2.1. High-Frequency Cable Modeling

Fig. 1 illustrates a typical SiC-based motor drive system consisting of a dc-link voltage source Vdc, a voltage source inverter (VSI), power cables, and an electric motor. Precise modeling of the power cables is essential for analyzing voltage reflection and the resulting overvoltage at the motor terminals. In this work, cable modeling is also employed to support the design of overvoltage suppression filters. Although the conventional second-order section model shown in Fig. 2 is commonly adopted due to its simplicity, it may not adequately represent frequency-dependent phenomena, such as dielectric loss, proximity effect, and skin effect. Moreover, variations in cable length and physical properties make accurate parameter identification difficult when conventional modeling approaches are applied. To overcome these limitations, the high-frequency cable model, shown in Fig. 3 is employed. This model introduces additional elements to represent dielectric losses (Rp2, Cp2) as well as proximity and skin effects (Ls2, Rs2) ^[18]. As a result, the cable behavior may be described more accurately over a wide frequency range. The model parameters are extracted from measured short-circuit and open-circuit impedances, denoted as ZSC and ZOC, using an impedance analyzer.

Based on the short-circuit measurements, the inductances in (1) are obtained from low- and high-frequency impedance characteristics. Similarly, the capacitances in (2) are determined using the open-circuit measurement results.

where Ls1 and Ls2 represent the series and parallel inductances, Rs1 and Rs2 denote the corresponding resistances, Cp1 and Cp2 are the parallel and series capacitances, and Gp1 (=1/Rp1) and Gp2 (=1/Rp2) represent the parallel and series conductances, respectively.

By incorporating dielectric, proximity, and skin effects, the proposed high-frequency cable model improves the predictive accuracy of resonant behavior, voltage overshoot, and waveform distortion at the motor terminals. This enhanced modeling accuracy directly contributes to more reliable filter designs and more effective overvoltage suppression in SiC inverter systems.

2.2. Overvoltage Cause of Motor Line to Line Voltage

The motor drive system shown in Fig. 1 may be represented by the equivalent circuit illustrated in Fig. 2. In this representation, the motor is modeled by an impedance Zm, which includes stator leakage inductance as well as parasitic resistance and capacitance between the stator windings and the motor frame. Resistive and capacitive coupling between the neutral point of the stator and the frame is also considered. The inverter is modeled as a PWM voltage source with an internal impedance Zs, which is typically neglected as the dc-link capacitor provides a low-impedance path during fast voltage transitions. The power cable is modeled as a transmission line, and instead of a conventional representation, the high-frequency cable model shown in Fig. 3 is applied to the equivalent circuit in Fig. 2. The cable model parameters are derived from the measured ZSC and ZOC.

The series impedance, Z, and shunt admittance, Y, of the cable, including high-frequency effects, are expressed in (3) and (4), respectively.

The propagation constant $\gamma$ of the transmission line is obtained as the square root of the product of Z and Y, as given in (5). This parameter determines the signal propagation characteristics along the cable.

The characteristic impedance Z0 of the high-frequency cable is defined as the ratio of the series impedance to the propagation constant, as shown in (6).

When impedance measurements are used, the characteristic impedance may also be expressed in the form given in (7). For a lossless transmission line approximation, the characteristic impedance simplifies to the expression based on the per-unit-length inductance and capacitance.

Voltage reflection in SiC inverter systems arises from the impedance mismatch between the power cable and the motor, leading to overvoltage oscillations at the motor terminals. This reflected-wave phenomenon (RWP) has been extensively studied using transmission line theory and reported in literature. Therefore, only a brief overview is provided here.

As illustrated in Fig. 4, the PWM voltage generated by the inverter propagates toward the motor along the transmission line. Due to the impedance mismatch between the motor impedance, Zm, and the cable characteristic impedance, Z0, part of the voltage is reflected at the motor terminal and travels back toward the inverter^[17]. When this reflected wave reaches the inverter, it is reflected again and propagates toward the motor. The reflection coefficients at the motor and inverter ends are defined by (8) and (9), respectively.

Because the inverter impedance. Zi is typically close to zero, and the motor impedance, Zm, is much larger than Z0, the reflection coefficients approach ideal values, i.e., $\Gamma_i \approx -1$ and $\Gamma_m \approx 1$. This indicates that voltage reflection at the inverter occurs with inverted polarity, whereas reflection at the motor occurs with the same polarity.

The rise time, tr, of the inverter output voltage is much shorter than the propagation delay time, tp, satisfying tr 《 tp. The propagation delay depends on the cable length, lc, and is expressed by (10) and (11).

where v is the propagation velocity.

As illustrated in Fig. 4, the voltage pulse generated by the inverter travels toward the motor while the motor terminal voltage remains at 0 V. After one propagation delay, the pulse reaches the motor terminal and generates a backward-reflected wave. Following one complete round-trip propagation, the motor terminal voltage is expressed by (12).

After multiple reflections, the voltage at the motor terminal evolves according to (13) until the oscillations gradually decay and settle at the steady-state dc-link voltage.

The damping of these oscillations is mainly governed by AC resistance and proximity effects in the cable and motor. The oscillation frequency under overvoltage conditions, shown in Fig. 4, is given by (14).

3.1. Overvoltage Mitigation Through Rise-Time Control

In motor drive systems employing long transmission cables, voltage reflection is inevitable due to propagation delay effects. Because the inverter output impedance is significantly smaller than the characteristic impedance of the cable, the reflection coefficients may be approximated as $\Gamma_i = -1$ and $\Gamma_m = 1$. As illustrated in Fig. 5, even relatively short cables may lead to considerable motor terminal overvoltage in SiC-based inverter systems owing to their fast dv/dt characteristics. One possible method to reduce overvoltage is to increase the inverter output rise time by adjusting gate resistance. However, effective suppression requires the rise time to exceed twice the cable propagation delay, which inevitably increases switching losses and compromises the advantages of wide-bandgap devices. To overcome this limitation, an RLC filter is employed in this study to suppress overvoltage while preserving high switching performance.

In the proposed design, a damping resistor is incorporated into the RLC filter to prevent resonant amplification. The resulting filter structure satisfies the dv/dt and insulation requirements specified by NEMA standards. As shown in Fig. 6, the RLC filter offers a simple and cost-efficient solution for mitigating motor terminal overvoltage. Although resonance may occur during RLC filter operation, this issue may be effectively addressed through appropriate parameter selection. Since the filter resonant frequency is above the switching frequency, the resonant current may not be directly controlled. Therefore, a damping resistor is introduced to dissipate the stored energy in the resonant circuit. Owing to its simplicity and effectiveness, the RLC filter shown in Fig. 6 has been widely adopted for reducing reflected-wave overvoltage at motor terminals.

3.2. Modeling and Analysis of the RLC Filter

Fig. 7(a) illustrates the idealized equivalent circuit of the proposed RLC filter, which is used to analyze the voltage response, power dissipation, and overvoltage suppression capability at the motor terminals. In this modeling approach, it is assumed that conduction occurs in only one phase leg at a given time, while switching transitions in the remaining legs are sufficiently separated. Under this assumption, the inverter output voltage is governed by the switching states and may be represented as a short-circuit condition. In the equivalent circuit of Fig. 7(a), the motor and cable are neglected because their impedances are assumed to be significantly larger than that of the filter around its resonant frequency. The inverter-side voltage applied to the filter is denoted as Vinv, and the corresponding filter output voltage is represented as Vout. The RLC filter parameters Rf, Lf, and Cf correspond to the damping resistance, inductance, and capacitance of each phase leg, respectively. For analytical convenience, the three-phase filter configuration in Fig. 7(a) may be reduced to an equivalent single-phase series RLC circuit, as shown in Fig. 7(b). Through appropriate series and parallel combinations, the equivalent resistance, inductance, and capacitance are obtained as expressed in (15). This transformation allows the filter dynamics to be analyzed using a simplified second-order system.

To evaluate the maximum dv/dt at the filter output, a step voltage input with an amplitude of Vdc is applied at the inverter side. This step input represents the ideal voltage transition generated by a fast-switching SiC inverter. To further simplify the analysis, the output terminals of the RLC filter are assumed to be open-circuited, as illustrated in Fig. 7(b). The dynamic behavior of the filter circuit is first described in the time domain and then transformed into the frequency domain using the Laplace transform. The resulting voltage and current relationships are expressed in (16). Based on these expressions, the output voltage Vout(s) and the filter current If(s) are derived as shown in (17) and (18), respectively^[19].

These equations yield a second-order transfer function, which is presented in (19) and (20).

When the damping ratio, $\zeta$, is selected as unity, the resonant terms in the transfer function are eliminated, leading to a simplified system response. Under this critically damped condition, the required damping resistance for the single-phase equivalent circuit is given by (21).

Assuming a step input at the inverter side, the frequency-domain expression of the output voltage is obtained as shown in (22), and the corresponding time-domain response is derived via the inverse Laplace transform, as expressed in (23).

To relate the peak voltage timing of the RLC filter output to the natural angular frequency, $\omega_o$, the characteristic roots of the second-order transfer function in (19) are obtained using the quadratic formula, as shown in (24). The resulting roots are summarized in (25).

By differentiating the time-domain voltage expression in (23), the time instant at which the output voltage reaches its maximum value is determined, as expressed in (26), where tpeak denotes the peak occurrence time.

When losses in the RLC filter are neglected, the peak time may be approximated as given in (27).

The rising edge of the output waveform is primarily governed by the natural frequency, $\omega_o$, and time. The ratio between the peak time tpeak and the rise time tr, defined as kT, is approximately 1.74684. Using this relationship, $\omega_o$ may be expressed as shown in (28).

Once $\omega_o$ is determined, the current transfer function of the RLC filter may be analyzed to select appropriate filter parameters, as expressed in (29). Because the filter is excited by a step voltage, the filter current may also be interpreted as a unit-step response.

Following a procedure analogous to the voltage analysis, the current response is expressed in (30), and the corresponding time-domain expression is obtained via the inverse Laplace transform, as shown in (31).

The time at which the filter current reaches its maximum value is given by $1/\omega_o$, and the peak current magnitude is expressed in (32), where if.peak denotes the maximum filter current.

Although the above analysis is carried out using a per-phase equivalent circuit, the derived expressions are directly applicable to a three-phase system. For a Y-connected configuration, the resistance, inductance, and capacitance values are scaled by factors of 2/3, 2/3, and 3/2, respectively, because the dc-link voltage is applied as a line-to-line quantity. The resulting RLC filter parameters for the three-phase Y-connected system are summarized in (33).

When the damping resistance of the RLC filter is selected to match the cable characteristic impedance, the system becomes overdamped, and resonant effects may be neglected from a control perspective. The initial value of ifpeak may be chosen arbitrarily; however, it should be adjusted by considering the inductance value in (33).

3.3. Modeling and Analysis of the LC Filter

An inverter output LC filter is presented in Fig. 8. A similar analytical method applied to an RLC filter may also be used for this type of filter. The transfer function for a single-phase LC circuit is expressed as (34).

Since the inverter output voltage applied to the filter is approximately a step input of Vdc/s, the filter’s output voltage is determined by (35).

Using the inverse Laplace transform, the time-domain dynamics of the filter output voltage may be represented by (36).

The inductor value for the LC filter is determined based on the current ripple passing through the filter and the inverter’s switching frequency. The inductor value may be calculated using (37).

where Vdc is dc-link voltage, fsw is the inverter’s switching frequency, and $\Delta I$ is the allowable current ripple. The current ripple is preferably set to 10-20 % of the rated load current.

To select the capacitor value, the natural frequency must be considered. The natural frequency should be significantly lower than the inverter’s switching frequency. It is typically chosen to be 1/5 to 1/10 of the switching frequency, and if the capacitance value is too large, the voltage drop across the filter may increase. The capacitor is selected by calculating using (38).