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Electric and Hybrid Drive Systems homework

HW3PPMSMBLDCmath2019.docx

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ECE 4/595 Electric Drive Systems Prof Ka C Cheok

Homework Assignment #3

Controlling the Stator Field of PMSM[footnoteRef:1] and FOC BLDC[footnoteRef:2] Motors [1: Permanent magnet phase synchronous ] [2: Field oriented controlled brushless direct current ]

Issued: 23 May ’19 Due: 30 May ‘19

3-phase Permanent Magnet Synchronous Motor (PMSM)

PMSM component & math model

A 3-phase inverter 3-phase PMSM with a mechanical load consists of the following components.

Variables

or desired stator voltage (software)

, 3-phase input (software) to 3-phase power inverter

, Output voltages of 3-phase power inverter

Stator currents

Back emf’s (hardware)

Voltage of the common node if the Y circuit

or Effective stator current

or Effective magnetic motive force (mmf)

Angle, speed and acceleration of effective mmf

Rotor angle, speed and acceleration

Rotor magnetic flux density.

Electromechanical torque generated between the rotor and stator

External torque experienced by the rotor

Parameters

Resistance and inductance in the branch of stator Y circuit

Number of turns in a stator winding. E.g..

Effective radius & length of stator winding (cylindrical shape)

Moment of inertia & damping coefficient of mechanical load

Peak value of the flux density in the rotor

Sequence of actions for PMSM

Steps a. through j. give you an understanding dynamical behavior of a PMSM. Let’s take a walk through it.

a.

Specify a desired stator voltage or , where . (Software)

b.

Use Inverse Clarke Transform to convert into a 3-phase voltage, where. (Software).

c.

Convert into pulse width modulation (PWM) duty cycles (%) that switch a 3-phase power inverter, whose PWM output voltages are capable of delivering high power (voltage current). E.g., 5Vdc 3A to 300Vdc 100 A. (Hardware)

d.

The drive voltage generates a stator current , obeying the electrical circuit law with back emf included as a feedback. (Hardware)

e.

The Clark transform phenomenon that converts to an effective stator current or . (Hardware)

f.

The magnetic motive force (mmf) would be generated in the stator winding. or . (Hardware)

g.

The rotor is a cylindrical permanent magnet that rotates to chase after the stator field. The rotor magnetic flux density is denoted by, which is a vector oriented according to rotor angle. is the peak value of the flux density.

h.

The torque generated between and is , where are the effective radius and length of the cylindrical stator/rotor shape.

i.

spins the mechanical shaft, gear and wheel and overcomes external load to produce rotor acceleration , rotor speedand rotor angle .

j.

and generates back emfs which acts as feedback generators in the stator circuit.

Homework Problem Set 1 – PMSM

Problem 1. A desired stator voltage vector volt is issued. What is the magnitude and the orientation angle of the vector? Sketch these relationships clearly on the axes provided to the right.

Problem 2. Convert into a 3-phase voltage vector, where all the voltages must be nonnegative, i.e., . Hint: Inverse Clarke Transform.

.

Problem 3. The stator current is Amps. What is the effective field oriented stator current ? Sketch the results on the axes on the right blank diagram. Hint: Clarke Transform.

Problem 4. Suppose we know that , and , what is the torque generated between the stator and the rotor?

Problem 5. Based on the diagram shown on the first page of this notes, complete the differential equations for the mechanical dynamics below.

Problem 6. Suppose we know that , , and , what is the back emfs generated in the stator by the rotor?

Problem 7. Based on the diagram shown at the beginning of this notes, complete the differential equations for the electrical stator circuits below. Note we have redundant notations:.

3-phase FOC BLDC Motor

BLDC Motor component & math model

A 3-phase field oriented controlled brushless DC (BLDC) motor is an extension of PMSM with additional software (see bottom half of diagram below) that orients the effective stator current so that the stator fieldis always perpendicular to the rotor field .

Additional variables

Measurements of stator currents

Calculated axis stator current based on measurements

Measurement of rotor angle

Direct & quadrature currents, projected by onto rotor axes.

Desired reference for direct & quadrature currents

Control values on theanalysis on the rotor shaft

Control values that are components on the axes.

Additional parameters

& Current control loop compensators or controller transfer functions

Proportional gains associated with current controller

Integral gains associated with current controller

Sequence of actions for BLDC motor

Steps k through o are the additional software formulas that implements the current feedback loop to ensure that the stator fieldis perpendicular to the rotor field

k.

Compute the field oriented stator current from measurement using the Clarke Transformation.

l.

Compute the dq current which represent the perpendicular components (projections or shadows) of onto the rotor shaft axes.

a. This matrix-vector multiplication is called the Park Transform.

b.

is the component (shadow) of along the direct (0o) direction of the rotor angle .

c.

is the component (shadow) of along the quadrature (90o) direction of the rotor angle .

m. Specify values for desired dq currents as follows:

d.

. For the BLDC motor, we would like to make zero or a small insignificant value.

e.

value representing the magnitude of the stator field current, and hence the stator magnetic field strength. E.g., or would cause the stator field current to point different directions with different magnitudes.

n.

Compute desired dq voltage , where & are current control loop compensator or controller transfer functions. E.g., a P-controller or a PI controller . values are components on rotor shaft axes.

o.

Compute desired voltage , which is then applied to Step a. The matrix-vector multiplication is called the Inverse Park Transformation. values are components on the axes and close the loop for the BLDC motor.

Problem Set 2 – BLDC Motor

Problem 8. The stator currents was measured and found to be A. Complete the formulate for calculating the effective field oriented stator current..

Problem 9. Suppose we know that and the rotor angle is . What is the direct-quadrature representation?

Problem 10. A P-controller with gain were to be used to compute the desired dq voltage. Suppose that the dq-current was computed to be and it is desired that . What is the value of ?

Problem 11. Why do we normally set in the current loop controller of the previous problem?

Problem 12. If and, what would be the stator voltage that would be applied to complete the control loop for the BLDC motor?

Problem 13: Many application notes and references such as

http://cdn.shopify.com/s/files/1/0115/3752/files/IndustrialVC_grande.jpg?1609

or a search for “images of field oriented control for BLDC motor” often reveal a diagram such as the following. Identify and relate the blocks with the problems for this assignment.

Problem………..?

Problem………..?

Problem………..?

Problem………..?

Problem………..?

Problem………..?

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