Model the dynamics of a three-phase permanent magnet synchronous machine with sinusoidal or trapezoidal back electromotive force, or the dynamics of a five-phase permanent magnet synchronous machine with sinusoidal back electromotive force

Fundamental Blocks/Machines

The Permanent Magnet Synchronous Machine block operates in either generator or motor mode. The mode of operation is dictated by the sign of the mechanical torque (positive for motor mode, negative for generator mode). The electrical and mechanical parts of the machine are each represented by a second-order state-space model.

The sinusoidal model assumes that the flux established by the permanent magnets in the stator is sinusoidal, which implies that the electromotive forces are sinusoidal.

The trapezoidal model assumes that the winding distribution and flux established by the permanent magnets produce three trapezoidal back EMF waveforms.

The block implements the following equations.

These equations are expressed in the rotor reference frame (qd frame). All quantities in the rotor reference frame are referred to the stator.

$$\frac{d}{dt}{i}_{d}=\frac{1}{{L}_{d}}{v}_{d}-\frac{R}{{L}_{d}}{i}_{d}+\frac{{L}_{q}}{{L}_{d}}p{\omega}_{m}{i}_{q}$$

$$\frac{d}{dt}{i}_{q}=\frac{1}{{L}_{q}}{v}_{q}-\frac{R}{{L}_{q}}{i}_{q}-\frac{{L}_{d}}{{L}_{q}}p{\omega}_{m}{i}_{d}-\frac{\lambda p{\omega}_{m}}{{L}_{q}}$$

$${T}_{e}=1.5p[\lambda {i}_{q}+({L}_{d}-{L}_{q}){i}_{d}{i}_{q}]$$

L | q and d axis inductances |

R | Resistance of the stator windings |

i | q and d axis currents |

v | q and d axis voltages |

ω | Angular velocity of the rotor |

λ | Amplitude of the flux induced by the permanent magnets of the rotor in the stator phases |

p | Number of pole pairs |

T | Electromagnetic torque |

The *L _{q}* and

$${L}_{ab}={L}_{d}+{L}_{q}+\left({L}_{q}-{L}_{d}\right)\mathrm{cos}\left(2{\theta}_{e}+\frac{\pi}{3}\right)$$

Θ_{e} represents the electrical angle.

The next figure shows the variation of the phase to phase inductance in function of the electrical angle of the rotor.

For a round rotor, there is no variation in the phase inductance.

$${L}_{d}={L}_{q}=\frac{{L}_{ab}}{2}$$

For a salient round rotor, the dq inductances are given by:

$${L}_{d}=\frac{\mathrm{max}({L}_{ab})}{2}$$

and

$${L}_{q}=\frac{\mathrm{min}({L}_{ab})}{2}$$

These equations are expressed in the rotor reference frame using an extended Park transformation (q1d1 and q2d2 frame). All quantities in the rotor reference frame are referred to the stator.

$$\frac{d}{dt}{i}_{d1}=\frac{1}{L}{v}_{d1}-\frac{R}{L}{i}_{d1}+\frac{{L}_{q}}{L}p{\omega}_{m}{i}_{q1}$$

$$\frac{d}{dt}{i}_{q1}=\frac{1}{L}{v}_{q1}-\frac{R}{L}{i}_{q1}-\frac{{L}_{d}}{L}p{\omega}_{m}{i}_{d1}-\frac{\lambda p{\omega}_{m}}{L}$$

$$\frac{d}{dt}{i}_{d2}=\frac{1}{L}{v}_{d2}-\frac{R}{L}{i}_{d2}$$

$$\frac{d}{dt}{i}_{q2}=\frac{1}{L}{v}_{q2}-\frac{R}{L}{i}_{q2}$$

$${T}_{e}=2.5p\lambda {i}_{q1}$$

L | Armature inductance |

R | Resistance of the stator windings |

i | q1 and d1 axis currents |

v | q1 and d1 axis voltages |

i | q2 and d2 axis currents |

v | q2 and d2 axis voltages |

ω | Angular velocity of the rotor |

λ | Amplitude of the flux induced by the permanent magnets of the rotor in the stator phases |

p | Number of pole pairs |

T | Electromagnetic torque |

These equations are expressed in the phase reference frame (abc
frame). Note that the phase inductance *L _{s}* is
assumed constant and does not vary with the rotor position.

$$\begin{array}{c}\frac{d}{dt}{i}_{a}=\frac{1}{3{L}_{s}}\left(2{v}_{ab}+{v}_{bc}-3{R}_{s}{i}_{a}+\lambda p{\omega}_{m}\left(-2{\Phi}_{a}^{\prime}+{\Phi}_{b}^{\prime}+{\Phi}_{c}^{\prime}\right)\right)\\ \frac{d}{dt}{i}_{b}=\frac{1}{3{L}_{s}}\left(-{v}_{ab}+{v}_{bc}-3{R}_{s}{i}_{b}+\lambda p{\omega}_{m}\left({\Phi}_{a}^{\prime}-2{\Phi}_{b}^{\prime}+{\Phi}_{c}^{\prime}\right)\right)\\ \frac{d}{dt}{i}_{c}=-\left(\frac{d}{dt}{i}_{a}+\frac{d}{dt}{i}_{b}\right)\\ {T}_{e}=p\lambda \left({\Phi}_{a}^{\prime}\cdot {i}_{a}+{\Phi}_{b}^{\prime}\cdot {i}_{b}+{\Phi}_{c}^{\prime}\cdot {i}_{c}\right),\end{array}$$

The electromotive force Φ^{'} is
represented by

and

L | Inductance of the stator windings |

R | Resistance of the stator windings |

i | a, b and c phase currents |

Φ | a, b and c phase electromotive forces |

v | ab and bc phase to phase voltages |

ω | Angular velocity of the rotor |

λ | Amplitude of the flux induced by the permanent magnets of the rotor in the stator phases |

p | Number of pole pairs |

T | Electromagnetic torque |

$$\begin{array}{c}\frac{d}{dt}{\omega}_{r}=\frac{1}{J}\left({T}_{e}-{T}_{f}-F{\omega}_{m}-{T}_{m}\right)\\ \frac{d\theta}{dt}={\omega}_{m,}\end{array}$$

| Combined inertia of rotor and load |

| Combined viscous friction of rotor and load |

| Rotor angular position |

| Shaft mechanical torque |

| Shaft static friction torque |

ω | Angular velocity of the rotor (mechanical speed) |

**Number of phases**Select between a three-phase machine model or a five-phase machine model. This parameter is disabled when the

**Back EMF waveform**parameter is set to`Trapezoidal`

, or when the**Rotor type**parameter is set to`Salient-pole`

.**Back EMF waveform**Select between the

`Sinusoidal`

and the`Trapezoidal`

electromotive force. This parameter is disabled when the**Number of phases**parameter is set to`5`

.**Rotor type**Select between the

`Salient-pole`

and the round (cylindrical) rotor. This parameter is disabled when the**Number of phases**parameter is set to`5`

or when the**Back EMF waveform**parameter is set to`Trapezoidal`

.**Mechanical input**Select the torque applied to the shaft, the rotor speed as a Simulink

^{®}input of the block, or to represent the machine shaft by a Simscape™ rotational mechanical port.Select

`Torque Tm`

to specify a torque input, in N.m., and change labeling of the block input to`Tm`

. The machine speed is determined by the machine Inertia J and by the difference between the applied mechanical torque Tm and the internal electromagnetic torque Te. The sign convention for the mechanical torque is when the speed is positive, a positive torque signal indicates motor mode and a negative signal indicates generator mode.Select

`Speed w`

to specify a speed input, in rad/s, and change labeling of the block input to`w`

. The machine speed is imposed and the mechanical part of the model (Inertia J) is ignored. Using the speed as the mechanical input allows modeling a mechanical coupling between two machines.The next figure indicates how to model a stiff shaft interconnection in a motor-generator set when friction torque is ignored in machine 2. The speed output of machine 1 (motor) is connected to the speed input of machine 2 (generator), while machine 2 electromagnetic torque output Te is applied to the mechanical torque input Tm of machine 1. The Kw factor takes into account speed units of both machines (pu or rad/s) and gear box ratio

`w2/w1`

. The KT factor takes into account torque units of both machines (pu or N.m) and machine ratings. Also, as the inertia J2 is ignored in machine 2, J2 referred to machine 1 speed must be added to machine 1 inertia J1.Select

`Mechanical rotational port`

to add to the block a Simscape mechanical rotational port that allows connection of the machine shaft with other Simscape blocks with mechanical rotational ports. The Simulink input representing the mechanical torque`Tm`

or the speed`w`

of the machine is then removed from the block.The next figure indicates how to connect an Ideal Torque Source block from the Simscape library to the machine shaft to represent the machine in motor mode, or in generator mode, when the rotor speed is positive.

**Preset Model**Provides a set of predetermined electrical and mechanical parameters for various permanent magnet synchronous motor ratings of torque (N.m), DC bus voltage (V), rated speed (rpm), and continuous stall torque (N.m).

The

**Preset Model**parameter is enabled only when the**Number of phases**parameter is set to`3`

, the**Back EMF waveform**parameter is set to`Sinusoidal`

, and the**Rotor type**parameter is set to`Round`

.Select one of the preset models to load the corresponding electrical and mechanical parameters in the entries of the dialog box. Select

`No`

if you do not want to use a preset model, or if you want to modify some of the parameters of a preset model.When you select a preset model, the electrical and mechanical parameters in the

**Parameters**tab of the dialog box become nonmodifiable (unavailable). To start from a given preset model and then modify machine parameters:Select the preset model that you want to initialize the parameters.

Change the

**Preset model**parameter value to`No`

. This does not change the machine parameters. It just breaks the connection with the particular preset model.Modify the machine parameters as you want, then click

**Apply**.

**Use signal names to identify bus labels**When this check box is selected, the measurement output uses the signal names to identify the bus labels. Select this option for applications that require bus signal labels to have only alphanumeric characters.

When this check box is cleared, the measurement output uses the signal definition to identify the bus labels. The labels contain nonalphanumeric characters that are incompatible with some Simulink applications.

**Stator phase resistance**The stator phase resistance Rs (Ω).

**Armature inductance (H)**The armature inductance of the sinusoidal model with round rotor (Ld is equal to Lq). This parameter is visible only when the

**Back EMF waveform**parameter is set to`Sinusoidal`

, and the**Rotor type**parameter set to`Round`

.**Inductances [Ld(H) Lq(H)]**The phase to neutral Ld (H) and Lq (H) inductances in the d-axis and q-axis of the sinusoidal model with salient-pole rotor. This parameter is visible only when the

**Back EMF waveform**parameter is set to`Sinusoidal`

, and the**Rotor type**parameter set to`Salient-pole`

.**Stator phase inductances Ls (H)**The stator-phase-to-neutral inductance Ls (H) of the trapezoidal model. This parameter is visible only when the

**Back EMF waveform**parameter is set to`Trapezoidal`

.**Specify**Lets you select the machine constant that you want to specify for block parameterization:

`Flux linkage established by magnets`

`Voltage Constant`

`Torque Constant`

Once you select a constant, you can enter its value in the appropriate parameter field, while the other two parameters become inaccessible.

**Flux linkage**The constant flux λ (Wb) per pole pairs induced in the stator windings by the magnets.

**Voltage constant**The peak line to line voltage per 1000 rpm. This voltage represents the peak open circuit voltage when the machine is driven as a generator at 1000 rpm.

**Torque constant**The torque per ampere constant. This constant assumes that the machine is driven by an inverter which provides a perfect synchronization between the current and the Back-EMF.

Sinusoidal model: A sine wave current is assumed (for more information, see

`ac6_example_simplified`

).Trapezoidal model: A square ware current is assumed (for more information, see

`ac7_example_simplified`

).

**Back EMF flat top area**The width of the flat top for a half period of the electromotive force Φ

^{'}(degrees) (only for trapezoidal machine).**Inertia, viscous damping, pole pairs, static friction**The combined machine and load inertia coefficient J (kg.m

^{2}), combined viscous friction coefficient F (N.m.s), pole pairs p, and shaft static friction*T*(N.m)._{f}If the

**static friction**parameter value is omitted or not specified, the block considers this value to be 0.**Initial conditions**Specifies the mechanical speed (rad/s), mechanical angle Θ

_{m}(degrees), and instantaneous stator current (A):Three-phase machine [w

_{m}, Θ_{m}, i_{a}, i_{b}]Five-phase machine [w

_{m}, Θ_{m}, i_{a}, i_{b}, i_{c}, i_{d}]Because the stator is wye-connected and the neutral point is isolated, the current i

_{c}in the three-phase machine is given by i_{c}= -i_{a}-i_{b}, and the current i_{e}in the five-phase machine is given by i_{e}= -i_{a}-i_{b}-i_{c}-i_{d}.

**Sample time (−1 for inherited)**Specifies the sample time used by the block. To inherit the sample time specified in the Powergui block, set this parameter to

`−1`

.**Rotor flux position when theta = 0**Lets you select the reference position of the rotor flux relative to the phase A axis.

Select

`Rotor 90 degrees behind phase A axis when theta = 0 (Modified Park)`

to choose the reference position of the rotor represented by:The modified Park transformation [4] is more convenient for vector control because the maximum phase induction occurs at theta = 0.

Select

`Rotor aligned with phase A axis when theta =0 (Original Park)`

to choose the reference position of the rotor represented by:

`Tm`

The Simulink input is the mechanical torque at the machine shaft. This input is normally positive because the Permanent Magnet Synchronous Machine block is usually used as a motor. If you choose to use the block in generator mode, you can apply a negative torque input.

`w`

The alternative block input (depending on the value of the

**Mechanical input**parameter) is the machine speed, in rad/s.- m
The Simulink output of the block is a vector containing measurement signals. The available signals depend on the model you selected. You can demultiplex these signals by using the Bus Selector block provided in the Simulink library.

Name

Definition

Units

Model

ias

Stator current is_a

A

all

ibs

Stator current is_b

A

all

ics

Stator current is_c

A

all

ids

Stator current is_d

A

5-Phase Sinusoidal

ies

Stator current is_e

A

5-Phase Sinusoidal

iqs

Stator current is_q

A

3-Phase Sinusoidal

ids

Stator current is_d

A

3-Phase Sinusoidal

iqs1

Stator current is_q1

A

5-Phase Sinusoidal

ids1

Stator current is_d1

A

5-Phase Sinusoidal

iqs2

Stator current is_q2

A

5-Phase Sinusoidal

ids2

Stator current is_d2

A

5-Phase Sinusoidal

vqs

Stator voltage Vs_q

V

3-Phase Sinusoidal

vds

Stator voltage Vs_d

V

3-Phase Sinusoidal

vqs1

Stator voltage Vs_q1

V

5-Phase Sinusoidal

vds1

Stator voltage Vs_d1

V

5-Phase Sinusoidal

vqs2

Stator voltage Vs_q2

V

5-Phase Sinusoidal

vds2

Stator voltage Vs_d2

V

5-Phase Sinusoidal

ea

Phase back EMF e_a

V

3-Phase Trapezoidal

eb

Phase back EMF e_b

V

3-Phase Trapezoidal

ec

Phase back EMF e_c

V

3-Phase Trapezoidal

ha

Hall effect signal h_a

^{*}logic 0-1

3-Phase, Sinusoidal and Trapezoidal

hb

Hall effect signal h_b

^{*}logic 0-1

3-Phase, Sinusoidal and Trapezoidal

hc

Hall effect signal h_c

^{*}logic 0-1

3-Phase, Sinusoidal and Trapezoidal

w

Rotor speed wm

rad/s

all

theta

Rotor angle thetam

rad

all

Te

Electromagnetic torque Te

N.m

all

The Hall effect signal provides a logical indication of the back EMF positioning. This signal is very useful to directly control the power switches. There is a change of state at each zero crossing of the phase to phase voltage. These signals must be decoded before being applied to the switches.

The Permanent Magnet Synchronous Machine block assumes a linear magnetic circuit with no saturation of the stator and rotor iron. This assumption can be made because of the large air gap usually found in permanent magnet synchronous machines.

When you use Permanent Magnet Synchronous Machine blocks in discrete systems, you might have to use a small parasitic resistive load, connected at the machine terminals, to avoid numerical oscillations. Large sample times require larger loads. The minimum resistive load is proportional to the sample time. Remember that with a 25 μs time step on a 60 Hz system, the minimum load is approximately 2.5% of the machine nominal power. For example, a 200 MVA PM synchronous machine in a power system discretized with a 50 μs sample time requires approximately 5% of resistive load or 10 MW. If the sample time is reduced to 20 μs, a resistive load of 4 MW is sufficient.

The `power_brushlessDCmotor`

example
illustrates the use of the Permanent Magnet Synchronous Machine block
in motoring mode with a closed-loop control system built entirely
with Simulink blocks. The complete system includes a six step
inverter block from the Simscape Power Systems™ library. Two control
loops are used; the inner loop synchronizes the pulses of the bridge
with the electromotive forces, and the outer loop regulates the motor's
speed, by varying the DC bus voltage. The mechanical torque applied
at the motor's shaft is originally 0 N.m (no load) and steps to its
nominal value (3 N.m) at t = 0.1 second.

Set the simulation parameters as follows:

Type: Fixed-step

Integrator type:

`Runge-Kutta`

,`ode4`

Sample time: 5e-6 (set automatically by the

**Model properties**)Stop time:

`0.2`

Set the **Flux distribution** parameter
to `Trapezoidal`

and run the simulation to
observe the motor's torque, speed, and currents. Change the **Back EMF flat top area parameter** of the trapezoidal
model from 120 to 0 and observe the waveform of the electromotive
force e_a.

The torque climbs to nearly 28 N.m and stabilizes rapidly to its reference value. The nominal torque is applied at t = 0.1 second and the controller reacts rapidly and increases the DC bus voltage to produce the required electric torque. Observe the saw tooth shape of the currents waveforms. This is caused by the six step controller, which applies a constant voltage value during 120 electrical degrees to the motor. The initial current is high and decreases during the acceleration to the nominal speed. When the nominal torque is applied, the stator current increases to maintain the nominal speed. The saw tooth waveform is also observed in the electromotive torque signal Te. However, the motor's inertia prevents this noise from appearing in the motor's speed waveform.

When the **Back EMF flat top area** parameter
of the trapezoidal model is changed from 120 to 0, the model reacts
exactly like the sinusoidal model. The electromotive force e_a is
purely sinusoidal and the torque ripple is less than the previous
case. The sinusoidal model requires a larger current to produce the
same torque. That reason is why the trapezoidal machine is used in
high torque applications, and the sinusoidal machine in precision
applications.

[1] Grenier, D., L.-A. Dessaint, O. Akhrif,
Y. Bonnassieux, and B. LePioufle. "Experimental Nonlinear Torque
Control of a Permanent Magnet Synchronous Motor Using Saliency." *IEEE ^{®} Transactions
on Industrial Electronics*, Vol. 44, No. 5, October 1997,
pp. 680-687.

[2] Toliyat, H.A. "Analysis and Simulation
of Multi-Phase Variable Speed Induction Motor Drives Under Asymmetrical
Connections." *Applied Power Electronics Conference
and Exposition*, Vol. 2, March 1996, pp. 586-592.

[3] Beaudart, F., F. Labrique, E. Matagne,
D. Telteux, and P. Alexandre. "Control under normal and fault
tolerant operation of multiphase SMPM synchronous machines with mechanically
and magnetically decoupled phases." *International
Conference on Power Engineering, Energy and Electrical Drives*,
March 2009, pp. 461-466.

[4] Krause, P.C., O. Wasynczuk, and S.D. Sudhoff. *Analysis
of Electric Machinery and Drive Systems*. IEEE Press, 2002.

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