Dr. Amr AbdAllah 1

Electric Machines IIIA

COURSE EPM 405A

FOR

4th Year Power and Machines

ELECTRICAL DEPARTMENT

Lecture 02

Dr. Amr AbdAllah 2

Reference Frame Theory Equations of Transformation: Change of Variables:

where

It can be shown that the inverse transformation is given by:

or

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Reference Frame Theory

Stationary Circuit Variables transformed to the arbitrary Reference Frame

Resistive Elements:

For a three phase resistive circuit we have: …………(2.1)Applying the arbitrary transformation

Then we have:

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Reference Frame Theory

Resistive Elements cont’d:

Multiplying by Ks from left, we get

…………(2.2)

Which means that the resistance matrix in the new arbitrary reference frame is given by:

…………(2.3)

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Reference Frame Theory

Resistive Elements cont’d:All power system equipment (transformers, capacitors banks, transmission lines, etc.) as well as the

stator windings of machines are designed to have same resistance. The resistance matrix in the abc frame is thus given as:

Which is the same resistance matrix in the new arbitrary reference frame since

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Reference Frame Theory

Resistive Elements cont’d:Conclusion is that the resistance matrix associated with the arbitrary reference frame is equal to

the resistance matrix associated with the actual variables if all phases are symmetrical (equal resistance).

For the unsymmetrical case it can be shown that the resistance matrix associated with the arbitrary reference variables will contain sinusoidal functions of except when equals zero (that is is zero or constant).

This means that the transformation yields constant resistances for the unbalanced case only if the reference frame is fixed to the actual frame (abc) where the unbalance physically exists.

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Reference Frame Theory

Stationary Circuit Variables transformed to the arbitrary Reference Frame

Inductive Elements:

For a three phase inductive circuit we have: …………(2.4)Where abcs is the vector of flux linkages in the

actual (abc) frame. p is the differential operator d/dt. For linear systems the flux linkages is the product of the inductance and current matrices. For the ac machines it will be noticed that the inductance matrix is function of the rotor position.

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Reference Frame Theory

Stationary Circuit Variables transformed to the arbitrary Reference Frame

Inductive Elements:Applying the arbitrary transformationthen we have:

Multiplying by Ks from left, we get

…………(2.5)

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Reference Frame Theory

Stationary Circuit Variables transformed to the arbitrary Reference Frame

Inductive Elements:

Therefore: …………(2.6) (Student Assignment)

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Reference Frame Theory

Stationary Circuit Variables transformed to the arbitrary Reference Frame

Inductive Elements:

Substituting (2.6) in (2.5)We get: …………(2.7)

Where

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Reference Frame Theory

Stationary Circuit Variables transformed to the arbitrary Reference Frame

Inductive Elements:(2.7) is expanded as:

The terms containing the speed (angular velocity of the arbitrary reference frame) are referred to as “Speed voltage”. It is clear that for =0 (that is the reference frame attached to the frame of the physical circuit) these terms don’t exist.

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Reference Frame Theory

Stationary Circuit Variables transformed to the arbitrary Reference Frame

Inductive Elements:For linear Magnetic circuits the flux linkage may be

expressed as:

Where Ls is the inductance matrix of the phases in the actual frame (abc) where the circuit physically exist.

The flux linkage in the arbitrary reference frame can thus be written as:

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Reference Frame Theory

Stationary Circuit Variables transformed to the arbitrary Reference Frame

Inductive Elements:For a balanced inductive circuit without inductive

magnetic coupling the inductive matrix associated with the actual reference frame of the physical circuit is a diagonal matrix with equal diagonal elements (and zero off diagonal elements) and accordingly the inductive matrix associated with the arbitrary reference frame is given as:

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Reference Frame Theory

Stationary Circuit Variables transformed to the arbitrary Reference Frame

Example: Figures a & b demonstrate thetransmission line and stator windings of electricmachines. Both circuits encounter resistive inductive circuit with mutual coupling between phases. It will be noticed that the stator windings of a symmetrical induction machine or a synchronous machine will have mutualinductance given by M=-0.5Lms and self

Inductance of the winding given byLs=Lls+Lms

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Reference Frame Theory

Stationary Circuit Variables transformed to the arbitrary Reference Frame

Example Cont’d: The RL balanced circuit is defined by:

Each phase voltage can be expressed as the Sum of the voltages across each element:

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Reference Frame Theory

Stationary Circuit Variables transformed to the arbitrary Reference Frame

Example Cont’d: Similarly in the arbitrary reference frame we have:

Since rs is a diagonal matrix with balance

Resistance in each phase then we get:

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Reference Frame Theory

Stationary Circuit Variables transformed to the arbitrary Reference Frame

Example Cont’d: The new inductance matrix in the arbitrary reference

frame can be calculated as: (Student Assignment)

Therefore the voltage equations in the arbitrary reference frame may be expressedas:

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Reference Frame Theory

Stationary Circuit Variables transformed to the arbitrary Reference Frame

Example Cont’d: The flux linkages are given as function of the new current

variables in the arbitrary reference frame by:

The figures below show the equivalent circuits which describes the voltage equations of the q, d and 0 axes

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Reference Frame Theory

Stationary Circuit Variables transformed to the arbitrary Reference Frame

Capacitive Elements:

For a three phase Capacitive circuit we have: …………(2.8)Which yield to:

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Reference Frame Theory

Stationary Circuit Variables transformed to the arbitrary Reference Frame

Capacitive Elements:Applying 2.6:

We get:

…………(2.9)

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Reference Frame Theory

Stationary Circuit Variables transformed to the arbitrary Reference Frame

Capacitive Elements:2.9 in expanded form results in:

The terms containing the speed (angular velocity of the arbitrary reference frame) are referred to as “Speed current” applying same terminology used for the inductive circuits.

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Reference Frame Theory

Stationary Circuit Variables transformed to the arbitrary Reference Frame

Capacitive Elements:For linear capacitive circuits the relation between the

charge and voltage is given as:

Where Cs is the capacitance matrix of the phases in the actual frame (abc) where the circuit physically exist.

The charge in the arbitrary reference frame can be written as:

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Reference Frame Theory

Commonly used Reference Frames

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Reference Frame Theory

Transformation between Reference FramesIn some derivations and analysis it is convenient to relate

variables in one frame to variables in another reference frame directly.

Let x denote the reference frame from whichthe variables are being transformed and y denote the reference frame to which the variables are being transformed; then

…………(2.10)

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Reference Frame Theory

Transformation between Reference FramesWe have, …………(2.11) Substitute 2.11 in 2.10 to get: …………(2.12)Also we have …………(2.13)Comparing 2.12 and 2.13 it can be shown that: …………(2.14)From which: …………(2.15)

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Reference Frame Theory

Transformation between Reference FramesThe desired transformation is obtained by substituting the

appropriate transformations into 2.13 hence:

(Student Assignment)

It can also be shown that:

(Student Assignment)

This transformation which is usually referred

to as Vector rotator is also obtainable by the

resolving of fqsx and fdsx (first row) into fqsy

and fdsy (second row).

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NEXT LECTURE

CHAPTER IISymmetrical Induction Machine Models