lecture 17 synchronous machines (1)...synchronous machines (1) acknowledgment-these handouts and...

Copyright © 2017 Hassan Sowidan

ECE 330

POWER CIRCUITS AND ELECTROMECHANICS

LECTURE 17

SYNCHRONOUS MACHINES (1)

Acknowledgment-These handouts and lecture notes given in class are based on material from Prof. Peter

Sauer’s ECE 330 lecture notes. Some slides are taken from Ali Bazi’s presentations

Disclaimer- These handouts only provide highlights and should not be used to replace the course textbook.

4/9/2018


SYNCHRONOUS MACHINES

• The main element in terms of generation of power.

• Range all the way from a few MVA to 1100 MVA.

• Can be operated as either a generator or motor.

• The large majority of applications are as generators.

• The three-phase generators have an AC winding on

the stator with a wye-connection.

• The rotor is excited by a DC field winding

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• It is connected to the prime movers, such as steam or

hydro-turbine.

• As motors, synchronous machines are less used

except at low power levels such as permanent

magnet synchronous motors (PMSM).

• In certain cases, synchronous machines at a high

rating are operated to act as power factor correcting

devices. 4/9/2018 3



• We will discuss the fundamental concepts of

deriving torque expressions and only the sinusoidal

steady-state operation using the equivalent circuit.

• For a proper understanding of a three-phase

machine, we will motivate it via the single- and two-

phase machines.

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Source: emadrlc.blogspot.com

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Source: pelectric.blogsky.com

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SALIENT POLE

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ROUND ROTOR

• Source: en.partzsch.de

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SINGLE-PHASE ROTATING MACHINE

The fundamental component of mutual inductance

will be .

In practical machines, the number of turns are

so positioned on stator and rotor that the higher

harmonics are minimized and the mutual inductance is

largely due to this fundamental component.

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0 1 2 /s rN N L cosM



The winding, instead of being concentrated, is

distributed.

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• We have already seen flux linkage derivation of

single-phase machines:

2

0 0

2

0 0

21 ( )

21 ( )

( ) ( ) cos( )

s s s s r r s s sr r

r r r s r s r r rs s

sr rs

N L i N N L i L i L i

N L i N N L i L i L i

L L M

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• The co-energy and torque are:

' ' ' '

0 0

2 2

' ( ,0, ) ( , , )

1 1( )

2 2

( )'sin( )

s ri i

m s s s r s r r

s s r r sr s r

e srms r s r

W i di i i d i

L i L i L i i

LWT i i i i M

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• The electrical differential equations are:

ss s s

rr r r

dv i R

dt

dv i R

dt

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• The mechanical differential equation is:

2

2

e

m

d dJ K B T T

dt dt

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Under sinusoidal excitation, the power becomes:

cos( )

cos( )

cos( )cos( )sin( )

( .)

cos( )cos( )sin( )

s s s

r r s

e e

m m m s r s r

m

m m s r s r m

i I t

i I t

dP T T I I M t t

dt

t is some arbitary const

P I I M t t t

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• Power can also be expressed as:

where:

1 2

3 4

sin ( ) sin ( )4

sin ( ) sin ( )m m s r

t tP I I M

t t

1 2

3 4

,

,

m s r m s r

m s r m s r

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Since a sinusoidal function can have no average value

Pm can have an average value only if for

, or 4, i.e.,

• If ,

A necessary condition for average power is that one of

the ’s is zero, and a sufficient condition is that

.

m s r 2 0

( )

sin( )

4

m s rm av

I I MP

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= 0i

= 1,2,3i =m s r

i sin 0


TWO-PHASE ROTATING MACHINE

There is average power when there is pulsating torque

due to other ’ s.

To eliminate this, we can have a two-phase machine.

In the two-phase machine, there is an additional

winding on both the stator and the rotor. The two-

phase machine creates a rotating magnetic field.

i

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• We look at the stator magnetic field of one- and two-

phase machines.

• With sinusoidal excitation, no rotating field as it is

• always maximum at ψ = 90o:

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π

H

ψ2π-π-2π 0

Nsis/(2g)

-Nsis/(2g)

Ns, is

Hψ

sin( )2

s sN iH

g

sin( )cos( )2

s ss

N IH t

g



• Two-phase machine: Rotating magnetic field!

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π

Hb

ψ2π-π-2π 0

Nsibs/(2g)

-Nsibs/(2g)

π

Ha

ψ3π-π 0

Nsias/(2g)

-Nsias/(2g)

2 223π2

Ns, ibs

Ha,Hbψ

Ns, ias



4/9/2018 21

cos ( ) sin ( )2 2

: cos , sin

cos( )cos( ) sin( )sin( )2

s as bs

s as s bss

as s s bs s s

s ss s

H H H

N i N iH

g g

Assume i I t i I t

N IH t t

g



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cos( )2

Revolving magnetic field

at 0, 0

,

Re

s ss

s

N IH t

g

t peak is at

t t peak is at t

volves continue clockwise

lecture 17 synchronous machines (1)...synchronous machines (1) acknowledgment-these handouts and...

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