Fitzgerald A.E. Electric Machinery

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116 CHAPTER 3

ElectromechanicaI-Energy-Conversion Principles

which act to squash or deform the rotor, play no significant role in the performance

of the motor and generally are not calculated.

To understand the behavior of rotating machinery, a simple physical picture is

quite useful. Associated with the rotor structure is a magnetic field (produced in

many machines by currents in windings on the rotor), and similarly with the stator;

one can picture them as a set of north and south magnetic poles associated with each

structure. Just as a compass needle tries to align with the earth's magnetic field, these

two sets of fields attempt to align, and torque is associated with their displacement

from alignment. Thus, in a motor, the stator magnetic field rotates ahead of that of

the rotor, pulling on it and performing work. The opposite is true for a generator, in

which the rotor does the work on the stator.

Various techniques have evolved to calculate the net forces of concern in the

electromechanical-energy-conversion process. The technique developed in this chap-

ter and used throughout the book is known as the

energy method

and is based on

the principle of

conservation of energy.

The basis for this method can be understood

with reference to Fig. 3.3a, where a magnetic-field-based electromechanical-energy-

conversion device is indicated schematically as a lossless magnetic-energy-storage

system with two terminals. The electric terminal has two terminal variables, a volt-

age e and a current i, and the mechanical terminal also has two terminal variables, a

force ffld and a position x.

This sort of representation is valid in situations where the loss mechanism can be

separated (at least conceptually) from the energy-storage mechanism. In these cases

the electrical losses, such as ohmic losses in windings, can be represented as external

elements (i.e., resistors) connected to the electric terminals, and the mechanical losses,

such as friction and windage, can be included external to the mechanical terminals.

Figure 3.3b shows an example of such a system; a simple force-producing device

with a single coil forming the electric terminal, and a movable plunger serving as the

mechanical terminal.

The interaction between the electric and mechanical terminals, i.e., the elec-

tromechanical energy conversion, occurs through the medium of the magnetic stored

ffld

-Jr- 0 /W~ , Ol-

+ + Winding

-1-(

v resistance e ¢

m m m C

Electrlca~ ~vtecnanical -- o o----

terminal terminal Lossless 1

winding

(a)

Magnetic cot

.......

.........

I \] Movable

..... ] magnetic plunger

(b)

Figure 3.3 (a) Schematic magnetic-field electromechanical-energy-conversion device; (b) simple

force-producing device.

3.2 Energy Balance 117

energy. Since the energy-storage system is lossless, it is a simple matter to write that

the time rate of change of Wild, the stored energy in the magnetic field, is equal to the

electric power input (given by the product of the terminal voltage and current) less

the mechanical power output of the energy storage system (given by the product of

the mechanical force and the mechanical velocity)"

dWfld dx

= ei

- ffld~

(3.7)

dt dt

Recognizing that, from Eq. 1.27, the voltage at the terminals of our lossless

winding is given by the time-derivative of the winding flux linkages

d~,

e = (3.8)

and multiplying Eq. 3.7 by

dt,

we get

dWnd = i d)~ -

fnd

(3.9)

As shown in Section 3.4, Eq. 3.9 permits us to solve for the force simply as a

function of the flux ~. and the mechanical terminal position x. Note that this result

comes about as a consequence of our assumption that it is possible to separate the

losses out of the physical problem, resulting in a lossless energy-storage system, as

in Fig. 3.3a.

Equations 3.7 and 3.9 form the basis for the energy method. This technique is quite

powerful in its ability to calculate forces and torques in complex electromechanical-

energy-conversion systems. The reader should recognize that this power comes at the

expense of a detailed picture of the force-producing mechanism. The forces them-

selves are produced by such well-known physical phenomena as the Lorentz force on

current carrying elements, described by Eq. 3.6, and the interaction of the magnetic

fields with the dipoles in the magnetic material.

3.2 ENERGY BALANCE

The principle of conservation of energy states that energy is neither created nor

destroyed; it is merely changed in form. For example, a golf ball leaves the tee with a

certain amount of kinetic energy; this energy is eventually dissipated as heat due to air

friction or rolling friction by the time the ball comes to rest on the fairway. Similarly,

the kinetic energy of a hammer is eventually dissipated as heat as a nail is driven into

a piece of wood. For isolated systems with clearly identifiable boundaries, this fact

permits us to keep track of energy in a simple fashion: the net flow of energy into the

system across its boundary is equal to the sum of the time rate of change of energy

stored in the system.

This result, which is a statement of the first law of thermodynamics, is quite

general. We apply it in this chapter to electromechanical systems whose predominant

energy-storage mechanism is in magnetic fields. In such systems, one can account for

118 CHAPTER 3

ElectromechanicaI-Energy-Conversion Principles

energy transfer as

(Energy input~ fMechanical )/Increase in energy~ fEnergy ~

from electric] - ~energy + ~stored in magnetic) + ~convertedJ

sources / \output \field \into heat /

(3.10)

Equation 3.10 is written so that the electric and mechanical energy terms have positive

values for motor action. The equation applies equally well to generator action: these

terms then simply have negative values. In either case, the sign of the heat generation

term is such that heat generation within the system results in a flow of thermal energy

out of the system.

In the systems which we consider here, the conversion of energy into heat occurs

by such mechanisms as ohmic heating due to current flow in the windings of the

electric terminals and mechanical friction due to the motion of the system compo-

nents forming the mechanical terminals. As discussed in Section 3.1, it is generally

possible to mathematically separate these loss mechanisms from the energy-storage

mechanism. In such cases, the device can be represented as a lossless magnetic-

energy-storage system with electric and mechanical terminals, as shown in Fig. 3.3a.

The loss mechanisms can then be represented by external elements connected to

these terminals, resistances to the electric terminals, and mechanical dampers to the

mechanical terminals. Figure 3.3a can be readily generalized to situations with any

number of electric or mechanical terminals. For this type of system, the magnetic

field serves as the coupling medium between the electric and mechanical terminals.

The ability to identify a lossless-energy-storage system is the essence of the

energy method. It is important to recognize that this is done mathematically as part

of the modeling process. It is not possible, of course, to take the resistance out of

windings or the friction out of bearings. Instead we are making use of the fact that a

model in which this is done is a valid representation of the physical system.

For the lossless magnetic-energy-storage system of Fig. 3.3a, rearranging Eq. 3.9

in the form of Eq. 3.10 gives

Welec --

Wmech -~-

Wil d (3.11)

where

dWelec = i d~. = differential electric energy input

Wmech "-- ffld

dx =

differential mechanical energy output

d Wfld

= differential change in magnetic stored energy

From Eq. 3.8, we can write

dWelec "-

ei dt

(3.12)

Here e is the voltage induced in the electric terminals by the changing magnetic stored

energy. It is through this reaction voltage that the external electric circuit supplies

power to the coupling magnetic field and hence to the mechanical output terminals.

Thus the basic energy-conversion process is one involving the coupling field and its

action and reaction on the electric and mechanical systems.

3.3 Energy in Singly-Excited Magnetic Field Systems 119

Combining Eqs. 3.11 and 3.12 results in

dWelec --

ei dt

= dWmech d-

dWfld

(3.13)

Equation 3.13, together with Faraday's law for induced voltage (Eq. 1.27), form

the basis for the energy method; the following sections illustrate its use in the analysis

of electromechanical-energy-conversion devices.

3.3 ENERGY IN SINGLY-EXCITED MAGNETIC

FIELD SYSTEMS

In Chapters 1 and 2 we were concerned primarily with fixed-geometry magnetic

circuits such as those used for transformers and inductors. Energy in those devices is

stored in the leakage fields and to some extent in the core itself. However, the stored

energy does not enter directly into the transformation process. In this chapter we are

dealing with energy-conversion systems; the magnetic circuits have air gaps between

the stationary and moving members in which considerable energy is stored in the

magnetic field. This field acts as the energy-conversion medium, and its energy is the

reservoir between the electric and mechanical systems.

Consider the electromagnetic relay shown schematically in Fig. 3.4. The resis-

tance of the excitation coil is shown as an external resistance R, and the mechanical

terminal variables are shown as a force

fnd

produced by the magnetic field directed

from the relay to the external mechanical system and a displacement x; mechanical

losses can be included as external elements connected to the mechanical terminal.

Similarly, the moving armature is shown as being massless; its mass represents me-

chanical energy storage and can be included as an external mass connected to the

mechanical terminal. As a result, the magnetic core and armature constitute a lossless

magnetic-energy-storage system, as is represented schematically in Fig. 3.3a.

This relay structure is essentially the same as the magnetic structures analyzed in

Chapter 1. In Chapter 1 we saw that the magnetic circuit of Fig. 3.4 can be described by

an inductance L which is a function of the geometry of the magnetic structure and the

R i

ii] x ~1

"-I

,.~ Mechanical

ffld source

-- t

Electrical

source

~, e

Lossless

coil

Magnetic core

~ Massless

magnetic

armature

Figure

3.4 Schematic of an electromagnetic relay.

120 CHAPTER 3

ElectromechanicaI-Energy-Conversion Principles

permeability of the magnetic material. Electromechanical-energy-conversion devices

contain air gaps in their magnetic circuits to separate the moving parts. As discussed in

Section 1.1, in most such cases the reluctance of the air gap is much larger than that of

the magnetic material. Thus the predominant energy storage occurs in the air gap, and

the properties of the magnetic circuit are determined by the dimensions of the air gap.

Because of the simplicity of the resulting relations, magnetic nonlinearity and

core losses are often neglected in the analysis of practical devices. The final results

of such approximate analyses can, if necessary, be corrected for the effects of these

neglected factors by semi-empirical methods. Consequently, analyses are carried out

under the assumption that the flux and mmf are directly proportional for the entire

magnetic circuit. Thus the flux linkages ~. and current i are considered to be linearly

related by an inductance which depends solely on the geometry and hence on the

armature position x.

~. = L(x)i

(3.14)

where the explicit dependence of L on x has been indicated.

Since the magnetic force ftd has been defined as acting from the relay upon the

external mechanical system and

d Wmech

is defined as the mechanical energy output

of the relay, we can write

dWmech = ftd

(3.15)

Thus, using Eq. 3.15 and the substitution dWelec = i d~., we can write Eq. 3.11 as

dWfld = i d)~ - ftd dx

(3.16)

Since the magnetic energy storage system is lossless, it is a

conservative system

and the value of Wild is uniquely specified by the values of Z and x; ~. and x are thus

referred to as

state variables

since their values uniquely determine the state of the

system.

From this discussion we see that Wild, being uniquely determined by the values

of ~ and

x, is the same regardless of how )~ and x are brought to their final values.

Consider Fig. 3.5, in which two separate paths are shown over which Eq. 3.16 can be

integrated to find Wild at the point (~.0, x0). Path 1 is the general case and is difficult to

~'o

Wnd (Z.O, %)

-2a Xo

Figure 3.5

Integration paths

for ~ld.

3.3 Energy in Singly-Excited Magnetic Field Systems 121

integrate unless both i and fnd are known explicitly as a function of)~ and x. However,

because the integration of Eq. 3.16 is path independent, path 2 gives the same result

and is much easier to integrate. From Eq. 3.16

Wnd(Xo, xo) = f dWnd + f dWnd

(3.17)

path 2a path 2b

Notice that on path 2a, d)~ = 0 and

ffld "--

0 (since ~. = 0 and there can be no magnetic

force in the absence of magnetic fields). Thus from Eq. 3.16, d Wfld = 0 on path 2a.

On path 2b, dx = 0, and, thus, from Eq. 3.16, Eq. 3.17 reduces to the integral of i d)~

over path 2b (for which x = x0).

fO L°

Wfld()~0, X0) =

i(~, x0) d~ (3.18)

For a linear system in which )~ is proportional to i, as in Eq. 3.14, Eq. 3.18 gives

j~o ;~ ~o x U 1 ~2

Wfld(~,,

X) --

i(X', x) dU = L(x) dU = 2 L(x) (3.19)

It can be shown that the magnetic stored energy can also be expressed in terms of

the energy density of the magnetic field integrated over the volume V of the magnetic

field. In this case

(J0")

Wild

H. dB' dV (3.20)

For soft magnetic material of constant permeability (B =/zH), this reduces to

/, (")

Wild= ~ dV (3.21)

The relay shown in Fig. 3.6a is made from infinitely-permeable magnetic material with a

movable plunger, also of infinitely-permeable material. The height of the plunger is much

greater than the air-gap length (h >> g). Calculate the magnetic stored energy Wad as a function

of plunger position (0 < x < d) for N = 1000 turns, g = 2.0 mm, d = 0.15 m, l = 0.1 m,

and i = 10 A.

II Solution

Equation 3.19 can be used to solve for Wnd when ~. is known. For this situation, i is held

constant, and thus it would be useful to have an expression for Wnd as a function of i and x.

This can be obtained quite simply by substituting Eq. 3.14 into Eq. 3.19, with the result

Wftd = ~ L(x)i 2

The inductance is given by

L(x) =

/Zo N2 Agap

EXAMPLE 3.2

122 CHAPTER 3

ElectromechanicaI-Energy-Conversion Principles

where Agap is the gap cross-sectional area. From Fig. 3.6b, Agap can be seen to be

agap = l(d - x) = ld (1- d)

Thus

and

L(x) = lz°N21d(1 - x/d)

1 N2lzold(1 - x/d)

w.. = ~

1 (10002)(4n x 10-7)(0.1)(0.15) ( x)

2(0.002) x 102 1 -

= 236 (1- d) J

)ractice Problem 3.:

The relay of Fig. 3.6 is modified in such a fashion that the air gaps surrounding the plunger are

no longer uniform. The top air gap length is increased to gtop -- 3.5 mm and that of the bottom

gap is increased to gbot

2.5 mm. The number of turns is increased to N = 1500. Calculate

the stored energy as a function of plunger position (0 < x < d) for a current of i = 5 A.

Solution

(x)

Wnd=88.5 1--~ J

Lossless

N-turn coil

(a) (b)

Figure 3.6 (a) Relay with movable plunger for Example 3.2. (b) Detail showing

air-gap configuration with the plunger partially removed.

3.4 Determination of Magnetic Force and Torque from Energy 123

In this section we have seen the relationship between the magnetic stored energy

and the electric and mechanical terminal variables for a system which can be repre-

sented in terms of a lossless-magnetic-energy-storage element. If we had chosen for

our example a device with a rotating mechanical terminal instead of a linearly dis-

placing one, the results would have been identical except that force would be replaced

by torque and linear displacement by angular displacement. In Section 3.4 we see

how knowledge of the magnetic stored energy permits us to solve for the mechanical

force and torque.

3.4 DETERMINATION OF MAGNETIC FORCE

AND TORQUE FROM ENERGY

As discussed in Section 3.3, for a lossless magnetic-energy-storage system, the mag-

netic stored energy

Wild

is a state function, determined uniquely by the values of

the independent state variables ~. and x. This can be shown explicitly by rewriting

Eq. 3.16 in the form

dWfld(~,, x) =

i d)~

- ffld

(3.22)

For any state function of two independent variables, e.g., F(Xl, x2), the total

differential of F with respect to the two state variables X l and x2 can be written

OF dxl + ~ dx2 (3.23)

dF(xl, x2) = ~ x2 x,

It is extremely important to recognize that the partial derivatives in Eq. 3.23 are each

taken by holding the opposite state variable constant.

Equation 3.23 is valid for any state function F and hence it is certainly valid for

Wnd; thus

0 Wild 0 Wild

dWfld(~., x) -- d~. + dx (3.24)

8~. x dx

Since k and x are independent variables, Eqs. 3.22 and 3.24 must be equal for all

values of d~ and dx, and so

i .__

own~(Z,x)

(3.25)

where the partial derivative is taken while holding x constant and

0 Wfld(/~, X)

fro = -- (3.26)

Ox z

in this case holding )~ constant while taking the partial derivative.

This is the result we have been seeking. Once we know

Wild as

a function of )~

and x, Eq. 3.25 can be used to solve for i (~., x). More importantly, Eq. 3.26 can be

used to solve for the mechanical force ffld (~, x). It cannot be overemphasized that the

partial derivative of Eq. 3.26 is taken while holding the flux linkages ~. constant. This

124 CHAPTER 3

ElectromechanicaI-Energy-Conversion Principles

is easily done provided W~d is a known function of ~. and x. Note that this is purely

a mathematical requirement and has nothing to do with whether Z is held constant

when operating the actual device.

The force ffld is determined from Eq. 3.26 directly in terms of the electrical state

variable ~.. If we then want to express the force as a function of i, we can do so by

substituting the appropriate expression for ~. as a function of i into the expression for

fna that is obtained by using Eq. 3.26.

For linear magnetic systems for which ~. =

L(x)i,

the energy is expressed by

Eq. 3.19 and the force can be found by direct substitution in Eq. 3.26

Ofl)~2~

_ ~.2

dL(x)

ffld --

Ox -2 L(x) ;~

2L(x) 2

(3.27)

If desired, the force can now be expressed in directly in terms of the current i simply

by substitution of ~. --

L(x)i

i 2 dL(x)

f~d = 2 dx

(3.28)

"XAMPLE 3.,

Table 3.1 contains data from an experiment in which the inductance of a solenoid was measured

as a function of position x, where x = 0 corresponds to the solenoid being fully retracted.

Table 3.1

Data for Example

3.3.

.|ll

x [cm] 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

L [mH] 2.8 2.26 1.78 1.52 1.34 1.26 1.20 1.16 1.13 1.11 1.10

Plot the solenoid force as a function of position for a current of 0.75 A over the range 0.2 <

x < 1.8 cm.

Solution

The solution is most easily obtained using MATLABJ First, a fourth-order polynomial fit of

the inductance as a function of x is obtained using the MATLAB function

polyfit.

The result is

of the form

L(x)

- a(1)x 4 + a(2)x 3 + a(3)x 2 + a(4)x + a(5)

Figure 3.7a shows a plot of the data points along with the results of the polynomial fit.

Once this fit has been obtained, it is a straight forward matter to calculate the force from

Eq. 3.28.

i 2 dL(x) i 2

f,d = 2 dx

= 2 (4a(1)x3 + 3a(2)x2 + 2a(3)x + a(4))

This force is plotted in Figure 3.7b. Note that the force is negative, which means that it is acting

in such a direction as to pull the solenoid inwards towards x = 0.

t MATLAB is a registered trademark of The MathWorks, Inc.

3.4 Determination of Magnetic Force and Torque from Energy 125

2.8

2.6

2.4

2.2

1.8

1.6

1.4

1.2

I I I I I I I I I

1 I I I I I I I I I

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x [cm]

(a)

-0.01

-0.02

-0.03

--0.04

--0.05

--0.06

--0.07

--0.08

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

X [cm]

(b)

Figure 3.7 Example 3.3. (a) Polynomial curve fit of inductance. (b) Force as a

function of position x for i = 0.75 A.