Dr. Amr AbdAllah 1

Electric Machines IIIA

COURSE EPM 405A

FOR

4th Year Power and Machines

ELECTRICAL DEPARTMENT

Lecture 03

Dr. Amr AbdAllah 2

Induction Machine Modeling Review of Magnetically coupled circuits:

Magnetically coupled electric circuits are central to the operation of transformers and electric machines.

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Induction Machine Modeling Review of Magnetically coupled circuits:

In case of transformers, stationary circuits are magnetically coupled for the purpose of changing the voltage and current levels.

In the case of electric machines circuits in relative motion are magnetically couples for the purpose of transferring energy between mechanical and electrical systems.

Review of Magnetically coupled circuits:

The flux linking each coil may be expressed as:

Because it is assumed that flux 1 links the equivalent turns of coil 1 (N1) and flux 2 links the equivalent turns of coil 2 (N2), then the flux linkages may be written as:

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Induction Machine Modeling

Review of Magnetically coupled circuits:

If saturation is neglected, the system is linear and the fluxes may be expressed:

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Induction Machine Modeling

Dr. Amr AbdAllah

Review of Magnetically coupled circuits:

Since and

Thus Self inductances

and

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Induction Machine Modeling

Dr. Amr AbdAllah

Review of Magnetically coupled circuits:Relating the magnetizing inductances of coil 1 and coil 2 we get:

It is obvious that L12=L21. The mutual inductances may be related to the magnetizing inductances as follows:

The flux linkage now can be written as:

where

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Induction Machine Modeling

Dr. Amr AbdAllah

Review of Magnetically coupled circuits:

Referring coil 2 variables to coil 1, we have:

Since power has to be unchanged by this substitution, that is:

then:

The flux linkage (voltage-second) is treated the same way as the voltage, that is:

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Induction Machine Modeling

Dr. Amr AbdAllah

Review of Magnetically coupled circuits:

Substituting the relations , , in

We can show using ( for Lm2) that:

where

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Induction Machine Modeling

Dr. Amr AbdAllah

Review of Magnetically coupled circuits:

The voltage equation become:

where

Suggest T equivalent circuit with coil 1 as reference

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Induction Machine Modeling

Dr. Amr AbdAllah

Inductances of Induction machine

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Induction Machine Modeling

Dr. Amr AbdAllah

Inductances of induction machine: Using the concepts introduced for the magnetic coupling of the

stationary 2 coil device, the self and mutual inductances of the different machine windings can be deduced.

The mutual inductances between the different phases is decided according to the angle between the magnetic axes of these windings.

In general we can write:LAB is maximum when the two magnetic axes are coinciding (i.e.

A=B) and is zero when the magnetic axes are 90° apart from each other.

L0 is the maximum value of the mutual inductance.

s

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Induction Machine Modeling

Dr. Amr AbdAllah

Inductances of induction machine: Self inductances of stator and rotor windings are

independent on rotor position r:In order to obtain the self inductance of a winding substitute

A=B , to get the so called useful self inductance and then add the leakage inductance of the winding. For phase a of the stator winding we have:

Similarly for the b and c phases of the stator windings we have:

For the rotor windings

While referred to stator:

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Induction Machine Modeling

Dr. Amr AbdAllah

Inductances of induction machine: Mutual inductances between stator windings are

independent on rotor position r:The mutual inductance between a and b phases of the

stator is given as:

Similarly:

Mutual inductances between rotor windings are independent on rotor position r:

and

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Induction Machine Modeling

Dr. Amr AbdAllah

Inductances of induction machine: Mutual inductances between stator and rotor windings

are dependent on rotor position r:The mutual inductance between a and b phases of the

stator is given as:

Similarly:

It can also be shown that:

Student assignment

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Induction Machine Modeling

Dr. Amr AbdAllah

Voltage equations of induction machine:The voltage equations in machine variables may be expressed as:

Where

The flux linkages can be expressed as:

Example:

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Induction Machine Modeling

Dr. Amr AbdAllah

Voltage equations of induction machine:

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Induction Machine Modeling

Dr. Amr AbdAllah

Voltage equations of induction machine:When expressing the voltage equations in machine variable

form it is convenient to refer all rotor variables to stator windings with appropriate turns ratio (Ns/Nr):

Since the magnetizing and mutual inductances are associated with the same magnetic flux path, then we can define:

since

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Induction Machine Modeling

Dr. Amr AbdAllah

Voltage equations of induction machine:Also since:

then defining

leads to:

where

The voltage equations expressed in terms of machine variables referred to stator may be written as:

where

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Induction Machine Modeling

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NEXT LECTURE

CHAPTER II (CONT’D)Symmetrical Induction Machine Models