An Improved Near-State Pulse-Width Modulation with Low Switching Loss for a Permanent Magnet Synchronous Machine Drive System

Abstract

Common-mode voltage (CMV) leads to the shaft voltage and shaft current by coupling the capacitor network in the permanent magnet synchronous machine (PMSM), which affects the reliability of the whole motor drive system. Based on the low-CMV modulation strategy for the PMSM drive system, this paper proposed an improved near-state pulse-width modulation (NSPWM) on switching loss. First, the generation mechanism for the switching signals of NSPWM was analyzed, and it was observed that there exists one phase of switches in an inactive state for every sector. Then, to reduce the switching loss of the NSPWM, this paper proposed an improved NSPWM modulation strategy based on power factor angle to adjust switching action, which ensures the switching tubes that have the biggest conduction current have no switching action. In addition, the switching loss analytic formula of the NSPWM was derived to prove the correctness of the proposed method for optimizing switching loss. Finally, the proposed modulation strategy was carried out in the simulation and experimental platform. Under the premise of good steady and dynamic performance, the results show that the proposed modulation strategy has less switching loss.

1. Introduction

Compared to current source inverters (CSIs) and other multilevel inverters, three-phase two-level voltage source inverters (VSIs) have the advantage of a simple structure and bidirectional energy conversion [1,2], so VSIs have been widely used in permanent magnet synchronous machine (PMSM) drive systems [3], and many modulation methods for VSIs have been proposed, such as SPWM, SVPWM, DPWM and dual-vector modulation [4,5]. However, there are some problems that endanger the safe and stable operation of this system. For example, the common-mode voltage (CMV) generated at the neutral point leads to the shaft voltage and shaft current by coupling the capacitor network in the motor [6], which affects the reliability of the whole PMSM drive system [7].

To reduce the CMV of the PMSM drive system fed by VSI, Reference [8] introduces CMV into the cost function and proposes a model predictive control scheme that takes into account CMV. An optimized zero-sequence voltage injection method is proposed [9], and the CMV amplitude can decrease to 50%. In [10], a kind of harmonic elimination pulse-width modulation with a common-mode voltage reduction ability is presented. However, the CMV reduction method proposed in [11] is suitable for multi-level inverters.

For two-level VSIs, a switching tube is introduced to the DC bus, which makes the AC part of the three-phase converter float in the zero state. Through improvements in structure and control strategy, the CMV remains unchanged in the zero state [12]. Reference [13] proposes a three-phase AC-side voltage-doubling topology with an intrinsic buck–boost unit, which suppresses the average CMV to zero by modulating the output phase voltage around its negative bus. However, the above two methods add the hardware circuit in the three-phase two-level VSI, which not only increases costs but also reduces efficiency and reliability. Therefore, reducing CMV through modulation is the most effective means.

Since the amplitude of CMV in zero state is Udc/2, which is 3 times of that in active state, reducing the common-mode voltage of the zero vector or eliminating the zero vector is the most effective modulation method [14]. References [15,16] control the dead time switching tube by current sector determination and switch function direct control, so the CMV amplitude is suppressed by changing the current freewheeling path. Although this method can reserve the zero state, the dead time at the commutation time will cause a zero current clamping effect. Active zero-state PWM (AZSPWM) adopts two opposite AVVs to replace the ZVV, so the high-frequency CMV can be effectively reduced. However, the ripple of the current is too great [17]. In comparison, near-state PWM (NSPWM) adopts three adjacent active vectors to synthesize the reference vector; it can reduce the high- and low-frequency CMV simultaneously, and the steady state under NSPWM is better than that under AZSPWM [18]. It should be noted that no matter where the reference vector is located, there are two vector synthesis methods, and there exists one phase of switches in an inactive state.

Based on the characteristic analysis of NSPWM, to reduce the switching loss, this paper proposed an improved NSPWM modulation strategy based on power factor to adjust the switching action. To ensure the switching tube that has the biggest conduction current has no switching action, the NSPWM with the least switching loss is proposed. In the simulation and experiment part, the effectiveness of the proposed method on steady-state, dynamic-state performance and switching loss optimization is shown.

This paper is organized as follows: Section 2 describes the topology and whole control system. Then, an improved NSPWM scheme is proposed in Section 3. Consequently, the effectiveness and superiority of the proposed scheme are validated by the simulation and experiment results in Section 4 and Section 5. Finally, the conclusion is drawn in Section 6.

2. Topology and Field-Oriented Control

2.1. Topology of PMSM Control System

According to [18], the NSPWM has a limited linearity region, which means it can only modulate the voltage in that the modulation index M is greater than 0.61. However, when PMSM operates at low speed, the NSPWM generates high CMV and does not modulate the reference voltage correctly. Therefore, to reduce the DC voltage, the buck converter is introduced in the DC side of the PMSM drive system.

Figure 1 shows the topology of the PMSM control system fed by a three-phase voltage source inverter with a buck converter, where $u_{in}$ represents the DC source voltage; $u_{dc}$ represents the DC-link voltage; $C_{dc}$, $S_0$, $D_0$ and $L_{dc}$ constitute the buck converter; $S_1$, $S_2$, $S_3$, $S_4$, $S_5$ and $S_6$ represent the switching tube of the three-phase VSI bridge.

2.2. Mathmatical Model of PMSM

According to [19], in the synchronous rotating frame ($d-q$ frame), the voltage state equations of the PMSM are shown as follows:

$$u_d=R_s{i}_d+\frac{d{\psi }_d}{dt}-\omega {\psi }_q$$

(1)

$$u_q=R_s{i}_q+\frac{d{\psi }_d}{dt}-\omega {\psi }_d$$

(2)

where $u_d$ and $u_q$ represent the stator voltage in the $d-q$ frame, $i_d$ and $i_q$ represent the stator current in the $d-q$ frame, $R_s$ is the stator resistance, $\omega$ is the synchronous angle frequency and ${\psi }_d$ and ${\psi }_q$ are the flux linkage in the $d-q$ axis.

The equations of electromagnetic flux linkage are shown as [20]

$${\psi }_d=L_d{i}_d+{\psi }_f$$

(3)

$${\psi }_d=L_q{i}_q$$

(4)

where $L_d$ and $L_q$ are the stator inductance in the $d-q$ frame and ${\psi }_f$ is the flux linkage of PMSM.

The electromagnetic torque equation is shown as

$$T_e=\frac{3}{2}{n}_p[{\psi }_f{i}_q+(L_d-L_q)i_d{i}_q]$$

(5)

where $T_e$ represents the electromagnetic torque and $n_p$ is the pole pairs.

2.3. Field-Oriented Control

Field-oriented control (FOC) is adopted in this paper; the whole diagram is shown in Figure 2. FOC mainly includes speed closed-loop and current closed-loop control. The given rotational speed is compared with the actual rotational speed, the deviation is passed through the speed PI controller and the $i_d$ and $i_q$ are compared with the actual value to obtain the deviation to the current PI controller. And the output of the current controller $u_d$ and $u_q$ are transformed into the $\alpha -\beta$ coordinate system. Through the modulation module output switching signals to the inverter, the inverter outputs three-phase voltage to directly drive the PMSM.

2.4. Conventional NSPWM

NSPWM adapts three neighbor voltage vectors to match the reference voltage. The three voltage vectors are the voltage vectors that are closest to the reference voltage vector and its two neighbors (Figure 3b). The utilized voltage vectors are changed every 60° throughout the space. As shown in Figure 3a, the voltage vector space is divided into six sectors. Defined with indices i, voltage vectors $V_{i-1}$, $V_i$, $V_{i+1}$ are utilized for region $A_i$.

Utilizing the three near-state voltage vectors, the volt–seconds balance equation and the PWM period constraint for NSPWM are given for region $A_i$ in the equations below.

$$V_{i-1}{t}_{i-1}+V_i{t}_i+V_{i+1}{t}_{i+1}=V_{ref}{T}_s$$

(6)

$$t_{i-1}+t_t+t_{i+1}=T_s$$

(7)

where $T_s$ is the PWM period.

Normalizing the voltage vector, the vector duty cycles can be found as $d_k=\frac{t_k}{T_s}$, where k is $i-1$, i, $i+1$. Utilizing the above equations, the duty cycles of the voltage vectors can be calculated for region $A_i$ as follows:

$$d_{t-1}=1-\frac{2\sqrt{3}}{\pi }{M}_{i}sin(\theta -\frac{(i-2)\pi }{3})$$

(8)

$$d_{t-1}=-1+\frac{3}{\pi }{M}_{i}cos(\theta -\frac{(i-2)\pi }{3})+\frac{3\sqrt{3}}{\pi }{M}_{i}sin(\theta -\frac{(i-2)\pi }{3})$$

(9)

$$d_{t-1}=1-\frac{3}{\pi }{M}_{i}cos(\theta -\frac{(i-2)\pi }{3})-\frac{\sqrt{3}}{\pi }{M}_{i}sin(\theta -\frac{(i-2)\pi }{3})$$

(10)

In the above equations, $\theta$ is the angle of the reference voltage vector with the $\alpha$ axis and $M_i$ is the modulation index which indicates the voltage utilization level $M_i=\frac{V_m}{\frac{2V_{dc}}{\pi }}$. Equations (8)–(10) yield a valid solution only in the colored region shown in Figure 3c, and in the white remaining regions, there is no solution.

The sequence of NSPWM is shown in

Table 1. Basic synthesized vector sequence for NSPWM in all sectors.

	${\mathit{A}}_1$	${\mathit{A}}_2$	${\mathit{A}}_3$	${\mathit{A}}_4$	${\mathit{A}}_5$	${\mathit{A}}_6$
1st Vector	$V_2$	$V_3$	$V_4$	$V_5$	$V_6$	$V_1$
2nd Vector	$V_1$	$V_2$	$V_3$	$V_4$	$V_5$	$V_6$
3rd Vector	$V_6$	$V_1$	$V_2$	$V_3$	$V_4$	$V_5$
4th Vector	$V_1$	$V_2$	$V_3$	$V_4$	$V_5$	$V_6$
5th Vector	$V_2$	$V_3$	$V_4$	$V_5$	$V_6$	$V_1$

for region definitions of Figure 3a.

3. Optimized NSPWM Strategy

3.1. Conventional NSPWM Redundancy Analysis

NSPWM adopts three near-basic voltage vectors to synthesize the reference voltage vector; it has one phase switch that remains clamped for every PWM period. For example, in $A_1$, the voltage vectors $u_6$, $u_1$ and $u_2$ are adopted, and phase A remains clamped open. Figure 4 shows the three-phase switching signals for all six sectors.

A conventional NSPWM can reduce switching loss by voltage clamping. However, the conventional NSPWM synthesis strategy does not optimize the switching loss. To analyze the switching loss, Figure 5 shows the three-phase reference voltage vector and three-phase current vector in the synchronous station frame ($\alpha$-$\beta$) through the Clarke transform in Equation (11).

$$i=\frac{2}{3}(i_a+i_b{e}^{j\frac{2\pi }{3}}+i_c{e}^{j\frac{2\pi }{3}})$$

(11)

In Equation (11), i represents the actual current vector, and $i_a$, $i_b$ and $i_c$ represent the current magnitude in phases a, b and c.

For PMSM, the current vector i lags reference voltage vector $U_{ref}$ by power factor angle ($\phi$), and the angle between $U_{ref}$ with $\alpha$ axis is $\theta$. The $U_{ref}$ location determines the basic synthesized vector group, but it still can be improved. For example, Figure 6 shows the two methods to synthesize the $U_{ref}$ when $U_{ref}$ is located near the sector boundary, but the current vector i could be used to optimize the switching loss. In Figure 5b, when $\phi$ = $\theta$, the phase A current is maximum. Therefore, to reduce the switching loss, phase A needs clamping, and the best-synthesized group is $V_6$, $V_1$, $V_2$. Actually, when $U_{ref}$ is located in $B_1$, the phase A current magnitude is the biggest phase current, so the best-synthesized vector group for $U_{ref}$ located in $B_1$ is $V_6$, $V_1$, $V_2$. In conclusion, when the inverter works in a non-unit power factor, the conventional NSPWM cannot best optimize the switching loss.

3.2. Improved NSPWM on Switching Loss

When an inverter switches, it will cause large switching loss, reduce the working efficiency and cause large heat dissipation. To reduce the switching loss of NSPWM, the improved NSPWM method is proposed. Compared with NSPWM, the proposed method adopts different sector division methods, which introduce angle variable $\alpha$ to dynamically adjust the sector division, as shown in Figure 7a. The synthesized voltage vector sequences for all sectors are shown in

Table 2. Basic synthesized vector sequence for the improved NSPWM in all sectors.

	${\mathit{B}}_1$	${\mathit{B}}_2$	${\mathit{B}}_3$	${\mathit{B}}_4$	${\mathit{B}}_5$	${\mathit{B}}_6$
1st Vector	$V_2$	$V_3$	$V_4$	$V_5$	$V_6$	$V_1$
2nd Vector	$V_1$	$V_2$	$V_3$	$V_4$	$V_5$	$V_6$
3rd Vector	$V_6$	$V_1$	$V_2$	$V_3$	$V_4$	$V_5$
4th Vector	$V_1$	$V_2$	$V_3$	$V_4$	$V_5$	$V_6$
5th Vector	$V_2$	$V_3$	$V_4$	$V_5$	$V_6$	$V_1$

, which are the same as those for the conventional NSPWM.

3.3. Linearity Region

Compared with the conventional NSPWM, the proposed improved NSPWM just optimizes the synthesized vector group; the basic vector synthesis strategy is the same as NSPWM. Therefore, the improved method still has the limited linearity region problem. From the synthesized principle, the linearity region is related to variable $\alpha$. For example, in Figure 7b, the basic vector group $u_6$, $u_1$, $u_2$ can only synthesize the reference voltage vector in the black-lined region, and the linearity region is the light blue region, which is expressed as

$$\frac{\pi }{6cos(\left|\alpha \right|+30°)}\le M_i\le 0.907$$

(12)

Therefore, only when $\alpha \in \left[-24.74°,24.74°\right]$, the improved NSPWM has per-fundamental linearity region. The larger the $\left|\alpha \right|$, the smaller the linearity region. So, choosing $\alpha$ not only relates to switching loss but also relates to the linearity region.

3.4. Switching Loss Analysis

In this paper, to evaluate the switching loss of the proposed method and the conventional NSPWM, the current ripple is neglected, and thus, the phase current $i_p$ can be assumed as

$$i_p=Icos(\theta -\phi )$$

(13)

In the equation above, I represents the amplitude of the fundamental current component, $\phi$ is the power factor angle and $\theta$ represents the electrical angle of the reference voltage.

From [21], the phase average switching loss in one unit for a one-carrier cycle is defined as

$$P_{sw\_p}=\frac{f_{sw}{u}_{dc}(E_{on}+E_{off})}{2\pi I_{rate}{U}_{rate}}[{\int }_a^b|i_p\left(\theta \right)|d\theta ]$$

(14)

In the equation above, $E_{on}$ and $E_{off}$ represent the switch-on energy loss and switch-off energy loss under $f_{sw}$ working frequency, $I_{rate}$ and $U_{rate}$ represent the rated current and rated voltage for the switch and ${\int }_a^b$ represent the interval that the switch does not clamp. For phase a, it has voltage clamping in sectors $A_1$ and $A_4$, so for the improved NPSWM, the switching loss for phase A in a one-carrier cycle is expressed as

$$P_{sw\_a}=\frac{f_{sw}{u}_{dc}(E_{on}+E_{off})}{2\pi I_{rate}{U}_{rate}}[{\int }_{\frac{\pi }{6}}^{\frac{5\pi }{6}}|i_a\left(\theta \right)|d\theta +{\int }_{\frac{7\pi }{6}}^{\frac{11\pi }{6}}|i_a\left(\theta \right)|d\theta ]$$

(15)

By simplifying the equation above, the switching loss for improved NSPWM can be obtained as follows:

$$P_{sw\_a}=\frac{I{f}_{sw}{u}_{dc}(E_{on}+E_{off})}{\pi I_{rate}{U}_{rate}}$$

(16)

It can be seen that the switching loss is proportional to the switching frequency, the DC-link voltage and the phase current. The same conclusion can be obtained in reference [22]. For comparison, in the conventional NPSWM, the switching loss in a one-carrier cycle is expressed as

$$P_{sw\_a}^{\prime }=\frac{f_{sw}{u}_{dc}(E_{on}+E_{off})}{2\pi I_{rate}{U}_{rate}}[{\int }_{\frac{\pi }{6}+\phi }^{\frac{5\pi }{6}+\phi }|i_a\left(\theta \right)|d\theta +{\int }_{\frac{7\pi }{6}+\phi }^{\frac{11\pi }{6}+\phi }|i_a\left(\theta \right)|d\theta ]$$

(17)

By simplifying the equation above, the switching loss for the conventional NSPWM can be obtained as follows:

$$P_{sw\_a}^{\prime }=\frac{I{f}_{sw}{u}_{dc}(E_{on}+E_{off})}{\pi I_{rate}{U}_{rate}}[2-cos\left(\phi \right)]$$

(18)

It is obvious that the conventional NSPWM switching loss is larger than the improved NSPWM ($P_{sw\_a}<P_{sw\_a}^{\prime }$). The correctness of the proposed method on optimizing the division of the sector by introducing variable $\alpha$ has been theoretically proved. For phases b and c, the switching loss expression is similar.

4. Simulation Results

The simulation models with thermal analysis of the proposed method and the conventional NSPWM are established in the Matlab Simulink(R2023b)/PLECS (4.8.4.) environment. PLECS is a particular tool working in Simulink to analyze the thermal environment of some equipment. The Simulink and PLECS models are both used in this model; the Simulink model mainly works on the signal processing and NSPWM control signal generation, while the PLCES model works mainly on the thermal analysis of the three-phase inverter. The steady state, dynamic performance and switching loss are verified by the simulation results; the PMSM parameters are shown in

Table 3. Parameters of PMSM.

Parameters	Description	Value
$R_s$ ($\mathrm{\Omega }$)	Stator resistance	1.443
$L_d$ (mH)	d-axis inductance	5.541
$L_q$ (mH)	q-axis inductance	5.541
${\psi }_f$ (Web)	Flux linkage	0.2852
$i_N$ (A)	Rated current	4.5
$n_N$ (r/min)	Rated speed	1000
$T_N$ (N·m)	Rated torque	10
$n_p$	Pole pairs	4
J ($\mathrm{kg}·{\mathrm{m}}^2$)	Rotational inertia	0.00194
$u_{dc}$ (v)	DC-bus voltage	270
$f_s$ (Hz)	Carrier frequency	10,000
$P_N$ (kW)	Rated power	1.0
$U_N$ (V)	Rated voltage	110

.

4.1. Steady-State Experiment

PMSM operates at a rated speed and rated load, and $u_{ref}$ is located in the at linearity region. $\alpha$ = $30$ was set for the proposed method; the speed, torque and phase A current waveforms are shown in Figure 8a,b for the conventional NSPWM and the proposed NSPWM, respectively. Because the proposed method just optimizes the division of sectors and does not change the vector synthesis strategy, the results show that both methods operate at good steady-state performance with 6.05% and 5.61% current THD and 0.2283 N·m and 0.2290 N·m torque ripple, respectively. The results prove that the proposed method has the same steady-state performance as the conventional NSPWM.

4.2. Dynamic-State Experiment

To verify the dynamic-state performance of the proposed method, $\alpha$ = $30$ was set, and PMSM was given a speed control step from static to rated speed without load. The comparison waveforms of speed, load and phase A current are shown in Figure 9a,b for the conventional NSPWM and the proposed method, respectively. The response time of the conventional NSPWM is 5.03 ms, and for the proposed method, it is 4.98 ms. The results show that the proposed method has the same dynamic performance as the conventional NSPWM.

4.3. Switching Loss Analysis

To analyze the switching loss of the IGBT, a thermal model was established in the PLECS environment and a thermal library was established based on the IGBT datasheet in Figure 10. Figure 11a,b show the three-phase inverter conductivity, switching loss and the phase current waveforms for the conventional NSPWM and the proposed method in a one-carrier cycle, respectively. The electrical angle waveforms of PMSM are shown at the top of Figure 11. The switching loss waveform comparison of the inverter show that the proposed method has less energy loss compared with the conventional NSPWM. This further proves that the proposed method optimizes the switching loss by optimizing the division of sectors.

5. Experiment Results

To show the correctness of the proposed method, the experimental platform was set up in Figure 12 according to

Table 4. Parameters of experimental platform.

Description	Value (Type)
IGBT module	FS400R07AE3
Film capacitor	400 µF/600 V
DC-link voltage	270V
DC source	PR300-4
Current sensors	ACS724-10AB
DSP	TMS320F28335
Carrier and sampling frequency	10 kHZ

. The parameters of the PMSM are presented in Table 3, and a magnetic powder brake with a rated torque of 10 N·m is used as the load of PMSM. The torque and speed are converted into analog signals with a refresh rate of 5 kHz through the DAC module.

5.1. Steady-State Performance

Experimental verification used the same operating conditions as the simulation. For the operational condition, 1000 rpm with a load of 10 N·m, $u_{ref}$ was located in the linear region of NSPWM. The steady experimental waveforms are shown in Figure 13a,b, and the current THD analysis results are shown in Figure 14a,b for the conventional NSPWM and the proposed method, respectively. The results show that the proposed method has the same steady-state performance as the conventional NSPWM method, operating with good current ripple.

The main harmonic components of the phase current are 5th, 7th, 11th and 13th orders, which are reflected in the 6th and 12th components in the torque. Meanwhile, the current THD of the proposed method is slightly smaller than that of the conventional NSPWM, so the electromagnetic torque THD of the proposed method is 9.62%, which is less than that of the conventional NSPWM.

5.2. Dynamic-State Performance

To verify the dynamic performance of the proposed method, the speed step and load step experiments were set up, from static to rated speed without load (Figure 15a,b) and load increasing from no load to rated load at rated speed (Figure 16a,b). The results show that the proposed improved NSPWM has a similar response time to the conventional NSPWM, which proves that the proposed improved NSPWM has a similar dynamic performance to the conventional NSPWM.

5.3. Switching Loss Analysis

The switching loss of the inverter in various operation conditions is collected, as well as for PMSMs operating at rated speeds with different load torques (Figure 17) and for PMSMs operating at rated load torques with different speeds (Figure 18). According to Equations (16) and (18), the switching loss is linearly proportional to the current peak value, so the mathematical switching loss results, the current peak value and the measured switching loss are all shown in these figures. Obviously, the switching loss of the proposed improved NSPWM is less than the conventional NSPWM under all operational conditions. The results show that, with the increase in load torque, the current magnitude increases, so the switching loss is also nearly linearly proportional to the increase. But the speed does not affect the current magnitude, so the switching loss remains nearly the same. Meanwhile, the theortical results from Equations (16) and (18) have little error with the experimental switching loss results, which prove the correctness of the derived switching loss analytic formula (Equations (16) and (18)) for the improved NSPWM and conventional NSPWM.

5.4. Common-Mode Voltage Analysis

To verify the common-mode voltage reduction performance of the proposed method, the steady-state CMV results are collected for the proposed method and SVPWM for comparison in Figure 19a,b. Because of the benefits of the voltage vector synthesizing strategy, the common-mode voltage waveform results show that the proposed improved NSPWM reduces the common-mode voltage peak value by three times compared with SVPWM.

6. Conclusions

To optimize the switching loss of the common-mode voltage suppressing scheme, this paper has proposed an improved near-state pulse-width modulation scheme. The proposed NSPWM optimizes the sector division, introducing a power factor angle to dynamically adjust the sector division and ensure the longest voltage clamping for the largest current. The feasibility and correctness are verified by the simulation and experimental results. The main conclusions can be summarized as follows:

• The proposed method ensures the switching tube that has the biggest conduction current has no switching action. So, the least switching loss can be obtained.
• According to the derived switching loss analytic formula, the proposed method has the least switching loss, and it does not relate to the power factor angle.
• The proposed method has the same steady-state performance and dynamic-state performance as the conventional NSPWM because it only introduces a power factor that does not change the vector synthesized strategy.