In modern electric vehicles (EVs), the high-performance electric drive system is a core component responsible for converting electrical energy from the battery into controlled mechanical torque. A prevalent topology for this system employs a three-phase, two-level voltage source inverter driving a permanent magnet synchronous motor (PMSM). To achieve a wide speed range, particularly for high-speed cruising, flux-weakening control is often implemented. This technique intentionally advances the current vector to weaken the stator flux linkage, allowing the motor to operate beyond its base speed where the back-electromotive force (back-EMF) would otherwise exceed the DC-link voltage. While effective, this operational mode introduces a significant safety risk: if the control system fails (e.g., due to a software fault or loss of sensor signals), the inverter’s freewheeling diodes can form an uncontrolled rectifier bridge. This can lead to excessive regenerative power flowing back into the battery pack or, if the battery contactor opens, cause destructive overvoltage on the DC-link capacitors.
To mitigate this critical failure mode, the Active Short Circuit (ASC) protection strategy is widely adopted in automotive electric drive systems. The fundamental principle is to forcibly turn on either all three upper or all three lower insulated gate bipolar transistors (IGBTs) of the inverter. This action creates a three-phase short-circuit across the motor terminals. The primary purpose is to ensure the anti-parallel diodes of the complementary switches remain reverse-biased, thereby preventing the uncontrolled rectification mode. Simultaneously, the shorted motor generates a small braking torque, which is generally acceptable for vehicle functional safety standards as it does not produce an unintended large acceleration or deceleration. However, a direct, instantaneous application of ASC (referred to as Direct-ASC) causes a step change in the terminal voltage, exciting a large transient current surge. This peak current can exceed the motor’s steady-state short-circuit current, posing risks of inverter overcurrent destruction and potential demagnetization of the permanent magnets.
To address this drawback, a refined method known as Linear Variable Modulation Ratio ASC (LVM-ASC) has been proposed. Instead of an immediate short, LVM-ASC transitions the system gradually from the uncontrolled rectifier state to the fully shorted state by modulating the effective short-circuit duty cycle over a defined transition time. This approach effectively dampens the current peak. While effective, a comprehensive analytical model detailing its dynamics and a systematic parameter design methodology has been lacking. This article aims to fill this gap by providing a quantitative analysis of the ASC transient, developing a discrete-time model for the LVM-ASC strategy, and proposing a parameter design method to enhance the flexibility and safety of protection strategies in automotive electric drive systems.
Transient Current Analysis in Direct Active Short Circuit
The analysis begins with the established model of a PMSM in the rotating d-q reference frame. Under standard assumptions (sinusoidal MMF distribution, negligible saturation and core losses, symmetric construction), the voltage equations are:
$$
\begin{align*}
u_d &= R_s i_d + L_d \frac{d i_d}{dt} – \omega L_q i_q \\
u_q &= R_s i_q + L_q \frac{d i_q}{dt} + \omega L_d i_d + \omega \psi_f
\end{align*}
$$
where \(R_s\) is the stator resistance, \(u_d, u_q\) and \(i_d, i_q\) are the d- and q-axis voltages and currents, \(L_d, L_q\) are the d- and q-axis inductances, \(\psi_f\) is the permanent magnet flux linkage, and \(\omega\) is the electrical angular velocity. This can be written in state-space form with the current vector \(\mathbf{i}_{dq} = [i_d, i_q]^T\) as the state variable:
$$
\dot{\mathbf{i}}_{dq} = \mathbf{A} \mathbf{i}_{dq} + \mathbf{B} \mathbf{u}
$$
with
$$
\mathbf{A} = \begin{bmatrix}
-\frac{R_s}{L_d} & \frac{\omega L_q}{L_d} \\
-\frac{\omega L_d}{L_q} & -\frac{R_s}{L_q}
\end{bmatrix}, \quad
\mathbf{B} = \begin{bmatrix}
\frac{1}{L_d} & 0 \\
0 & \frac{1}{L_q}
\end{bmatrix}, \quad
\mathbf{u} = \begin{bmatrix}
u_d \\
u_q – \omega \psi_f
\end{bmatrix}
$$
Upon triggering a Direct-ASC (e.g., turning on all lower switches), the applied voltage vector becomes \(\mathbf{u}(0^+) = [0, -\omega \psi_f]^T\). For a constant speed \(\omega\), the system is linear time-invariant after the fault. The solution for the current response, given an initial condition \(\mathbf{i}_{dq}(0) = [i_{d0}, i_{q0}]^T\), is:
$$
\mathbf{i}_{dq}(t) = e^{\mathbf{A}t}[\mathbf{i}_{dq}(0) – \mathbf{i}_{dq}(\infty)] + \mathbf{i}_{dq}(\infty)
$$
The steady-state short-circuit current is \(\mathbf{i}_{dq}(\infty) \approx [-\psi_f / L_d, 0]^T\). Assuming high electrical frequency such that \(\omega L_{d,q} >> R_s\), the eigenvalues of \(\mathbf{A}\) are predominantly complex, and the state transition matrix can be approximated. The resulting transient current trajectories in the d-q plane are approximated by decaying elliptical spirals centered at the steady-state point \(E(-\psi_f/L_d, 0)\), starting from the initial operating point \(S(i_{d0}, i_{q0})\).
The key insight is that the peak transient current magnitude \(I_P\) is largely governed by the magnitude of the initial stator flux linkage vector \(|\boldsymbol{\psi}_0|\) at the moment of fault, where \(\psi_d(0)=L_d i_{d0}+\psi_f\) and \(\psi_q(0)=L_q i_{q0}\). An approximate analytical expression for the first and major current peak is:
$$
I_P \approx \frac{|\boldsymbol{\psi}_0|}{L_d} e^{-\frac{R_s}{\omega} \frac{L_d+L_q}{L_d L_q} \frac{\delta}{\omega}}, \quad \text{where } \delta = \arctan\left(\frac{L_q i_{q0}}{L_d i_{d0} + \psi_f}\right)
$$
This analysis for the electric drive system reveals that deeper flux-weakening operation (smaller \(|\boldsymbol{\psi}_0|\)) prior to the fault results in a lower transient current peak during ASC. Conversely, operation near the maximum torque per ampere (MTPA) trajectory or in the generating region at high speed can lead to dangerously high current peaks, jeopardizing the integrity of the electric drive system.
Discrete-Time Modeling of the LVM-ASC Strategy
The LVM-ASC strategy improves upon Direct-ASC by introducing a controlled transition. The system toggles between the diode rectification state and the three-phase short-circuit state within each switching period \(T_s\). The duty cycle for the short-circuit state, \(D(t)\), is ramped linearly from 0 to 1 over a transition time \(T_r\):
$$ D(t) = \begin{cases}
0, & t < 0 \\
t/T_r, & t \in [0, T_r) \\
1, & t \ge T_r
\end{cases}
$$
Modeling this hybrid dynamics is challenging due to the nonlinear nature of the diode rectification mode. An approximate model is developed using a fundamental-wave approach and piecewise linearization.
Fundamental-Wave Model of Diode Rectification Mode
When all IGBTs are off and the motor back-EMF is sufficiently high, the inverter acts as an uncontrolled three-phase diode bridge. Assuming continuous conduction, the phase voltages imposed on the motor are six-step waveforms determined by the signs of the phase currents. For a symmetric fundamental-wave motor current \(\mathbf{i}_{abc1}(t)\) with amplitude \(I_s(t)\), the fundamental component of the differential-mode voltage can be derived as:
$$
\mathbf{u}_{abc1}(t) \approx -\frac{2}{\pi} \frac{U_{dc}}{I_s(t)} \mathbf{i}_{abc1}(t)
$$
where \(U_{dc}\) is the DC-link voltage. Transforming to the d-q frame yields a non-linear relationship:
$$
\mathbf{u}_{dq1} = -\frac{2}{\pi} \frac{U_{dc}}{|\mathbf{i}_{dq}|} \mathbf{i}_{dq}
$$
Substituting this into the machine state equation gives the system model during the diode rectification intervals:
$$
\dot{\mathbf{i}}_{dq} = \mathbf{A} \mathbf{i}_{dq} + \mathbf{B} \left( -\frac{2}{\pi} \frac{U_{dc}}{|\mathbf{i}_{dq}|} \mathbf{i}_{dq} + \mathbf{u}_f \right)
$$
where \(\mathbf{u}_f = [0, -\omega \psi_f]^T\). This represents a nonlinear, non-affine system.
Discrete State-Space Equation via Piecewise Linearization
To model the switching behavior of LVM-ASC, the system dynamics over one PWM period are analyzed. Within a period where the short-circuit duty is \(D(k)\), the interval is split:
- Short-Circuit Stage (Duration \(D T_s\)): The dynamics are linear with \(\mathbf{u} = [0, -\omega \psi_f]^T\).
- Diode Rectification Stage (Duration \((1-D) T_s\)): The dynamics follow the nonlinear equation above.
The nonlinear function \(f(\mathbf{i}_{dq}) = -\frac{2}{\pi} \frac{U_{dc}}{|\mathbf{i}_{dq}|} \mathbf{i}_{dq}\) is linearized around the state at the beginning of the PWM period, \(\mathbf{i}_{dq}(k)\):
$$
f(\mathbf{i}_{dq}) \approx f(\mathbf{i}_{dq}(k)) + \mathbf{F}(k) (\mathbf{i}_{dq} – \mathbf{i}_{dq}(k))
$$
where \(\mathbf{F}(k) = \frac{\partial f}{\partial \mathbf{i}_{dq}} \big|_{\mathbf{i}_{dq}(k)}\). This allows approximating the diode rectification stage as a linear time-invariant system over that sub-interval. Applying the exact solution for the linear short-circuit stage and the approximate solution for the linearized diode stage, a discrete-time state update equation from sample \(k\) to \(k+1\) is derived:
$$
\mathbf{i}_{dq}(k+1) \approx \hat{\mathbf{A}}(k) \mathbf{i}_{dq}(k) + \hat{\mathbf{B}}(k) \mathbf{u}_f + \hat{\mathbf{g}}(k)
$$
The matrices \(\hat{\mathbf{A}}(k)\), \(\hat{\mathbf{B}}(k)\), and \(\hat{\mathbf{g}}(k)\) are functions of the duty cycle \(D(k)\), the linearization point \(\mathbf{i}_{dq}(k)\), machine parameters, and \(U_{dc}\). They incorporate matrix exponentials of the system matrices for both operational stages. This discrete model enables the prediction of the current trajectory \(\mathbf{i}_{dq}(k)\) for a given transition profile \(D(k)\) (defined by \(T_r\)), which is crucial for analyzing peak currents and designing parameters.
Parameter Design for the Transition Time
The primary goal of the LVM-ASC strategy in the electric drive system is to limit the peak phase current \(I_{s}^{max}\) below a safe threshold (e.g., the inverter’s maximum current rating) while also minimizing unwanted effects such as the total regenerative energy pumped back into the DC link during the transition, which could stress the battery system. The transition time \(T_r\) is the key design parameter.
The energy fed back to the DC-link during the transition, \(\Delta E_p\), is approximated from the average AC-side power in the diode rectification stages:
$$
\Delta E_p \approx \sum_{k=0}^{N-1} \left[1 – D(k)\right] \cdot \frac{3}{\pi} U_{dc} T_s |\mathbf{i}_{dq}(k)|, \quad \text{with } N = T_r / T_s
$$
An optimal control problem can be formulated:
$$
\begin{aligned}
& \min_{T_r} \quad J = \Delta E_p \\
& \text{subject to:} \quad \max_{t} |\mathbf{i}_{dq}(t)| \leq I_{s}^{max} \\
& \qquad \qquad 0 \leq D(t) \leq 1
\end{aligned}
$$
Given the monotonic relationship observed via simulation between \(T_r\) and the two key metrics—peak current \(I_{s}^{max}\) decreases with longer \(T_r\), while feedback energy \(\Delta E_p\) increases—a practical design method emerges. For a given safety limit \(I_{s}^{max}\), the optimal \(T_r^*\) is the smallest transition time that still satisfies the current constraint. This can be found efficiently using the discrete model within an iterative search routine (e.g., the secant method), predicting the peak current for a candidate \(T_r\) until the predicted peak meets the target constraint within a tolerance. This method provides a systematic, model-based approach to tune the LVM-ASC strategy for a specific electric drive system.
Simulation and Experimental Verification
The proposed models and design method were verified through simulation and experiment on a 100 kVA automotive electric drive system prototype. The PMSM parameters are listed in Table 1.
| Component | Parameter | Value |
|---|---|---|
| PMSM | Pole Pairs | 3 |
| Flux Linkage \(\psi_f\) | 0.1067 Wb | |
| Stator Resistance \(R_s\) | 0.019 Ω | |
| d-axis Inductance \(L_d\) | 0.486 mH | |
| q-axis Inductance \(L_q\) | 1.25 mH | |
| Inverter | DC-link Voltage \(U_{dc}\) | 150 V |
| Switching Frequency \(f_s\) | 10 kHz | |
| Max. AC Current | 600 A (peak) |
Simulations were conducted from high-speed flux-weakening points. The discrete model’s predictions for current trajectories under different \(T_r\) values showed good agreement with detailed switching simulations, especially for \(T_r > 5\) ms. The error increased for very fast transitions due to the linearization inaccuracy during severe transients. A parameter sensitivity analysis, summarized in Table 2, showed the model error is more sensitive to variations in \(L_d\) and \(\psi_f\) than to \(L_q\) and \(R_s\).
| Parameter Changed | Impact on Model Accuracy |
|---|---|
| \(L_d\) (\(\pm 20\%\)) | Significant change in error plane. Affects steady-state current \(i_d(\infty)\). |
| \(\psi_f\) (\(\pm 20\%\)) | Significant change in error plane. Affects steady-state current \(i_d(\infty)\). |
| \(L_q\) (\(\pm 20\%\)) | Minor change in error plane. |
| \(R_s\) (\(\pm 50\%\)) | Minor change in error plane. |
The relationship between \(T_r\), peak current \(I_s^{max}\), and feedback energy \(\Delta E_p\) was simulated and measured. As predicted, \(I_s^{max}\) decreased monotonically with increasing \(T_r\), while \(\Delta E_p\) increased. Applying the parameter design method with a target peak current of 230.5 A (approx. 105% of steady-state short-circuit current) yielded the optimal transition times listed in Table 3. The experimental results confirmed the effectiveness of the design, with both peak current and feedback energy closely matching the predictions.
| Speed (rpm) | Designed \(T_r^*\) (ms) | Predicted \(I_s^{max}\) (A) | Measured \(I_s^{max}\) (A) | Error (%) | Predicted \(\Delta E_p\) (J) | Measured \(\Delta E_p\) (J) | Error (%) |
|---|---|---|---|---|---|---|---|
| 3000 | 30.00 | 230.4 | 236.4 | 2.5 | 317.2 | 294.2 | 7.8 |
| 4000 | 37.24 | 230.4 | 234.7 | 1.8 | 449.7 | 433.9 | 3.6 |
Conclusion
This article presented a comprehensive analysis and design methodology for active short-circuit protection in vehicle electric drive systems. The transient current peak during a Direct-ASC event was quantitatively analyzed using state-space methods, revealing its dependence on the pre-fault stator flux linkage. To mitigate this peak, the LVM-ASC strategy was investigated. A discrete-time model for its current dynamics was developed using a fundamental-wave approximation and piecewise linearization, enabling accurate trajectory prediction for various transition times. Based on this model, a practical parameter design method was proposed to select the transition time \(T_r\) that minimizes DC-link feedback energy while respecting the inverter’s current limit. Simulation and experimental results on a high-performance electric drive system validated the accuracy of the model and the effectiveness of the design approach. The methodologies provided enhance the safety and flexibility of fault management strategies in modern automotive electric drive systems, contributing to more robust and reliable electric vehicle powertrains.