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Automotive Innovation

, Volume 1, Issue 3, pp 226–236 | Cite as

Modeling and Testing of the Multi-pole Field of a Motor for Pure Electric Vehicles

  • Dongchen Qin
  • Lei ChengEmail author
  • Tingting Wang
  • Yingjia Wang
  • Yaokai Wang
Article

Abstract

From the principles of electromechanical energy conversion and electromagnetic torque generation, our study evaluated the mathematical model of the electromagnetic torque and the vector control method of motors. An analysis of motor types indicates that the electromechanical energy conversion component is interchangeable. Three distinct types of motor structures, namely DC, induction, and synchronous, are possible, all three being commonly used in pure electric vehicles. For each motor type, simulation models were developed using Modelica, a modeling language for object-oriented multi-domain physical system. A test model of each motor type was configured in the MWorks simulation platform. With a representative motor, specifically the permanent-magnet DC motor, the asynchronous induction motor, and the permanent-magnet synchronous motor, mechanical properties were simulated and analyzed. The simulation results show that the characteristics of each motor model are consistent with the theoretical and engineering performance of the representative motor. Therefore, modeling, motor control, and performance testing of a unified multi-pole-field motor, which is used in pure electric vehicles, have been achieved.

Keywords

Electric motor vehicle Multi-pole field Unified modeling Electromechanical energy conversion 

1 Introduction

At present, environmental pollution as well as resource demands and energy shortages are increasing, and with that, pure electric vehicles have gradually risen in prominence in current development schemes within the automotive industry. Their rapid technological development has focused on new and better performing drive motors that are commonly used in such vehicles. These motors are electromechanical devices that convert electric energy from a battery into mechanical energy to drive the electric vehicle. The motor and its control system are at the heart of the electric vehicle [1].

Globally, much research work by experts and scholars has gone into configuring motor models for pure electric vehicles and performing simulations to improve motor performances. Using Simulink within the MATLAB environment, Wu et al. [2] developed modular diagrams, a S-function model, and a differential equation editor (DEE) model of the inductive motor according to its mathematical model based on a two-phase fixed reference frame, solving the high-order nonlinear problems encountered in modeling and simulating. Using the MATLAB/Simulink dynamic simulation platform, Liao et al. [3] modeled and simulated a permanent-magnet brushless DC motor of electric vehicles using the bonding graph diagram and provided a reference model of the whole electric vehicle system to analyze the driving system. Ma [4] modeled thermal aspects of a permanent-magnet drive motor and produced the cloud diagram of the temperature distribution, thereby providing a reliable basis for motor design. In regard to the permanent-magnet synchronous motor (PMSM), Wang [5] analyzed using ANSYS the sensorless control strategy, which is based on adaptive vector filtering. The current situation concerning electric motor research is that a unified model and simulation methods, as well as control methods, are still lacking, and this applies to all types of motors.

In recent years, Modelica, a language for modeling complex object-oriented multi-domain physics systems, has undergone rapid development and has been widely used in aviation, aerospace, automotive, power, electricity, new energy, and other fields [6, 7]. Based on non-causal modeling ideas, Modelica adopts mathematical equations (equalities) and object-oriented structures and supports the reuse of models. In general, the language allows for a unified mechanism based on the generalized Kirchhoff principle that supports attributes such as class, inheritance, equations, components, connectors, and connections. The language can therefore be used to describe electrical, mechanical, thermodynamic, and control features and other aspects [8, 9]. In addition, Modelica’s standard library already has a large number of reusable domain models and basic components. The electrical component library contains a wealth of mathematical models describing electrical components, thereby enabling the unified modeling and simulation of different types of motors. In recent decades, many scholars have adopted Modelica in their research on motors. Deng [10] developed an asynchronous motor model and performed simulations and analysis using Modelica in MWorks, thereby achieving a unified multi-pole field modeling of induction motors. Chen [11] established a PMSM for a pure electric vehicle in Dymola also using Modelica and performed simulations to confirm its correctness and feasibility. Similarly, within the Dymola/Modelica platform, Yun et al. [12] modeled a synchronous machine and developed a method to perform simulations of the motor in a modular fashion on a finite vehicle. For our study, this work has provided a great deal of guidance.

The basic principles of electromechanical energy conversion and electromagnetic torque generation of an electromechanical conversion device provide the basis for unified modeling, vector control, and performance test simulations of different types of motor drive systems. Mathematical models of the electromagnetic torque of these drive systems and vector control methods have been studied. A Modelica modeling of a permanent-magnet DC (PMDC) motor, an induction motor, and a PMSM is undertaken that exploits the various aspects of the language. Moreover, by taking advantage of the reusable components in Modelica’s library, the mechanical properties of all three motor types are evaluated in the MWorks simulation platform. This study provides a new and effective method for a unified analysis of the multi-domain models, control methods, and performance of pure electric vehicle motors.

2 Electromechanical Conversion Principle and Generation of Electromagnetic Torque

2.1 Electromechanical Conversion Principle

The process of converting electrical energy transforming into mechanical energy is called electromechanical conversion. The operation of the motor is usually based on coupling a current-carrying circuit and the motor’s changing magnetic field. The conductor produces a magneto-motive force, and a circuit current is produced when the conductor and the ferromagnetic component are subjected to an electromagnetic force. The magnetic flux associated with the circuit changes due to changes or displacements in the current. Moreover, the change in flux produces an electromotive force in the circuit. Electromechanical energy conversion is based on different physical principles, specifically [13]:
  1. (1)

    Ampere’s law governing the magnetic field generated by the current-carrying conductor;

     
  2. (2)

    Faraday’s law of electromagnetic induction relating the rate of change in magnetic flux to the electromotive force induced in the circuit.

     
  3. (3)

    Lorentz’s force, which is the electromagnetic force acting on a moving charge in the presence of an electric field and a magnetic field.

     
  4. (4)

    Kirchhoff’s law concerning the relationships between voltages and the sum of currents in a circuit, and the relationship between the flux and the magneto-motive force in the magnetic circuit.

     
Given these physical principles and the physical structure of the motor, the motor device can be regarded as a magnetic circuit composed of stator, rotor, and air gap (Fig. 1).
Fig. 1

Electromechanical conversion device structure diagram

Coil A is embedded in the stator groove to form a stator winding, and coil B is fitted into the rotor groove to form a rotor winding. The cores of the stator A and rotor B are considered to be made of a ferromagnetic material. A uniform air gap, denoted g, is designed between the stator windings and the rotor windings. (The total air gap is \( \delta = 2g \).) S is the magnetic field axis, that is, the amplitude of the magnetic field along the radial line. r is the magnetic field axis of the rotor winding B, that is, the radial current of the radial fundamental wave generated by the forward current iB in the air gap. Taking the S-axis as the space reference axis, then \( \theta_{\text{r}} \) is the rotor position angle (electrical angle) determined by the rotor’s counterclockwise rotation.

To help derive the mathematical model of the motor, an electromechanical device based on the simplified electromechanical conversion principle is assumed as follows [14]:
  1. (1)

    The air gap between the rotor and the rotor is assumed uniform, and edge effects from the magnetic field in the air gap are not taken into account;

     
  2. (2)

    The magnetic resistance and dielectric loss of the rotor core magnetic circuit are ignored;

     
  3. (3)

    Electrical and magnetic energy losses and mechanical friction losses are negligible;

     
  4. (4)

    The number of turns in the stator and rotor windings are the same (\( N_{\text{A}} = N_{\text{B}} \)).

     
Furthermore, the magnetic field energy is assumed to be stored in the air gap. For our electromechanical device (Fig. 1), the magnetic energy \( W_{\text{m}} \) is a function of stator flux \( \psi_{\text{A}} \), rotor flux \( \psi_{\text{B}} \), and rotation angle \( \theta_{\text{r}} \); the magnetic energy \( W^{\prime}_{\text{m}} \) is a function of currents iA and iB and electrical angle \( \theta_{\text{r}} \). That is, mathematically,
$$ W_{\text{m}} = W_{\text{m}} \left( {\psi_{\text{A}} ,\psi_{\text{B}} ,\theta_{\text{r}} } \right), $$
(1)
$$ W^{\prime}_{\text{m}} = W^{\prime}_{\text{m}} \left( {i_{\text{A}} ,i_{\text{B}} ,\theta_{\text{r}} } \right). $$
(2)
As changes in flux and rotor position cause a change in magnetic energy dWm (full differential), then
$$ {\text{d}}W_{\text{m}} = \frac{{\partial W_{\text{m}} }}{{\partial \psi_{\text{A}} }}{\text{d}}\psi_{\text{A}} + \frac{{\partial W_{\text{m}} }}{{\partial \psi_{\text{B}} }}{\text{d}}\psi_{\text{B}} + \frac{{\partial W_{\text{m}} }}{{\partial \theta_{\text{A}} }}{\text{d}}\theta_{\text{r}} , $$
(3)
which can be rewritten as
$$ {\text{d}}W_{\text{m}} = i_{\text{A}} {\text{d}}\psi_{\text{A}} + i_{\text{B}} {\text{d}}\psi_{\text{B}} + \frac{{\partial W_{\text{m}} }}{{\partial \theta_{\text{A}} }}{\text{d}}\theta_{\text{r}} . $$
(4)
Similarly, as changes in the rotor current and rotor position are caused by changes in magnetic energy delay, the differential equations (DDE) can be expressed as
$$ \begin{aligned} {\text{d}}W^{\prime}_{\text{m}} & = \frac{{\partial W^{\prime}_{\text{m}} }}{{\partial i_{\text{A}} }}{\text{d}}i_{\text{A}} + \frac{{\partial W^{\prime}_{\text{m}} }}{{\partial i_{\text{B}} }}{\text{d}}i_{\text{B}} + \frac{{\partial W^{\prime}_{\text{m}} }}{{\partial \theta_{\text{r}} }}{\text{d}}\theta_{\text{r}} \\ & = \psi_{\text{A}} {\text{d}}i_{\text{A}} + \psi_{\text{B}} {\text{d}}i_{\text{B}} + \frac{{\partial W^{\prime}_{\text{m}} }}{{\partial \theta_{\text{r}} }}{\text{d}}\theta_{\text{r}} . \\ \end{aligned} $$
(5)
The first two terms in Eqs. (4) and (5) correspond to the magnetic field energy of the stator and rotor windings, respectively. The third term describes the change in magnetic energy arising from the angular displacement of the rotor and indicates that the rotor rotates from changes in the energy stored in the air gap. That is, a portion of the magnetic field energy is converted into mechanical energy. In addition, the rotor is subjected to an electromagnetic torque te when turned through angle dθr. The mechanical torque produced by the electromagnetic torque is dWmech. Using the principle of energy conservation, the electromagnetic torque is determined from
$$ {\text{d}}W_{\text{e}} = {\text{d}}W_{\text{m}} + {\text{d}}W_{\text{mech}} = {\text{d}}W_{\text{m}} + t_{\text{e}} {\text{d}}\theta_{\text{r}} . $$
(6)
When the flux linkage is constant, the rotor is affected by the electromagnetic torque when a small angular displacement of the rotor causes the magnetic energy to change. The direction of the electromagnetic torque is the direction of decrease in magnetic energy in the constant flux. Given the two winding flux linkages and the angle as an independent variable, the torque equation is
$$ \begin{aligned} \, t_{\text{e}} {\text{d}}\theta_{\text{r}} & = {\text{d}}W_{\text{e}} - {\text{d}}W_{\text{m}} = (i_{\text{A}} {\text{d}}\psi_{\text{A}} + i_{\text{B}} {\text{d}}\psi_{\text{B}} ) \\ & \quad - \left( {i_{\text{A}} {\text{d}}\psi_{\text{A}} + i_{\text{B}} {\text{d}}\psi_{\text{B}} + \frac{{\partial W_{\text{m}} }}{{\partial \theta_{\text{r}} }}{\text{d}}\theta_{\text{r}} } \right). \\ \end{aligned} $$
(7)
That is,
$$ t_{\text{e}} = - \frac{{\partial W_{\text{m}} \left( {\psi_{{{\text{A}}^{\prime } }} ,\psi_{{{\text{B}}^{\prime } }} ,\theta_{\text{r}} } \right)}}{{\partial \theta_{\text{r}} }}. $$
(8)
On applying a constant current constraint, the small angular displacement of the rotor causes a change in the magnetic energy. The torque direction is the direction in which the magnetic flux is increased under constant current conditions. At this instant, the rotor receives an electromagnetic torque
$$ \begin{aligned} t_{\text{e}} {\text{d}}\theta_{\text{r}} & = {\text{d}}W_{\text{e}} - {\text{d}}W_{\text{m}} \\ & = i_{\text{A}} {\text{d}}\psi_{\text{A}} + i_{\text{B}} {\text{d}}\psi_{\text{B}} - {\text{d}}(i_{\text{A}} + i_{\text{B}} + {\text{d}}W^{\prime}_{\text{m}} ). \\ \end{aligned} $$
(9)
That is,
$$ t_{\text{e}} = \frac{{\partial W^{\prime}_{\text{m}} (i_{\text{A}} ,i_{\text{B}} ,i_{\text{r}} )}}{{\partial \theta_{\text{r}} }}. $$
(10)
In Eqs. (8) and (10), to obtain the derivatives of Wm and \( W^{\prime}_{\text{m}} \) with respect to \( \theta_{\text{r}} \), the flux (or current) is held constant—this being just a mathematical constraint. Then, imposing hypothesis (2), the magnetic energy of the electromechanical device (Fig. 1) can be expressed as
$$ W_{\text{m}} = W^{\prime}_{\text{m}} = \frac{1}{2}L_{\text{A}} i_{\text{A}}^{2} + \frac{1}{2}L_{\text{B}} i_{\text{B}}^{2} + \frac{1}{2}L_{\text{AB}} (\theta_{\text{r}} )i_{\text{A}} i_{\text{B}} , $$
(11)
where the mutual inductance LAB is a function of the angle \( \theta_{\text{r}} \). That is, the stored magnetic field energy changes with the angular displacement of the rotor. Substituting Eq. (11) into (10), the electromagnetic torque obtains,
$$ t_{\text{e}} = i_{\text{A}} i_{\text{B}} \frac{{\partial L_{\text{AB}} (\theta_{\text{r}} )}}{{\partial \theta_{\text{r}} }} = - i_{\text{A}} i_{\text{B}} M_{\text{AB}} \sin \theta_{\text{r}} , $$
(12)
where MAB = LmA = LmB.
When the rotor rotates, the rotor angular displacement causes the magnetic energy in the air gap to change, and some of the magnetic field can be released and converted into mechanical energy. In the rotation time dt of the rotor, the power input from the power supply to winding A and winding B is
$$ {\text{d}}W_{\text{e}} = \psi_{\text{A}} {\text{d}}i_{\text{A}} + \psi_{\text{B}} {\text{d}}i_{\text{B}} + 2i_{\text{A}} i_{\text{B}} \frac{{\partial L_{\text{AB}} (\theta_{\text{r}} )}}{{\partial \theta_{\text{r}} }}{\text{d}}\theta_{\text{r}} , $$
(13)
where the first two terms are the electric power absorbed by the transformer electromotive force because of changes in iA and iB. The latter is the energy absorbed by the electromotive force of motion as the rotor rotates.
According to Eq. (12), the mechanical energy converted by the magnetic field in time dt is
$$ {\text{d}}W_{\text{mech}} = t_{\text{e}} {\text{d}}\theta_{\text{r}} = i_{\text{A}} i_{\text{B}} \frac{{\partial L_{\text{AB}} (\theta_{\text{r}} )}}{{\partial \theta_{\text{r}} }}{\text{d}}\theta_{\text{r}} . $$
(14)

In summary, the rotor rotation causes changes in the magnetic energy in the air gap. In this way, the stator winding, the rotor winding, and the gap between the poles of the coupled field conspire to release energy from the magnetic field and convert it to mechanical energy. This electromechanical conversion takes place in the air gap between the magnetic circuit (stator) and the circuit (rotor). The resultant induced electromotive force is necessary for the coupled field to absorb electrical energy from the power supply, and this electromotive force is key to the electromechanical energy conversion. The electromagnetic torque and potential generated by the motion of the rotor in the coupled field constitute an electromechanically coupled pair, which is central in the electromechanical energy conversion and a fundamental element of motor modeling and simulation tests in this study.

2.2 Generation of Electromagnetic Torque

The generation of electromagnetic torque in the motor occurs in two ways. One is the rotor’s rotation caused by changes in the air gap permeability that produces the electromagnetic torque, known as the reluctance torque. The other is the rotor winding excitation generated by the electromagnetic torque, known as the excitation torque [14].
  1. (1)
    Excitation torque: When the sub-torque is set in the positive clockwise direction, Eq. (12) can be rewritten as
    $$ t_{\text{e}} = \frac{1}{{L_{\text{mB}} }}\left( {L_{\text{mB}} i_{\text{B}} } \right)\left( {L_{\text{mA}} i_{\text{A}} } \right)\sin \theta_{\text{r}} = \frac{1}{{L_{\text{mB}} }}\psi_{\text{mA}} \psi_{\text{mB}} \sin \theta_{\text{r}} , $$
    (15)
    where Lm = MAB = LmA = LmB; \( \psi_{\text{mA}} \) and \( \psi_{\text{mB}} \) are the excitation fluxes generated by windings A and B, respectively.

    Equation (15) expresses the principle of the excitation torque, that is, the electromagnetic torque is the result of the interaction between the stator and rotor magnetic fields, these two fields being sinusoidal. The relative position of the two axes determines the size and direction of the moment. When \( \theta_{\text{r}} \) = 90°, the electromagnetic torque reaches its maximum.

     
  2. (2)
    When the rotor is of salient-pole type, a reluctance torque is generated because of an uneven air gap. When \( \theta_{\text{r}} \) = 0° (or 180°), the air gap is at a maximum; the rotor is at this instant in the position in which the stator winding has a self-inductance equal to the direct-axis (d-axis) inductance Ld. When \( \theta_{\text{r}} \) = 90° (or 270°), the rotor cross shaft coincides with the stator winding axis, and the air gap permeability is at a minimum. The self-inductance of the stator winding determines the crossed inductance Lq. During the rotation of the rotor, the stator winding inductance LA varies between Ld and Lq. Assuming that LA varies sinusoidally with rotor angle \( \theta_{\text{r}} \), then
    $$ L_{\text{A}} (\theta_{\text{r}} ) = L_{0} + \Delta L\cos 2\theta_{\text{r}} , $$
    (16)
    where \( L_{0} = \frac{1}{2}\left( {L_{d} + L_{q} } \right);\Delta L = \frac{1}{2}\left( {L_{d} - L_{q} } \right) \). Equation (11) can then be expressed as
    $$ W_{\text{m}} = W^{\prime}_{\text{m}} = \frac{1}{2}L_{\text{A}} (\theta_{\text{r}} )i_{\text{A}}^{2} , $$
    (17)
    which when substituted into Eq. (10) yields electromagnetic torque,
    $$ t_{\text{e}} = - \Delta Li_{\text{A}}^{2} \sin 2\theta = - \frac{1}{2}\left( {L_{d} - L_{q} } \right)i_{\text{A}}^{2} \sin 2\theta_{\text{r}} . $$
    (18)
     
Hence, with the clockwise direction being the positive direction of the torque, the electromagnetic torque is
$$ t_{\text{e}} = \frac{1}{2}\left( {L_{d} - L_{q} } \right)i_{\text{s}}^{2} \sin 2\theta_{\text{r}} , $$
(19)
where \( \theta_{\text{r}} \) is specified in the counterclockwise direction of the rotor with the positive direction of the torque having the same sense as \( \theta_{\text{r}} \). In Eq. (19), when the stator current iA does not change, the maximum value of the reluctance torque depends on the difference between Ld and Lq. Unlike the excitation electromagnetic torque, the maximum value of the reluctance torque is not the position where the rotor d-axis is orthogonal to the stator s-axis, but the space phase angle between the rotor d-axis and the stator s-axis is 45°, 135°, ….

2.3 Control of Electromagnetic Torque

In the electric drive system, the electric motor provides a driving torque applied to a load. Load control of the movement may be achieved by controlling the electromagnetic torque of the motor or the transmission speed of the drive system. Design of the speed control system is necessary when precision and effective control of the electromagnetic torque is needed to achieve control of the speed. If the electrical system needs to control the rotation angular displacement, a position servo system (servo control system) is necessary [14].

For a motor drive system of a pure electric vehicle, it can be simplified as a motor that directly drags the load of the electrical drive system. According to the kinetic principle, the mechanical equation of motion can be written as
$$ t_{\text{e}} = J\frac{{{\text{d}}\varOmega_{\text{r}} }}{{{\text{d}}t}} + R_{\varOmega } \varOmega_{\text{r}} + t_{\text{L}} , $$
(20)
where te is the electromagnetic torque, J the system moment of inertia (including the rotor), \( R_{\varOmega } \) the damping system, \( \varOmega_{\text{r}} \) the rotor mechanical angular velocity, and tL the load torque. Combined with Eq. (20), the equation determining the position of the system obtains,
$$ t_{\text{e}} = J\frac{{{\text{d}}^{2} \varOmega_{\text{r}} }}{{{\text{d}}t^{2} }} + R_{\varOmega } \frac{{{\text{d}}^{2} \theta_{\text{r}} }}{{{\text{d}}t}} + t_{\text{L}} . $$
(21)

From Eqs. (20) and (21), we can see that the control of the motor’s speed and position can be realized by controlling the torque (te − tL). In other words, we need to first build an effective and accurate speed control system and servo system for a high-performance electromagnetic motor.

In this study, using Modelica, modeling and simulation tests of three types of motors that are commonly used in pure electric vehicles, specifically the DC motor, induction motor, and PMSM, were performed.

3 Motor Vector Control and Modelica Model

The electric motor needs to perform electromechanical energy conversion continuously and constantly, which demands an average electromagnetic torque [15].

3.1 DC Motor

3.1.1 Electromagnetic Torque Vector Control of a DC Motor

From the physical structure of a DC motor, the stator winding A (Fig. 1) is changed to a stator field winding, and the rotor winding B is changed to a commutator winding. The rotor magnetic field generated by current D is no longer rotating. Hence, the basic principle of the DC motor can be represented as
$$ t_{\text{e}} = i_{\text{A}} i_{\text{B}} M_{\text{AB}} \sin \theta_{\text{r}} = \psi_{f} i_{\text{a}} . $$
(22)

From the perspective of electromechanical energy conversion, when the rotor winding rotates about the main magnetic pole, the electrical energy is continuously converted into mechanical energy, making the rotor the “center” of energy conversion.

For multi-pole DC motors, Eq. (22) can be written as,
$$ t_{\text{e}} = p\psi_{f} i_{\text{a}} \sin \theta_{\text{r}} = p\psi_{f} \times i_{\text{a}} , $$
(23)
where p is the number of electrodes. Equation (23) indicates that to achieve control of the electromagnetic torque, one needs to control simultaneously the amplitude of the spatial vectors \( \psi_{f} \) and ia, as well as the spatial phase between them. Therefore, vectorial control is required. For DC motors, \( \psi_{f} \) and ia are spatially orthogonal when the brush is geometrically neutral, i.e.,
$$ t_{\text{e}} = p\psi_{f} i_{\text{a}} . $$
(24)

The electromagnetic torque of the DC motor depends linearly on current ia. Therefore, for the servo control of the DC motor, high-quality control can be attained through the current.

3.1.2 Modelica Model of the PMDC Motor

Given the physical structure of the DC motor, the excitation mode of the DC motor is replaced by a constant current source device to equalize the permanent-magnet excitation. With the mathematical model of the motor and the vector control method of the DC motor, we can change the electromechanical conversion device (Fig. 1) to a permanent-magnet DC (PMDC) motor structure.

The resistance and inductance of the armature are modeled directly after configuring the armature pins. Then, the excitation of the permanent magnet is modeled using a constant equivalent excitation current feeding the AirGapDC model in Modelica’s library. The model of the motor takes into account the following loss effects:
  1. (1)

    Joule heat losses in the temperature-dependent armature winding.

     
  2. (2)

    Brush losses in the armature circuit.

     
  3. (3)

    Friction losses.

     
  4. (4)

    Core losses (only eddy current losses, no hysteresis losses).

     
  5. (5)

    Stray load losses.

     
The model of the PMDC motor is illustrated in Fig. 2.
Fig. 2

Modelica model of the PMDC motor

3.2 Induction Motor

The induction motor is one kind of asynchronous motor. From the physical structure of the induction motor, the stator winding A and rotor winding B (Fig. 1) are both three-phase symmetrical windings. In addition, a − x, b − y, and c − z are shorted together [16].

3.2.1 Electromagnetic Torque and Vector Control of Induction Motor

Introducing the space vector product, the vector form of the electromagnetic torque, Eq. (12), for the three-phase induction motor can be written as
$$ t_{\text{e}} = - L_{\text{m}} i_{\text{s}} i_{\text{r}} \sin \theta_{\text{sr}} = - L_{\text{m}} i_{\text{s}} \times i_{\text{r}} . $$
(25)

This shows that the electromagnetic torque of the induction motor can be expressed as the vector product of current vectors of the stator and rotor windings. As S is the spatial angle from vector A to vector B, then the electromagnetic torque is a space vector.

With multi-pole pairs found in induction motors, the vector equation of the electromagnetic torque can be expressed as
$$ \begin{aligned} t_{\text{e}} & = - pL_{\text{m}} i_{\text{s}} \times i_{\text{r}} \\ & = - p\psi_{{{\text{s}}g}} \times i_{\text{r}} = - p\frac{1}{{L_{\text{m}} }}\psi_{{{\text{s}}g}} \times \psi_{{{\text{r}}g}} , \\ \end{aligned} $$
(26)
which shows that the electromagnetic torque is the result of the interaction between the excitation magnetic fields generated by the stator and rotor windings.

3.2.2 Modelica Model of the Induction Motor

From the foregoing description of the induction motor and the principle of electromagnetic torque generation and its vector control method, an induction motor model was configured using various basic components in Modelica’s domain library related to electronics. The resistances and stray inductances of the stator and rotor are modeled directly as stator and rotor phases. Using a space phasor transformation and a stator-fixed AirGap model, the machine models take the following loss effects into account:
  1. (1)

    Joule heat losses in the temperature-dependent stator winding.

     
  2. (2)

    Joule heat losses in the temperature-dependent rotor winding.

     
  3. (3)

    Friction losses.

     
  4. (4)

    Core losses (only eddy current losses, no hysteresis losses).

     
  5. (5)

    Stray load losses.

     
The model of the three-phase asynchronous induction machine motor with slip-ring rotor is illustrated in Fig. 3.
Fig. 3

Modelica model of an induction motor

3.3 PMSM

From the analogous physical structure of the induction motor, the stator winding A (Fig. 1) is changed to symmetrical three-phase windings A–X, B–Y, and C–Z. A three-phase symmetrical current is introduced and then a concentrated winding B is inserted into the rotor slot embedded in the permanent magnet (or distributed winding) as an excitation magnet [17, 18]. The structure is representative of a three-phase PMSM.

3.3.1 Electromagnetic Torque and Vector Control of the PMSM

Typically, the motor torque refers to the torque acting on the rotor and is identical to the torque acting on the stator. From Eq. (12), we obtain
$$ t_{{{\text{e}}1}} = i_{\text{s}} i_{f} L_{\text{m}} \sin \beta . $$
(27)
When the rotor structure is of salient-pole type, the reluctance torque generated by the synchronous motor is given by
$$ t_{{{\text{e}}2}} = \frac{1}{2}\left( {L_{d} - L_{q} } \right)i_{\text{s}}^{2} \sin 2\beta . $$
(28)
Hence, the total electromagnetic torque for the three-phase PMSM is
$$ t_{{{\text{e}}3}} = i_{f} i_{\text{s}} L_{\text{m}} \sin \beta + \frac{1}{2}\left( {L_{d} - L_{q} } \right)i_{\text{s}}^{2} \sin 2\beta . $$
(29)
Therefore, for the multi-pole pair of a salient-pole synchronous motor, the electromagnetic torque is
$$ t_{{{\text{e}}4}} = p\psi_{\text{s}} i_{\text{s}} , $$
(30)
where \( \psi_{\text{s}} \) is the stator flux vector. From the analysis above, vector control of the electromagnetic torque of the synchronous motor is achieved by the direct control of the amplitude, and control of the angular phase difference achieves precise control of the torque. For this control, it is necessary to detect the rotor position \( \theta_{\text{r}} \) at all times. After determining \( \theta_{\text{r}} \), the phase of the stator current vector is in the ABC axis then can be determined.

3.3.2 Modelica Model of PMSM

A PMSM is generated from the electric excitation synchronous motor by changing the excitation circuit of the electric excitation motor to a permanent magnet. If a magnetic field similar to that for the structure of the electric excitation synchronous motor is produced, a PMSM is formed. In configuring a Modelica model for the PMSM, we reused the three-phase induction motor stator and other physical structure models. We used Modelica’s standard library files, such as “PermanentMagnet,” “DamperCage,” as well as other basic components to establish the PMSM model.

The resistance and stray inductance of the stator are modeled directly from the stator phases and then applying a space phasor transform and a rotor-fixed AirGap model. For the rotor’s squirrel cage, resistance and stray inductance are modeled using the two axes of the rotor-fixed coordinate system. The permanent-magnet excitation is modeled by a constant equivalent excitation current feeding the d-axis. The machine models take into account the following energy loss effects:
  1. (1)

    Joule heat losses in the temperature-dependent stator winding.

     
  2. (2)

    Joule heat losses (optional, when enabled) in the temperature-dependent damper cage.

     
  3. (3)

    Friction losses.

     
  4. (4)

    Core losses (only eddy current losses, no hysteresis losses).

     
  5. (5)

    Stray load losses.

     
  6. (6)

    Permanent-magnet losses.

     
The model of the three-phase permanent-magnet synchronous induction machine is illustrated in Fig. 4.
Fig. 4

Modelica model of a PMSM

4 Modelica Model Testing and Motor Analysis

The correctness of the unified mathematical model and the vector control method of each motor type was verified. Given the electromechanical conversion devices, the feasibility of the three types of motors can be improved. In addition, functional tests of the Modelica motor model were performed in the MWorks simulation environment. Simulation tests and analyses were carried out in regard to different characteristics of the PMDC motor, induction motor, and PMSM.

4.1 PMDC Motor

For the DC motor start-up, when the need for large torque characteristics is greatest, we built a test model to test the mechanical properties of the PMDC motor obtained from the Modelica model. A ramp voltage is applied to the DC motor armature to start and load the test. The simulation model is illustrated in Fig. 5, and the simulation parameter settings are listed in Table 1.
Fig. 5

PMDC performance test

Table 1

Simulation settings for the PMDC Motor

Name

Description

Value

Units

V a

Nominal armature voltage

100

V

I a

Nominal armature current

100

A

J s

Stator’s moment of inertia

0.29

kg m2

J r

Rotor’s moment of inertia

0.15

kg m2

n

Nominal speed

1425

rpm

T a

Nominal torque

63.66

Nm

P

Nominal mechanical output

9.5

kW

η

Efficiency

95.0

%

Testing of the motor starts at 0.2 s when the DC voltage begins. Voltage ramping lasts for 0.8 s. At 1.5 s, loading commences to test the speed of the model. Torque and other mechanical properties of the output are shown in Fig. 6. The output curves are the time variations of torque and speed obtained from the PMDC motor model test. From these curves, we see that as the voltage increases, the field accelerates the rotational inertia of the rotor, and the torque quickly increases to a maximum. The torque and speed quickly stabilize when the voltage reaches a stable state. The voltage is disturbed but stabilizes again when the load is applied. The simulation results show that the test model of the PMDC motor performs well mechanically, thus meeting electric vehicle requirements of a large and stable start-up torque from its DC motor.
Fig. 6

PMDC torque and speed curves

4.2 Induction Motor

In engineering practice, the current of an induction motor during direct start-up changes considerably. It may reach as high as 4–7 times that of the rated current, but the start-up torque is not sufficient. In addition, extra heat is generated because of internal losses, and therefore, selecting the starting mode requires a well-reasoned assessment. In this study, the Y–Δ start method is used to start up the asynchronous induction motor (ASIM). The model is illustrated in Fig. 7, and the simulation settings are listed in Table 2.
Fig. 7

ASIM Y-Δ start-up performance test

Table 2

Simulation settings for the Y–Δ start induction motor

Name

Description

Values

Unit

V a

Nominal armature voltage

100

V

I a

Nominal armature current

100

A

J s

Stator’s moment of inertia

0.29

kg m2

J r

Rotor’s moment of inertia

0.29

kg m2

n

Nominal speed

1440

rpm

T a

Nominal torque

161.4

Nm

f

Nominal frequency

50

Hz

P

Nominal mechanical output

24.35

kW

η

Efficiency

92.7

%

The Y–Δ start-up test of the ASIM motor begins at 0.1 s. To show the start-up process, the test mode changes the supply voltage from the Y connection to the delta connection. At this instant, torque and speed have reached certain values. The simulation results are shown in Fig. 8.
Fig. 8

ASIM Y–Δ start-up performance test

From the trend in the current evolution, the current amplitude is seen to be well controlled within three times of the existing value and ultimately reaches the nominal speed, thereby meeting the specifications required for a pure electric vehicle.

4.3 PMSM Test

Because of its small size, light weight, energy efficiency, and other advantages, PMSMs are perceived publicly as “low-carbon” motors. The PMSM has attracted increasing attention for its synchronous motor running characteristics and with its control technology maturing continually.

With the mathematical model and the vector control method of the PMSM, a theoretical analysis was successfully carried out. For the purpose of controlling the electromagnetic torque, PMSM vector control using the current was attained.

In testing the model for the PMSM, the rotor was accelerated under a quadratic-speed-dependent load from standstill. The simulation results are shown in Fig. 9; the simulation settings are listed in Table 3. By controlling the current vector to control the PMSM speed, we obtained the mechanical properties shown in Fig. 10.
Fig. 9

PMSM performance test model

Table 3

Simulation settings for the PMSM

Name

Description

Values

Unit

V a

Nominal armature voltage per phase

100

V

p 0

Number of pole pairs p

2

 

I a

Nominal armature current

100

A

J s

Stator’s moment of inertia

0.29

kg m2

J r

Rotor’s moment of inertia

0.29

kg m2

n

Nominal speed

1440

rpm

T a

Nominal torque

181.4

Nm

f

Nominal frequency

50

Hz

P

Nominal mechanical output

28.50

kW

η

Efficiency

92.7

%

Fig. 10

PMSM performance test curves

In performing the simulation to test the PMSM model, we began with a load start-up, at 0 s that lasted 1.2 s and increased the load gradually. From the torque and rotational speed curves, we see that during loading, both torque and speed increased steadily with no obvious disturbance. This steadiness indicates that the output torque quickly reaches equilibrium under loading. This behavior complies with the power demand of the motor-driven vehicle and demonstrates the correctness of the PMSM model.

5 Conclusions

  1. (1)

    According to the basic principle of electromechanical energy conversion, the motors of the pure electric vehicles are divided into three structural types, including the DC motor, induction motor, and synchronous motor. Using the theory of motor-based vector control, three types of motor models are configured with a Modelica and are used to simulate the respective mechanical properties.

     
  2. (2)

    A unified modeling methods of three different motor types are provided by exploiting the multi-domain physical system-modeling language Modelica. The simulation results show that the PMDC motor performs well in mechanical properties and meets a large and stable startup torque requirements of pure electric vehicles. The current amplitude of the asynchronous induction motor (ASIM) are well controlled within 3 times of the existing value and ultimately reaches the nominal speed, thereby ASIM also meets the specifications requirement of a pure electric vehicle. During loading, both torque and speed of the PMSM model are increasing steadily with no obvious disturbance. And the output torque quickly reaches the equilibrium point under loading. This performance of the PMSM complies with the power demand of the motor-driven vehicle. The simulation results demonstrate the correctness of the PMDC, ASIM and PMSM model in Modelica.

     
  3. (3)

    This highlights the benefits of Modelica’s model reusing strategy in achieving this unified multi-domain modeling and providing design and performance testing of a motor. Also the new modeling and testing method are given for research and simulation of electric motor of pure electric vehicle.

     

Given the basic methods of unified modeling, vector control, and simulations of the three motor types, applications of the models in regard to actual engineering practice need further research and experimentation.

Notes

Acknowledgements

The authors thank their colleagues at the Zhengzhou University for fruitful discussions.

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Copyright information

© China Society of Automotive Engineers (China SAE) 2018

Authors and Affiliations

  1. 1.College of Mechanical EngineeringZhengzhou UniversityZhengzhouChina

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