Electro-Mechanical Energy Conversion

Lecture 1

Consider the following plunger system

x

V

R

e

i

• The voltage e exists only under dynamic

conditions, ie when there is a change in current

(transformer voltage) or a change in air-gap resulting

in a change in inductance (mechanical voltage).

Otherwise e = 0.

•

dt

di

dt

dL

dt

λde

dt

dLi

dt

diL

dt

dLi

dtie diLidLi2

Energy supplied through the coil

Assume that initially that the air gap length is kept at x. A voltage is then impressed on the coil and the current allowed to grow until it stabilizes at value I. The stored energy is given by integrating above equation with no change in inductance

diLiWfld

2

21 IL

Now release the plunger so that it moves by a distance .

x

ANμ

I

NHAμ

I

φN

I

λL 2

00

x

Change in inductance

xx

ANμL

2

2

0

xSince is negative the change in inductance is positive (L increases).

Change in stored energy .

The total energy supplied by the coil when the plunger has moved by distance

iiLiLWfld2

21

x

diLidLi2dtie

iiLLi2or

Comparing change in stored energy and energy supplied it is seen that more than half of the supplied energy is added to the stored energy. The rest is used as the work done to move the plunger through distance .x

Difference between energy supplied and change in stored energy = 2

21 iL

2

21 iL x

dx

dLi2

21 xF

dx

dLiF 2

21

2

22

021

x

AINμ

AHBF21

dx

dF 2

21

It is easily verified that other expressions for F are:

Where is the reluctance of the air-gap.

Now consider a rotating device:

r

x

l

Effective area of air-gap = lr

x

lrN

x

ANL

22

20

20Then, inductance

If the rotor is rotated through an angle then

x

lrNL

2

20Change in inductance

(Note that the angle decreases. Thus is negative and the inductance is less).

iiLiLWfld2

21

As before change in stored energy

iiLLi2

And energy supplied by the coil when the rotor was rotated by an angle is

Difference between energy supplied and change in stored energy is equal to work done

2

21 iL θ

θd

dLi2

21

θTBut work done

θd

dLiT 2

21

x

lrINμ

2

22

021

The energy is actually extracted since is negative. The system acts as a generator supplying energy to the coil. The work done forces the rotor to move through an angle (against the reluctance torque).

d

d2

21

It can be verified that T can also be expressed as

It is important to note the following

Motor action:Electrical energy supplied mechanical energy + change in field storage

Generator action:Electrical energy delivered mechanical energy + change in field storage

The previous examples considered single excited system, ie only one coil. The resulting forces or torques are called reluctance forces and reluctance torques because they arise from the change of reluctance with gap distance or with angle

dx

d

d

d

Double Excited Systems

A double excited system has two energized coils. Most motors and generators are double excited. Consider the general case of two energized coils carrying currents I1, I2.

e1

e2

i1

i2

Emf equations for both coils are

)()( 2111 iMdt

diL

dt

de

)()( 1222 iMdt

diL

dt

de

To find the energy stored in the field let us first energize coil 1 up to current I1, with current in coil 2 = 0 (coils are stationary; hence inductances are constant).Stored energy

dtedtieWfld 02111

2

1121 IL

Next, we keep current in coil 1 fixed at I1 and energize coil 2 up to I2.

Total energy stored in field

dtiedtIeWfld 22112

2

2221

21 ILIIM

21

2

22212

1121

21IIMILILWWW fldfldfld

Now let coil 2 rotate by an angle . Any inductance which is position dependent will change with the angle. Hence the change in stored energy is:

Or

21

2

22212

1121 iiMiLiLWfld

21221211 iiMiLiiMiL

θiiθd

dMi

θd

dLi

θd

dLWfld 21

2

22

212

11

21

21221211 iiMiLiiMiL

The energy supplied by the two coils during this rotation is

tietieW 2211

MiiLiMiiLi 212

2

2211

2

1

21221211 iiMiLiiMiL

θiiθd

dMi

θd

dLi

θd

dL21

2

222

11 2

21221211 iiMiLiiMiL

The difference between energy supplied and change in stored energy constitutes the work done. Hence

θiiθd

dMi

θd

dLi

θd

dLθT 21

2

22

212

11

21

21

2

22

212

11

21 ii

θd

dMi

θd

dLi

θd

dLT

Or

The first two terms are known as the reluctance torques since they depend upon single excitation only. The third term is the excitation torque and requires that both coils are excited