5.6: Introduction of Permanent Magnetic Motor Principle, Drive, and Control

Learning Objectives

After completing this section, you will be able to:

1. Explain the working principle of a permanent magnetic synchronous motor (PMSM).
2. Describe the vector control of PMSM along with the rotational \(d\)-\(q\) coordinate, the fixed stator (\(α\), \(β\)) coordinate, and the stator \((a,\ b,\ c)\) coordinate, and associated coordinate transformations.
3. Provide an explanation of the PMSM PWM (pulse-width modulated) drive circuit.

A synchronous electric motor is an AC (alternative current) electric motor; its rotational shaft is synchronized with the frequency of the supply current at the steady state. In addition, its rotation period equals an integer number of AC cycles. Synchronous motors use so-called electromagnets. These are energized by applying voltage to the stator windings to create a rotating magnetic field synchronized in time with the oscillations of the current. The electric machine rotor with permanent magnets or electromagnets turns in step with the stator field at the same rate. As a result, it provides the second synchronized rotating magnet field.

Synchronous and induction motors are widely used AC motors. Because they do not rely on induction to produce the rotor’s magnetic field, synchronous motors rotate at a rate locked to the line frequency. By contrast, induction motors require slip between the rotor rotating speed and line frequency. This is because the rotor must rotate at a frequency slightly slower than the AC alternations in order to induce current in the rotor. Small synchronous motors are often used in timing devices such as in synchronous clocks, timers, tape recorders, and precision servomechanisms that must operate at a precise speed. In these cases, the speed accuracy is dependent on the supply power line frequency controlled in interconnected grid systems or by DC-to-AC inverters.

In fact, synchronous motors also include synchronous generators when their rotors are driven by the external torque. The two major types of synchronous motors are distinguished by how the rotor is magnetized: non-excited and direct-current excited. In non-excited motors, the rotor is made of steel. It rotates in step with the stator’s rotating magnetic field, so it has an almost-constant magnetic field through it. The external stator field magnetizes the rotor, inducing the magnetic poles needed to turn it. The rotor is often made of a highly retentive steel such as cobalt steel.

Permanent-magnet synchronous motors (PMSMs) use permanent magnets embedded in the rotor to create a constant magnetic field. The stator carries windings connected to an AC electricity supply in order to produce a rotating magnetic field (as in asynchronous motors). At synchronous speed, the rotor poles lock to the rotating magnetic field. Note that PMSMs are similar to brushless DC motors, where Neodymium magnets are widely used with possible ferrite magnets. Machines that use ferrite magnets have lower power and torque densities when compared with neodymium machines [46].

Almost all PMSMs use three-phase alternating current to set up a rotating magnetic field (RMF). In these cases, the frequency and power are controlled by the power electronics (or DC-AC converters) that respond to the command speed/acceleration. In a PMSM, stators contain numerous parallel slots stuffed with interconnected loops of copper windings. These can be bulky loops of round copper wire or tidy hairpin-shaped copper insertions with square cross-sections that increase both fill density and direct wire-to-wire contact within the grooves. Denser windings improve torque capability, and tidier interlacing at the ends amounts to less bulk and a smaller overall package.

Figure 5.41 shows the working principle for a three-phase controlled two-pole PMSM. In the top plot, the three-phase voltages (Phases A, B, and C) are applied to the three windings of the PMSM. As a result, a rotation magnetic field (RMF) is formed because the three-phase power has a 120-degree phase apart. The applied three-phase voltages and the corresponding windings are color coded in Figure 5.41. The bottom plot shows the magnetic field direction (red arrow at the center) and the corresponding rotor position with two half-circle strips representing the permanent magnets. As the magnetic field rotates in a clockwise direction due to applied three-phase voltages, the rotor rotates in the same direction (clockwise) and is synchronized with the rotating magnetic field. This confirms the synchronous operation of the motor.

PMSMs are typically controlled using so-called direct torque control (DTC) [12] or field-oriented control (FOC) [47]. In this section we will mainly address the field-oriented control (FOC). The field-oriented control, also called vector control, is a variable-frequency drive (VFD) control method in which the stator currents of a three-phase AC (or brushless DC) electric motor are identified as two orthogonal components. These components can be visualized with a vector; one component is defined as the motor magnetic flux and the other as the torque. The control system of the motor drive calculates the corresponding current component references from the flux as well as the torque references provided by the speed control command. Typically, proportional-integral (PI) controllers (Section 3.4) are used to keep the measured current components at their reference values. The PWM (pulse-width modulation) of the variable-frequency drive defines the switch-commands for these transistors (switches), according to the desired stator voltage references.

The electrical PMSM model in the \(d\)-\(q\) reference frame can be expressed by equations \ref{5.65} and \ref{5.66}.

\[  \left[ \begin{array}{c} u_d \\ u_q \end{array} \right]=\left[ \begin{array}{cc} R_s+L_d\mathcal{D} & -ωL_q \\ ωL_d & R_s+L_q\mathcal{D} \end{array} \right]\left[ \begin{array}{c} i_d \\ i_q \end{array} \right]+\left[ \begin{array}{c} 0 \\ ωψ_f \end{array} \right] \label{5.65}\]

\[  τ_{em}=n_pψ_fi_q \label{5.66}\]

where \(u_d\) and \(u_q\) are the \(d\)- and \(q\)-axis terminal voltages, respectively; \(i_d\) and \(i_q\) are the \(d\)- and \(q\)-axis armature currents, respectively; and \(L_d\) and \(L_q\) are the \(d\)- and \(q\)-axis inductance, respectively. In addition, \(R_s\) is the armature resistance; \(ω\) is the electrical angular velocity (or motor angular speed); \(ψ_f\) is the permanent magnet flux-linkage; \(n_p\) is the number of pole pairs of the PMSM; and \(\mathcal{D}\) is a differential operator such that the following is true.

\[ \mathcal{D}\left[\cdot \right]=\dfrac{d\left[\cdot \right]}{dt}, \ e.g., \mathcal{D}i_q=\dfrac{di_q}{dt} \label{5.67}\]

Inductors \(L_d\) and \(L_q\) are equal for non-salient motors. As indicated by equation \ref{5.66}, the motor torque is determined only by \(i_q\) and has nothing to do with \(i_d\). In order to get better motor efficiency, \(i_d\) is controlled to be zero during operation, which reduces the copper (energy) loss [46].

Figure 5.42 shows a simplified PMSM control circuit (or an inverter circuit converting DC power to AC). The circuit uses six MOSFETs (metal-oxide-semiconductor field-effect transistors). Note that these MOSFETs are controlled by PWM signals, where the lower PWM signal (e.g., \({PWM}_{Q_2}\)) has an opposite level of voltage (see \({PWM}_{Q_1}\) in Figure 5.42). Three stator windings are connected together at one end. The other three are controlled by the inverter. A power MOSFET is a voltage-controlled transistor designed for high-frequency operations. It has a very small load voltage drop when it is turned on. However, the saturation temperature sensitivity limits its application in high power applications. Thus, the IGBT (insulated-gate bipolar transistor) should be used for high power applications. However, IGBT has a high voltage drop due to utilizing of bipolar transistor, leading to high conduction loss.

Vector control (VC) is an elegant control method for PMSMs, where field-oriented control (FOC) theory is used to control space vectors of magnetic flux, current, and voltage. It is possible to set up the coordinate system to decompose the vectors into a magnetic field-generating part (\(q\)) and a torque-generating part (\(d\)). The structure of the motor controller (vector control controller) is then almost the same as for a separately excited DC motor, which simplifies the control of PMSM. The VC technique was developed specifically to achieve a similarly dynamic performance in PMSM. Figure 5.43 presents a control system block diagram that is widely used for speed and torque control with current (\(i_d,\ i_q\)) feedback.

Figure 5.43 shows the basic structure of vector control of the PMSM. First, there are two feedback blocks. One of these, labeled Hall, is the hall-effect sensor (see Section 6.2 for details of its working principle) used for motor-axle position and speed measurements. The other, is for stator current measurements. Denoted by the circles drawn on the stator current wires, it is used for feedback control of (\(i_d,\ i_q\)) for torque and flux regulation. The pulse train signals from these hall-effect encoders are to be processed in the block labeled Position & speed estimation. Here, the signals are converted to the motor rotor position and speed signals. Section 6.2 will note that the hall-effect sensor outputs are two square-pulse trains with their period inversely proportional to the rotational speed \({\dot{θ}}_{est}.\) The rotor position \(θ_{est}\) can be obtained by counting the signal pulse train with known initial position (or using multichannel hall-effect pulse trains).

Next, we have the block named \(\left(i_a,i_b,i_c\right)\ \)to \((i_α,\ i_β)\). Based upon the Kirchhoff current law (summation of currents entering a junction equals to that of leaving it), we have the following.

\[ i_a+i_b+i_c=0, \ e.g., i_c=-i_a-i_b \label{5.68}\]

As a result, even though only stator currents \(i_a\) and \(i_b\) are measured, state current \(i_c\) can be calculated based on equation \ref{5.68}. Now, we need to convert the measured currents \(i_a\), \(i_b\), and \(i_c\) into the two-phase current space denoted by the (\(α,\ β\)) coordinate. This transformation is also called the Clark transformation [48]. The Clark transformation utilizes the balanced three-phase components \(\left(i_a,i_b,i_c\right)\) in the \((a,\ b,\ c)\) reference frame and outputs the balanced two-phase orthogonal components \((i_α,i_β)\) in the stationary \((α,\ β)\) reference frame, demonstrated in Figure 5.44.

From Figure 5.44, the coordinate transformation from \(\left(i_a,i_b,i_c\right)\) to \((i_α,i_β)\) reference frame can be completed utilizing the following equation.

\[ \left[ \begin{array}{c} i_α \\ i_β \end{array} \right] =k\left[ \begin{array}{ccc} 1 & -\dfrac{1}{2} & -\dfrac{1}{2} \\ 0 & \dfrac{\sqrt{3}}{2} & -\dfrac{\sqrt{3}}{2} \end{array} \right]\left[ \begin{array}{c} i_a \\ i_b \\ i_c \end{array} \right] \label{5.69}\]

Using equation \ref{5.68} and condition \(i_α{=i}_a\), equation \ref{5.69} can be simplified as below.

\[ \left[ \begin{array}{c} i_α \\ i_β \end{array} \right] =\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \dfrac{1}{\sqrt{3}} & -\dfrac{1}{\sqrt{3}} \end{array} \right]\left[ \begin{array}{c} i_a \\ i_b \\ i_c \end{array} \right]=T_{clark}\left[ \begin{array}{c} i_a \\ i_b \\ i_c \end{array} \right] \label{5.70}\]

Returning to Figure 5.43, after the current signals \(i_a,i_b,i_c\) are transferred to \((i_α,i_β)\) coordinate, we next use the block labeled \((i_α,i_β)\) to \((i_d,i_q)\) to transfer the current signals \(i_α,i_β\) in the stationary coordinate to the \(d\)-\(q\) coordinate in order to obtain \(i_d,i_q\). This transformation is also called the Park transformation [48]. Note that vector control is performed entirely in the \(d\)-\(q\) coordinate system to simplify the control of PMSMs.

Figure 5.45 shows the coordinate relationship between \((i_α,i_β)\) and \((i_d,i_q)\) based upon the Park transformation shown below.

\[ \left[ \begin{array}{c} i_d \\ i_q \end{array} \right] =\left[ \begin{array}{cc} \cos θ_f\ } & \sin θ_f\ } \\ -\sin θ_f\ } & \cos θ_f\ } \end{array} \right]\left[ \begin{array}{c} i_α \\ i_β \end{array} \right]=T_{park}\left[ \begin{array}{c} i_α \\ i_β \end{array} \right] \label{5.71}\]

where \(θ_f\) is the rotational field angle and \(i_s\) is the rotational current vector. Because we are dealing with the synchronous motor, \(θ_f\) is equal to the rotor position that can be measured and estimated using the hall-effect sensor signals. In practical applications, \(θ_{est}\) will be used in place of \(θ_f\) in Figure 5.43.

Next, from Figure 5.43, the given reference current signals \(i_{d\_ref}\) and \(i_{q\_ref}\) are subtracted by the estimated signals \(i_d\) and \(i_q\), respectively. The resulting error signals are used as the input to the two PI controllers (see Section 3.4) to generate the two control voltage signals \(u_d\) and \(u_q\), respectively.

The control magnitudes \(u_d\) and \(u_q\) are in the rotational \(d\)-\(q\) coordinate. Therefore, they need to be transferred back to the static \((α,\ β)\) coordinate; see the \((u_d,u_q)\) to \((u_α,u_β)\) block in Figure 5.43. This transfer can be achieved by reversing the process described in equation \ref{5.71} as follows.

\[ \left[ \begin{array}{c} u_α \\ u_β \end{array} \right] =T^{-1}_{park}\left[ \begin{array}{c} u_d \\ u_q \end{array} \right]=\left[ \begin{array}{cc} \cos θ_f\ } & -\sin θ_f\ } \\ \sin θ_f\ } & \cos θ_f\ } \end{array} \right]\left[ \begin{array}{c} u_d \\ u_q \end{array} \right] \label{5.72}\]

The block in Figure 5.43 labeled SVPWM (space vector pulse width modulation) generates control signals for these switches on the DC-AC inverter (see an example in Figure 5.42). Thus, precise output voltage and current waveforms can be generated. The purpose of the SVPWM is to convert the static voltages \(u_α,u_β\) to the \((a,\ b,\ c)\) coordinate using the inverse Clark transformation \(T^+_{clark}\) and then follow the process described in references [49] and [50], where \({\left[\cdot \right]}^+\) is the pseudo inverse of matrix \(\left[\cdot \right].\) As a result, the pseudo inverse \(T^+_{clark}\) can be calculated using the singular value decomposition (SVD) as below.

\[\ T_{clark}=U\cdot S\cdot V^*=\left[ \begin{array}{cc} U_1 & U_2 \end{array} \right]\ \left[ \begin{array}{cc} Σ & 0 \\ 0 & 0 \end{array} \right]\left[ \begin{array}{c} V^*_1 \\ V^*_2 \end{array} \right], Σ>0\ \label{5.73}\]

where “*” represents the complex conjugate transpose. Based on SVD results, the pseudo inverse of \(T_{clark}\) can be calculated as below.

\[T^+_{clark}=V_1\cdot Σ^{-1}\cdot U^*_1, Σ>0\ \label{5.74}\]

With \(T^+_{clark}\) defined in equation \ref{5.74}, the desired voltage signals \(u_a\), \(u_b\), and \(u_c\) in \((a,\ b,\ c)\) coordinate can be calculated as follows.

\[\left[ \begin{array}{c} u_a \\ u_b \\ u_c \end{array} \right]=T^+_{clark} \left[ \begin{array}{c} u_α \\ u_β \end{array} \right] \label{5.75}\]

The PWM control signals \({PWM}_{Q1}\) to \({PWM}_{Q6}\) (see Figure 5.43 for detail) can be generated by a certain kind of DSP (digital signal processor). Its mathematical principle can be found in reference [51].

Last, the Inverter in Figure 5.43 is a device converting the PWM command from the SVPWM block to generate motor control voltage signals. This conversion results in the motor current \(i_a\), \(i_b\), and \(i_c\), where a sample inverter circuit can be found in Figure 5.42.

As a summary, a permanent magnet synchronous motor (PMSM) can be considered similar to a brushless direct current (BLDC) motor with a permanent magnet rotor and wound stators. For example, a three-phase stator layout is designed to create sinusoidal flux distribution in the air gap; it makes the back electromotive force (BEMF) sinusoidal with motor rotation period equal to an integer number of the current oscillations frequency.

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