Analysis and Test Research for Three-Phase Short-Circuit of Permanent Magnet Synchronous Motor for an Electric Vehicle

Abstract

Permanent magnet synchronous motor (PMSM) is widely used in the electric vehicle as a drive motor. For specific application requirements of the electric vehicle motor system, there are new faults and application characteristics in motor system’s three-phase short-circuit. Based on the $dq$ axis coordinate system mathematical model of PMSM, this paper derives the $dq$ axis current transient equations of three-phase transient short-circuit with load or without load in initial state. Combined electric vehicle PMSM operating characteristics, based on the typical working conditions, this paper analyzes the influences of motor demagnetization risk in different conditions by comparing the transient current changing process simulation results of different initial transient $dq$ axis currents. Then according to the characteristics of the three-phase transient short-circuit, designs and builds a test platform, and verifies the correctness of the analysis and mathematical simulation models by testing. It provides an important theoretical analysis and test verification methods for analyzing the electric vehicle PMSM three-phase short-current function and determining the risk of demagnetization.

9.1 Introduction

Electric vehicles is an important developing trend to cut off emission, nowadays, mainstream OEMs have invested heavily for this kind of future vehicles. Permanent magnet synchronous motor (PMSM) which has been widely used in the electric vehicle covers many advantages such as high torque and power density, wide range of flux-weakening control and easily maintenance. However, the PMSM system of the vehicle will also bring some new security features such as three-phase short-circuit. Research on three-phase short-circuit of traditional motor system has been done for years in China and abroad, but there are no literatures in-depth study of PMSM three-phase short-circuit for electric vehicles.

There are new fault handling and application characteristics on three-phase short-circuit in the field of electric vehicles. On the one hand, the three-phase short-circuit as a kind of fault may cause the demagnetization of permanent magnets or partial demagnetization [1]. The rising winding temperature and vibration noise caused by short-circuit may lead to the damage of other parts of the vehicle. On the other hand, if critical fault such as the inverter IGBT breakdown or insulation failure on high voltage system happens suddenly during the vehicle moving, the inverter can execute a three-phase active short-circuit to decouple the motor and transmission system in order to avoid the risk of the motor over-speed. This function can guarantee that the inverter thin-film capacitor and power devices are not be broken down by over voltage and also can braking the vehicle and shut down the engine safely by the anti-drag torque [2]. The three-phase short-circuit may be either one kind of fault or one kind of active safety function. The vehicle configurations, motor electromagnetic and heat dissipation solutions, and the safety requirements for electronic control system are all needed to be considered as a safety function. Above all, the three-phase short-circuit condition analysis is crucial for electric vehicles’ safety.

Before designing a three-phase active short-circuit function or analyzing the failure modes of the system after three-phase short-circuit, it should be confirmed that there is no risk of reversible demagnetization at the condition of three-phase transient short-circuit. Considering the maximum negative -axis current when the motor working, the demagnetization current limit and safety factor should be set during the design, but there is no security check for motor transient short-circuit conditions [3]. Thus, it is necessary for electric vehicle motor design to analyze the negative-axis current at the three-phase transient short-circuit condition and confirm the current within the maximum range of the demagnetization current by testing. Electric vehicle drive system increases the new motor system, which has different degrees of coupling relationship with engine, transmission and so on as a new source of power. Since the working condition is extremely complex, the motor is usually designed as high output and wide speed range in order to match the power train system. When we are analyzing the three-phase transient short-circuit, the vehicle condition such as low speed start, driving generating etc. shall be considered to evaluate the effect caused by the transient short-circuit demagnetization current. The three-phase transient short-circuit test platform and research method is aimed at steady state or low power permanent magnet synchronous motor [4]. There is a problem of transient test bench speed fluctuation caused by the transient impact [5]. Then to simulate electric vehicle short-circuit condition on the bench, it is worth to study how to solve the speed fluctuation problem caused by three-phase transient short-circuit and ensure the safety of the system and dynamic synchronous sampling.

Based on the $dq$-axis model of vector control, this paper derives the $dq$-axis current equations of the electric vehicle motor system at the time of three-phase transient short-circuit. And based on different initial working conditions, this paper analyzes the condition where the maximum demagnetization current occurs during the three-phase transient short-circuit. This paper also gives out the simple simulation algorithm to verify the maximum demagnetization current, and verifies the correctness of the transient short-circuit current equations through the test.

9.2 Three-Phase Transient Short-Circuit Model

Usually, the short-circuit current’s curve change over time can be calculated by CAE simulation software in engineering design. In this paper, the process of PMSM transient short-circuit can be analyzed by vector control of the $dq$-axis model, which can be used to analyze the steady-state and transient process. The algorithm also has a salient feature that is fast operation and evaluation.

Before establishing the mathematical model, the follow assumptions shall be done.

• (1) Ignore the rotor core reluctance, excluding the eddy current and hysteresis loss.

• (2) The conductivity of permanent magnet material is zero, and the magnetic permeability inside the permanent magnet is consistent with the air.

• (3) Permanent magnet excitation magnetic field generates a sine wave induced electromotive force in the phase windings.

• (4) Permanent magnet magnetic field and three-phase armature reaction magnetic field is sinusoidal in the air gap.

• (5) Ignore the effect of the magnetic saturation and the temperature to the motor parameters, the motor parameters are constant.

In this paper, interior permanent magnet synchronous motor (IPMSM) is used to establish a transient short-circuit model of permanent magnet synchronous motors. The model also applies to surface mounted permanent magnet synchronous motor (SPMSM). Interior permanent magnet synchronous motor’s $L_{d}$ and $L_{q}$ is not equal. Currently, most vehicles with permanent magnet synchronous motor are interior, mainly because in the case of the same current, the output torque can be increased by the reluctance torque.

Figure 9.1 shows a synchronous rotating $dq$-axis, the U, V, W phase current is transformed into a stationary $dq$-axis current by the coordinate transformation. Voltage equation is given by

$$ \left\{ {\begin{array}{*{20}l} {U_{d} = R_{S} i_{d} + L_{d} \frac{{di_{d} }}{dt} - \omega_{e} L_{q} i_{q} } \hfill \\ {U_{q} = R_{S} i_{q} + L_{q} \frac{{di_{q} }}{dt} - \omega_{e} \left( {L_{d} i_{d} + \psi_{f} } \right)} \hfill \\ \end{array} } \right. $$

(9.1)

Fig. 9.1

Synchronous rotating $dq$ axis

$U_{d} ,U_{q}$ :

Stator voltage $dq$-axis component,

$i_{d} ,i_{q}$ :

Stator voltage $dq$-axis component,

$R_{s}$ :

Stator phase resistance,

$\psi_{f}$ :

Flux linkage generated by permanent magnet,

$L_{d} L_{q}$ :

Stator winding $dq$-axis inductance,

$\omega_{e}$ :

$\omega_{e} = p_{n} \times \omega_{r} ,p_{n} ;$ Stator current electrical frequency,

$p_{n}$ :

Motor pole pairs,

$\omega_{r}$ :

Rotor angular velocity.

Firstly, let’s analyze the situation when $dq$-axis current is zero at the time of the three-phase short-circuit. The $dq$-axis transient short-circuit current and voltage is given by

$$ \left\{ {\begin{array}{*{20}l} {i_{d} \left( {0_{ - } } \right) = 0,\quad i_{q} \left( {0_{ - } } \right) = 0} \hfill \\ {U_{d} \left( {0_{ + } } \right) = 0,\quad U_{q} \left( {0_{ + } } \right) = 0} \hfill \\ \end{array} } \right. $$

(9.2)

By Laplace transform the differential equations formula (9.1) can be transformed into complex frequency domain equation as (9.3).

$$ \left\{ {\begin{array}{*{20}l} {R_{S} i_{d} \left( s \right) + L_{d} si_{d} \left( s \right) - \omega_{e} L_{q} i_{q} \left( s \right) = 0} \hfill \\ {R_{S} i_{q} \left( s \right) + L_{d} si_{q} \left( s \right) - \omega_{e} L_{d} i_{d} \left( s \right) + \psi_{f} = 0} \hfill \\ \end{array} } \right. $$

(9.3)

The formula (9.4) can be calculated by solving Eq. (9.3).

$$ \left\{ {\begin{array}{*{20}l} {i_{q} \left( s \right) = \frac{{ - \frac{r}{{L_{d} L_{q} }}\omega \psi_{f} - \frac{r}{{L_{q} }}\omega \psi_{f} s}}{{s\left( {s^{2} + s\left( {\frac{r}{{L_{d} }} + \frac{r}{{L_{q} }}} \right) + \frac{{r^{2} }}{{L_{d} L_{q} }} + \omega^{2} } \right)}}} \hfill \\ {i_{d} \left( s \right) = \frac{{ - \frac{1}{{L_{d} }}\omega^{2} \psi_{f} }}{{s\left( {s^{2} + s\left( {\frac{r}{{L_{d} }} + \frac{r}{{L_{q} }}} \right) + \frac{{r^{2} }}{{L_{d} L_{q} }} + \omega^{2} } \right)}}} \hfill \\ \end{array} } \right. $$

(9.4)

The denominator can be decomposed into:

$$ s\left( {s^{2} + s\left( {\frac{r}{{L_{d} }} + \frac{r}{{L_{q} }}} \right) + \frac{{r^{2} }}{{L_{d} L_{q} }} + \omega^{2} } \right) = s\left( {s - s_{1} } \right)\left( {s - s_{2} } \right) $$

where

$$ s_{1} ,s_{2} = - a \pm j\sqrt {b^{2} - a^{2} } = - \frac{r}{2}\left( {\frac{1}{{L_{d} }} + \frac{1}{{L_{q} }}} \right) \pm j\sqrt {\frac{{r^{2} }}{{L_{d} L_{q} }} + \omega^{2} - \frac{{r^{2} }}{4}\left( {\frac{1}{{L_{d} }}} \right.\left. { + \frac{1}{{L_{q} }}} \right)^{2} } $$

$$ a = \frac{r}{2}\left( {\frac{1}{{L_{d} }} + \frac{1}{{L_{q} }}} \right),\quad b = \sqrt {\frac{{r^{2} }}{{L_{d} L_{q} }} + \omega^{2} } ,\quad c = - \frac{1}{{L_{d} }}\omega^{2} \psi_{f} $$

Solve the original function of the formula (9.5).

$$ \left\{ {\begin{array}{*{20}l} {i_{d} \left( s \right) = - \frac{1}{{L_{d} }}\omega^{2} \psi_{f} \frac{1}{{s\left( {s - s_{1} } \right)\left( {s - s_{2} } \right)}}} \hfill \\ {i_{d} \left( s \right) = \frac{c}{{b^{2} }}\left( {\begin{array}{*{20}l} {\frac{1}{s} - \frac{s + a}{{\left( {s + \left. a \right)} \right.^{2} + b^{2} - a^{2} }}} \hfill \\ {\;\; - \frac{a}{{\left( {s + \left. a \right)} \right.^{2} + b^{2} - a^{2} }}} \hfill \\ \end{array} } \right)} \hfill \\ \end{array} } \right. $$

(9.5)

The original function is:

$$ \begin{aligned} i_{d} \left( t \right) & = \frac{c}{{b^{2} }}\left[ {1 - e^{ - at} \left( {\begin{array}{*{20}l} {\cos \sqrt {b^{2} - a^{2} } \cdot t} \hfill \\ {\;\; + \frac{a}{{\sqrt {b^{2} - a^{2} } }}\sin \left( {\sqrt {b^{2} - a^{2} } \cdot t} \right)} \hfill \\ \end{array} } \right)} \right] \\ & = \frac{c}{{b^{2} }}\left[ {1 - \frac{b}{{\sqrt {b^{2} - a^{2} } }}e^{ - at} \sin \left( {\begin{array}{*{20}l} {\sqrt {b^{2} - a^{2} } \cdot t} \hfill \\ {\;\; + \arctan \left( {\frac{{\sqrt {b^{2} - a^{2} } }}{a}} \right)} \hfill \\ \end{array} } \right)} \right] \\ \end{aligned} $$

(9.6)

Similarly

$$ \begin{aligned} i_{q} \left( s \right) & = \frac{{ - \frac{r}{{L_{d} L_{q} }}\omega \psi_{f} - \frac{r}{{L_{q} }}\omega \psi_{f} s}}{{s\left( {s - s_{1} } \right)\left( {s - s_{2} } \right)}} = - \frac{d}{{b^{2} }}i_{d} \left( s \right) - \frac{\omega }{{L_{q} }} \cdot \psi_{f} \cdot \frac{1}{{\left( {s - s_{1} } \right)\left( {s - s_{2} } \right)}} \\ & = - \frac{d}{{b^{2} }}i_{d} \left( s \right) - \frac{e}{{b^{2} }} \cdot \frac{{b^{2} }}{{\left( {s - s_{1} } \right)\left( {s - s_{2} } \right)}} \\ & = - \frac{d}{{b^{2} }}i_{d} \left( t \right) - \frac{e}{{b^{2} }} \cdot \frac{1}{{\sqrt {b^{2} - a^{2} } }} \cdot e^{ - at} \sin \left( {\sqrt {b^{2} - a^{2} } \cdot t} \right) \\ \end{aligned} $$

(9.7)

where

$$ d = - \frac{r}{{L_{d} L_{q} }}\omega \psi_{f} ,e = \frac{\omega }{{L_{d} }}\psi_{f} $$

The formula (9.7) is a transcendental function. The maximum −i_d current cannot be calculated from this formula. But the response curve can be obtained by computer in engineering generally, so to obtain the maximum −i_d current. This work will be described in detail in the next chapter.

$$ \frac{c}{{b^{2} }} = - \frac{{\omega^{2} \psi_{f} L_{q} }}{{r^{2} + \omega^{2} L_{d} L_{q} }},\quad \frac{d}{{b^{2} }} = - \frac{{r\omega \psi_{f} }}{{r^{2} + \omega^{2} L_{d} L_{q} }} $$

(9.8)

Formula (9.8) are the steady-state final values of $dq$-axis current respectively, and are also the analytical expressions of the $dq$-axis steady-state short-circuit.

When the initial values of $i_{d} ,i_{q}$ is not zero, that is, the motor system has output torque at the time of transient short-circuit. The differential Eq. (9.1) has the following initial conditions.

$$ \left\{ {\begin{array}{*{20}l} {i_{d} \left( {0_{ - } } \right) \ne 0,\quad i_{q} \left( {0_{ - } } \right) \ne 0} \hfill \\ {U_{d} \left( {0_{ + } } \right) = 0,\quad U_{q} \left( {0_{ + } } \right) = 0} \hfill \\ \end{array} } \right. $$

(9.9)

Solve the Laplace transform equations containing the initial value. The solution are:

$$ \left\{ {\begin{array}{*{20}l} {i_{d} \left( s \right) = \frac{{i_{d} \left( {0_{ - } } \right) \cdot s^{2} }}{{s\left( {s - s_{1} } \right)\left( {s - s_{2} } \right)}} + \frac{{\left[ {\frac{{\omega_{e} L_{q} }}{{L_{d} }} \cdot i_{q} \left( {0_{ - } } \right) + \left( {\frac{r}{{L_{q} }} - \frac{r}{{L_{d} }}} \right) \cdot i_{d} \left( {0_{ - } } \right)} \right] \cdot s}}{{s\left( {s - s_{1} } \right)\left( {s - s_{2} } \right)}} - \frac{{\left[ {\frac{{\omega_{e} r}}{{L_{d} }} \cdot i_{q} \left( {0_{ - } } \right) + \frac{{\omega_{e}^{2} }}{{L_{d} }}\psi_{f} + \frac{{r^{2} }}{{L_{d} L_{q} }} \cdot i_{d} \left( {0_{ - } } \right)} \right]}}{{s\left( {s - s_{1} } \right)\left( {s - s_{2} } \right)}}} \hfill \\ {i_{q} \left( s \right) = \frac{{i_{q} \left( {0_{ - } } \right) \cdot s^{2} }}{{s\left( {s - s_{1} } \right)\left( {s - s_{2} } \right)}} + \frac{{\left[ { - \frac{r}{{L_{q} }} \cdot i_{q} \left( {0_{ - } } \right) - \frac{{\omega_{e} \psi_{f} }}{{L_{q} }} + \frac{{i_{q} \left( {0_{ - } } \right)}}{{L_{d} r}} - \frac{{\omega_{e} L_{d} }}{{L_{q} }}} \right] \cdot s}}{{s\left( {s - s_{1} } \right)\left( {s - s_{2} } \right)}} - \frac{{\frac{{\omega \psi_{f} r}}{{L_{d} L_{q} }} + \frac{\omega r}{{L_{q} }} \cdot i_{d} \left( {0_{ - } } \right)}}{{s\left( {s - s_{1} } \right)\left( {s - s_{2} } \right)}}} \hfill \\ \end{array} } \right. $$

(9.10)

The above formula can be written is given by

$$ \left\{ {\begin{array}{*{20}l} {i_{d} \left( s \right) = \frac{{A \cdot s^{2} + B \cdot s + C}}{{s\left( {s - s_{1} } \right)\left( {s - s_{2} } \right)}}} \hfill \\ {i_{q} \left( s \right) = \frac{{A^{{\prime }} \cdot s^{2} + B^{{\prime }} \cdot s + C^{{\prime }} }}{{s\left( {s - s_{1} } \right)\left( {s - s_{2} } \right)}}} \hfill \\ \end{array} } \right. $$

(9.11)

where

$$ \left\{ {\begin{array}{*{20}l} {A = i_{d} \left( {0_{ - } } \right) \cdot s^{2} } \hfill \\ {B = \left[ {\frac{{\omega_{e} L_{q} }}{{L_{d} }} \cdot i_{q} \left( {0_{ - } } \right) + \left( {\frac{r}{{L_{q} }} - \frac{r}{{L_{d} }}} \right) \cdot i_{d} \left( {0_{ - } } \right)} \right]} \hfill \\ {C = \left[ {\frac{{\omega_{e} r}}{{L_{d} }} \cdot i_{q} \left( {0_{ - } } \right) + \frac{{\omega_{e}^{2} }}{{L_{d} }}\psi_{f} + \frac{{r^{2} }}{{L_{d} L_{q} }} \cdot i_{d} \left( {0_{ - } } \right)} \right]} \hfill \\ \end{array} } \right. $$

(9.12)

$$ \left\{ {\begin{array}{*{20}l} {A^{{\prime }} = i_{q} \left( {0_{ - } } \right)} \hfill \\ {B^{{\prime }} = \left[ { - \frac{r}{{L_{q} }} \cdot i_{q} \left( {0_{ - } } \right) - \frac{{\omega_{e} \psi_{f} }}{{L_{q} }} + \frac{{i_{q} \left( {0_{ - } } \right)}}{{L_{d} r}} - \frac{{\omega_{e} L_{d} }}{{L_{q} }}} \right]} \hfill \\ {C^{{\prime }} = - \frac{{\omega \psi_{f} r}}{{L_{d} L_{q} }} + \frac{\omega r}{{L_{q} }} \cdot i_{d} \left( {0_{ - } } \right)} \hfill \\ \end{array} } \right. $$

(9.13)

$i_{d}$ and $i_{q}$ can be obtained as

$$ \left\{ {\begin{array}{*{20}l} {i_{d} \left( s \right){ = }A \cdot \frac{s}{{\left( {s - s_{1} } \right)\left( {s - s_{2} } \right)}} + B \cdot \frac{1}{{\left( {s - s_{1} } \right)\left( {s - s_{2} } \right)}} + C \cdot \frac{1}{{s\left( {s - s_{1} } \right)\left( {s - s_{2} } \right)}}} \hfill \\ {i_{q} \left( s \right) = A^{{\prime }} \cdot \frac{s}{{\left( {s - s_{1} } \right)\left( {s - s_{2} } \right)}} + B^{{\prime }} \cdot \frac{1}{{\left( {s - s_{1} } \right)\left( {s - s_{2} } \right)}} + C^{{\prime }} \cdot \frac{1}{{s\left( {s - s_{1} } \right)\left( {s - s_{2} } \right)}}} \hfill \\ \end{array} } \right. $$

(9.14)

$$ \left\{ {\begin{array}{*{20}l} \begin{aligned} i_{d} \left( t \right) & = \frac{A}{{\sqrt {b^{2} - a^{2} } }} \cdot \left[ \begin{aligned} - ae^{ - at} \sin \left( {\sqrt {b^{2} - a^{2} } \cdot t} \right) \hfill \\ + e^{ - at} \left( {\sqrt {b^{2} - a^{2} } } \right) \cdot \cos \left( {\sqrt {b^{2} - a^{2} } \cdot t} \right) \hfill \\ \end{aligned} \right] + \frac{B}{{\sqrt {b^{2} - a^{2} } }} \cdot \left[ {e^{ - at} \sin \left( {\sqrt {b^{2} - a^{2} } \cdot t} \right)} \right] \\ & \quad + C \cdot \left[ {1 - \frac{1}{{\sqrt {b^{2} - a^{2} } }} \cdot e^{ - at} \sin \left( \begin{aligned} \sqrt {b^{2} - a^{2} } \cdot t \hfill \\ + \arctan \left( {\frac{{\sqrt {b^{2} - a^{2} } }}{a}} \right) \hfill \\ \end{aligned} \right)} \right] \\ \end{aligned} \hfill \\ \begin{aligned} i_{q} \left( t \right) & = \frac{{A^{{\prime }} }}{{\sqrt {b^{2} - a^{2} } }} \cdot \left[ \begin{aligned} - ae^{ - at} \sin \left( {\sqrt {b^{2} - a^{2} } \cdot t} \right) \hfill \\ + e^{ - at} \left( {\sqrt {b^{2} - a^{2} } } \right) \cdot \cos \left( {\sqrt {b^{2} - a^{2} } \cdot t} \right) \hfill \\ \end{aligned} \right] + \frac{{B^{{\prime }} }}{{\sqrt {b^{2} - a^{2} } }} \cdot \left[ {e^{ - at} \sin \left( {\sqrt {b^{2} - a^{2} } \cdot t} \right)} \right] \\ & \quad + C^{{\prime }} \cdot \left[ {1 - \frac{1}{{\sqrt {b^{2} - a^{2} } }} \cdot e^{ - at} \sin \left( \begin{aligned} \sqrt {b^{2} - a^{2} } \cdot t \hfill \\ + \arctan \left( {\frac{{\sqrt {b^{2} - a^{2} } }}{a}} \right) \hfill \\ \end{aligned} \right)} \right] \\ \end{aligned} \hfill \\ \end{array} } \right. $$

(9.15)

The condition $dq$-Axis current initial value is not zero and the condition that $dq$-axis current initial value is zero have the same final steady-state values.

By the above expression we can also see, the three-phase transient short-circuit process is related to the inductance, the resistance and the flux linkage and other parameters of the motor, and also, related to the $i_{d} ,i_{q}$ initial state, of which the process is more complex, there are more influenced dynamic parameters. In the actual engineering design, the complete system response curve can be drawing by computer solving. So not only the dynamic process of the known parameters’ motor can be calculated fast and the risk of demagnetization can be assessed, but also the effect to the transient short-circuit process and demagnetization current under different $i_{d}$, $i_{q}$, $\psi_{f}$ and other parameters can be compared.

9.3 Transient Short-Circuit Simulation Analysis Based on the Actual Operation Condition

We have analyzed the three-phase transient short-circuit in the last chapter, which gives a analytical expression of $i_{d} ,i_{q}$ short circuit current. From the expression, it can be found that the changing process of the transient short-circuit is related to motor $dq$-axis inductance, permanent magnet flux and other motor parameters, and also will be effected by the speed and the initial value of $i_{d} i_{q}$ current at the moment of short-circuit state. Ignore the magnetic saturation, and assume that the motor paraments do not change in this process. The transient short-circuit time is very short during the actual vehicle moving, considering the vehicle inertia and other factors, the simulation process can be approximated that the motor speed is constant before and after the short circuit. The operating condition at the moment of short-circuit determine the initial value of $i_{d} i_{q}$ current. Next, combining the electric vehicle common conditions, the three-phase short-circuit transient current will be simulated.

MTPA (maximum torque current ratio) of Electric vehicle PMSM is achieved by vector control. During the actual using, regardless of the flux weakening region or the non flux weakening region, the basic principle of motor control algorithm is realizing MTPA control by adjusting the $i_{d} i_{q}$ current ratio to increase the magnetic reluctance torque. PMSM torque equation is:

$$ T_{e} = 1.5 \times P_{n} \times \left[ {\psi_{f} i_{q} + (L_{d} - L_{q} )i_{d} i_{q} } \right] $$

(9.16)

Motor system working conditions is related to the requirement of the electric vehicle’s dynamic performance closely. And the requirements of hybrid electric vehicles and pure electric vehicle are very different [6]. For embedded PMSM, usually have:

• (1) Drive motor converts electrical power into mechanical energy to provide a driving force. Positive torque is corresponding to the vehicle’s acceleration and climbing condition, and the q-axis current is positive, and d-axis current is negative, so the motor output torque is positive. Figure 9.2 shows, along with the MTPA curve, $i_{d}$ increases, $i_{d}$ decreases, positive torque increases.

  Fig. 9.2

  

  MTPA stator current vector trajectory

• (2) Q-axis current is negative, d-axis current is also negative, the motor output torque is negative, which is corresponding to power generating, braking energy recovery and other conditions. Figure 9.2 shows, $i_{d}$ and $i_{d}$ both increase, negative toque increase.

Use MTPA to simulate the changing process of the $i_{d}$$i_{q}$ current at the moment of transient short-circuit. The parameters of the star-connection motor used for simulating and testing are as follow.

Under motor system follow-up working conditions, which is without load ($i_{d} = 0$, $i_{q} = 0$), compare the short-circuit current at 1000, 2000, 3000, 4000 rpm, as shown in Fig. 9.3.

Fig. 9.3

Transient short-circuit current simulation charts at different speed

Compare the transient short-circuit current of different working condition at 1000 rpm. There are low-speed accelerating condition,$i_{q}$ = 200 A, $i_{d}$ = −100 A (142 Nm), low-speed climbing condition, $i_{q}$ = 300 A, $i_{d}$ = −250 A (291 Nm), low-speed braking energy recovery, $i_{q}$ = −200 A, $i_{d}$ = −100 A (−142 Nm), low-speed emergency braking, $i_{q}$ = −300 A, $i_{d}$ = −250 A (−296 Nm). Compare the transient short-circuit current of high-speed accelerating and high-speed braking energy recovery simultaneously (Figs. 9.4 and 9.5).

Fig. 9.4

Transient short-circuit current simulation charts at different torque

Fig. 9.5

Transient short-circuit current simulation charts at high speed

From the above simulation we can see that:

• (1) At the different speed, without load short-circuit, with the higher speed, the more oscillation periods of $dq$-axis current appear. But the transient process time is almost the same, the maximum demagnetization-current is almost the same too.

• (2) At the same speed, with load short-circuit, regardless of driving or generating condition, with the load increasing, the demagnetization-current increases, the transient process time is almost the same.

• (3) Under the same load, at different speed, with the higher speed, the demagnetization-current increases, and the more the oscillation periods appear.

9.4 Test Platform Design and Test Results Analyze

In order to verify the correctness of the analysis of PMSM transient three-phase short-circuit model theory in Chap. 2, and support the simulation in Chap. 3, this paper has completed the design of transient short-circuit test platform and the transient short-circuit test on the test platform. The unit under test (UUT) is a C-level hybrid vehicle permanent magnet synchronous motor (IPMSM). The parameters of the UUT are shown in Table 9.1, in Chap. 3.

Table 9.1 The parameters of PMSM tested unit

By the above two chapters’ analyzing we know the process of transient short-circuit is very complex and short. Transient current shock will make the motor torque shock and also has some destructive effect. So we need to consider the high response speed and stability to resist the impact of destruction during test platform designing.

Figure 9.6 shows the test platform, which includes dynamometer, battery simulator, elastic shaft, torque sensor, resolver and position acquisition devices, $dq$-axis current calculating unit, the tested motor and inverter, execution unit of three-phase transient short-circuit. The dynamometer as the test load should be set in speed mode during the test and the speed can be changed. The tested motor and inverter are running in torque mode by controlling the inverter’s $dq$-axis current to change the output torque. A flexible coupling is installed between the tested motor and the dynamometer for eliminating the shock of the system running speed caused by the transient short-circuit torque. Compared to the tested motor, a larger inertia dynamometer is selected, in order to reduce the effect of the shock to the system stability.

Fig. 9.6

The structure of the torque fluctuation detection system

The three-phase transient short-circuit execution unit mainly contains a high-power IGBT and its drive hardware circuit. The three-phase synchronous short-circuit process is triggered by hardware. The current and the position signal acquisition unit is consist of several high-response, high-precision sensors. The current and the angle signal is delivered to the $dq$-axis current calculation unit in real time, of which the output $dq$-axis current operation rate is 50 kHz. The tested inverter must have over-current protection to prevent the IGBT over burning caused by the over-current in the process of the transient short-circuit.

The dynamometer is set to 1000 rpm, Fig. 9.7 is the phase current waveform and the motor resolver position signal waveform during transient short-circuit. The pole pairs of the resolver is the half of the tested motor’s. The current has a great shock in the process of short-circuit, but the resolver signal is linear well, which proves the speed of the test system has no significant change during the shock, and the actual speed is consistent with simulation conditions before and after the transient short-circuit. As the current and rotor position is known, the $dq$-axis current can be calculated by Clark park in real time.

Fig. 9.7

Phase current waveform and the motor resolver position signal waveform after transient short-circuit

In this paper, in order to verify the correctness of the theoretical analysis and compare the simulated date and the actual test data, the following three conditions have be selected, 1000 rpm @56 Nm ($i_{d}$ = −35 A, $i_{q}$ = 95 A), 2000 rpm @120 Nm ($i_{d}$ = −55 A, $i_{q}$ = 195 A), 4000 rpm @56 Nm ($i_{d}$ = −35 A, $i_{q}$ = 95 A). The comparison charts of the simulation results and the test results under different conditions is shown in Fig. 9.8.

Fig. 9.8

The comparison charts of the simulation results and the test results under different conditions

By comparing the simulated waveforms and the measured waveform of the above three conditions, the consistency of simulated and actual $i_{d}$, $i_{q}$ current transient response is very high, the simulated steady-state time and the oscillation process are very consistent with the actual situation, and the maximum instantaneous value of the negative $i_{d}$ current is also consistent basically. In these three conditions, the maximum simulation values of the simulation are: −522.4, −616.2, −613.7 A, the maximum measured values are: −499.8, −597.7, −600 A. The deviation of the maximum demagnetization current is within 5%. The motor after the test has no abnormal changes of mechanical and electrical characteristics. There is no demagnetization phenomenon after confirming the back EMF of the motor without load.

Next we will analyze the reason for the deviation of the simulation and the measurement, we can find that even the short-circuit process has been steady, the current also has obvious oscillations of d, q-axis (shown in Fig. 9.8) during the actual test. There are two main reasons as follows.

• (1) Since the simulation has assumed the permanent magnetic field induces a sine wave of electromotive force in the phase windings and the three-phase steady-state short-circuit current is fully sinusoidal current. But actually, in addition to the fundamental wave the induced electromotive force also contains a certain amount of harmonics, and the three-phase steady-state short-circuit current also has certain amount of harmonics, so there is current oscillation during the actual test. Non-sinusoidal induced electromotive force will have some impact on the simulation of the transient process.

• (2) There is a cross-saturation effect of permanent magnet synchronous motor inductance, which the parameters of $L_{q}$, $L_{q}$, $\psi_{f}$ are influenced by the current actual value of $i_{d}$, $i_{q}$, as shown in Fig. 9.9. Because the test temperature is difficult to keep constant during the test, the resistance parameters and flux parameters are also influenced by temperature. Dynamic change of motor parameter will influence both the transient state and the steady state.

  Fig. 9.9

  

  Inductance variation curve with $i_{d}$, $i_{q}$

9.5 Conclusion

The following conclusions can be got from this paper by analyzing an simulation.

• (1) Three-phase transient short-circuit analysis is very important to PMSM. The simulation analysis based on $dq$-axis models shows that the oscillation of the $dq$ transient short-circuit $dq$-axis current gradually decays to steady short-circuit condition. The $- i_{d}$ current during the oscillation process may cause the demagnetization of permanent magnets. Simulation method can be used for fast calculating the dynamic process with the known parameters’ motor and assessing the risk of demagnetization, and also for comparing the effect to the transient short-circuit process and demagnetization current under different $L_{d}$, $L_{q}$, $\psi_{f}$ and other parameters.

• (2) The PMSM maximum $- i_{d}$ current at the time of the transient short-circuit is larger than the steady-state short-circuit. The maximum $- i_{d}$ current and $dq$-axis current dynamic process is related to the operating conditions (the initial value of $- i_{d}$,$i_{d}$) and the speed at the time of transient short-circuit. The simulation results show that a higher motor speed and a greater load will cause a greater short-circuit $i_{d}$ current.

• (3) The transient short-circuit test bed designed in this paper can maintain the speed stability with the transient impact during the test. And the synchronization of each channel’s signal is good. The tests show $dq$-axis current transient response simulation is consistent with test results, with the exact parameters of the motor, the current simulation method is applicable to PMSM transient short-circuit analysis.

• (4) The transient changes of there EMF harmonic, $L_{d}$, $L_{q}$, and $\psi_{f}$ will affect the short-circuit process and the steady-state value. Nest, further analysis of transient $dq$ inductance will help to analyze the dynamic process and demagnetization risk accurately.