lecture 15 electromechanical systems (1) 1… · sauer’s ece 330 lecture notes. some slides are...

Copyright © 2017 Hassan Sowidan

ECE 330

POWER CIRCUITS AND ELECTROMECHANICS

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.

10/8/2017

LECTURE 15

ELECTROMECHANICAL SYSTEMS (1)


ELECTROMECHANICAL SYSTEMS

• Focus on magnetic circuits in which

there is a moving member.

• It can be either a translational or rotational system.

• Source: clrwtr.com Source: stackexchange.com

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The purpose is to present techniques of systematic

derivation of mathematical models (differential

equations) for lumped-parameter electromechanical

systems. The theory is outlined for N electrical coils

and M mechanical variables.

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• A set of electrical coils that interact to produce the

force or torque on the mechanical system.

• The force or torque of electric origin will be derived

from fundamental principles of stored energy in the

magnetic field.

• both the currents in the electrical coils and the force

or torque are time varying.

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• A basic electromechanical system has an electrical

part, a mechanical part, and electromechanical

coupling. This coupling is assumed to be lossless.

• Systems with linear motion see a force f and a

displacement x, while rotational systems see a torque

T and a displacement θ.

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v, i f, x


EXAMPLES

• Linear(translational) motion:

• Rotational motion:

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I

Ngxlp μ

μ

Xo

Core mean length: lc

Core area: Ac

Coil

Stator

Rotor


FLUX LINKAGE AND VOLTAGE

• The flux linkage λ can be a function of the

displacement x and the current i.

• The voltage becomes:

• If λ=L(x) i,

• If L(x)=L= constant,

10/8/2017 7

.d di dx

vdt i dt x dt

( )( ) .

d di L x dxv L x i

dt dt x dt

.d di

v Ldt dt


MULTI-PORT SYSTEMS

• The force of electric origin at the mechanical ports

can be functions of all the variables.

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v1, i1 f1, x1

vN , iN fM , xM

1 2 1 2,.....( , , ,..... , ), 1.....k k N Ni i i x x x k N

1 1

, 1,....NN M

j jk k kk

j jj j

di dxdv k

dt i dt x dt

1 2 1 2,( , ,..... , , ..... )e e

i i N Mf f i i i x x x


LINEAR SYSTEM EXAMPLE

Basic configuration is used for tripping circuit

breakers, operating valves, etc. It consists of:

• Fixed structure made of highly permeable material.

• A movable plunger of non magnetic sleeve.

,

• H exist only in the air gap

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N

g

x

w

g

2w

H1

H3

H2

i

,

=B H =

= 0H in the iron


LINEAR SYSTEM EXAMPLE

• Ampere’s loops:

d is the depth of the structure

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N

g

x

w

g

2w

H1

H3

H2

i

3 1

2 1

1 1 3 3 2 2

1 2

1 2

2 2

0

0

2 2 0

2(2 )

o o

oo

H g H g

H x H g N i

B A B A B A

H wd wd H

NiH H

g x

wd NiH wd

g x


LINEAR MOTOR EXAMPLE

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2

2

2 2

2

2( )

( )( )

2 2

o

o o

N

L i

w d NL x

i g x

di dL x dxv L x i

dt dx dt

w d N w d Ndi dxv i

g x dt dtg x

Transformer

voltage

Speed

voltage

This problem can be easily

solved using the magnetic

equivalent circuit !


READING MATERIAL

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• Reading material: Sections 4.1-4.3.

• Next time: Continue Section 4.3.

lecture 15 electromechanical systems (1) 1… · sauer’s ece 330 lecture notes. some slides are...

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