1. Introduction

Silicon carbide (SiC)-based power devices provide significant benefits over their traditional silicon counterparts, including the ability to operate at higher switching frequencies and elevated temperatures^{[1, 2]}. However, the faster switching behavior of SiC-based power devices causes issues that impact connected electric motors^{[3- 5]}. The reflected voltage phenomenon arises from an impedance mismatch between the inverter output and the motor terminals in motor drive systems^[6]. The double pulsing effect (DPE) occurs when the reflected voltage from the previous pulse remains undamped before the arrival of the next switching pulse^[7]. The overvoltage caused by DPE can reach two to four times the DC bus voltage, which exceeds the limits defined by the National Electrical Manufacturers Association (NEMA)^[8]. Several factors affect the magnitude of the overvoltage, such as cable length, the degree of impedance mismatch between the motor and the cable, and the pulse damping duration^[9]. Prior studies have proposed various countermeasures to mitigate overvoltage, including output filter implementation and the optimization of switching patterns^[10].

The dv/dt filters are used to mitigate overvoltage^{[11, 12]}. However, filter design optimization is essential to maintain efficiency in motor drive applications as filters add bulk and cost to the system^{[13- 15]}. Applying PWM techniques can reduce system costs and mitigate motor overvoltage more effectively compared to filters^{[16, 17]}. The maximum minimum pulse technique (MMPT) mitigates overvoltage by limiting the duty ratio to a specific threshold^[18]. The pulse elimination technique (PET) mitigates overvoltage by switching to either the maximum or minimum duty ratio in the regions where overvoltage occurs^{[19, 20]}. Nevertheless, both MMPT and PET cause distortions of the output voltage, thereby degrading current quality. Conventional DPWM, including 30°, 60°, and 120°, is a modulation technique that reduces the switching frequency by operating only two of the three phases to minimize power loss^{[21, 22]}. These conventional DPWM methods achieve higher efficiency than the space vector PWM (SVPWM) in high-power and high-frequency applications^{[23- 26]}. 60° DPWM is particularly effective in mitigating overvoltage as the discontinuous region is located in the peak phase voltage region, where the probability of overvoltage occurrence is highest. However, the discontinuous region of 60° DPWM is fixed, which results in higher total harmonic distortion (THD) and greater harmonic components of the output current compared to SVPWM^[27].

Adjusting the discontinuous region according to the DPE occurrence is essential to improve the output characteristics in the conventional 60° DPWM^[28]. The proposed method determines the required discontinuous region for overvoltage mitigation based on the P&O algorithm. Unlike the 60° DPWM, the proposed method eliminates unnecessary discontinuous regions from the perspective of overvoltage mitigation, thereby improving output characteristics. The discontinuous region is set to 120° within one period of the reference voltage, similar to 60° DPWM.

This paper proposes a mitigation of motor overvoltage using P&O alogrithm-based discontinuous DPWM in motor drive systems. The voltage reflection is analyzed using transmission line theory in motor drive systems with long transmission lines, and the mechanism of the DPE is studied. The proposed method detects overvoltage occurrence and adjusts the discontinuous region to mitigate motor overvoltage events. The variation characteristics of the discontinuous region are analyzed with respect to the MI. The validity of the proposed method is verified through simulation results.

2. Motor Overvoltage in Motor Drive Systems with a Long Cable

The overvoltage phenomenon is examined through the application of transmission line theory in motor drive systems. The mechanism of the double pulsing effect is clarified by analyzing the propagation of switching signals and the resulting voltage reflections along the transmission line.

2.1 Analysis of the Reflected Voltage Using Transmission Line Theory

The voltage reflection on the transmission line with an inverter and the motor are analyzed using transmission line theory. The mechanism of double pulsing effect is clarified by tracing reflections of the inverter switching edges. Fig. 1(a) illustrates a motor drive system in which the inverter and motor are connected by a cable. Fig. 1(b) presents the RLGC per-unit-length model of an infinitesimal line segment. Applying Kirchhoff’s voltage law and current law, Fig. 1(b) yields the equations for v(x, t) and i(x,t), given in equation (1) and equation (2).

(1)

$V(z)=(R+j\omega L)\Delta zI(z) + V(z+\Delta z)$

(2)

$I(z)=(G+j\omega C)\Delta zV(z+\Delta z) + I(z+\Delta z)$

The equations of the transmission line are as follows:

(3)

$\frac{dV(z)}{dz} = \lim_{\Delta z\to 0} \frac{V(z+\Delta z)-V(z)}{\Delta z} = -(R+j\omega L)I(z)$

(4)

$\frac{dI(z)}{dz} = \lim_{\Delta z\to 0} \frac{I(z+\Delta z)-I(z)}{\Delta z} = -(G+j\omega C)V(z)$

The second-order wave equations (3) and (4) lead to the voltage and current wave. Each solution is decomposed into incident and reflected components. The +z term denotes the incident wave propagating in the +z direction, and the −z term denotes the reflected wave propagating in the −z direction. The voltage and current wave is presented in equation (5) and equation (6), respectively.

(5)

$V(z) = V_0^+ e^{-rz} + V_0^- e^{rz}$

(6)

$I(z) = I_0^+ e^{-rz} - I_0^- e^{rz}$

The characteristic impedance Z₀ of Fig. 1(b) is presented with the cable parameters R, L, G, and C. Z₀ is expressed as shown in equation (7). Impedance mismatch at inverter and motor terminals generates incident and reflected waves. The reflection coefficients at the inverter and motor are defined in equation (8). The reflection coefficients specify the ratio of the reflected waves to the incident wave at each terminal. In Fig. 1(b), R models series resistance due to thermal loss during current flow, L signifies series inductance caused by the magnetic field, G represents shunt conductance related to leakage current loss, and C specifies shunt capacitance caused by the electric field when voltage is applied to the cable. Fig. 2 shows the equivalent cable model used for connection with an inverter and the motor. The series branch parameters R_s-2 and L_s-2 account for parameters related to dielectric loss, and R_p-2 and C_p-2 represent parameters considering the proximity effect and the skin effect.

(7)

$Z_0 = \frac{V_0^+}{I_0^+} = \sqrt{\frac{R+j\omega L}{G+j\omega C}}$

where Z_inv, Z_mot, and Z₀ mean impedances of the inverter and motor and characteristic impedance respectively.

(8)

$\Gamma_{inv} = \frac{Z_{inv}-Z_0}{Z_{inv}+Z_0}, \Gamma_{mot} = \frac{Z_{mot}-Z_0}{Z_{mot}+Z_0}$

The propagation time t_p is the travel time of a voltage pulse from an inverter to the motor terminals, as given in equation (9). The propagation velocity v is determined by the per-unit-length inductance L and capacitance C. The characteristic impedance is Z₀, which reduces when R and G are negligible.

(9)

$t_p = \frac{l}{v} = l\sqrt{LC}$

Fig. 1. Motor drive system and cable: (a) Motor drive system with a long cable (b) The equivalent circuit of a transmission line

Fig. 2. Equivalent cable model accounting for high-frequency effects

2.2 Voltage Oscillation Caused by Reflected Waves

Fig. 3 illustrates the process of voltage oscillation with respect to the propagation time t_p between the inverter output and the motor terminals. Region (1) in Fig. 3 denotes the interval after the inverter steps to V_dc and before t_p elapses, during which the voltage wave propagates from the inverter output to the motor terminals. In this interval, the inverter-side voltage V_ab rises to V_dc, while the motor-side voltage V_mot remains at 0 V as the wave has not reached the motor terminals. Region (2) in Fig. 3 shows the instant the wave arrives at the motor after t_p and reflects toward the inverter. The reflected voltage at the motor terminals is scaled by the reflection coefficient Γ_mot, yielding the peak voltage V_max, as shown in equation (10).

(10)

$V_{max} = (1+\Gamma_{mot})V_{dc}$

Following Region (2), multiple reflections propagate between the inverter and the motor, with amplitudes decaying according to reflection coefficients. The minimum motor-terminal value V_min is given in equation (11).

(11)

$V_{min} = (1+\Gamma_{mot} + \Gamma_{mot}\Gamma_{inv} + \Gamma_{mot}^2\Gamma_{inv}) \cdot V_{dc}$

The voltage oscillations at the motor terminals due to voltage reflection phenomenon are generalized and represented by equations (12) and (13).

(12)

$V_{u(n)} = V_{dc} - \left\{ \sum_{x=0}^{2n-1} \Gamma_{mot}^x \Gamma_{inv}^x \cdot (1+\Gamma_{mot}) \cdot V_{dc} \right\}$

(13)

$V_{d(n)} = \left\{ \sum_{x=0}^{2n-2} \Gamma_{mot}^x \Gamma_{inv}^x \cdot (1+\Gamma_{mot}) \cdot V_{dc} \right\} - V_{dc}$

Fig. 3. Voltage reflection phenomenon with respect to propagation time

Fig. 4. Line-to-line voltage at the motor terminals induced by voltage reflection.

2.3 The Occurrence Mechanism of the Double Pulsing Effect

Fig. 4 illustrates the line-to-line voltages at both the inverter and the motor terminal during the occurrence of the DPE. The application time of the zero-voltage vector affects the probability of the DPE occurrence. This phenomenon occurs when the reflected voltage at the motor terminals has not fully decayed before the arrival of the next pulse, causing the motor line-to-line voltage to exceed twice the DC bus voltage. The minimum on-time of the zero-voltage vector required to prevent the DPE, denoted as t_lim, and the maximum duty ratio at which the DPE does not appear, denoted as Duty_lim, are defined as follows:

(14)

$t_{lim} = nt_{osc} + \frac{t_{osc}}{2}$

(15)

$Duty_{lim} = \frac{V_{ref}}{V_{dc}/2} = \frac{T_s-t_{lim}}{T_s} = \frac{T_s-nt_{osc}+\frac{t_{osc}}{2}}{T_s}$

Fig. 5 presents the enlarged waveform within one control period T_s, when the DPE occurs. When the magnitude of the phase-a reference voltage exceeds Duty_lim, the application time of the zero-voltage vector is shorter than t_lim. When the application time of the zero-voltage vector is shorter than t_lim, the fluctuation of the line-to-line voltage cannot be fully damped and overlaps with the next pulse, thereby increasing the likelihood of DPE occurrence. Therefore, the probability of DPE occurrence increases proportionally with the magnitude of the reference voltage.

Fig. 5. Expanded waveform of the reference voltage during DPE occurrence

3. Adjustment of Discontinuous Region Based on the P&O Algorithm

The offset voltage required to generate the discontinuous region is analyzed in 60° DPWM. The proposed method mitigates overvoltage by utilizing a discontinuous region based on 60° DPWM. The optimal discontinuous region required for overvoltage mitigation is determined using the P&O algorithm.

3.1 The Characteristics of 60°DPWM

60° DPWM is widely employed in high-power systems to balance the tradeoff between switching frequency and power loss. To minimize loss, the discontinuous region is positioned at the peak of the phase voltage. The offset voltage V_sn required to implement 60° DPWM is defined by equations (16) and (17). v_x,max and v_x,min represent the maximum and minimum values among the three-phase reference voltage. V_sn is added according to predefined conditions so that each phase maintains a discontinuous region.

(16)

$V_{sn} = \frac{V_{dc}}{2} - v_{x,max} \ (if \ v_{x,max} + v_{x,min} \ge 0)$

(17)

$V_{sn} = \frac{V_{dc}}{2} - v_{x,min} \ (if \ v_{x,max} + v_{x,min} < 0)$

Fig. 6 illustrates one cycle of the reference voltage in 60° DPWM. Region (a) corresponds to the interval where the upper switch of phase-a remains is held at the maximum duty ratio, preventing switching. Region (b) represents the interval where the lower switch of phase-a is held at the minimum duty ratio to disable switching. Outside the discontinuous regions, the reference voltage is modified by adding the offset voltage to maintain the magnitude of the fundamental voltage. The placement of discontinuous regions affects both the switching loss and the harmonic performance. The discontinuous regions are arranged near the peak phase current to reduce switching loss within one period of the reference voltage. For each phase, the discontinuous region extends up to 120°, within a single cycle of the reference voltage.

Fig. 6. Reference voltage of 60° DPWM

3.2 Proposed Method Using the P&O Algorithm

The P&O algorithm operates based on experimental data without requiring prior information. Therefore, the P&O algorithm is suited for embedded controllers and cost-sensitive applications. In the proposed method, the P&O algorithm is utilized to dynamically regulate the discontinuous region for mitigating the DPE in comparison with 60° DPWM. The discontinuous region is adjusted according to the detected occurrence of motor overvoltage.

Fig. 7 presents a flowchart of the operating principle in the proposed method. The magnitude of the discontinuous region is updated based on the presence of the DPE. When the discontinuous region exceeds 60°, the fluctuation component of the discontinuous region is constrained to zero in the subsequent control cycle to prevent a further increase.

Fig. 7. Flowchart of the proposed method

4. Simulation Results

The two-level inverter system is modeled by PSIM software, and the performance of the proposed method is verified through imulations. The simulations are performed under the conditions shown in Table 1. Table 2 represents the parameters used for cable modeling. The simulation results illustrate the optimal discontinuous region where the DPE is mitigated according to MI. The mitigation performance and output characteristics of the proposed method and 60° DPWM are compared based on simulation results.

Fig. 8(a) illustrates the simulation result for the case without applying the proposed method and shows the region where the DPE occurs. Region (a) of Fig. 8(a) depicts the region where the reference voltage exceeds Duty_lim, resulting in the occurrence of the DPE. Fig. 8(b) shows an enlarged view of region (a) where the DPE occurs. The peak value of the line-to-line voltage measured at the motor terminals reaches 737.13 V.

Fig. 9(a) and Fig. 9(b) illustrate the simulation results where the proposed method is applied to mitigate the DPE under different MI conditions. Region (1) of Fig. 9(a) demonstrates the process of determining the discontinuous region that synchronizes with the overvoltage region varying with MI. The discontinuous region derived in region (1) is 17°. The derived discontinuous region is applied to the inverter system, resulting in the mitigation of overvoltage in region (2). The discontinuous region required for overvoltage mitigation is obtained as 51.2° in Fig. 9(b). Consequently, the time required to identify the appropriate discontinuous region for mitigating overvoltage increases as MI increases.

Fig. 10 presents a comparison of the output characteristics between the proposed method and 60° DPWM. Based on the simulation results, the proposed method enhances output performance by minimizing unnecessary discontinuous regions that do not contribute to mitigating overvoltage, unlike 60° DPWM.

The proposed method requires sensing of the motor-side line-to-line voltage to determine the occurrence of DPE. Therefore, a high-performance voltage sensor with a bandwidth sufficient to detect the oscillation time of the reflected voltage is required.

Fig. 8. Simulation results of the DPE occurrence region (a) before applying the proposed method (b) enlarged waveform of region (a)

Fig. 9. Simulation results of the proposed method (a) MI = 0.91 (b) MI = 1

Fig. 10. Simulation results of the proposed method and 60° DPWM

5. Conclusion

This paper proposes a mitigation of motor overvoltage using P&O-based discontinuous DPWM in motor drive systems. Voltage reflection is investigated based on transmission line theory. The proposed method detects overvoltage by a voltage sensor and modulates the discontinuous region to mitigate overvoltage. The dependence of the discontinuous region is derived according to the MI. The effectiveness of the proposed method is validated through simulation results.