Torque-load angle characteristic and speed control of Permanent magnet AC motor

August 28, 2026
2 hours to complete
Workbench v1.5

Introduction

In this experiment, a permanent-magnet AC motor's load-angle variation as a function of electromagnetic torque is characterized. This is followed by closed-loop speed control using hysteretic current control.

Theoretical background

One of the major disadvantages of DC motors is the need for commutator brushes, which constantly wear out. This requirement is eliminated in PMAC motors by using sinusoidally distributed windings that create a rotating magnetic field under balanced 3Φ sinusoidal stator currents. A simplified radial cross-section of a two-pole PMAC machine is shown below, illustrating the sinusoidal stator-conductor density distribution.

Internal cross-section of a permanent-magnet synchronous motor showing sinusoidally distributed stator windings

The stator-flux space vector Φ_s, produced by currents in the stator windings, interacts with the rotor flux Φ_r, produced by permanent magnets mounted on the rotor, creating the electromagnetic torque T_em given by:

T_{\mathrm{em}} \propto \Phi _{s} \times \Phi _{r} \times \sin(\delta ) \propto I_{s} \times \Phi _{r} \times \sin(\delta )\tag{1}

where δ is the load angle, which is the angle between the stator-flux and rotor-flux space vectors.

The stator current space vector I_s is given by the Clarke transform:

I_{s} = I_{a} + I_{b} \times e^{j2\pi /3} + I_{c} \times e^{j4\pi /3}\tag{2}

For a constant stator-current magnitude, the variation of electromagnetic torque as a function of load angle (Eqn. 1) is shown below:

PMAC electromagnetic torque varying sinusoidally as the load angle increases

The maximum T_em occurs when the flux induced by the stator current is at 90° to the rotor flux. If the steady-state load torque were to exceed this value, the motor would enter an unstable operating condition and rapidly come to a standstill.

To generate a constant steady-state torque and keep the rotor spinning under a constant load, the flux induced by the stator current is maintained at an angle between 0 and π/2 with respect to the rotor flux. This is achieved by exciting the stator windings with balanced 3Φ sinusoidal voltages, producing a rotating stator-flux space vector whose rotational speed is given by:

\omega _{\mathrm{sync}} = \frac{2}{P} \times \omega _{s} = \frac{2}{P} \times 2\pi f\tag{3}

where ω_sync is the synchronous rotor-flux speed, which equals the mechanical rotor speed ω_mech at steady state; f is the frequency of the terminal voltage; and P is the number of rotor poles.

If the load torque were greater than the maximum electromagnetic torque that can be generated, as identified in the figure above, the load angle δ would increase beyond π/2. At that point, the motor would enter an unstable operating region in which the generated torque would continue decreasing, leading to a further increase in the load angle and, in turn, a further reduction in electromagnetic torque. This cycle would continue until the motor stalled.

Open-loop speed control

The per-phase equivalent circuit of a PMAC motor under steady-state conditions is shown below:

PMAC motor equivalent circuit

The back-emf E_b is proportional to the rotational speed:

E_{b} = k_{e} \times \omega _{\mathrm{mech}}\tag{4}

From Eqn. 3, the speed of a PMAC motor depends solely on the input-voltage frequency. The magnitude of the applied voltage controls the maximum electromagnetic torque. At startup, the rated motor voltage cannot be applied directly because the back-emf is zero, which would lead to high inrush currents that could potentially damage the motor. The same principle applies at low speeds, where there would not be sufficient back-emf buildup to limit the current resulting from the applied voltage. At high speeds, applying less than the required voltage would lead to insufficient stator current and, in turn, insufficient peak electromagnetic torque. If this torque is lower than the load torque, the motor will stall, as explained earlier. This leads to a high short-circuit current because the back-emf becomes zero. To avoid either extreme, V/f (V-by-f) control is used for open-loop speed control, in which the applied stator-voltage magnitude V_a is controlled in proportion to the applied stator-voltage frequency f_a. This relationship is given by:

V_{a} = \frac{V_{\mathrm{rated}}}{f_{\mathrm{rated}}} \times f_{a}\tag{5}

This control scheme is summarized in the following figure. As shown, under nominal operating conditions, the applied voltage and frequency are proportional to one another. At speeds above the rated speed, the voltage is limited to the rated voltage V_rated. This could be due either to a limitation in the available voltage or to the need to avoid damaging the motor windings. At very low speeds, a constant voltage V_min is applied to overcome the high initial static friction and the voltage drop across the stator-winding resistance.

V by f control, showing the linear relation between applied voltage and frequency under nominal operating condition

Closed-loop speed control

Unlike the DC motors covered in previous experiments, the PMAC motor's speed does not change with load torque because it is determined solely by the applied stator-voltage frequency. However, this does not mean that the motor is operating at optimal efficiency. The optimal operating point is where the load angle δ is maintained at π/2, regardless of the desired speed and load-torque conditions. This ensures that the desired electromagnetic torque, equal to the load torque at steady state, is generated with the lowest stator current. This results in lower losses and, therefore, higher efficiency.

To achieve this, an inner current-control loop is added to the outer speed-control loop, as in DC motor closed-loop speed control. In the case of the DC motor, the output of the outer speed-loop controller was the reference rotor current. In the case of PMAC motor control, its output is the magnitude of the desired 3Φ stator current, I_m. The stator-current space-vector angle is maintained at π/2 greater than the rotor-flux space-vector angle Φ_r, as discussed above. The 3Φ reference current is given by:

I_{a,\mathrm{ref}} = I_{m} \times \sin\left(\Phi _{r} + \frac{\pi }{2}\right)\tag{6}

I_{b,\mathrm{ref}} = I_{m} \times \sin\left(\Phi _{r} + \frac{\pi }{2} - \frac{2\pi }{3}\right)\tag{7}

I_{c,\mathrm{ref}} = I_{m} \times \sin\left(\Phi _{r} + \frac{\pi }{2} - \frac{4\pi }{3}\right)\tag{8}

where $\Phi _{r} = \frac{P}{2}∫ \omega _{\mathrm{mech}} dt$ .

The desired reference current is generated using hysteresis control. In this method, if the actual phase current falls below the lower hysteresis limit around the reference current, the corresponding motor-phase terminal is connected to the positive DC bus by turning on the top switch of the corresponding inverter leg to increase the phase current. If the actual phase current rises above the upper hysteresis limit, the corresponding motor-phase terminal is connected to the negative DC bus by turning on the bottom switch of the corresponding inverter leg to reduce the phase current. This method is summarized in the figure below:

PMAC motor outer speed-control PI loop and inner hysteresis current-control loop

In the following section, the PMAC motor will be run at different speeds and under different load-torque conditions using V/f control, and the load-angle variation will be characterized. This will be followed by running the motor under closed-loop speed control using hysteresis current control.

Real-time torque-load-angle characterization

Assemble the PMAC motor–DC motor set as shown below:
1. DC generator (the one with an A-quad-B encoder mounted on its back).
2. PMAC motor.
3. Oldham coupler. Tighten if loose using M2.5 hex wrench.
4. Oldham coupling disk. Attach it to the motor and generator coupling.
5. Safety enclosure. The coupling unit resides inside the safety enclosure.
6. M4 hex socket-head screws - 8.
7. M2.5 hex wrench. Use it to tighten the motor and generator onto the safety enclosure.
8. 2048-line A-quad-B encoder. Attach the encoder cable to the A-quad-B encoder.
The wiring color code followed is:

DC (Inverter input):
1. DC +ve - ● (Red)
2. Ground - ● (Green)
3. ```
DC -ve  - ● (Black)
```
AC (Inverter output):
1. Phase A - ● (Black)
2. Phase B - ● (Red)
3. Phase C - ● (Blue)
The inverter output terminals are located on the front panel of the three-inverter module. Connect the PMAC motor terminals to the Inverter 1 terminals such that the color of both these terminals matches one another. Connect the DC generator -ve (black terminal) to Inverter 2 A phase (black terminal) and DC generator +ve (red terminal) to Inverter 2 B phase (red terminal). The DC generator might have a third cable with green terminal. This is the ground wire and must be left unconnected.

Warning
Ensure that the DC power supply is turned off before making the connections. If the DC power supply does not have a dedicated ground terminal, that connection can be left floating. If present, it is strongly recommended that it be connected to the three-inverter module's ground terminal to mitigate any electrical hazard.
Connect the three-inverter to the computer via USB. Connect the generator speed feedback to the three-inverter's Encoder 1 DSUB connector in the back panel.
Open Workbench and pin the Explorer and Properties docks.
Navigate to and open the PMVbyF project file in the Experiment6\PMLoadAngle folder, usually found in the following location: C:\Program Files (x86)\Sciamble\WorkBench v1\Examples\CUSPLab\BasicDrives.
Expand the project and open the RotorInit model file shown below:

The incremental encoder does not output the rotor's actual absolute position. Instead, it outputs a pulse when the rotor turns through a certain angle. Because the initial rotor position cannot be determined using the incremental encoder, it is forcibly initialized to a predetermined position. This is achieved by briefly connecting the phase-A winding to V_dc and the phase-B and phase-C windings to 0, as done in the RotorInit model file. This ensures that the initial rotor magnetic-flux axis is aligned with the stator phase-A winding axis.
Open the project properties by double-clicking the project node in Explorer, and set Start model/Function to RotorInit.
Turn ON the DC power supply and set the voltage to 40 V.
Ensure that the button on the top dock is pressed to enable real-time mode. Click the button.
After about 5 s, click the button to stop the model.
Open the VbyF model file within the project in the Explorer dock. The model file is shown below:

The tools highlighted in purple form the closed-loop current PI controller, which controls the load torque applied by the DC generator coupled to the PMAC motor. This is the same method used in previous experiments to emulate a load torque. The tools highlighted in green compute the difference between the rotor-flux position, obtained from the position encoder, and the stator-current space vector I_s, as specified by Eqn. 2. The computed difference is constrained to be between ±180°. While this experiment is running, if the applied load torque (the Tload slider) exceeds the PMAC motor's peak electromagnetic torque, the rotor stalls, and this could lead to high short-circuit currents that could potentially damage the motor. This is prevented by the logic implemented by the tools in black, which disables the inverter if the rotor speed is less than 2 rad/s and the applied stator frequency is greater than 5 Hz.

Finally, the tools highlighted in blue generate the PWM signals for the inverter connected to the PMAC motor. The frequency of the applied terminal voltage is controlled by the slider. The internal implementation of the VByF subsystem is shown below. As shown, the frequency f is scaled by a V/f gain of 0.014 V/Hz. Within the saturation block, the maximum phase voltage is limited to 20 V, within the motor's rated voltage, and the minimum voltage is limited to 1.25 V, which is required to overcome the initial static friction.
Open the project properties by double-clicking the project node in Explorer, and set Start model/Function to VbyF.
Ensure that the button on the top dock is pressed to enable real-time mode. Click the button.
Move the Frequency slider gradually to 30 Hz.

Increase the applied load using the Tload slider in steps of 0.005 Nm. At each step, note the load angle in the deltaTheta display and the rotor speed ω_mech. When the load torque exceeds the maximum electromagnetic torque that can be synthesized for the applied terminal voltage, the motor stalls. In this case, reduce the load torque to 0 Nm and the frequency to 0 Hz. Then gradually increase the frequency to 30 Hz. Now decrease the load torque in steps of -0.005 Nm, and record the load angle and rotor speed.

T_l (Nm)	ω_mech (rad/s at f = 30 Hz)	δ (° at f = 30 Hz)	ω_mech (rad/s at f = 40 Hz)	δ (° at f = 40 Hz)	ω_mech (rad/s at f = 50 Hz)	δ (° at f = 50 Hz)
-0.12
⋮
0.0
⋮
0.12

Note

The maximum steady-state load angle is 90°. If the display gives a value above this, ignore the reading. This happens because the current space vector's angle is computed from currents measured by the ADCs connected to the current sensors. These ADCs generally have offsets that cause the measured values to differ slightly from the actual values. This causes the computed space-vector position to differ slightly from the actual current space-vector position.

Reduce the load torque back to 0 Nm and the frequency back to 0 Hz. Repeat the previous two steps for 40 Hz and 50 Hz.
Click the button to stop the model.
Turn OFF the DC power supply.

This is the torque-load-angle characteristic of the PMAC motor. In the next part of the experiment, a closed-loop speed controller with an inner hysteresis current-control loop is implemented to maintain a constant speed and a load angle of π/2 for maximum efficiency. The speed-controller design is similar to that of the DC motor speed controller because the per-phase equivalent circuit is the same. The model is run in simulation using a mathematical model of a PMAC motor, and then the same model is run in real time and the results are observed. The details of the motor-model design are part of the advanced electric drives lab and are not discussed here. For further reading, refer to Vector control of PMSM.

Simulation of closed-loop speed control

Open Workbench and pin the Explorer and Properties docks.
Navigate to and open the PMHysteresisControl project file in the Experiment6\PMCurrentControl folder, usually found in the following location: C:\Program Files (x86)\Sciamble\WorkBench v1\Examples\CUSPLab\BasicDrives.
Expand the project and open the Hysteresis model file shown below:

The top section of the model consists of the PMAC motor model, preceded by the inner hysteresis current controller, which is in turn preceded by the outer speed PI controller. The bottom section of the model contains the DC generator model, with a PI controller that regulates the current and thereby controls the load torque. The motor reference speed is gradually increased from 0 to 150 rad/s at t = 1 s. At t = 3 s, the DC generator reference current is stepped to 0.2 A to generate a load torque of 0.04 Nm.
Ensure that is NOT pressed. Click the button.
Observe the PMAC motor speed and stator current response.

Real-time closed-loop speed control

Open the project properties by double-clicking the project node in Explorer, and set Start model/Function to RotorInit.
Turn ON the DC power supply and set the voltage to 26 V. Hysteresis control adjusts the output current by rapidly switching between the DC voltage levels. The smaller the difference between these levels, the lower the switching rate.

Note
The DC power-supply voltage must be set to 26 V, not 40 V.
Ensure that the button is pressed. Click the button.
After about 5 s, click the button to stop the model.
Reset the Start model/Function to Hysteresis.
Ensure that the button is pressed. Click the button.
After about 10 s, click the button to stop the model.
Observe the motor speed.
Turn OFF the DC power supply and disconnect all connections, including the USB cable.

This concludes the experiment on torque-angle characterization, open-loop V/f control, and closed-loop hysteresis speed control of the PMAC motor. In most real-world applications, vector control of PMAC motors is used instead of the hysteresis control covered in this experiment because the latter produces high harmonic content in the stator current. These increased harmonics cause greater torque pulsations, audible noise and vibration, and thermal losses. In addition, hysteresis control requires the power electronics to handle a very high switching frequency that varies inversely with the motor leakage inductance. Thus, it is especially unsuitable for motors with very low time constants, such as the one used here. Vector control of a PMAC motor is covered in the advanced drives lab. For further reading, refer to Vector control of PMSM.

Lab report and reading assignment

Plot the result of T_load (y-axis) as a function of the load angle δ (x-axis) for 30, 40, and 50 Hz under open-loop V/f control. Mark the regions in which the PMAC motor is in motoring mode and generation mode.
What was the load angle for all three frequencies at 0 Nm load torque? If it was nonzero, explain why.
At what load torque does the load angle become 0? What is the PMAC motor's electromagnetic torque at this point?
Attach a plot of ω_mech under open-loop speed control for all the different frequencies.
What is the steady-state load angle δ under the closed-loop hysteresis current control used in this experiment? Explain why this value was chosen.
Attach plots of ω_mech under closed-loop speed control for step changes in reference speed and load torque, from both simulation and real-time operation.
Attach a plot of the PMAC motor's 3Φ stator currents under closed-loop speed control from the simulation. Zoom in on the interval from 2.95 to 3.05 s.
What would happen if a PMAC motor were connected directly to its rated voltage at the rated frequency, starting from standstill?