒☛✡✙✥☛✡✪✔☛✯✴✵✪✱✄✼☛❁⑦✔✌❭❉ ☞✿ ❉✚❈✏➂ ❪ ❣ ✣ ❺✺✹✐❷✒⑨✜✹✐⑨ ❼✒❶ ✑ ❹ ✛✹❹ ✮✥❹ ✛ ❬✖ ❸✹❹ ✹ ✢✏✓ ✇✥ ❶✟❷✒⑨✔✓✒❼✒❶ ✤ ✛ ❷✡✢✥❶ ✝✛ ❹✐❶❙⑨❘⑨ ✹✔✕ .SENS ❋✁☎✄✝✆✻✞❂✠☞☛✍✌✘☛✍✄✼✞✒✑✔✓☞✕ ✱❻✎ ✂ ❻ ✑ ✧✰✢ ✚✮❘❺✌✓✪✩ ✍ ☎ ✖ ✁☎✬☞✳✎✞✍✄✝✚q◆ ✦✧✳✘✞✍✄✼✪✔✞✍②✘✪✔✑✫☛✒✑✔☛✡✙✥☛✡✪✔☛✯✴✵✪✱✄✼☛❁✬✘☛✍✌☞✰✔✄✝★❀✺✒✞✍✄✏☛✉❇✵☛✡✙☞✁✩✄✏☛✯✴✏✰✲☛❂✞✍✌✘✞✒✑✔✪✶✮✯✞ ❇✵☛✍✌✥✮✯✪✶✰✲✪✱✳✘✪✱✰▼✓✍✸✲✪✔✑✔✁☎✄➄⑦✔✌①✄✏✞✍✬✘✁✩✄✹✰✂✺✍★①✰✲✁☎✸✲✪✥✬✘✞✍✄✼✞✍✆✟☛✍✰✔✄✏✪✫✪✘✺✒✪✶✄✏✺✍★✘✪✶✰✔★✘✑✱★✘✪ ❣ Exemplu: .SENS V(9) V(4,3) I(VCC) .STEP ✁ ❶❋❺✺✹✔❹ ✑✩❺✦✥✇❶ ④❺ ✵❹ ☎ 58 Forme generale: .STEP [LIN] <varname> <start> <end> <incr> .STEP [OCT][DEC] <varname> <start> <end> <points> .STEP <varname> LIST <value>* [LIN], [OCT],[DEC], LIST - se variaza variabila <varname> liniar, pe ❇✵✞✍★①⑦✔✌①✬☞★☞✌✘✺✍✰▼☛✒✑✔☛ ✖ ✳✘✞✒✑✱★✘☛✛✚q◆❿❇✇✬✎☛✒✺✒✪✔▲✹✪✫✺✒✞✍✰✲☛✭⑦✫✌✷✑✫✪✐❇✏✰✲✓ octave, pe❇✏✰✲✞✍decade ✄✹✰✐✚ ☛✍✌✘✙✩✚✾✦✧✳✎✞✒✑✔✁☎✄✼✪✔✑✔☛❂✙☞☛✭⑦✫✌✘✺✒☛✍✬☞★☞✰④✴✵✪❙❇P▲ ✾✍✄✵✴P✪✶✰❙✞✒✑✫☛✭✳✘✞✍✄✼✪✔✞✍②✘✪✫✑✔☛✒✑✔✁☎✄ ✖ ✖ <incr> = incrementul ✓ <points> = num rul de puncte Exemple: .STEP VIN -.25 .25 .05 .STEP LIN I2 5mA -2mA 0.1mA .STEP RES RMOD(R) 0.9 1.1 .001 .STEP TEMP LIST 0 20 27 50 80 .STEP PARAM X 1 5 0.1 .SUBCKT - Definirea unui subcircuit Formele generale: .SUBCKT <name> [<node>*] [PARAMS: <par>[=<val>]* ] <name> = numele subcircuitului <node>* = nodurile extreme ale subcircuitului. Niciunul nu poate fi zero. PARAMS: <par> [ = <val>]* - setarea unor parametri Exemple: .SUBCKT OPAMP 1 2 101 102 .SUBCKT FILTER IN OUT PARAMS: CENTER, WIDTH=10KHz .TEMP - Temperatura Forma generala: .TEMP <value>* ✴ <value>* = valorile temperaturii pentru care se dore te analiza circuitului Exemple: .TEMP 125 .TEMP 0 27 125 ✑ ✎ ⑧ ✑ ❿❺✺✐✹ ❷✒⑨✜✹✐⑨✜✹❙⑨❘❶❙❼✛❹❙❾✹⑨❙❶❘❷✛❸✝❹✔❹✔✘❙❼ ✛ ✓✯❺✩❶✔✕ ❾✝❼✡✓ ✇✥ ❶✟❷✒⑨✔✓✒❼✒❶ ✤ ✛ ❷✡✢✥❶ ✝✛ ❹✐❶❙⑨❘⑨ ✂ ❋✁☎✄✝✆✻✞❂✠☞☛✍✌✘☛✍✄✼✞✒✑✔✓☞✕ ⑧ ✧ ✢ ✚✮❘❺✌✓✪✩✫✧✂❹ ✸✕ ✓✒✪❷ ✩ ✖ ✁☎✬☞✳✘✞✍✄✇✚✾✦✧✳✘✞✍✄✼✪✔✞✍②✘✪✔✑✫✞✡✙☞☛❂✪✔☛✯✴✵✪✱✄✼☛ ✖ ✪✱✬❘❇✏✄✼✺✛✚✾✦✧✌☞★☞✆✟☛✒✑✔☛✡✠✥☛✍✌✘☛✍✄✏✞✍✰▼✁☎✄✹★✎✑✱★✘✪✎✞✡✺✒✞✍✄✝★✘✪✥✳✘✞✒✑✔✁✥✞✍✄✏☛❁✄✏☛✍✬✥✄✏☛✛✮✯✪✱✌✥✰✲✓✭✳✎✞✍✄✏✪✔✞✍②✎✪✔✑✔✞❂✙☞☛✡✪✶✌☞✰✔✄ ✑ ✎ ☎ ☎ Exemple: .TF V(5) VIN 59 .TF I(VDRIV) ICNTRL .TRAN - Analiza de regim tranzitoriu ✓ Forma general : .TRAN[/OP] <pstep> <ftime> [<noprint> [<ceiling>]] [SKIPBP] ✬✘✞✯❇✏★✘✑✎✙☞☛❁✰✲✪✱✆✷✬❏✬✎☛✍✌☞✰✔✄✝★✷✺✒✞✍✄✼☛✉❇P☛❂✞✒▲✝✪✐✴✵☛✒✞✛✮✯✓✭✄✼☛✛✮ ★✘✑✱✰✲✞✍✰✫★✘✑ <pstep> = ✦✧✰✲✪✱✆✷✬☞★✘✑✎▲✝✪✱✌✘✞✒✑✥✬✘☛✍✌☞✰✔✄✝★❀✺✒✞✍✄✏☛✉❇✵☛✡✙✥✁☎✄✏☛✯✴✏✰✲☛❂✞✍✌✘✞✒✑✔✪❩✮✯✞✭✰✔✄✼✞✍✌✥✮✯✪✱✰▼✁☎✄✏✪✫☛ ✖ ▲✐✰▼✪✱✆✻☛✛✚ ✖ ✌✘✁☎✬☞✄✼✪✱✌☞✰✐✚ ✦✧✰✲✪✶✆❭✬☞★✎✑✘✪✶✌✘✪✱✰✲✪✔✞✒✑✥✬✘☛✍✌✥✰✔✄✹★❀✺✒✞✍✄✼☛❈❇✵☛❂✙☞✁☎✄✼☛✯✴✏✰✲☛✡✞✒▲✝✪✐✴✵✞✍✄✼☛✒✞✭✄✼☛✛✮ ★✘✑✱✰✲✞✍✰▼☛✒✑✔✁☎✄ ✖ ✺✒☛✒✪✔✑✔✪✱✌✎✠✩✚ ✦ limita superioara a pasului de timp; daca nu se specifica se considera ceiling=ftime/50 pentru un circuit care contine elemente dinamice sau ceiling=pstep pentru un circuit rezistiv. ✺✒✁☎✌✘✙✥✪✱✸✲✪✔✪✔✑✫✁☎✄✽✪✱✌✘✪✶✰✲✪✔✞✒✑✔☛❂✙☞☛✒▲✝✪✱✌✘✪✱✰✲☛❂✪✱✌❀✑✔✪✱✌✎✪✔✪✔✑✔☛❂✙☞✪✱✳✘☛✍✄✵❇✵☛✒✑✔✁☎✄✽☛✒✑✫☛✍✆✻☛✍✌☞✰▼☛ [SKIPBP] = folosirea ✙ ☛❂✺✒✪✱✄✼✺✍★✘✪✱✰④❇✵✞✍★✷✺✒✞✒✑✫✺✍★✘✑✔✞✍✰✲☛❂▲✹✁✥✑✔✁❵❇✵✪✱✌✘✙✷❇✵☛✍✰✲✓✍✄✼✪✔✑✔☛❂✙☞✪✱✌❀✑✔✪✱✌✘✪✫✞✡✙☞☛❂✺✒✁☎✆✟✞✍✌✘✙☞✞ ❣ ❈✏➂ ☞ Exemple: .TRAN 1nS 100nS .TRAN/OP 1nS 100nS 20nS SKIPBP .TRAN 1nS 100nS 0nS .1nS .WATCH - ⑧⑩⑨✚✓✒❶✎❹ ✩❺✌✒✓ ❼✯❺✶✓✒❼ ❬⑨✜✹ ✛P✺❺ ✛✇❼✪✹✷✢✏✓ ✏✥ ❶✟❾ ✢☞✓✒❽ ❺ ❶❙⑨❘❽◗✡❼ ✓✛❹✐❷ ✑ ✑ ✖✦✥✇❶❁✹✛ ❹✐❽ ❋⑨ ❋ ✹ ❺✩❶❋❺✺✹✔❹ ❬❼✛❹❘❷✛❹✷✓✒❷✒⑨✘❹ ✛✏⑨✜✹✐⑨✘❹ ☎ ✑ ❋✁☎✄✝✆✻✞❂✠☞☛✍✌✘☛✍✄✼✞✒✑✔✓ : .WATCH [DC] [AC] [TRAN] <<varname> <l, h>>* [DC], [AC], [TRAN] = tipul analizei ✌☞★☞✆✟☛✒✑✔☛✭✳✎✞✍✄✏✪✔✞✍②✎✪✔✑✔☛✒✪❙■✎✬ ✾✍✌✘✓❂✑✔✞❁✰✔✄✼☛✒✪☞✳✎✞✍✄✏✪✔✞✍②✎✪✔✑✔☛❜❑❴✴✵✪✘✑✫✪✱✆✻✪✶✰✲☛✒✑✔☛ <varname> <l, h> = ❇✵✞✒✑✔☛❂✪✱✌✘▲✝☛✍✄✏✪✫✁☞✞✍✄✏✓✉✴✵✪❙❇✇★✥✬✘☛✍✄✏✪✫✁☞✞✍✄✏✓ Exemple: .WATCH .WATCH DC V(1) (1,5) I (R5) TRAN VC(2) (-0.8, 0.8) IL(L) (0.1mA, 0.6mA) ✳❜⑨❘✔❶ ✯✩❹✐❽◗❼✯❺✁✔✹ ❹✐❶✎❹✔❹ ✹ ✢☞✓ ✥✇❶✧❾✲❹✂✵❹✐✡❼ ✓✒⑨✜✹✔✘❘❼❜❹✐✁❼ ✵❹ ✓✒❼ .WIDTH - ✞❂✠☞☛✍✌✘☛✍✄✼✞✒✑✔✓☞✕ ❋✁☎✄✝✆✻ ✑✎ ✲ ✧✔✮❙✺❺ ✹ ✩ ✘ ✳ ✒ ✞ ✶ ✑ ✚ ❖ ✦ ☞ ✌ ★☞✆✟✓✍✄✝★✘✑❴✙☞☛❭✺✒✁✥✑✔✁☞✞✍✌✘☛✷▲✹✪❩❫❵✞✍✰❴✬✘☛✍✌☞✰✔✄✝★ ✑✔✪✱✌✘✪✔✪✫✑✔☛❭✙✥✪✱✌ ▲✹✪✐✴✵✪✔☛✍✄✝★✘✑❴✙☞☛✷✪✔☛✯✴✵✪✱✄✼☛ ❣ ❉❋✁☞✞✍✰✲☛✷▲✹✪➄⑦✔✌✥✰✔✄✏☛ ✖ ✄ ✆ ✎ ✁ ✌ ✎ ✁ ✣❄ ❪❴❫❵☛✍✆✷✬✘✑✔☛☞✕ ❊❜✈ ➃➄✦ ❣ ❈✕✍❿➃ ✂ ✂ ✴✵✪⑩❚✛❨☞❨ ❣ s ✞✒✑✔✁☞✞✍✄✼☛✒✞❁✬☞✄✏☛✒✙✥☛✒▲✹✪✱✌✎✪✱✰✲✓❂☛✯❇✇✰▼☛☎✺❄ ❣ ✺❄ 60 CAP 3. LUCRARI DE LABORATOR LUCRAREA 1 ✂✁☎✄✆✞✝✞✁✠✟☛✡✌☞✍✁☎✎✏✁✒✑✓✄✔✡✌✕✏✡✖✞✝✞✄✏✡✍✎✆✟✗✔✘✙✎✆✟☛✁☎✎✚✝✔✝ ✺✱ ✁✲ ✳ ✿❵✞✉❇P☛✉❇✵✪✱✆✷★✘✑✔☛✛✮✯☛❂✺✍★①❉❀☞✿ ❉✚❈✏➂ ✂ ✑ ✆ ✗ ❪✢★☞✄✝✆✟✞✍✰✲✁☞✞✍✄✼☛✒✑✔☛✡✺✒✪✶✄✏✺✍★✘✪✶✰✲☛❈❇✵✪❙❇✵✓❈❇✵☛❂☛✛❫✿✬✘✑✔✪✫✺✒☛✭✄✼☛✛✮ ★✘✑✱✰✲✞✍✰▼☛✒✑✔☛☞✕ ❪ ✦ ❉ s❭❱ ❪✜✛ ✦ ❉ ✈✣✢ ❱ ✜❪ ✥❙✦ ❚ ❲✿s✷❱✌❈✼❇ ❨❿✦ ❚ t❭❱ ❱❿✄★✛❙✦ ✟ ❱ ✄ ✦ ✟ Ω Ω ✄ ✦✢❨ ❱❿✄✫✘✪ ✦✢❨ ❱ Ω Ω ✂✪✫✠ ❣ ❚ ❣ ❚ ❣ ❪ ☎ ☎ ☎ ✁ ✎ ❇✱ ✗✎ ✛✞ ❑◗❪✤✢❘✦ ❚ s❭❱ ②✥❑◗❪✜✛❙✦✾❲☞s❭❱ ✺✛❑ ✩✄ ✛✂✦✾❲ ❱ ✦✾❲✿s✷❱ ❪ ✦✾❲❋◆ ❈ ❱ ❈✧✦ ✦ ❚ t✷❱❬✄ ✧ ✦ ✄ ✦ ❚ ✄ ✦✧✄ ✦ ❚ ☎❱ ✄✧❘✢ ✦✾❲ Ω❱ ☎ Ω Ω Ω ✂✪✫✠ ❣ ❚ ❣ ❲ ❣ ✭✬ ✼✗✳ ✎ ✎ ✳✦✲ ❘❉ ✄ ❣ ❚ ❣ ❚ ❣ ✿✥☛ ✁✩②❘❇✵☛✍✄✹✳✘✞ ✺✒✞❖✺✒✪✱✄✏✺✍★✎✪✱✰✔★✘✑✡✌☞★ ✞✍✄✼☛✢②☞★✘✺✒✑✫☛ ▲✹✁✩✄✹✆✟✞✍✰✲☛✢✌☞★✥✆✻✞✒✪✉✙☞✪✱✌➀❇✏★☞✄✵❇✵☛ ✙☞☛✢✰▼☛✍✌❘❇✵✪✱★☞✌✘☛ ❇✵✞✍★ ❇✵☛✒✺✍✰✲✪✶★☞✌✘✪①▲✹✁✩✄✹✆✟✞✍✰✲☛ ✌☞★☞✆✟✞✒✪❏✙✥✪✱✌ ❇✇★✥✄P❇✵☛ ✙☞☛ ✺✍★✥✄✏☛✍✌☞✰ ❣ ➂✭✪✱✄✼✺✍★✘✪✱✰✔★✘✑①✑✔✪✶✌✘✪✔✞✍✄ ✞✍✄✼☛ ✙✥☛✒✺✒✪❏✁ ❇✵✁☞✑✱★✥✰✲✪✔☛❖★☞✌✘✪✔✺✒✓ ❣ ✿✥✁✥✑✱★☞✰✲✪✔✞ ✙✥☛✍✰✲☛✍✄✹✆✟✪✱✌✘✞✍✰▼✞✡✺✍✫ ★ ✿☞❉✚❈✏➂ ❪ ❇✵☛❁✳✘☛✍✄✏✪✫▲✹✪✔✺✒✞❂✺✍★①②✘✪✔✑✔✞✍✌✥✰✔★✘✑✥✬☞★☞✰✲☛✍✄✼✪✔✑✔✁✩✄④★☞✰✲✪✔✑✫✪✶✮✯✞✍✌✘✙①✙☞✞✍✰✲☛✒✑✔☛❂✙☞✪✱✌❀▲✝✪✐❇✵✪✔☛✍✄✝★✘✑➄◆ ❣ ✁✩★☞✰ ❣ ❉❘✄ ❣ ❚ ❣ ❲ ❣ ❈✹✌ ✺✒✞✛✮ ★✘✑❋★☞✌✥★✘✪✤✺✒✪✱✄✼✺✍★✘✪✱✰ ✑✔✪✱✌✎✪✔✞✍✄ ✺✍★ ❇✏★☞✄✵❇✵☛❏✺✒✁✩✆✻✞✍✌✘✙✥✞✍✰✲☛✷❇P☛ ✬✘✁✥✞✍✰✲☛ ✺✒✞ ✙☞☛✍✰✲☛✍✄✝✆✻✪✶✌✘✞✍✌☞✰✔★✘✑✂✆✻✞✍✰✔✄✼✪✔✺✒☛✒✪ ❇✵✪✐❇✏✰✲☛✍✆✷★✘✑✱★✎✪✾✙☞☛ ☛✒✺✍★✘✞✍✸✲✪✔✪✾✞✒✑✾✺✒✪✱✄✼✺✍★✘✪✱✰✫★✘✑✱★✘✪◗❇✵✞ ❇✵☛➀✞✍✌✥★✘✑✔☛✛✮✯☛ ✬✘☛✍✌☞✰✔✄✝★ ✞✍✌☞★☞✆✟✪✱✰✲☛ ✳✘✞✒✑✫✁☎✄✏✪✧✞✒✑✔☛ ✬✘✞✍✄✼✞✍✆✻☛✍✰✔✄✼✪✔✑✔✁✩✄ ✙☞☛ ✺✒✁✩✆❭✆✟✞✍✌✘✙☞✓ ❣ ❉❋☛✍✌☞✰✔✄✝★ ✞ ✙✥☛✍✰✲☛✍✄✹✆✟✪✱✌✘✞✢☛✛❫❵✪✐❇✏✰✲☛✍✌✥✰✲✞ ✴✵✪ ★☞✌✘✪✫✺✒✪✱✰✲✞✍✰✲☛✒✞ ❇✵✁☞✑✱★☞✰✲✪✫☛✒✪✭✺✒✪✱✄✏✺✍★✎✪✱✰✔★✘✑✱★✎✪✭✙☞✪✱✌ ▲✝✪✔✠☎★☞✄✼✞ ❚ ❣ ❲ ❣ ✬✘☛✍✌☞✰✫✄✹★ ✺✒✞✛✮ ★✥✄✏✪✔✑✫☛❿✞✛❑✼✜✿②✥❑ ✴✵✪✂✺✛❑ ❇✵☛❿✙☞☛✍✰▼☛✍✄✹✆✟✪✱✌✘✞❿✠✥☛✍✌✘☛✍✄✏✞✍✰▼✁☎✄✹★✎✑✂☛✒✆✺ ✘☛ ✪✱✳✎✞✒✑✔☛✍✌☞✰➄✙✥☛❂✰✲☛✍✌❘❇✵✪✱★☞✌✎☛❿✞✒✑✂✺✒✪✱✄✼✺✍★✘✪✱✰✫★✘✑✱★✘✪❋✙☞✪✱✌☞✰✔✄✼☛❂②✘✁✩✄✹✌✘☛✒✑✫☛❜t ✴✵✪ ❣ ➁❜☛✛✮✯✪✲❇✇✰▼☛✍✌☞✰✲✞❿☛✒✺✆✘☛ ✪✶✳✘✞✒✑✔☛✍✌☞✰✲✞✡⑦✔✌☞✰✔✄✼☛❜②✎✁☎✄✹✌✎☛✒✑✔☛❈✺✒✪✱✄✼✺✍★✘✪✱✰✫★✘✑✱★✘✪✘✬✘✞✯❇✵✪✱✳✘✪✶✮✯✞✍✰✝✜ R ✬✎✁☞✞✍✰✲☛❈▲✝✪❙✙☞☛✍✰✲☛✍✄✝✆✻✪✱✌✎✞✍✰✲✓❈✙☎★✥✬✘✓❈✺✍★☞✆ ★✥✄✹✆✟☛✒✞✛✮✯✓ ❣ ❿☛✒✁✥✞✍✄✏☛✒✺✒☛❂✞✒✺✒☛✯❇✏✰❙✺✒✪✶✄✏✺✍AB★✘✪✶0✰ ✺✒✁✩✌☞✰✲✪✱✌✘☛✻✁ ❇✏★☞✄✵❇✵✞❀✺✒✁☎✆✟✞✍✌✘✙☞✞✍✰▼✞✯✜❙✬✘☛✍✌☞✰✔✄✝★ ✺✒✞✒✑✫✺✍★✘✑✱★✘✑⑩✑✱★✘✪ ❇✵☛❭✞✍✬✘✑✫✪✔✺✒✞ R AB 0 ⑦✫✌☞✰✔✄✼☛❆t ❇✵✫ ✪ ★✥✌ ✺✍★✥✄✏☛✍✌☞✰ ✙☞☛ ❚ t ✺✒✞ ⑦✔✌ ▲✝✪✔✠☎★☞✄✼✞❆❲ ❣ ❚ ❣ ➁❜☛✛✮ ★✘✑✶✰✲✞ ☎ ✑ ✄ ✁ ✁ ✌ ✁ ✍ ✂✪✔✠ ❣ ❲ ❣ ❚ ❣ ✰▼☛✍✌❘❇✵✪✱★☞✌✘☛✒✞❁✈ ✙☞✪✱✌❀✺✒✞✍✄✼☛❈❇✵☛❂✙☞☛✍✰✲☛✍✄✝✆✻✪✱✌✎✓ R AB 0 = ✿❵☛❂✙☞☛✍✰✲☛✍✄✝✆✻✪✱✌✎✞✡✠☞☛✍✌✎☛✍✄✏✞✍✰✲✁✩✄✹★✘✑✎☛✒✺✆☛✘✪✶✳✘✞✒✑✔☛✍✌☞✰✂✙☞☛❁✰✲☛✍✌❘❇✵✪✱★☞✌✎☛✡✪✱✌✥✰✔✄✏☛❁②✘✁☎✄✝✌✘☛✒✑✔☛❁t❇ ❣ ❉❋☛✍✌☞✰✔✄✝★❀✙☞☛✍✰✲☛✍✄✝✆✻✪✶✌✘✞✍✄✏☛✒✞❁✄✼☛✛✮✯✪✐❇✏✰✲☛✍✌☞✰✲☛✒✪✎☛✒✺✆☛✘✪✱✳✎✞✒✑✔☛✍✌☞✰✲☛❜➁✯✮✱✰❜❇P☛❁✬✘✞✯❇✵✪✱✳✘✪❩✮✯☛✒✞✛✮✯✓✡✺✒✪✱✄✼✺✍★✘✪✱✰✫★✘✑ ❣ 61 U ❣ 1 ➁❜☛✛✮ ★✘✑✱✰✲✞❂☛✒✺✍★✘✞✍✰✲✪✫✪✔✑✔☛ ✂ I1 + I3 = I2 2I1 − 2I 3 = 0 ✁ ✄ ✂ 2I = 2I + U 1 ➂ ★ 3 I = 1A ❿✪✱✌①★✘2✑✶✰✲✪✱✆✟☛✒✑✔☛✡☛✒✺✍★✎✞✍✰✲✪✔✪✥✄✏☛✛✮ ★✎✑✱✰✲✞❁✈✉✦✰❄ ✄✼☛✯❇✏✬✘☛✒✺✍✰✲✪✱✳➄✕ ✍ R AB 0 = ❉❋☛✍✌☞✰✔✄✝★❀✞✡✙✥☛✍✰✲☛✍✄✹✆✟✪✱✌✘✞ U AB0 U = 0Ω 1A ❇P☛✉❇✵✺✍✄✏✪✶★✷☛✒✺✍★✎✞✍✰✲✪✔✪✔✑✔☛❂✺✒✪✱✄✼✺✍★✘✪✱✰✔★✎✑✱★✘✪✇✕ I5 = 0 ✞ ☎ I2 = I4 ☎ ☎ I1 + I 3 = I 2 ☎ ✝ I1 + I 2 + 2 I 4 = 2 ☎ ☎ I 1 + I 2 = 2 + 2 I 1 − U AB 0 ☎ ☎ ✆ ➁❜☛✛✮ ★✎✑✱✰✲✞ ☛ I3 = 1 ☛ ✟ I1 = − ✟ I2 = I4 ✟ ✟ I 4 = 1 + I1 ✡ ✟ ✟ ✟ I4 = ✡ ☞ ✟ I1 + I 4 + 2 I 4 = 2 ✟ ✟ ✠ U AB 0 = 2 I 1 + 2 I 4 1 A 4 3 A 4 ✠ ✟ ✟ 2 6 U AB 0 = − + = 1V 4 4 ➁❜☛✛✮ ★✎✑✱✰✲✞❂✠☞☛✍✌✘☛✍✄✼✞✍✰✲✁☎✄✝★✘✑✎☛✒✺✆☛✘✪✱✳✘✞✒✑✔☛✍✌✥✰❙✙✥☛✭✰✲☛✍✌❙❇P✪✶★☞✌✘☛❂✑✔✞✭②✎✁☎✄✹✌✎☛✒✑✔☛✭t✁ ✞✛❑ ▼❈ ✌❀✞✒✺✒☛✯❇✏✰✂✺✒✞✛✮ U −E =0 ❇✵✪ AB0 3 ✵❇ ✪❴✺✒0✪✱⋅✄✼I✺✍★✘= ✪✱0✰✔★✎✑✤✞✍✄✏☛①✁✾✪✱✌✘▲✝✪✱✌✘✪✶✰✲✞✍✰✲☛①✙☞☛✷❇✵✁☞✑✱★☞✰✲✪✫✪❴■✲✁☎✄✼✪✔✺✒☛ ❑✵❱ I ∈R 62 − 2 I 4 = 2 I 1 − U AB 0 ②❵❑✰❈▼✌❀✞✒✺✒☛✯❇✏✰✂✺✒✞✛✮ U ✄✼☛✛✮ AB0 ★✘✑✱✰▼✞ − E4 ≠ 0 0⋅I ≠ 0 ❇✵✪✎✺✒✪✱✄✏✺✍★✎✪✱✰✔★✘✑✥✌☞★❀✞✍✄✼☛❈❇✵✁☞✑✱★✥✰✲✪✔☛☎❱ ✺✛❑ ❈▼✌❀✞✒✺✒☛✯❇✏✰✂✺✒✞✛✮ I= U AB0 R4 ✵❇ ✪❖✬✘☛✍✌☞✰✫✄✹★ ★✥✌✘✪✔✺✒✞ ❣ R4 ≠ 0 ✺✒✪✱✄✼✺✍★✘✪✱✰✔★✎✑ ✞✍✄✼☛ ✁ ❇✵✁☞✑✶★☞✰✲✪✔☛ ✂✁☎✄✝✆✟✞✂✠☛✡✌☞✎✍✌✠☛✄✝☞✌✆✏✄✑✞✓✒✔✄✕✍✗✖ ✿✥☛❁✳✘✁☎✄➄★☞✰✲✪✔✑✔✪❩✮✯✞✭★☞✄✝✆✟✞✍✰✲✁☞✞✍✄✼☛✒✑✔☛✡✪✶✌❘❇✏✰✔✄✹★✎✺✍✰✲✪✱★☞✌✘✪✏✕ a) Descrierea elementelor de circuit - resistor R<name> <+node> <-node> [<model>] <value> R<name> = numele rezistorului <+node> , <-node> = nodurile intre care se conecteaza <model> = numele modelului (optional) <value> = valoarea rezistentei, in Ω ✣ ✘✝✙✛✚✜✘✝✢✤✣✦✥✛✧✛✢✩★✪✢✜✥✛✧✫✢✜✥✂✬✭✢✮✧✛✢✮✯✜✙✛✚✜✢✜✥✂✬ I<name> <+node> <-node> [[DC] <value>] [AC <mag> [<phase>]] [ <transient specifications> ] <+node>, <-node> = nodurile pozitiv si negativ [[DC] <value>]= semnal de curent continuu de tip Is = <value> , in A. - ✘✝✙✛✚✜✘✝✢✤✣✦✥✛✧✛✢✩★✪✢✜✥✛✧✫✢✜✥✂✬✭✢ de tensiune V<name> <+node> <-node> [[DC] <value>] [AC <mag> [<phase>]] [ <transient specifications> ] <+node>, <-node> = nodurile pozitiv si negativ [[DC] <value>] = semnal de tensiune continua de tip E = <value> , in V. - ✘✝✙✫✚✜✘✝✢ comandate liniar - sursa de tensiune comandata in tensiune (VCVS) 63 E<name> <+node> <-node> <+control> <-control> <gain> <+node> ,<-node> = nodurile pozitiv si negativ ale sursei <+contro>,<-control> = nodurile pozitiv si negativ ale portii de comanda <gain> = E/Uc - sursa de tensiune comandata in curent (CCVS) H<name> <+node> <-node> <vname> <gain> <+node>, <-node> = nodurile pozitiv si negativ <vname> = sursa de tensiune al carei curent reprezinta marimea de comanda <gain> = E/Ic - sursa de curent comandata in tensiune (VCCS) G<name> <+node> <-node> <+control> <-control> <gain> <+node> ,<-node> = nodurile pozitiv si negativ ale sursei <+contro>,<-control> = nodurile pozitiv si negativ ale portii de comanda <gain> = Is/Uc - sursa de curent comandata in curent (CCCS) F<name> <+node> <-node> <vname> <gain> <+node>, <-node> = nodurile pozitiv si negative ale sursei <vname> = sursa de tensiune al carei curent reprezinta marimea de comanda <gain> = Is/Ic b) Descrierea tipului de analiza de curent continuu .OP - realizeaza determinarea punctului de functionare in current continuu si tipareste valorile potentialelor nodurilor si ale curentilor prin sursele de tensiune in fisierul *.out. .DC [LIN] <varname> <start> <end> <incr> [<nest>] .DC [OCT][DEC] <varname> <start> <end> <points> [<nest>] .DC <varname> LIST <values [<nest>] <varname> = numele sursei a carei valoare se variaza; [LIN],[OCT],[DEC] = variatie de tip liniar, pe octave, sau decadica a valorii sursei; LIST = variatie dupa o lista de valori <value>; <start>,<end> = valorile de inceput si sfsrsit ale parametrului considerat; <incr> = increment; 64 <points> = numarul de puncte in intervalul considerat; <nest> = cea de-a doua sursa a carei valoare se variaza (optional) ; LUCRAREA 2 CIRCUITE NELINIARE DE CURENT CONTINUU 1. PROBLEME 1.1.) Sa se simuleze cu PSPICE urmatorul circuit : E1= 1 .. 10 V, r1= 1 kΩ, r2= 2 kΩ Sa se reprezinte grafic caracteristica de transfer v(2) = f(V(1)) si sa se indice utilitatea acestui circuit. 1.2.) Sa se simuleze cu PSPICE urmatorul circuit: r1= 3.0 Ω, E1=10.0 V a) Sa se aprecieze numarul de solutii al circuitului prin metoda dreptei de sarcina. b) Sa se rezolve circuitul plecand de la urmatoarele valori pentr V(2) (se foloseste comanda .NODESET) : 0.0V ; 3.0V ; 7.0V ; 65 2. INTERPRETAREA REZULTATELOR La problema 1.1. se obtine cu SPICE urmatoarea caracteristica de transfer: Se constata ca pentru valori mai mari sau egale cu 6V ale tensiunii de intrare V(1), tensiunea de iesire V(2) este egala cu 4V. Rezulta ca circuitul este un stabilizator de tensiune. La punctul a) al problemei 1.2 se traseaza dreapta de sarcina cu ecuatia E = R i + u respectiv: 10 = 3 i + u . Dreapta se intersecteaza cu axele in punctele A de coordonate (0, 3,33) si B de coordonate (10, 0). Se constata ca dreapta de sarcina se intersecteaza cu caracteristica rezistorului neliniar in punctele corespunzatoare valorilor U1 = 11765 , V , U 2 = 3,2174 V si U 3 = 4,6102 V . Rezulta ca circuitul are trei solutii, deci un numar impar de solutii in conformitate cu teorema de existenta de la paragraful 2.4.1.2. din curs. Programul SPICE rezolva circuitul utilizand metoda Newton-Raphson. In cazul in care circuitul are mai multe solutii, prin aceasta metoda se determina numai una dintre ele si anume aceea care este “mai apropiata” de aproximatia initiala. In problema, aproximatia initiala se introduce cu instructiunea NODESET. Atribuind acestuia valorile 0V, 3V si 7V se obtin succesiv cele trei solutii. Totodata se verifica faptul ca daca aproximatia initiala nu este specificata, programul SPICE ii atribuie acesteia valoarea nula V (0) = 0 si in acest caz se determina prima solutie. 3. INSTRUCTIUNI SPICE Se utilizeaza urmatoarele linii pentru descrierea circuitului: 66 1) Descrierea unei diode D<name> <+node> <-node> <model>[<area>] <+node> ,<-node> = nodurile pozitiv si negativ <model> = numele modelului <area> = numarul de diode montate in paralel (daca acest parametru nu este precizat se considera area =1); 2) Descrierea unui model .MODEL <name> <type> [<param>=<value> ] <name>= numele modelului ( de exemplu cel declarat in linia de descriere a diodei); <type>= tipul modelului (pentru dioda este D); <param> = <value> - reprezinta setarea unor parametrii ai modelului. Principalii parametri ai diodei sunt: Parametru Nume IS curent de saturatie Unitate de masura Valoare predefinita A RS rezistenta serie 1.0E-14 Ω CJ0 BV capacitaea tensiunea jonctiunii inversa de la tensiune strapungere nula F V 0 0 ∝ IBV curentul la tensiunea BV A 1.0E-3 3) Descrierea surselor comandate neliniar: - comanda de tip polinom: E<name> <+node> <-node> POLY(<value>) < <+control> <-control> >* + < <coeff> >* F<name> <+node> <-node> POLY(<value>) < <vname> >* < <coeff> >* G<name> <+node> <-node> POLY(<value>) < <+control> <-control> >* + < <coeff> >* H<name> <+node> <-node> POLY(<value>)< <vname> >* < <coeff> >* <value> = dimensiunea polinomului (1;2;3) < coeff> = coeficientii polinomului Exemplu: - polinom de dimensiune 1 e = a0 + a1uc + a2uc2 + ... - comanda de tip o expresie analitica: E<name> <+node> <-node> VALUE=[<exp>] 67 G<name> <+node> <-node> VALUE=[<exp>] <exp> - o expresie analitica; - comanda de tip tabel (intre puncte se considera extrapolarea liniara): E<name> <+node> <-node> TABLE [<exp>] < (inval), (outval) >* G<name> <+node> <-node> TABLE [<exp>]= < (inval), (outval) >* Cu acest ultim tip de surse comandate se modeleaza rezistoarele neliniare cu caracteristici liniare pe portiuni (PWL) . LUCRAREA 3 CIRCUITE CU EXCITATIE SINUSOIDALA CARE FUNCTIONEAZA LA SEMNALE MICI SAU LA SEMNALE MARI 1. PROBLEME 1.1.) Sa se simuleze cu PSPICE urmatorul circuit: Rb = 1MΩ ; Rc = 4kΩ; E1 = 5V; Vcc = 9V; e2 = E2 sin (ωt); TB: β= 200 ; Cje = 2pF; Cjc = 2pF; Vje = 0.6V; Vjc = 0.6V; a) Sa se reprezinte grafic caracteristica de transfer in curent continuu Ve = f (Vi), unde Vi ∈[0,5E 1 ] , iar Ve este tensiunea pe rezistorul Rc. b) Sa se faca analiza de curent alternativ la semnal mic a circuitului pentru E2=1.0 V si respectiv E2= 20.0 V. Sa se reprezinte grafic Ve=f (frecv) , frecventa variind decadic intre 0.1 Hz si 100Mhz. Sa se discute valabilitatea rezultatelor obtinute prin analiza de semnal mic pentru cele doua valori ale lui E2, timand seama de caracteristica determinate la punctul a). 2. INTERPRETAREA REZULTATELOR La punctul a) al problemei, din analiza caracteristicii de transfer in curent continuu Ve = f (Vi ) prezentata in figura 1, rezulta ca aceasta prezinta doua portiuni liniare distincte: 68 1 0 V 5 V 0 V 0 V 5 V 1 0 V 1 5 V 2 0 V 2 5 V V 1 v (5 )-v (4 ) Fig. 1 prima portiune pentru Vi ≤ 11,25V si a doua portiune pentru Vi > 11,25V . Acest rezultat se datoreaza faptului ca tranzistorul are caracteristici neliniare de intrare si de iesire atunci cand functioneaza la semnal mare. Prima portiune liniara a carcateristicii u e = f (ui ) corespunde conditiilor de functionare la semnal mic. La punctul b) al problemei in urma analizei in curent alternativ a circuitului rezulta Ve = f ( frecv ) pentru E 2 = 10 . V , fig. 2 si E 2 = 20.0 V , fig. 3. 1 .0 V 0 .5 V 0V 1 0 0 m H z 1 .0 H z 100H z 10K H z V (5 )-V (4 ) F re q u e n c y Fig. 2 69 1 .0 M H z 100M H z 2 0 V 1 0 V 0 V 1 0 0 m H z 1 .0 H z 1 0 0 H z 1 0 K H z 1 .0 M H z 1 0 0 M H z V (5 )-V (4 ) F re q u e n c y Fig. 3 Se constata ca alura celor doua curbe este identica, curba din figura 3 reproducand la o alta scara curba din figura 2. Luand in considerare caracteristica determinata la punctul a) rezulta ca rezultatul obtinut la punctul b) pentru E 2 = 20.0 V nu este valabil, deoarece pentru Vi > 11,25 V circuitul functioneaza la semnal mare, deci modelul liniar la semnal mic nu este valabil. 3. INSTRUCTIUNI SPICE Se vor utiliza urmatoarele linii: a) descrierea tranzistorului bipolar: Q<name> <c> <b> <e> [<subs>] <model> <c> = nodul corespunzator colectorului <b> = nodul corespunzator bazei <e> = nodul corespunzator emitorului <subs> = nodul corespunzator substratului (optional) <model> = numele modelului b) descrierea modelului tranzistorului bipolar : .MODEL <name> <type> [<param>=<value> ] <name>= numele modelului (declarat in linia de descriere a tranzistorului) <type>=tipul modelului (pentru tranzistor bipolar este PNP sau NPN): <param> = <value> - reprezinta setarea unor parametrii ai modelului. Principalii parametri ai tranzistorului bipolar (modelul Gummel-Poon) sunt: Parametru Nume IS curent de BF amplificarea RB rezistenta 70 RE rezistenta RC rezistenta Unitate de masura Valoare predefinita Parametru Nume Unitate de masura Valoare predefinita saturatie A 1.0E-16 in curent β - bazei Ω 100 emitorului Ω 0 colectorului Ω 0 0 Vje tensiune de deschidere BE V Vjc tensiune de deschidere BC V Cje capacitatea BE (jonctiune nepolarizata) F Cjc capacitatea BC (jonctiune nepolarizata) F 0.75 0.75 0 0 c) descrierea analizei in curent alternativ a circuitului echivalent de semnal mic: .AC [LIN][OCT][DEC] <points> <start> <end> [LIN][OCT][DEC] = variatia liniara, pe octave, sau decadica a frecventei; <points> = numarul de puncte in intervalul considerat; <start>, <end> = capetele intervalului de frecventa considerat; Pentru sursele independente forma generala este: V<name> <+node> <-node> AC <mag> [<phase>] I<name> <+node> <-node> AC <mag> [<phase>] <+node>, <-node> = nodurile pozitiv si negativ AC <mag> [<phase>] = semnal de curent alternativ de tipul x = <mag> * sin (2 πf t + phase), unde x este tensiune sau curent si f este frecventa specificata in linia de comanda.AC 71 LUCRAREA 4 CIRCUITE LINIARE DE CURENT ALTERNATIV 1. PROBLEME 1.1.) Sa se simuleze cu PSPICE urmatorul circuit: Se dau: r1=10Ω; r2=1Ω; r3=2Ω; c2=1.5mF; L1=50mH; L2=25mH; L3=50mH; M=25mH; e(t)=30 sin(2πft + π/2); is(t)=2 sin(2πft); f ∈ (10Hz,200Hz) Se cere sa se calculeze curentul prin r2 (modulul, argumentul, partea reala si partea imaginara) pentru f=50Hz si sa se verifice concordanta acestor rezultate. 1.2.) Sa se traseze cu PSPICE caracteristicile de amplitudine si de faza ale amplificarii in tensiune Au= Ve/e pentru urmatoarele filtre in banda de frecvente 1Hz-1MHz. Sa se indice tipul fiecarui filtru. Fig.a) r1 =1kΩ; c1 = 8nF; c2 =11.5nF L1 = 3mH;L2 =12mH; e(t)=1 sin (ωt) 72 Fig. b) r1 =1,5 kΩ; c1 = 10nF; c2 = 8uF L1 = 2mH;L2 =12mH; e(t)=1 sin (ωt) Fig.c) r1 =1,5 kΩ; c1 = 10nF; c2 = 8uF L1 = 2mH; c3=1nF e(t)=1 sin (ωt) ; Rs=1.5kΩ Fig. d) r1 =1,5 kΩ; c1 = 0,1uF; c2 = 40uF L1 = 10mH; L2 =20mH; c3=1uF e(t)=1 sin (ωt) ; L3=4mH 2. INTERPRETAREA REZULTATELOR La problema 1.1. chestiunile care se vor urmari sunt: - modul de descriere a elementelor de circuit, bobine, bobine cuplate si condensatoare, in circuitele de curent alternativ; - modul de obtinere si tiparire a marimilor electrice, curenti si tensiuni, la o frecventa precizata; La problema 1.2., pentru fiecare punct, se va trasa caracteristica de amplificare si se vor obtine urmatoarele tipuri de filtre: trece jos, trece sus, trece banda si opreste banda. 3. INSTRUCTIUNI SPICE Se vor utiliza urmatoarele linii: a) descrierea bobinelor: L<name> <+node> <-node> [<model>]<value> [IC=<initial>] <+node> <-node>= nodurile intre care se conecteaza model = numele modelului (optional) 73 value = valoarea inductivitatii (in H) IC=<initial>= conditia initiala (pentru curent) (se foloseste pentru .TRAN cu optiunea SKIPBP) b) descrierea cuplajului K<name> L<name> < L<name> >* <coupling> L<name> < L<name> >* = bobinele intre care exista cuplaj <coupling> = factorul de cuplaj ( K ≤ M / L1 L2 ) c) descrierea condensatorului C<name> <+node> <-node> [<model>]<value>[IC=<initial>] <+node> ,<-node> = nodurile pozitiv si negativ <value> = valoarea capacitatii in F IC=<initial> = tensiunea initiala, in V, pe condensator (se foloseste pentru .TRAN cu optiunea SKIPBP) d) descrierea analizei in curent alternativ a circuitului echivalent de semnal mic: .AC [LIN][OCT][DEC] <points> <start> <end> [LIN][OCT][DEC] = variatia liniara, pe octave, sau decadica a frecventei; <points> = numarul de puncte in intervalul considerat; <start>, <end> = capetele intervalului de frecventa considerat; Pentru sursele independente forma generala este: V<name> <+node> <-node> AC <mag> [<phase>] I<name> <+node> <-node> AC <mag> [<phase>] <+node>, <-node> = nodurile pozitiv si negativ AC <mag> [<phase>] = semnal de curent alternativ de tipul x = <mag> * sin (2 πf t + phase), unde x este tensiune sau curent si f este frecventa specificata in linia de comanda .AC d) Comanda .PRINT pentru analiza de tip .AC .PRINT AC VM(1) VP(5) IR(r1) VI(5,4) VM(1) = amplitudinea potentialului nodului 1 (modulul numarului complex); VP(5) = faza potentialului nodului 5 (argumentul numarului complex); IR(r1) = partea reala a curentului prin r1; VI(5,4) = partea imaginara a tensiunii intre nodurile 5 si 4. 74 LUCRAREA 5 SUBCIRCUITE. ANALIZA IN REGIM TRANZITORIU 1. PROBLEME 1.) Sa se determine caracteristica amplificarii A= U2 / U1 pentru urmatorul circuit cu amplificatoare operationale: a) E1=10-3sin(2πft) (V), f∈(0.1Hz , 10MHz) Pentru amplificatorul operational se va utiliza urmatorul model, care va constitui un subcircuit: ri = 1MΩ; re=0.1mΩ; Ce=0.1pF b) sa se simuleze in regim tranzitoriu acelasi circuit cu f=500Hz si E1 = 1mV , E1 = 10V ; sa se vizualizeze tensiunea de iesire sis a se explice rezultatele obtinute. 2.) Sa se simuleze in regim tranzitoriu urmatorul circuit : 75 Rb = 1MΩ ; Rc = 4kΩ; E1 = 5V; Vcc = 9V; e2 = E2 sin (ωt); f=10kHz; TB: β= 200 ; Cje = 2pF; Cjc = 2pF; Vje = 0.6V; Vjc = 0.6V; Sa se faca analiza pentru E2=1.0 V si respectiv E2= 20.0 V. Sa se reprezinte grafic Ve.Sa se compare formele de unda obtinute pentru cele doua valori ale lui E2 .Sa se discute valabilitatea rezultatelor obtinute la lucrarea L3 pentru acelasi circuit. 2. INTERPRETAREA REZULTATELOR La problema 1. chestiunile care se vor urmari sunt: - Modul de utilizare a subcircuitelor; - Analiza in curent alternativ (.AC) excitand circuitul in gama de frecvente 0,1 Hz - 10 Mhz; - Se va verifica prin analiza in regim tranzitoriu ca pentru amplitudinea de 1 mV la intrare se obtine acelasi rezultat ca la analiza in curent alternativ deci circuitul functioneaza la semnale mici. Daca tensiunea de intrare are amplitudinea de 10V forma de unda la iesire nu este sinusoidala deci circuitul functioneaza la semnal mare si rezultatele obtinute cu analiza de semnal mic (AC) nu sunt valabile. La problema 2. simuland circuitul in regim tranzitoriu se obtin pentru E2=1.0 V figura 1 si pentru E2= 20.0 V figura 2. 10V 5V 0V 1 0 0 us 150us V (5 )-V (4 ) 200us 2 5 0 us V (2 ) T im e Fig. 1 76 3 0 0 us 350us 400us 40V 0V -4 0 V 100us 150us V (5 )-V (4 ) 200us 250us 300us 350 us 400us V (2 ) T im e Fig. 2 Din figura 2 rezulta ca forma de unda a tensiunii Ve pe rezistorul de sarcina Re, care la o anumita scara reprezinta curentul de colector, este distorsionata. Acest lucru se produce deoarece pentru circuitul neliniar cu tranzistor se poate folosi modelul liniar in curent alternativ numai pentru functionarea la semnale mici. E2=20.0 V corespunde functionarii la semnale mari. 3. INSTRUCTIUNI SPICE Se folosesc urmatoarele linii: a) Semnalul sinusoidal amortizat este descris astfel: Pentru sursele independente forma generala este: V<name> <+node> <-node> SIN (voff vampl freq td df phase) I<name> <+node> <-node> SIN (ioff iampl freq td df phase) I= ioff + iampl*expi-(t-td)*df*sin(2*3.14*freq*(t-td)+3.14*Phase/180) unde td < t < TSTOP Marime ioff iampl freq td df Phase Semnificatie Valoare predefinita 1/TSTOP 0.0 0.0 0.0 Valoare de offset Amplitudinea Frecventa Timp de intarziere Factor de amortizare Faza Unitate de masura A A Hz s s-1 grade b) comanda de analiza in regim tranzitoriu este: .TRAN <pstep> <ftime> [<noprint> [<ceiling>]][SKIPBP] <pstep> =pasul de timp pentru tiparirea rezultatelor; <ftime> = timpul final; <noprint> = timpul de la care se incepe tiparirea (vizualizarea) rezultatelor; 77 <ceiling> =limita superioara a pasului de timp; daca nu se specifica se considera ceiling=ftime/50 pentru un circuit care contine elemente dinamice sau ceiling=pstep pentru un circuit rezistiv. SKIPBP = utilizarea conditiilor initiale impuse (U=<value> pentru condensator si I=<value> pentru bobina, care sunt specificate in liniile care definesc elementele respective). c) definirea unui subcircuit .SUBCKT <name> [<nodes>*] [PARAMS: <par>[=<val>]* ] ..... (descrierea elementelor) .ENDS Utilizarea subcircuitului : X<name> [<nodes>]* <sname> [PARAMS: <<par>=<val>*>] <name> = numele modelului (definit in linia .SUBCKT) <sname> = numele elementului modelat cu subcircuitul respectiv; <nodes> = nodurile prin care subcircuitul se conecteaza cu circuitul 78 LUCRAREA 6 REGIMUL TRANZITORIU AL CIRCUITELOR NELINIARE 1. PROBLEME 1.) Pentru circuitul din figura sa se faca analiza Fourier a tensiunii de iesire pentru E2 =1 mV si E2 =10 V incazul in care frecventa sursei e2 este f=1 kHz. Rb = 20 kΩ ;C=100 uF, Rc = 100 Ω; E1 = 5V; Vcc = 9V; e2 = E2 sin (ωt); TB: β= 200 ; Cje = 200 pF; Cjc = 200 pF; Vje = 0.6V; Vjc = 0.6V; 2.) Sa se simuleze convertorul dc - dc : L=20mH ; C=1.25uF; R=400Ω ; E=100V; Comutator: - frecventa de comutatie f=10kHz ; - durata de inchidere reprezinta 50% din perioada de comutatie; - rezistente : Rinchis = 1Ω ; Rdeschis = 1e+06Ω ; D - dioda avand modelul predefinit de PSPICE; Sa se vizualizeze tensiunea de comanda a comutatorului(pe un interval de 5 perioade de comutatie) si tensiunea la iesirea convertorului(atat pe intreaga durata a regimului tranzitoriu cat si 3 perioade din regimul permanent). Sa se traseze spectrul de frecvente pentru tensiunea de comanda a comutatorului si pentru tensiunea la iesirea convertorului. 2. INTERPRETAREA REZULTATELOR 79 La problema 1., realizand analiza Fourier a tensiunii de iesire rezulta fig. 1 pentru E2=1.0 mV si fig. 2 pentru E2=10.0 V. Din analiza celor doua figuri rezulta ca tensiunea de iesire in cazul E2=10.0 V, care are forma de unda distorsionata, contine o gama larga de armonici superioare. 5.0V 2.5V 0V 0Hz 5KHz V(5)- V(4) 10KHz 15KHz 20KHz 25KHz 30KHz Frequency Fig. 1 1.0V 0.5V 0V 0Hz 10KHz 20KHz 30KHz V(5,4) Frequency Fig. 2 La problema 2. chestiunile care se vor urmari sunt: - modul de utilizare a comutatorului; - descrierea semnalelor de tip puls pentru surse in regim tranzitoriu; -trasarea spectrului de frecvente al unei marimi periodice 80 40KHz 50KHz 1100mV 800mV 400mV 0V 4.5ms V(4) 4.6ms 4.8ms 5.0ms Time Fig.3 In fig. 3 se prezinta semnalul generat cu functia pulse, iar in fig. 4 se prezinta semnalul la iesirea convertorului. Spectrele tensiunii de comanda si pentru tensiunea de iesire sunt prezentate in fig. 5 - 7. Figura 7 contine un detaliu al spectrului tensiunii de iesire care permite evidentierea armonicelor frecventei de comanda. 300V 200V 100V 0V 0s 2.0ms 4.0ms V(3) Time Fig.4 81 5.0ms 800mV 400mV 0V 0Hz 100KHz 200KHz 300KHz 400KHz V(4) Frequency Fig. 5 Se observa ca spectrul tensiunii de comanda este discret (contine numai componentele armonice ale frecventei de comutatie si componenta de current continuu) deoarece aceasta marime este periodica. Tensiunea la iesirea convertorului nefiind periodica (deoarece contine si componente tranzitorii care nu s-au amortizat) spectrul acesteia nu este discret. 200V 100V 0V 0Hz 10KHz 20KHz 30KHz 40KHz V(3) Frequency Fig.6 Pentru trasarea spectrului tensiunii de iesire s-au considerat numai ultimile 5 perioade din raspuns (o forma de unda aproapiata de o functie periodica). 82 1.0V 0.5V 0V 9KHz 25KHz 50KHz 75KHz 100KHz V(3) Frequency Fig.7 3. INSTRUCTIUNI SPICE Se vor utiliza urmatoarele linii: a) Semnalul de tip puls pentru surse in regim tranzitoriu este descris de: PULSE(i1 i2 td trise tfall pw per) pentru sursa de curent PULSE(v1 v2 td trise tfall pw per) pentru sursa de tensiune Marime i1 i2 td trise tfall pw per Semnificatie Valoare predefinita 0.0 TSTEP TSTEP TSTOP TSTOP Valoare initiala Valoare de varf Timp de intarziere Timp de crestere Timp de descrestere Durata impulsului Perioada 83 Unitate de masura A A s s s s s b) comutatorul comandat in tensiune S<name> <+node> <-node> <+control> <-control> <model> <+node> <-node> = nodurile intre care se conecteaza; <+control> <-control> = nodurile tensiunii de comanda; <model> = numele modelului; .MODEL <model> VSWITCH ( RON=... , ROFF=... , VON= ... , VOFF= ... ) Valori predefinite : RON=1Ω ; ROFF=1MΩ (rezistenta echivalenta a comutatorului in pozitiile inchis si deschis); VON=1V; VOFF=0V (valorile tensiunii de comanda la care comutatorul se inchide si se deschide) ; d) Analiza Fourier a unui semnal este descrisa de linia: .FOUR <freq> <output var>* <freq> = frecventa considerata fundamentala ; <output var> = semnalele pentru care se face analiza Fourier; 84 LUCRAREA 7 COMPORTAREA CALITATIVA A CIRCUITELOR DE ORDINUL II 1. PROBLEME 1.) Sa se simuleze circuitul cu conditiile initiale si valorile din tabelul 1 : L=1H ; C=1F; Caz 1 2 3 4 5 R [Ω] 3 -3 1 -1 1e-9 Conditii initiale (uc(0),il(0)) (1,1) (-1,1) (1,-1) (-1,-1) (1,1) (-1,1) (1,-1) (-1,-1) (1,1) (-1,1) (1,-1) (-1,-1) (1,1) (-1,1) (1,-1) (-1,-1) (1,1) (-1,1) (1,-1) (-1,-1) Tabelul 1 Sa se indice in cazurile 1 .. 5 tipul comportarii calitative a circuitului in jurul punctului de echilibru. 2. a) Sa se simuleze circuitul din figura pentru i (0) = 6A. Sa se vizualizeze i(t) si curba i = f(V(1)) si sa se explice rezultatul obtinut. b) Intre nodurile 1 si 0 se conecteaza un condensator cu C= 1mF avand conditia initiala Uc0= 8V. Sa se vizualizeze traiectoria in planul fazelor (iL – uC ). Sa se compare rezultatele obtinute pentru eroarea relative RELTOL cu valoarea predefinita( 1e-05) si valoarea impusa de 1e-07. 2. INTERPRETAREA REZULTATELOR 85 1. Circuitul din problema 1. este un circuit liniar, dinamic, de ordinul doi, cu excitatie nula, cu ecuatiile: uC + Ri + U L = 0 ✁✄✂ iC = iL = i Tinand cont de ecuatiile de functionare ale bobinei si condensatorului ☎ ✟ ✆ ✝✞ U L = Li L ✝ i C = CU C ✟ rezulta ecuatiile de stare ale circuitului: ✠ ✡ ✡ ☛☞ ✌ ✌ ✠ UC iL 0 ☞ ✍✏✎ =☞ ☛☞ − 1 L 1 ✡ ✌ C ✎☛ UC ✍ R ✎☞ − ✍ ✎ iL L ✎ Pentru a studia comportarea calitativa a unui circuit liniar cu excitatia nula se determina valorile proprii s1 si s2 ale matricei A a ecuatiei x✑ = Ax . Valorile proprii s1 si s2 sunt solutiile ecuatiei: ✔ ✗ 1 ✕ ✒ 0−s ✕ ✒ C det ✕ =0 ✒ 1 R ✖ − ✓ − −s L L respectiv s2 + R 1 s+ =0 L L ⋅C care pentru L=1H si C=1F devine s 2 + Rs + 1 = 0 Punctele de echilibru se determina rezolvand sistemul de ecuatii Ax=0 adica ✛ ✘ ✛ ✘ ✙✚ ✤ In cazul problemei ∆ = ✥✦ 0 1 −1 − R a11 a 21 a12 x1Q ✜✣✢ ✙✚ ✜ ✢ = 0 a 22 x 2 Q ✧ ★✏✩ si ∆ = 1 deci ∆ ≠ 0 si sistemul admite numai solutia banala si originea este singurul punct de echilibru adica x1Q = x 2Q = 0 . Evolutia circuitului plecand de la o stare initiala(uC(0) si iL(0)) poate fi reprezentata printr-o curba numita traiectorie de faza in planul de coordonate uC si iL numit planul fazelor. Evolutia circuitului corespunzatoare mai multor stari initiale poate fi reprezentataprintr-o multime de traiectorii in planul fazelor. Aceste traiectorii formeazaun portret de faza. Considerand perechile de valori U C 0 si I l 0 din tabelul 1 se determinatraiectoriile corespunzatoare starilor initiale date, care formeaza portrete de faza in cele 4 cazuri considerate. In functie de valorile rezistentei R din tabelul 1 rezulta 4 cazuri. Cazul1. R = 3 Ω si valorile proprii sunt: s1 = −3 + 5 2 s2 = deci s2 < s1 < 0 si originea este un nod stabil. 86 −3 − 5 2 Cazul 2. R = −3 Ω si valorile proprii sunt: s1 = 3+ 5 2 s2 = 3− 5 2 deci s2 > s1 > 0 si originea este un nod instabil. Cazul 3. R = 1 Ω si valorile proprii sunt: s1 = − 1 3 +j 2 2 s2 = − 1 3 −j 2 2 1 2 deci constanta de atenuare α = − si pentru α < 0 , punctul de echilibru este un focar stabil si corespunde raspunsului periodic amortizat. Cazul 4. R = −1 Ω si valorile proprii sunt: s1 = deci constanta de atenuare α = 1 2 1 3 +j 2 2 s2 = 1 3 −j 2 2 si in acest caz α > 0 , punctul de echilibru este un focar instabil. 87 Solutia x(t) a ecuatiei de stare x = Ax , are doua componente: uC(t) si iL(t). Cazul 5 R 0 si valorile proprii sunt s1, 2 = ± j si originea este centru ✁ 2. Pentru circuitul dinamic neliniar de la problema 2. punctul a), parcursul dinamic este: unde punctele Q1(1,-1) si Q2(-1,1) sunt puncte de impas iar originea 0 este un nod instabil. Simuland circuitul cu PSPICE se constata ca traiectoria evolueaza pe curba plecand din punctul corespunzator conditiei initiale iL(0) si se opreste in punctul de impas intalnit Q1(1,-1). La punctul b) al problemei 2. se constata ca leg nd la bornele circuitului un condensator C, circuitul se comporta ca un oscilator de relaxare. In cazul circuitului real nu este necesara introducerea in circuit a condensatorului C, rolul acestuia fiind indeplinit de capacitatea parazita dintre bornele circuitului. Cel de-al doilea punct al problemei 2. are ca scop ilustrarea preciziei analizei efectuata cu PSPICE in functie de diferite valori date pentru eroarea relativa RELTOL. ✂ 88 RELTOL=1e-05 RELTOL=1e-07 3. INSTRUCTIUNI SPICE .OPTIONS [<fopt>*] [<vopt>=<value>*] - setarea unor optiuni: Flag Options ACCT summary & accounting EXPAND show subcircuit expansion LIBRARY list lines from library files LIST output summary NODE output netlist NOECHO suppress listing NOMOD suppress model parameter listing NOPAGE suppress banners OPTS output option values Value Options ABSTOL best accuracy of currents CHGTOL best accuracy of charges CPTIME CPU time allowed DEFAD MOSFET default AD DEFAS MOSFET default AS DEFL MOSFET default L 89 DEFW MOSFET default W GMIN minimal conductance, any branch ITL1 iteration number - DC & bias point blind limit ITL2 iteration number - DC & bias point guess limit ITL4 iteration number - transient per-point limit ITL5 iteration number - transient total, all points LIMPTS maximal number of points for print/plot NUMDGT number of digits in output file PIVREL relative magnitude for matrix pivot PIVTOL absolute magnitude for matrix pivot RELTOL relative accuracy of voltages and currents TNOM default temperature TRTOL transient accuracy adjustment VNTOL best accuracy of voltages WIDTH printing width in output file 90 LUCRAREA 8 PROPRIETATI CALITATIVE IN CIRCUITELE ELECTRICE NELINIARE IN REGIM TRANZITORIU 1. PROBLEME Sa se simuleze circuitul cu urmatorii parametri: - dioda D : IS=8.3e-15A; RS=9.6Ω; TT=4e-06s (timp de tranzit) ; CJO=300pF ; M=0.4 (gradient); VJ=0.75V (potential de jonctiune); R = 15Ω ; L = 10mH; e(t)= E0sin (ωt) ; f=100kHz. Se vor utiliza un pas de afisare de T/2000=5.e-09 s si o valoare RELTOL=1e-6. Sa se reprezinte grafic dependenta i(t) = f ( e(t)) si sa se interpreteze rezultatele in urmatoarele situatii: a) E0 = 0.5 V . Se vor simula 40 perioade , dintre care se vor afisa ultimele 4. b) E0 = 1.5 V . Se vor simula 40 perioade, dintre care se vor afisa ultimele 4. c) E0 = 6.5 V . Se vor simula 60 de perioade, dintre care se vor afisa ultimele 20. 2. INTERPRETAREA REZULTATELOR In urma simularii se determina raspunsul I(L1) pentru cele trei cazuri: a) pentru E0=0,5 V rezulta un raspuns periodic de frecventa excitatiei. 91 400uA 400uA 0A 0A -400uA 360us I(l1) 370us 380us 390us 400us -400uA -500mV I(l1) Time 0V 500mV V(1) Fig. 1. E0=0,5 V Comportarea circuitului in acest caz este o comportare obisnuita deoarece pentru amplitudini relativ mici ale excitatiei comportarea diodei poate fi aproximata prin modelul de semnal mic. b) pentru E0=1,5 V se observa raspunsul subarmonic(periodic de frecventa ω /2). 1.0mA 1.0mA 0.5mA 0.5mA 0A 0A -0.5mA 360us I(l1) 370us 380us 390us -0.5mA -2.0V I(l1) 400us Time -1.0V 0V 1.0V 2.0V V(1) Fig. 2. E0=1,5 V c) pentru E0=6,5 V se observa comportarea haotica a circuitului(raspuns neperiodic la excitatie periodica). 92 4.0mA 4.0mA 2.0mA 2.0mA 0A 0A -2.0mA 400us I(l1) 450us 500us 550us -2.0mA -8.0V I(l1) 600us -4.0V 0V V(1) Time Fig. 3. E0=6,5 V 93 4.0V 8.0V CAP 4. SOLUTIILE PROBLEMELOR LUCRAREA 1 Problema 1 ✁✂✂ ✄✆☎✞✝✟✄✡✠☛☎✌☞✎✍☛✏✆✑✒✄✆✝✓☎✞✔✕☞✖☎✌✗✓✘ ✚✙✜✛✣✢✌✙✥✤✦✛✣✧✕★✂✩✪✙✜✙✟✫☛✬✮✭✰✯✂✢ ✬ ✱ ✝ ✬✲✬✆✳✵✴ ✝ ✳✲✶✸✷✸✹ ✝ ✶✵✴✺✳✵✹ ✝✼✻ ✶ ✻ ✴ ☎ ✴✺✶✲✳✽✬ ✏ ✶✵✴ ✻ ✶✾✳✸✿ ❀ ✬✲✬❁✷✲✿ ❀ ✹ ✻ ✷❂✬❃✳ ✗ ✔ ✪ ✍✺✄❄❀ ✬✓✷❂✬❅✷❂✬ ✱ ✔❆✝✟✗✪❇✓✏ ✱ ✏❈✘❉✍ ✱ ✱ ✂✂✁ ✑❋❊❍● ✫■✫ ✑✂☎❑❏▲✘✺● ✫ ❇✓☎▼●✸✑◆✑✒✗ ✫ ✠❉☞✖☎✌✗✓✘ ✘✾✗✼✍❙✏ ❀❉✗ ✫ ☞❈●❱❏✪✏ ✘✾✗✼✍❙✏ ❀❉✗ ✫ ☞❈●❉❏✼✏ ☞❈✏■❊❖✔❆✏■✝❁●❉☞❈✠☛✝✓✏◗P ✘✾✗✪✍☛✏ ✿ ✁✷ ✷✂✷✂✷ ❲ ✳✁❳ ❘ ✻ ✷✂✷✂✷ ❲ ✴❩❳❭❬❨✬ ✻ ✷✂✷✂✷ ❲ ✱ ✱ ✱ ❪ ❳ ❃ ✬ ✳ ✂ ✷ ✂ ✷ ✁ ✷ ✷ ❲ ✻ ✱ ✫ ❀❉✗ ☞❈●❱❏✪✏❫✑✒✗✟✠✺✝✟✄✆✏❴✄✡✠✺✝✓✝✓✏❵✘✲☞✡✑ ❲ ✬❨❳ ✘✺●✺❊❖✏ ❀ ✬ ❀ ✹ ✶✁❳ ❀❱✗ ✫ ☞❈●❉❏✼✏ ✹ ✁✷ ✷✂✷✂✷ ✱ ✄✡✠☛✝✓✝✓✏❈✘✾☞ ❬ ✻ ✂✷ ✷✂✷ ✏ ❬▼✷✖✬ ❬ ✻ ✱ ✷✂✷✂✷ ✏ ▼❬ ✷✖✬ ✱ ☞■✗❙☞❈● ✫ ✔✕✗✼❛❜✏■✝❝✍☛☎❞✑❆✑✂☎❡✔❢●❉☞✖☎✌✗✓✘ ✬ ✷ ❣✻ ✏■❤ ✷✖✬ ❛❜●❉☞■☞✡✑ ✱ ✁✂✂ ❉ ❀ ✗ ✫ ☞❈●❱❏✪✏ ❬ ✄✓✗▲✘✾☞✖✝✟✗ ✫■✫ ✏■✍✐❀❉✗ ✫ ☞❈●❱❏✪✏❫✑✒✗✟✠✺✝✟✄✆✏❁✑ ✏ ✶ ✘❉●☛❊❖✏ ❀ ❬✒ ✑ ✗✟✠✺✝✟✄✆✏ ❬☎ ✑✁✗✪✠☛✝✟✄✆✏ ❬❨✬ ✴ ✻ ✷ ✏■❤ ✷✖✬ ✬ ✻ ✱ ✷✂✷ ✏■❤ ✷✁✷ ✱ ❥❨✗✼❇❜✄✓✗▲✘✾✄ ✫ ✠☛✍☛✏❦✍ 94 ✳✒❘ ✂✷ ✷✂✷ ✍❙✏❚❏❯✄ ✱ ✘✲✗✪✍☛✏ ❀❱✗ ✫ ☞❈●❉❏✪✏ Problema 2 ✁✂✂a. ✄✆☎✞✝✟✄✡✠☛☎✌☞✎✍☛✏✆✑✒✄✆✝✓☎✞✔✕☞✖☎✌✗✓✘ pct ✙✜✛✣✢ ✙✥✤ ✛✣✧✕★✂✩✼✙✜✙✟✫❙✬❃✭✰✯✂✢ ✳ ✱ ✝ ✬✓✷❂✬✲✬ ✝ ✳✲✳✵✴ ✬ ✝ ✴✲✴❱✷❱✳ ☎ ✬✓✷❱✳✽✬ ❀ ✬✆✳✽✬✆✳ ✴ ❀ ✬▲❬ ✳ ✁ ✶✲✬✆✶✶ ✷❂ ✬ ❀ ✾ ✶ ✸ ✶ ❱ ✷ ✳ ✝ ✱ ✱ ❲ ✗ ✔ ✪ ✏❈✘❉✍ ✂✂✁ ✑❋❊❍● ✫■✫ ✑✂☎❑❏▲✘✺● ✫ ❇✓☎▼●✸✑◆✑✒✗ ✫ ✠❉☞✖☎✌✗✓✘ ✘✾✗✼✍❙✏ ❀❉✗ ✫ ☞❈●❱❏✪✏ ✘✾✗✼✍❙✏ ❀❉✗ ✫ ☞❈●❉❏✼✏ ✬❨❳❭❬ ✳ ✁✷ ✷✂✷✂✷ ❲ ✳✁❳ ✷ ✷✁✷✂✷✂✷ ❲ ✴❩❳❭❬✞✴ ✷✁✷✂✷✂✷ ✱ ✱ ✱ ❀❉✗ ✫ ☞❈●❱❏✪✏❫✑✒✗✟✠✺✝✟✄✆✏❴✄✡✠✺✝✓✝✓✏❵✘✲☞✡✑ ✘✺●✺❊❖✏ ✘✾✗✪✍☛✏ ❲ ✶✁❳ ❀❱✗ ✫ ☞❈●❉❏✼✏ ✳✒❘ ✂✷ ✷✂✷ ✍❙✏❚❏❯✄ ✱ ✘✲✗✪✍☛✏ ❀❱✗ ✫ ☞❈●❉❏✪✏ ✬ ✁✷ ✷✂✷✂✷ ✱ ✄✡✠☛✝✓✝✓✏❈✘✾☞ ❀ ✬ ❀ ✶ ❬ ✳ ✂✷ ✷✂✷ ✏■❤ ✂✷ ✷ ✶ ✱ ✻ ✂✷ ✷ ■ ✏ ❤ ✷✂✷ ✱ ☞■✗❙☞❈● ✫ ✔✕✗✼❛❜✏■✝❝✍☛☎❞✑❆✑✂☎❡✔❢●❉☞✖☎✌✗✓✘ ✂✂✁ ☞❈✏■❊❖✔❆✏■✝❁●❉☞❈✠☛✝✓✏◗P ❬ ✻ ✂✷ ✷ ✏ ❬▼✷✖✬ ❛❜●❉☞■☞✡✑ ✱ ✗✪✔❆✏■✝❁●❉☞✖☎ ✘✲❏ ✔✕✗✼☎ ✘✲☞✎☎ ✘✄✂ ✗✪✝✓❊❍●❱☞❈☎✌✗✓✘ ☞✖✏■❊❖✔❆✏❦✝❁●❱☞❈✠☛✝✓✏✽P ✁✂✂ ✄✡✠☛✝✓✝✓✏❈✘✾☞ ❬ ✄✓✗✓✘✲☞❈✝✟✗ ✫■✫ ✏■✍✐❀❱✗ ✫ ☞❈●❉❏✪✏❝✑✁✗✪✠☛✝✟✄✆✏❁✑ ✘❉●☛❊❖✏ ❀ ❬✒ ✑ ✗✟✠✺✝✟✄✆✏ ❬☎ ✑✁✗✪✠☛✝✟✄✆✏ ✁ ✬ ✶ ✷✁✷✂✷ ✏■❤ ✷✂✷ ❬ ✶ ✻ ✱ ✷✂✷ ✏■❤ ✷✁✷ ✱ pct b. ✁✂✂ ✄✆☎✞✝✟✄✡✠☛☎✌☞✎✍☛✏✆✑✒✄✆✝✓☎✞✔✕☞✖☎✌✗✓✘ ✙✜✛✣✢ ✙✥✤ ✛✣✧✕★✂✩✼✙✜✙✟✫❙✬❃✭✰✯✂✢ ✳ ✱ ✝ ✬✓✷❂✬✲✬ ✝ ✳✲✳✵✴ ✬ ✝ ✴✲✴❱✷❱✳ ☎ ✬✓✷❱✳✽✬ ❀ ✬✆✳✽✬✆✳ 95 ✳✒❘ ✂✷ ✷✂✷ ✍❙✏❚❏❯✄ ✱ ✬✆✶ ✴ ❀ ✬▲❬ ✳ ✲ ✶ ❀ ✶ ✷❱✳ ✝ ✶✾✶✸✷❱✳ ✁ ✱ ✱ ✗✪✔ ✏❈✘❉✍ ✄✞✠✟ ✩☛✡ ✛ ☞ ✂ ✞ ✍✛ ✌✏✎❦✯ ✛ ✂ ✧✕✙✑✌ ✟ ✙✥✤ ✟ ✥✌ ✧ ✛ ✄✂ ✠☎ ✓✟ ✧✔✌ ✝ ✏✩ ❦✎ ✧✰✢✌✛ ✩✜★✖✕❱✩✥✢✌✩✆✗ ✏■✝✓✝✟✗✼✝ ✞❬ ❬ ✁ ✄ ✄✂✆☎ ✩✥✢ ✝ ✩ ✂ ✙✜✩✓✯✂✢ ✫✒✌✏✎✌✧ ✂✠ ★✁✩ ❀❉✗ ✫ ☞❈●❱❏✪✏ ✘✾✗✼✍❙✏ ✬❨❳✟✬❅✷ ✂✷ ✷ ■ ✏ ❤ ✷✁✿ ✱ ❲ ☞✙✘ ✩✏✎ ✩ ☎✠✆✟ ✧✔✌ ✝ ✩✏✎✒✚✛✌✜✛ ✟ ✩✜★❉✧ P ✟✣✢ P P ✁ ✢❡✤ P ✕ ✛✧✘ ✂✓✂ ✛ ✂ ✝ ❉ ✱ P ✗✼✔✕☞✖☎✌✗✓✘ ❲ ❀❱✗ ✫ ☞❈●❉❏✼✏ ✘✾✗✪✍☛✏ ✴❣❳✟✬❅✷ ✂✷ ✷ ■ ✏ ❤ ✷✂✿ ✱ ❲ ✶✂❳ ✘✲✗✪✍☛✏ ❀❱✗ ✫ ☞❈●❉❏✪✏ ✳ ✂✷ ✷✁✷✂✷ ✱ ✄✂✓☎ ✩✥✢ ✝ ✆✩ ✗ ✙✥✤✁✢❡✢ ✩ ✂ ✧✤✎✥✚✤✌✜✛ ✟ ✩✜★✺✧ ☎ ❲ ❀ ❨✬ ❳ ☎ ❲ ❀ ✶✁❳ ☎ ❲ ✬❨❳ ✢ ✙ ✬❅✩ ❅✬ ✩ ❅✬ ✩ P ☞✙✘ ✩✏✎ ✩✜✎✌✤✂✯✂✯ ☞ ✢ ✳✂❳✟✬✥✷ ✁✷ ✷ ■ ✏ ❤ ✷✂✿ ✱ ❲ ❀ ❲ ✬❃❳ ❀ ❲ ✳✂❳ ❀ ❲ ✴❣❳ ❀❉✗ ✫ ☞❈●❉❏✼✏ ✘✾✗✼✍❙✏ ✬❅✩ ❬❃✬❅✩ ✬❅✩ ❤ ✷✖✬❅✷❉✭✆✬❅✩ ❤ ✷✖✬❅✷❉✭✆✬❅✩ ❤ ✷✖✬❅✷❉✭✆✬❅✩ ✙ ❤ ✷❦✬❅✷ ❤ ✷❦✬❅✷ ❤ ✷❦✬❅✷ ✦✂✆☎ ✩✥✢ ✝ ✆✩ ✗ ✷ ✬❅✷❉✭✆✬❅✩ ❤ ❤ ✖ ❤ ✷✖✬❅✷❉✭❵❬❃✬✥✩ ❤ ✷✖✬❅✷❉✭✆✬❅✩ ❤ ✷❦✬❅✷ ❤ ✷✖✬❅✷ ✷❦✬❅✷ ✑✒☞❈✏■✔ ❏✪❊❖☎ ✘ ❥❨✗✼❇❜✄✓✗▲✘✾✄ ✫ ✠☛✍☛✏❦✍ ☞■✗☛☞❵● ✫ ❥❨✗✪❇ ☞✖☎❡❊❖✏ ✱ ✷✖✬ ✁✂✂c. ✄✆☎✞✝✟✄✡✠☛☎✌☞✎✍☛✏✆✑✒✄✆✝✓☎✞✔✕☞✖☎✌✗✓✘ pct ✚✙✜✛✣✢✌✙✥✤✦✛✣✧✕★✂✩✪✙✜✙✟✫☛✬✮✭✰✯✂✢ ✳ ✱ ✝ ✬✓✷❂✬✲✬ ✝ ✳✲✳✵✴ ✬ ✝ ✴✲✴❱✷❱✳ ☎ ✬✓✷❱✳✽✬ ❀ ✬✆✳✽✬✆✳ ✬✆✶ ✴ ❀ ✬▲❬ ✳ ❀ ✶✲✶✸✷❉✳ ✝ ✶✲✶✸✷❱✳ ✁ ✱ ✱ ✗ ✔ ✪ ✏❈✘❉✍ ✂✂✁ ✑❋❊❍● ✫■✫ ✑✂☎❑❏▲✘✺● ✫ ❇✓☎▼●✸✑◆✑✒✗ ✫ ✠❉☞✖☎✌✗✓✘ ☞❈✏■❊❖✔❆✏■✝❁●❉☞❈✠☛✝✓✏◗P 96 ✳✒❘ ✂✷ ✷✂✷ ✍❙✏❚❏❯✄ ✱ ❀❉✗ ✫ ☞❈●❱❏✪✏ ✘✾✗✼✍❙✏ ❲ ✘✾✗✼✍❙✏ ❀❉✗ ✫ ☞❈●❉❏✼✏ ✬❨❳ ❬ ✳ ✂✷ ✷✂✷ ✏ ❬❨✬❃✳ ❲ ✳✂❳ ✳ ✷✂✷✁✷✂✷ ❲ ✴❣❳ ✱ ✱ ❀❱✗ ✫ ☞❈●❉❏✪✏❝✑✁✗✪✠☛✝✟✄✆✏❍✄✡✠☛✝✓✝✓✏❈✘✾☞✡✑ ✘✺●✺❊❖✏ ❀❱✗ ✫ ☞❈●❉❏✼✏ ✘✾✗✪✍☛✏ ✬ ✂✷ ✷✁✷✂✷ ✱ ❲ ✶✂❳ ✄✡✠☛✝✓✝✓✏❈✘✾☞ ❀ ✬ ❬ ✳ ✂✷ ✷✂✷ ✏ ❬❨✬❃✳ ✱ ☞■✗❙☞❈● ✫ ✔✕✗✼❛❜✏■✝❝✍☛☎❞✑❆✑✂☎❡✔❢●❉☞✖☎✌✗✓✘ ✶ ✂✷ ✷ ✏ ❬❨✬❃✳ ❛❜●❉☞■☞✡✑ ✱ ✁✂✂ ✄✡✠☛✝✓✝✓✏❈✘✾☞ ❬ ✄✓✗✓✘✲☞❈✝✟✗ ✫■✫ ✏■✍✐❀❱✗ ✫ ☞❈●❉❏✪✏❝✑✁✗✪✠☛✝✟✄✆✏❁✑ ✘❉●☛❊❖✏ ❀ ❬ ✑✒✗✟✠✺✝✟✄✆✏ ☎ ❬ ✑✁✗✪✠☛✝✟✄✆✏ ✬ ✶ ✷✁✷✂✷ ✏ ❬❃✬❨✳ ❬ ✻ ✂✷ ✱ ✷✂✷ ✏ ❬▼✷✖✬ ✱ ❥❨✗✼❇❜✄✓✗▲✘✾✄ ✫ ✠☛✍☛✏❦✍ ✁ LUCRARAEA 2 Problema 1 ✁✂✂ ✄✆☎✞✝✟✄✡✠☛☎✌☞✎✍☛✏✆✑✒✄✆✝✓☎✞✔✕☞✖☎✌✗✓✘ *Circ cu dioda Zener cdz1.cir R1 1 2 1k R2 2 0 2k D 0 2 Dioda_Zener *Modelul diodei *Bv=4 tens de strapungere inversa .MODEL Dioda_Zener D(Is=10f Rs=0 + Bv=4 Ibv=10m Cjo=0) *Tensiunea de polarizare V1 1 0 1 *Analiza in curent continuu .DC LIN V1 -10 15 .1 .PROBE .END **** Diode MODEL PARAMETERS Dioda_Zener IS 10.000000E-15 BV 4 IBV .01 97 ✬ ✂✷ ✷✁✷✂✷ ✱ ✘✲✗✪✍☛✏ ❀❱✗ ✫ ☞❈●❉❏✪✏ 4.0V 2.0V 0V -2.0V -10V V(2) -5V 0V 5V 10V 15V V(1) Problema 2 ✁✂✂a. ✄✆☎✞✝✟✄✡✠☛☎✌☞✎✍☛✏✆✑✒✄✆✝✓☎✞✔✕☞✖☎✌✗✓✘ pct *circuit cu rezistenta neliniara cnl1.cir R1 1 2 3 V1 1 0 10 G1 2 0 TABLE {V(2)}= +(0,0), (2,5), (4,0.5), (8,9) .NODESET V(2)=0.0 .OP .END **** SMALL SIGNAL BIAS SOLUTION NODE VOLTAGE NODE VOLTAGE TEMPERATURE = 27.000 DEG C NODE VOLTAGE ( 1) 10.0000 ( 2) 1.1765 VOLTAGE SOURCE CURRENTS NAME V1 CURRENT -2.941E+00 TOTAL POWER DISSIPATION 2.94E+01 WATTS **** VOLTAGE-CONTROLLED CURRENT SOURCES NAME I-SOURCE G1 2.941E+00 98 NODE VOLTAGE ✁✂✂b. ✄✆☎✞✝✟✄✡✠☛☎✌☞✎✍☛✏✆✑✒✄✆✝✓☎✞✔✕☞✖☎✌✗✓✘ pct *circuit cu rezistenta neliniara cnl1.cir R1 1 2 3 V1 1 0 10 G1 2 0 TABLE {V(2)}= +(0,0), (2,5), (4,0.5), (8,9) .NODESET V(2)=3.0 .OP .END **** SMALL SIGNAL BIAS SOLUTION NODE VOLTAGE NODE VOLTAGE TEMPERATURE = 27.000 DEG C NODE VOLTAGE NODE VOLTAGE ( 1) 10.0000 ( 2) 3.2174 VOLTAGE SOURCE CURRENTS NAME V1 CURRENT -2.261E+00 TOTAL POWER DISSIPATION 2.26E+01 WATTS **** VOLTAGE-CONTROLLED CURRENT SOURCES NAME G1 I-SOURCE ✁✂✂c. pct 2.261E+00 ✄✆☎✞✝✟✄✡✠☛☎✌☞✎✍☛✏✆✑✒✄✆✝✓☎✞✔✕☞✖☎✌✗✓✘ *circuit cu rezistenta neliniara cnl1.cir R1 1 2 3 V1 1 0 10 G1 2 0 TABLE {V(2)}= +(0,0), (2,5), (4,0.5), (8,9) .NODESET V(2)=7.0 .OP .END **** SMALL SIGNAL BIAS SOLUTION TEMPERATURE = 27.000 DEG C 99 NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE ( 1) 10.0000 ( 2) 4.6102 VOLTAGE SOURCE CURRENTS NAME CURRENT V1 -1.797E+00 TOTAL POWER DISSIPATION 1.80E+01 WATTS **** VOLTAGE-CONTROLLED CURRENT SOURCES NAME G1 I-SOURCE 1.797E+00 ✁✂✂d. ✄✆☎✞✝✟✄✡✠☛☎✌☞✎✍☛✏✆✑✒✄✆✝✓☎✞✔✕☞✖☎✌✗✓✘ pct *circuit cu rezistenta neliniara cnl1.cir R1 1 2 3 V1 1 0 10 G1 2 0 TABLE {V(2)}= +(0,0), (2,5), (4,0.5), (8,9) *.NODESET V(2)=7.0 .OP .END **** SMALL SIGNAL BIAS SOLUTION NODE VOLTAGE NODE VOLTAGE TEMPERATURE = 27.000 DEG C NODE VOLTAGE ( 1) 10.0000 ( 2) 1.1765 VOLTAGE SOURCE CURRENTS NAME CURRENT V1 -2.941E+00 TOTAL POWER DISSIPATION 2.94E+01 WATTS **** VOLTAGE-CONTROLLED CURRENT SOURCES NAME G1 I-SOURCE 2.941E+00 100 NODE VOLTAGE LUCRAREA 3 Punctul a. **** CIRCUIT DESCRIPTION *etb31.cir circ cu tranz bipolare *.options reltol=1e-6 itl5=0 numdgt=8 V1 2 0 DC 5 V2 5 0 DC 9 R1 2 3 1MEG R2 5 4 4K Q1 4 3 0 TB .MODEL TB NPN(Is=10.f Bf=200 Cje=2p + Cjc=2p Vje=.6 Vjc=.6) .DC LIN V1 0 25 .05 *.PRINT DC V(R2) .PROBE .END **** BJT MODEL PARAMETERS TB NPN IS 10.000000E-15 BF 200 NF 1 BR 1 NR 1 CJE 2.000000E-12 VJE .6 CJC 2.000000E-12 VJC .6 CN 2.42 D .87 JOB CONCLUDED 101 10V 5V 0V 0V 5V 10V 15V v(5)-v(4) V1 Punctul b **** CIRCUIT DESCRIPTION *etb32.cir circ cu tranz bipolare V1 2 1 DC 5 V10 + DC 0 + AC 1 0 V2 5 0 DC 9 R1 2 3 1MEG R2 5 4 4K Q1 4 3 0 TB .MODEL TB NPN(Is=10.f Bf=200 Cje=2p + Cjc=2p Vje=.6 Vjc=.6) *.DC LIN V1 0 25 .05 .AC DEC 100 .1 100MEG .OP .PROBE .END **** BJT MODEL PARAMETERS TB NPN IS 10.000000E-15 BF 200 NF 1 BR 1 NR 1 CJE 2.000000E-12 VJE .6 CJC 2.000000E-12 VJC .6 CN 2.42 102 20V 25V D **** .87 SMALL SIGNAL BIAS SOLUTION NODE VOLTAGE NODE VOLTAGE ( 1) 0.0000 ( 2) 5.0000 ( 3) ( 5) 9.0000 TEMPERATURE = 27.000 DEG C NODE VOLTAGE .6515 ( 4) 5.5212 VOLTAGE SOURCE CURRENTS NAME CURRENT V1 V V2 -4.349E-06 -4.349E-06 -8.697E-04 TOTAL POWER DISSIPATION 7.85E-03 WATTS **** BIPOLAR JUNCTION TRANSISTORS NAME Q1 MODEL TB IB 4.35E-06 IC 8.70E-04 VBE 6.51E-01 VBC -4.87E+00 VCE 5.52E+00 BETADC 2.00E+02 GM 3.36E-02 RPI 5.95E+03 RX 0.00E+00 RO 1.00E+12 CBE 3.49E-12 CBC 9.64E-13 CJS 0.00E+00 BETAAC 2.00E+02 CBX/CBX2 0.00E+00 FT/FT2 1.20E+09 103 NODE VOLTAGE 800mV 400mV 0V 100mHz 1.0Hz V(5)- V(4) 100Hz 10KHz Frequency **** CIRCUIT DESCRIPTION *etb32.cir circ cu tranz bipolare V1 2 1 DC 5 V10 + DC 0 + AC 20 0 V2 5 0 DC 9 R1 2 3 1MEG R2 5 4 4K Q1 4 3 0 TB .MODEL TB NPN(Is=10.f Bf=200 Cje=2p + Cjc=2p Vje=.6 Vjc=.6) *.DC LIN V1 0 25 .05 .AC DEC 100 .1 100MEG .OP .PROBE .END **** BJT MODEL PARAMETERS TB NPN IS 10.000000E-15 BF 200 NF 1 BR 1 NR 1 CJE 2.000000E-12 VJE .6 CJC 2.000000E-12 VJC .6 CN 2.42 D .87 104 1.0MHz 100MHz **** SMALL SIGNAL BIAS SOLUTION TEMPERATURE = 27.000 DEG C NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE ( 1) 0.0000 ( 2) 5.0000 ( 3) ( 5) 9.0000 .6515 ( 4) 5.5212 VOLTAGE SOURCE CURRENTS NAME CURRENT V1 V V2 -4.349E-06 -4.349E-06 -8.697E-04 TOTAL POWER DISSIPATION 7.85E-03 WATTS **** BIPOLAR JUNCTION TRANSISTORS NAME Q1 MODEL TB IB 4.35E-06 IC 8.70E-04 VBE 6.51E-01 VBC -4.87E+00 VCE 5.52E+00 BETADC 2.00E+02 GM 3.36E-02 RPI 5.95E+03 RX 0.00E+00 RO 1.00E+12 CBE 3.49E-12 CBC 9.64E-13 CJS 0.00E+00 BETAAC 2.00E+02 CBX/CBX2 0.00E+00 FT/FT2 1.20E+09 105 20V 10V 0V 100mHz 1.0Hz V(5)- V(4) 100Hz 10KHz 1.0MHz 100MHz Frequency LUCRARAEA 4 Problema 1 **** CIRCUIT DESCRIPTION *circ lin c.a. CLA1.CIR R1 2 3 10 R2 3 5 1 R3 3 4 2 L1 1 2 50m L2 6 7 25m L3 0 4 50m K1 l1 l3 0.5 K2 l2 l3 0.707 C2 5 6 1.5m V_SIN 7 0 +DC 0 +AC 30 90 I_SIN 0 1 +DC 0 +AC 2 0 .AC LIN 1 50 50 .PRINT AC IM(R2) IP(R2) IR(R2) II(R2) .PROBE .END **** SMALL SIGNAL BIAS SOLUTION TEMPERATURE = 27.000 DEG C NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE ( 1) 0.0000 ( 2) 0.0000 ( 3) 0.0000 ( 4) 0.0000 ( 5) 0.0000 ( 6) 0.0000 ( 7) 0.0000 VOLTAGE SOURCE CURRENTS 106 NAME V_SIN CURRENT 0.000E+00 TOTAL POWER DISSIPATION 0.00E+00 WATTS **** AC ANALYSIS TEMPERATURE = 27.000 DEG C FREQ IM(R2) IP(R2) IR(R2) II(R2) 5.000E+01 1.138E-01 -6.592E+01 4.645E-02 -1.039E-01 Problema 2a **** CIRCUIT DESCRIPTION *CIRC LIN DE CA FIL1 R1 1 2 1K L1 2 3 3M L2 3 4 12M C1 3 0 8N C2 4 0 11.5N V1 1 0 +DC 0 +AC 1 0 +SIN(0 1 50 0 0 0) .AC LIN 100 1 1MEG .PROBE .END **** SMALL SIGNAL BIAS SOLUTION TEMPERATURE = 27.000 DEG C NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE ( 1) 0.0000 ( 2) 0.0000 ( 3) 0.0000 ( 4) 0.0000 VOLTAGE SOURCE CURRENTS NAME CURRENT V1 0.000E+00 TOTAL POWER DISSIPATION 0.00E+00 WATTS 107 1.0V 0.5V 0V 1.0Hz V(4) 100Hz 10KHz 1.0MHz Frequency Problema 2b **** CIRCUIT DESCRIPTION *CIRC LIN DE CA FIL2 R1 1 2 1.5K L1 3 0 2M L2 4 0 12M C1 2 3 10N C2 3 4 8U V1 1 0 +DC 0 +AC 1 0 +SIN(0 1 50 0 0 0) .AC LIN 100 1 2MEG .PROBE .END **** SMALL SIGNAL BIAS SOLUTION NODE VOLTAGE NODE VOLTAGE TEMPERATURE = 27.000 DEG C NODE VOLTAGE ( 1) 0.0000 ( 2) 0.0000 ( 3) 0.0000 ( 4) 0.0000 VOLTAGE SOURCE CURRENTS NAME CURRENT V1 0.000E+00 TOTAL POWER DISSIPATION 0.00E+00 WATTS 108 NODE VOLTAGE 1.0V 0.5V 0V 1.0Hz V(4) 100Hz 10KHz 1.0MHz 100MHz Frequency Problema 2c **** CIRCUIT DESCRIPTION *CIRC LIN DE CA FIL3 R1 1 2 1.5K RS 4 0 1.5K L1 2 3 2M C1 2 0 10N C2 3 4 8U C3 4 0 1N V1 1 0 +DC 0 +AC 1 0 +SIN(0 1 50 0 0 0) .AC DEC 100 1 2MEG .PROBE .END **** SMALL SIGNAL BIAS SOLUTION NODE VOLTAGE NODE VOLTAGE TEMPERATURE = 27.000 DEG C NODE VOLTAGE ( 1) 0.0000 ( 2) 0.0000 ( 3) 0.0000 ( 4) 0.0000 VOLTAGE SOURCE CURRENTS NAME V1 CURRENT 0.000E+00 TOTAL POWER DISSIPATION 0.00E+00 WATTS 109 NODE VOLTAGE 500mV 250mV 0V 1.0Hz V(4) 100Hz 10KHz 1.0MHz 100MHz Frequency Problema 2d **** CIRCUIT DESCRIPTION *CIRC LIN DE CA FIL4 R1 1 2 1.5K L1 5 0 10M L2 2 3 20M L3 4 0 4M C1 2 5 .1U C2 2 3 40U C3 3 4 1U V1 1 0 +DC 0 +AC 1 0 +SIN(0 1 50 0 0 0) .AC DEC 100 1 2MEG .PROBE .END **** SMALL SIGNAL BIAS SOLUTION NODE VOLTAGE NODE VOLTAGE TEMPERATURE = 27.000 DEG C NODE VOLTAGE ( 1) 0.0000 ( 2) 0.0000 ( 3) 0.0000 ( 4) 0.0000 ( 5) 0.0000 VOLTAGE SOURCE CURRENTS NAME CURRENT V1 0.000E+00 110 NODE VOLTAGE TOTAL POWER DISSIPATION 0.00E+00 WATTS 1.0V 0.5V 0V 1.0Hz V(3) 100Hz 10KHz Frequency LUCRAREA 5 Problema 1 **** CIRCUIT DESCRIPTION *est51.cir CIR cu AO/ SUBcirc V1 1 0 + DC 0 + AC 1m 0 + SIN(0 1m 500 0 0 0) * R1 1 2 1 C1 1 2 1 R2 2 3 1 C2 3 6 1 R3 2 4 1 R4 4 5 1 R5 5 6 1 C5 5 6 1 R6 1 7 1 C6 7 5 1 * X1 [0 2 4] AO X2 [0 5 6] AO 111 1.0MHz 100MHz * .SUBCKT AO 2 1 4 Ri 1 2 1MEG Re 3 4 .1m Ce 4 2 .1p E 3 2 TABLE { V(2)-V(1)} = (-.0015,-15)(0,0)(.0015,15) .ENDS * .AC DEC 100 .1 10MEG *.tran 1n 4.m .probe .options reltol=1e-6 itl5=0 numdgt=8 .end **** SMALL SIGNAL BIAS SOLUTION NODE VOLTAGE NODE VOLTAGE TEMPERATURE = 27.000 DEG C NODE VOLTAGE NODE VOLTAGE ( 1) 0.00000000 ( 2) 0.00000000 ( 3) 0.00000000 ( 4) 0.00000000 ( 5) 0.00000000 ( 6) 0.00000000 ( 7) 0.00000000 ( X1.3) 0.00000000 ( X2.3) 0.00000000 VOLTAGE SOURCE CURRENTS NAME CURRENT V1 0.000E+00 TOTAL POWER DISSIPATION 0.00E+00 WATTS 1.0mV 0.5mV 0V 100mHz 1.0Hz V(6) 100Hz 10KHz Frequency 112 1.0MHz 1.0 0.5 0 100mHz 1.0Hz V(6)/ V(1) 100Hz 10KHz 1.0MHz Frequency *est51.cir CIR cu AO/ SUBcirc V1 1 0 + DC 0 + AC 1m 0 + SIN(0 1m 500 0 0 0) * R1 1 2 1 C1 1 2 1 R2 2 3 1 C2 3 6 1 R3 2 4 1 R4 4 5 1 R5 5 6 1 C5 5 6 1 R6 1 7 1 C6 7 5 1 * X1 [0 2 4] AO X2 [0 5 6] AO * .SUBCKT AO 2 1 4 Ri 1 2 1MEG Re 3 4 .1m Ce 4 2 .1p E 3 2 TABLE { V(2)-V(1)} = (-.0015,-15)(0,0)(.0015,15) .ENDS * *.AC DEC 100 .1 10MEG .tran 1n 4.m .probe .options reltol=1e-6 itl5=0 numdgt=8 .end **** INITIAL TRANSIENT SOLUTION TEMPERATURE = 27.000 DEG C 113 NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE ( 1) 0.00000000 ( 2) 0.00000000 ( 3) 0.00000000 ( 4) 0.00000000 ( 5) 0.00000000 ( 6) 0.00000000 ( 7) 0.00000000 ( X1.3) 0.00000000 ( X2.3) 0.00000000 VOLTAGE SOURCE CURRENTS NAME CURRENT V1 0.000E+00 TOTAL POWER DISSIPATION 0.00E+00 WATTS 1.0mV 0V -1.0mV 0s 1.0ms 2.0ms V(6) Time **** CIRCUIT DESCRIPTION *est51.cir CIR cu AO/ SUBcirc V1 1 0 + DC 0 + AC 1m 0 + SIN(0 10 500 0 0 0) * R1 1 2 1 C1 1 2 1 R2 2 3 1 C2 3 6 1 R3 2 4 1 R4 4 5 1 R5 5 6 1 C5 5 6 1 R6 1 7 1 C6 7 5 1 * X1 [0 2 4] AO 114 3.0ms 4.0ms X2 [0 5 6] AO * .SUBCKT AO 2 1 4 Ri 1 2 1MEG Re 3 4 .1m Ce 4 2 .1p E 3 2 TABLE { V(2)-V(1)} = (-.0015,-15)(0,0)(.0015,15) .ENDS * *.AC DEC 100 .1 10MEG .tran 1n 4.m .probe .options reltol=1e-6 itl5=0 numdgt=8 .end **** INITIAL TRANSIENT SOLUTION NODE VOLTAGE NODE VOLTAGE TEMPERATURE = 27.000 DEG C NODE VOLTAGE NODE VOLTAGE ( 1) 0.00000000 ( 2) 0.00000000 ( 3) 0.00000000 ( 4) 0.00000000 ( 5) 0.00000000 ( 6) 0.00000000 ( 7) 0.00000000 ( X1.3) 0.00000000 ( X2.3) 0.00000000 VOLTAGE SOURCE CURRENTS NAME CURRENT V1 0.000E+00 TOTAL POWER DISSIPATION 0.00E+00 WATTS 10mV 0V -10mV 0s 1.0ms 2.0ms V(6) Time Problema 2 **** CIRCUIT DESCRIPTION 115 3.0ms 4.0ms *etb52.cir circ cu tranz bipolare .options reltol=1e-6 V1 1 0 DC 5 C3 2 3 100u *V3 3 0 AC 1m SIN(0 1m 1K 0 0) V3 3 0 AC 10 SIN(0 10 1K 0 0) V2 5 0 DC 9 R1 1 2 20K R2 5 4 100 Q1 4 2 0 TB .MODEL TB NPN(IS=10.F BF=200 CJE=200p + CJC=200p) *.DC LIN V1 0 25 .05 *.AC DEC 100 .1 100MEG .TRAN .1u 10m .OP .PROBE .END **** BJT MODEL PARAMETERS TB NPN IS 10.000000E-15 BF 200 NF 1 BR 1 NR 1 CJE 200.000000E-12 CJC 200.000000E-12 CN 2.42 D .87 **** SMALL SIGNAL BIAS SOLUTION TEMPERATURE = 27.000 DEG C NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE ( 1) 5.0000 ( 2) ( 5) 9.0000 .7521 ( 3) 0.0000 ( 4) 4.7521 VOLTAGE SOURCE CURRENTS NAME V1 CURRENT -2.124E-04 116 NODE VOLTAGE V3 V2 0.000E+00 -4.248E-02 TOTAL POWER DISSIPATION 3.83E-01 WATTS **** OPERATING POINT INFORMATION TEMPERATURE = 27.000 DEG C **** BIPOLAR JUNCTION TRANSISTORS NAME Q1 MODEL TB IB 2.12E-04 IC 4.25E-02 VBE 7.52E-01 VBC -4.00E+00 VCE 4.75E+00 BETADC 2.00E+02 GM 1.64E+00 RPI 1.22E+02 RX 0.00E+00 RO 1.00E+12 CBE 3.35E-10 CBC 1.09E-10 CJS 0.00E+00 BETAAC 2.00E+02 CBX/CBX2 0.00E+00 FT/FT2 5.89E+08 **** INITIAL TRANSIENT SOLUTION TEMPERATURE = 27.000 DEG C NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE ( 1) 5.0000 ( 2) .7521 ( 3) 0.0000 ( 4) 4.7521 ( 5) 9.0000 VOLTAGE SOURCE CURRENTS NAME V1 V3 V2 CURRENT -2.124E-04 0.000E+00 -4.248E-02 TOTAL POWER DISSIPATION 3.83E-01 WATTS 117 NODE VOLTAGE 4.50V 4.25V 4.00V 0s 2ms V(5)- V(4) 4ms 6ms 8ms 10ms Time 10V 0V -10V 0s 2ms V(5)- V(4) 4ms 6ms Time LUCRAREA 6 Problema 1 **** CIRCUIT DESCRIPTION *etb61.cir circ cu tranz bipolare .options reltol=1e-6 V1 1 0 DC 5 118 8ms 10ms C3 2 3 100u V3 3 0 AC 1m SIN(0 1m 1K 0 0) *V3 3 0 AC 10 SIN(0 10 1K 0 0) V2 5 0 DC 9 R1 1 2 20K R2 5 4 100 Q1 4 2 0 TB .MODEL TB NPN(IS=10.F BF=200 CJE=200p + CJC=200p) *.TRAN .1u 10m .TRAN .1u 10m 5m .FOUR 1K 10 V(5,4) *.OP .PROBE .END **** BJT MODEL PARAMETERS TB NPN IS 10.000000E-15 BF 200 NF 1 BR 1 NR 1 CJE 200.000000E-12 CJC 200.000000E-12 CN 2.42 D .87 **** INITIAL TRANSIENT SOLUTION TEMPERATURE = 27.000 DEG C NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE ( 1) 5.0000 ( 2) ( 5) 9.0000 .7521 ( 3) 0.0000 ( 4) 4.7521 NODE VOLTAGE VOLTAGE SOURCE CURRENTS NAME CURRENT V1 V3 V2 -2.124E-04 0.000E+00 -4.248E-02 TOTAL POWER DISSIPATION 3.83E-01 WATTS **** FOURIER ANALYSIS TEMPERATURE = 27.000 DEG C 119 FOURIER COMPONENTS OF TRANSIENT RESPONSE V(5,4) DC COMPONENT = 4.247607E+00 HARMONIC FREQUENCY FOURIER NORMALIZED PHASE NORMALIZED NO (HZ) COMPONENT COMPONENT (DEG) PHASE (DEG) 1 2 3 4 5 6 7 8 9 10 1.000E+03 2.000E+03 3.000E+03 4.000E+03 5.000E+03 6.000E+03 7.000E+03 8.000E+03 9.000E+03 1.000E+04 1.625E-01 1.589E-03 5.077E-04 1.470E-04 3.713E-04 1.359E-04 2.854E-04 1.450E-04 2.411E-04 1.767E-04 1.000E+00 9.777E-03 3.125E-03 9.045E-04 2.285E-03 8.364E-04 1.756E-03 8.922E-04 1.484E-03 1.088E-03 7.960E-01 -8.396E+01 -5.091E+01 1.237E+02 7.320E+01 -1.204E+02 -1.695E+02 -4.044E+00 -4.427E+01 1.118E+02 0.000E+00 -8.476E+01 -5.171E+01 1.229E+02 7.240E+01 -1.212E+02 -1.703E+02 -4.840E+00 -4.506E+01 1.110E+02 TOTAL HARMONIC DISTORTION = 1.092516E+00 PERCENT 5.0V 2.5V 0V 0Hz 5KHz V(5)- V(4) 10KHz 15KHz Frequency 120 20KHz 25KHz 30KHz 1.0V 0.5V 0V 0Hz 2KHz V(5)- V(4) 4KHz 6KHz Frequency **** CIRCUIT DESCRIPTION *etb61.cir circ cu tranz bipolare .options reltol=1e-6 V1 1 0 DC 5 C3 2 3 100u *V3 3 0 AC 1m SIN(0 1m 1K 0 0) V3 3 0 AC 10 SIN(0 10 1K 0 0) V2 5 0 DC 9 R1 1 2 20K R2 5 4 100 Q1 4 2 0 TB .MODEL TB NPN(IS=10.F BF=200 CJE=200p + CJC=200p) *.TRAN .1u 10m .TRAN .1u 10m 5m .FOUR 1K 10 V(5,4) *.OP .PROBE .END **** BJT MODEL PARAMETERS TB NPN IS 10.000000E-15 BF 200 NF 1 BR 1 NR 1 CJE 200.000000E-12 CJC 200.000000E-12 121 8KHz 10KHz CN 2.42 D .87 **** INITIAL TRANSIENT SOLUTION TEMPERATURE = 27.000 DEG C NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE ( 1) 5.0000 ( 2) ( 5) 9.0000 .7521 ( 3) 0.0000 ( 4) 4.7521 NODE VOLTAGE VOLTAGE SOURCE CURRENTS NAME CURRENT V1 V3 V2 -2.124E-04 0.000E+00 -4.248E-02 TOTAL POWER DISSIPATION 3.83E-01 WATTS **** FOURIER ANALYSIS TEMPERATURE = 27.000 DEG C FOURIER COMPONENTS OF TRANSIENT RESPONSE V(5,4) DC COMPONENT = 2.677038E-01 HARMONIC FREQUENCY FOURIER NORMALIZED PHASE NORMALIZED NO (HZ) COMPONENT COMPONENT (DEG) PHASE (DEG) 1 2 3 4 5 6 7 8 9 10 1.000E+03 2.000E+03 3.000E+03 4.000E+03 5.000E+03 6.000E+03 7.000E+03 8.000E+03 9.000E+03 1.000E+04 5.345E-01 5.316E-01 5.269E-01 5.203E-01 5.120E-01 5.019E-01 4.903E-01 4.771E-01 4.625E-01 4.465E-01 1.000E+00 9.947E-01 9.858E-01 9.736E-01 9.580E-01 9.392E-01 9.173E-01 8.927E-01 8.653E-01 8.355E-01 1.600E-01 -8.959E+01 -1.794E+02 9.083E+01 1.041E+00 -8.875E+01 -1.785E+02 9.168E+01 1.897E+00 -8.788E+01 0.000E+00 -8.975E+01 -1.795E+02 9.067E+01 8.813E-01 -8.891E+01 -1.787E+02 9.152E+01 1.737E+00 -8.804E+01 TOTAL HARMONIC DISTORTION = 2.791763E+02 PERCENT 122 1.0V 0.5V 0V 0Hz 10KHz 20KHz 30KHz 40KHz 50KHz 8KHz 10KHz V(5,4) Frequency 1.0V 0.5V 0V 0Hz 2KHz 4KHz 6KHz V(5,4) Frequency Problema 2 **** CIRCUIT DESCRIPTION *etR61.cir CIR cu SWITCH options reltol=5.e-7 itl5=0 numdgt=8 V1 1 0 + DC 100 * V4 4 0 PULSE (0 1 50.u 1.p 1.p .05m .1m ) * L1 1 2 20m C1 3 0 1.25u R1 3 0 400 * d1 2 3 dioda .model dioda d 123 * S1 2 0 4 0 S1 * .MODEL S1 VSWITCH ( RON=1, ROFF=1MEG, + VON=1, VOFF=0 ) * .TRAN 10u 5m *.TRAN 10u 5m 4.5m *.FOUR 10.K V(3),V(4) .PROBE .END **** Diode MODEL PARAMETERS dioda IS 10.000000E-15 **** Voltage Controlled Switch MODEL PARAMETERS S1 RON ROFF VON VOFF **** 1 1.000000E+06 1 0 INITIAL TRANSIENT SOLUTION NODE VOLTAGE NODE VOLTAGE TEMPERATURE = 27.000 DEG C NODE VOLTAGE ( 1) 100.00000000 ( 2) 100.00000000 ( 3) 99.20229970 ( 4) 0.00000000 VOLTAGE SOURCE CURRENTS NAME CURRENT V1 V4 -2.481E-01 0.000E+00 TOTAL POWER DISSIPATION 2.48E+01 WATTS 124 NODE VOLTAGE 1100mV 800mV 400mV 0V 4.5ms V(4) 4.6ms 4.8ms 5.0ms Time 300V 200V 100V 0V 0s 2.0ms 4.0ms 5.0ms V(3) Time 800mV 400mV 0V 0Hz 100KHz 200KHz V(4) Frequency 125 300KHz 400KHz 200V 100V 0V 0Hz 10KHz 20KHz 30KHz 40KHz V(3) Frequency 1.0V 0.5V 0V 9KHz 25KHz 50KHz V(3) Frequency 126 75KHz 100KHz LUCRAREA 7 Problema 1 **** CIRCUIT DESCRIPTION *circ in reg tranz cetr1.cir .options reltol=1e-6 itl5=0 numdgt=8 NOMOD NOPAGE NOECHO R1 0 1 3 L1 1 2 1 IC=-.10E+01 C1 2 0 1 IC=.10E+01 .TRAN 10M 15 SKIPBP .PRINT TRAN V(2),I(L1) .PROBE .END **** TIME TRANSIENT ANALYSIS V(2) TEMPERATURE = 27.000 DEG C I(L1) 1.0000000E-02 9.9011134E-01 -9.8027822E-01 2.0000000E-02 9.8041165E-01 -9.6102829E-01 3.0000000E-02 9.7091674E-01 -9.4228926E-01 4.0000000E-02 9.6166173E-01 -9.2414761E-01 5.0000000E-02 9.5251726E-01 -9.0628115E-01 6.0000000E-02 9.4349691E-01 -8.8872298E-01 7.0000000E-02 9.3484698E-01 -8.7208478E-01 8.0000000E-02 9.2619704E-01 -8.5544659E-01 9.0000000E-02 9.1754711E-01 -8.3880839E-01 1.0000000E-01 9.0942287E-01 -8.2347094E-01 ……………………………………………………………. 1.4960000E+01 2.3808301E-03 -9.0939618E-04 1.4970000E+01 2.3716028E-03 -9.0587170E-04 1.4980000E+01 2.3623756E-03 -9.0234722E-04 1.4990000E+01 2.3531484E-03 -8.9882274E-04 1.5000000E+01 2.3439212E-03 -8.9529826E-04 127 1.0 0 -1.0 0s 5s I(L1) 10s 15s V(2) Time 1.0A 0A -1.0A 0V 0.2V 0.4V 0.6V I(L1) V(2) **** CIRCUIT DESCRIPTION *circ in reg tranz cetr1.cir .options reltol=1e-6 itl5=0 numdgt=8 NOMOD NOPAGE NOECHO R1 0 1 3 L1 1 2 1 IC=.10E+01 C1 2 0 1 IC=.10E+01 .TRAN 10M 15 SKIPBP .PROBE .END 128 0.8V 1.0V 1.0A 0A -1.0A 0s 5s I(L1) 10s 15s 1.0V 1.5V V(2) Time 1.0A 0A -1.0A 0V 0.5V I(L1) V(2) **** CIRCUIT DESCRIPTION *circ in reg tranz cetr1.cir .options reltol=1e-6 itl5=0 numdgt=8 NOMOD NOPAGE NOECHO R1 0 1 3 L1 1 2 1 IC=.10E+01 C1 2 0 1 IC=-.10E+01 129 .TRAN 10M 15 SKIPBP .PROBE .END 1.0A 0A -1.0A 0s 5s I(L1) 10s 15s V(2) Time 1.0A 0A -1.0A -1.0V -0.8V I(L1) -0.6V -0.4V V(2) **** CIRCUIT DESCRIPTION *circ in reg tranz cetr1.cir .options reltol=1e-6 itl5=0 numdgt=8 NOMOD NOPAGE NOECHO R1 0 1 -3 L1 1 2 1 IC=.10E+01 C1 2 0 1 IC=.10E+01 130 -0.2V 0V .TRAN 1M 2 SKIPBP .PROBE .END 5.0A 2.5A 0A 0s I(L1) 0.2s V(2) 0.4s 0.6s 0.8s 1.0s Time 5.0A 2.5A 0A 0V 0.5V 1.0V 1.5V I(L1) V(2) 131 2.0V 2.5V 3.0V 20KA 15KA 10KA 5KA 0A 0V 2.0KV 4.0KV 6.0KV 8.0KV I(L1) V(2) **** CIRCUIT DESCRIPTION *circ in reg tranz cetr1.cir .options reltol=1e-6 itl5=0 numdgt=8 NOMOD NOPAGE NOECHO R1 0 1 3 L1 1 2 1 IC=.10E+01 C1 2 0 -1 IC=.10E+01 .TRAN 1M 15 SKIPBP .PROBE .END 1.0A 0A -1.0A 0s I(L1) 0.5s V(2) 1.0s Time 132 1.5s 2.0s 1.0A 0A -1.0A 0V 0.5V 1.0V 1.5V 2.0V I(L1) V(2) **** CIRCUIT DESCRIPTION *circ in reg tranz cetr1.cir .options reltol=1e-6 itl5=0 numdgt=8 NOMOD NOPAGE NOECHO R1 0 1 1 L1 1 2 1 IC=.10E+01 C1 2 0 1 IC=.10E+01 .TRAN 1M 15 SKIPBP .PROBE .END 2.0A 0A -2.0A 0s I(L1) 0.2s V(2) 0.4s 0.6s Time 133 0.8s 1.0s 1.0A 0A -1.0A -0.5V I(L1) 0V 0.5V 1.0V 1.5V V(2) **** CIRCUIT DESCRIPTION *circ in reg tranz cetr1.cir .options reltol=1e-6 itl5=0 numdgt=8 NOMOD NOPAGE NOECHO R1 0 1 -1 L1 1 2 1 IC=-.10E+01 C1 2 0 1 IC=-.10E+01 .TRAN .1M 15 SKIPBP .PROBE .END 200A 100A 0A -100A -200A 0s I(L1) 2s V(2) 4s 6s Time 134 8s 10s 5.0A 2.5A 0A -2.0V I(L1) -1.0V 0V 1.0V 2.0V V(2) 150A 100A 50A 0A -50A -75V -50V 0V I(L1) V(2) **** CIRCUIT DESCRIPTION *circ in reg tranz cetr1.cir .options reltol=1e-6 itl5=0 numdgt=8 NOMOD NOPAGE NOECHO R1 0 1 1n L1 1 2 1 IC=.10E+01 C1 2 0 1 IC=.10E+01 .TRAN .1M 15 SKIPBP .PROBE 135 50V .END 2.0A 0A -2.0A 0s I(L1) 2s V(2) 4s 6s 8s 10s Time 2.0A 0A -2.0A -1.5V -1.0V I(L1) -0.5V 0V V(2) Problema 2 **** CIRCUIT DESCRIPTION *circ nelin de ord 2 cnd2.cir L1 0 1 1 IC=6 G1 1 0 TABLE {V(1)}= + (-10,-8),(-1,1),(1,-1),(10,8) .TRAN .1M 7 SKIPBP .PROBE .END 136 0.5V 1.0V 1.5V 2.0A 0A -2.0A -4.0V I(L1) -2.0V 0V 2.0V 4.0V 6.0V 2.0V 4.0V 6.0V V(1) *circ nelin de ord 2 cnd2.cir L1 0 1 1 IC=6 G1 1 0 TABLE {V(1)}= + (-10,-8),(-1,1),(1,-1),(10,8) C1 1 0 1M IC=8 .TRAN .1M 7 SKIPBP .PROBE .END 2.0A 0A -2.0A -4.0V I(L1) -2.0V 0V V1(C1) 137 *circ nelin de ord 2 cnd2.cir L1 0 1 1 IC=6 G1 1 0 TABLE {V(1)}= + (-10,-8),(-1,1),(1,-1),(10,8) C1 1 0 1M IC=8 .TRAN .1M 7 SKIPBP .PROBE .OPTIONS RELTOL=1e-7 .END 2.0A 0A -2.0A -4.0V I(L1) -2.0V 0V 2.0V V1(C1) 138 4.0V 6.0V LUCRAREA 8 Problema 1a **** CIRCUIT DESCRIPTION *el81.cir circ el nelin reg tran .options reltol=1e-6 itl5=0 numdgt=8 r1 1 2 15 l1 2 3 10m V1 1 0 sin(0 0.5 1.e+5 0 0 0) d 3 0 dioda .model dioda d(Is=8.3e-15 Rs=9.6 + Cjo=300p Vj=.75 + M=.4 Tt=4.e-06) .tran 5n 400u 360u *.tran 2u 200u 100u 1u .probe .end **** **** Diode MODEL PARAMETERS dioda IS 8.300000E-15 RS 9.6 TT 4.000000E-06 CJO 300.000000E-12 VJ .75 M .4 INITIAL TRANSIENT SOLUTION NODE VOLTAGE NODE VOLTAGE TEMPERATURE = 27.000 DEG C NODE VOLTAGE ( 1) 0.00000000 ( 2) 0.00000000 ( 3) 0.00000000 VOLTAGE SOURCE CURRENTS NAME CURRENT V1 0.000E+00 139 NODE VOLTAGE 400uA 0A -400uA 360us I(l1) 370us 380us 390us 400us Time 400uA 0A -400uA -500mV I(l1) 0V 500mV V(1) 140 Problema 1b v=1.5V 1.0mA 0.5mA 0A -0.5mA 360us I(l1) 370us 380us 390us 400us Time 1.0mA 0.5mA 0A -0.5mA -2.0V I(l1) -1.0V 0V 1.0V V(1) 141 2.0V Problema 1c v=6.5V 4.0mA 2.0mA 0A -2.0mA 400us I(l1) 450us 500us 550us 600us Time 4.0mA 2.0mA 0A -2.0mA -8.0V I(l1) -4.0V 0V 4.0V V(1) 142 8.0V ✂✁☎✄✝✆✞✁✠✟☛✡✌☞✎✍✑✏✒✁✔✓ [✕ ] ✖ ✗ ✙✛✚✢✜✔✣✥✤✧✦✥★✪✩✬✫✭✗✯✮✱✰✳✲✌✤✵✴✵✶✷✲✸✴✞✚☎✲✪✣✥✤✧✦✥★✌✩✛✹✻✺✼✶✾✽✾✣✎✿❀✣❁✣✎✿❀✣✥✦✎✴✔❂✥✰✷✴❃✣✥❄✌✲✸✚❀✦✎✚✢✚❅✗❇❆✭✶✷❂✎✴✔✣❈✶❊❉❋✗❍●❍✣✥✰✳❂✎✚❅✶✻✦✎✚☎❂✥✦✥★■✚✢✴✔✣✎✿❀✰✳❂ ✘ ✣✎✿☎✣✥✦✎✴❃❂✎✚❀✦✥✣✱✹❑❏✬▲✸✚✢✴✔★✪❂❈✶▼❆◆❂✎✚❀✲✸✴❃✣✥✦✥❄P❖❀◗❘✲✪▲✌✰✳❂✼●❇✚✢❙❚✰❱❯❃✩✷✺✬★✌✦✥★✌❂✥✣✥❲✵✴✠✚❀✩✸✕❨❳✳❳✳❩✳✩✷❉❋❬✑✺❍✙❭❳✳❪✷❫✳❴✔❳✳❩✷❵■❛✷❫✳❴✔❛✷❴❃❛■✗ [2] ✫✭✗✛✮✱✰✳✲✌✤✵✴✵✶✷✲✸✴✞✚☎✲✪✣✥✤✧✦✥★✌✩✛✖✘✗ ✙✛✚❜✜❃✣✥✤✧✦✥★❝✹❞❏❍✿❀✣✥✦✎✴✔❂✥✰✷✴✔❄✪✣✥✲✸✚❀✦❈❡❢❙❣✣✥✲■✴✔❂✥★❤✫✭✶✷✦✥★✸✿✢✴✵✶✾✴❃✣❈✶❞▲✌✣✐◗❥★✸✴❃✰❧❦♠✶❨✴✞✚❀✦❈❡❞❲✒✚ ✮❥✶✾✿❀✦✥★✸✿❅✶✾✴✔✰✑✶✷❂✥✣❈✹✷✩❧✦✥★✪❂✥✤◆❙❚★✸♥✪✿❜✚❀✦❈✶✾✴☛❙❣✣✯✚☎✲■✴✔✣✥❂✥✲✪✣✎✴✔✩ ❄✸✴✞✴✠❙◆♦q♣✢♣❅r✞✣✥❂✥❂❈✶✷❂✎✚✞✗s✿❀❦t✲❣✗s❙❣★❣✗✉❂✥✰❨♣❅✤✵✴✔★✌▲✪✣✥✲✸✴✞✚ [3] ✖❑✗ ✙✛✚✢✜✔✣✥✤✧✦✥★✪✩✼✫✭✗✈✮✛✰❧✲✪✤✒✴✵✶✷✲■✴✠✚❀✲✌✣✥✤✧✦✥★❚✹❢❏❍✿❀✣✥✦✎✴✔❂✥✰✷✴✔❄✪✣✥✲✸✚❀✦❈❡❢❙❣✣✥✲■✴✔❂✥★❤✫✭✶✷✦✥★✸✿✢✴✵✶✾✴✔✣❈✶❢❏✬✲✪✣✥❂✥✇✳✣✎✴✠✚❀✦❈❡✳✹❱✩✛✦✥★✌❂✥✤ ❙❣★✸♥✪✿✢✚☎✦❈✶✾✴☛❙❣✣✯✚❀✲✸✴✔✣✥❂✥✲✪✣✎✴✔✩ ❄✸✴✞✴✠❙①♦②♣✢♣☎r✠✣✥❂✥❂❈✶✷❂✎✚❅✗③✿❀❦t✲❣✗s❙❣★❚✗q❂✥✰❨♣☎✤✒✴✔★✪▲✌✣✥✲✸✴✞✚ [4] A ✗ Vladimirescu✩✑✹④❬❧❆⑤❉⑥✮✈❏⑦✹▼❏✬▲■✚✢✴✔★✌❂❈✶✱✴✔✣✥❄✌✲■✚❀✦❈❡❧✗✾✺✬★✪✦✥★✌❂✥✣✥❲✒✴✞✚☎✩✸✕✾❳✳❳✳❳❧✩✷❉❋❬❧✺❍✙❭❳✳❪❱❫✳❴✞❫❚✕⑧❴✒✕❈⑨✳⑨✑❛✷❴✔❳✑✗ [5] L. O. Chua, C. A. Desoer, E. S. Kuh, ✹❨⑩❍✚❀✲✪✣❈✶❱❂❶✶✷✲✌▲❁✲✪✰❧✲■✿✢✚☎✲✪✣❈✶✷❂✂✦✎✚❀❂✥✦✥★✸✚✢✴❃✤⑥✹❷✖❞✦❈✗❹❸❺❂❈✶✾❻✱❴✠❼❽✚✢✿❜✿❀✩ ✕❨❳✳❩✳❪✑✗ 143