MAXIMUM EFFICIENCY OF AN ELECTROMAGNET WITH A LINEAR MAGNETIC SYSTEM V. I. Malinin, A. N. Ryashenstev, and A. I. Tolstik

Electromagnetics are often used in reciprocating motion drives of vibration pulse systems in mining and construction processes, such a s t h e hammering of piles, drilling of holes, and pressing [I]. High-efficiency devices help save energy. Information on the maximum attainable efficiency of an electromagnet is important for technology analysis and design of electromagnetic pulse systems. Static efficiency (observed in a steady operation mode) does not determine the energy efficiency of machines that operate in dynamic modes. The following energy characteristics are more informative in this case: the energy efficiency during the time of motion, the dynamic efficiency (Nd), and the mean dynamic efficiency during the time of motion [2]. In [2] it has been demonstrated that dynamic efficiency is connected with the transfer function of the electromechanical device relative in velocity terms by the inverse Laplace transform. The transform can be used to estimate the energy cost for the various operation speeds of the electromechanical device. For an electromagnet, the dynamic efficiency (the subscript d will be omitted in the subsequent expressions) is defined as

qd(l ) where F ( t ) ,

v(t),

u(t),

i(t),

Pp(t),

Pz(t)

F (li.r ItJ

,, It~-i(~

Pp(tl

/'ztt) "

are the instant

(1) values

of t r a c t i o n ,

velocity,

intensity, current, useful power, and spent power (from the power main) at a particular point in the course of the motion. For the maximum efficiency we take the maximum attainable ~ in a certain operation mode for a process approaching the zero loss in the electromagnet. The transient curve of the dynamic characteristic of magnetization of an electromagnet is known to be determined by complex electromechanical processes. Various shapes of this curve are observed. The typical modalities of the dynamic magnetization characteristics (according to [3, 4]) are illustrated in Fig. i. In principle, the following dynamic operation modes are recognized on the basis of these characteristics: I) motion during the growth of the current and increasing magnetic flux linkage (~); 2) motion at constant current and increasing magnetic flux linkage; 3) motion with decreasing current and increasing flux linkage; 4) motion during increasing current and constant linkage; 5) motion with decreasing current and decreasing flux linkage; and 6) motion with constant current and decreasing flux linkage. For an analysis of dynamic efficiency, we make use of the law of energy conservation (with the usual assumption that the eddy currents and the hysteresis effect can be disregarded) and the second Kirchhoff law. On the basis of the second Kirchhoff law, we write l~idl = i 2 R d l + eidl. i . e . ,

where e is the instantaneous value of electric energy consumed, the loss in resistance R, and the electromagnetic according to the conservation law, we

d | l ' , = dQ + dIl~m,

(2)

the electromotive force and Wc, Q, and Wean are the the copper wires of the electromagnet winding with energy, respectively. From a different point of view, write

Institute of Mining, Siberian Branch, Academy of Sciences of the USSR, Novosibirsk. Translated from Fiziko-Tekhnicheskie Problemy Razrabotki Poleznykh Iskopaemykh, No. 4, pp. 72-76, July-August, 1991. Original article submitted May 7, 1988.

0038-5581/91/2704-0333512.50

9 1992 Plenum Publishing Corporation

333

i// IU

Fig. i.

of magnetization of electromagnets.

Dynamic characteristics

(3)

dW~ = dQ + dA + dW,,

since energy consumed fromt he power main minus the loss in the copper wires is spent to perform mechanical work (dA) and to modify the magnetic field energy (dWM). Comparing (2) and (3), we obtain the relationship between the electromagnetic and the magnetic energy

(4)

dWem = dA + dlV.,,. Here dWem = eidt = i d a ( i ,

x),

Therefore,

d A = Fdx,

the traction force should be defined

as

d|V~. -- i--:~q' di + . OW __ dW~. .

1: - - id~F (i. .r)

d~

With the aid of a power balance, law as

d.,"

ai T

t~

(5) is rewritten on the basis of the second Kirchhoff

dW . i)~ eli d~g d r i2]~ + u i = i " R + i--Tff = i2lt + t-~t - ~ + id.--7- d'--'{ =

it

allows

us t o e s t i m a t e

(5)

~lx

the dynamic efficiency

i i)~lt eli

"--T-~ +

for any operational

i OW "~x~ t:"

mode.

For the first operational mode, the differential of the magnetic field energy is expressed as

dH'p~ = A W ~ , = ~ +i

iPF

(i--, Ai)(q' + A ~ F ) - - + q r i = .0qr ,

I

di + =;-I--_ ~J.r u.r + +~F_

di.

According to (5), we define the traction force as F! ~ I.,) t~ll Olg ~,l. eli a----:-;--, , 1)

"r

21

qr drdi .

(6)

The power balance in this mode is written as 1~ = PQ +

Substituting

the value

i OW eli

,,-T -J? + i,5~g 0.-7" u"

of i(8~?/0x) from (6)

into

(7),

(7)

we o b t a i n '

1 ~ = PO + ~]r -JF di + 2 F v = I'Q + I'M + 2 1 ' ~

w h e r e PQ i s t h e i n s t a n t a n e o u s magnetic field energy.

thermal

loss

(8)

power and PM i s t h e i n s t a n t a n e o u s

power o f t h e

Thus, dynamic efficiency in this mode is expressed as

~h The maximum d y n a m i c e f f i c i e n c y

PQ i I'M

2P z

i s nlmax = 0 . 5 .

I n t h e s e c o n d mode t h e m a g n e t i c

field

dlV>,., = Alis, = - 71 i(W + Aq r ) -

334

1

2

energy differential

is different

~ q~i = + i d q r =--:r-,1 i ' oTwf :~a.t - b ' - 4 -1t T ..roarx, .

.-w-

because

The electromagnetic

force is now defined as I

t

. b W di

.OW

F2=-T, ~TVa.-7-. + ~ - ,

dW~,,,

tD-=.a.

-,

The dynamic efficiency takes the value

1 ~h

PO

2

"2p z"

Importantly, the electromagnetic force should be calculated only for a magnetic field energy differential in the respective operational mode. A specific feature of that mode is the fact that the traction force is created at the expense of an increased magnetic field energy. The third mode is characterized by the following magnetic field differential: I

dn'~'~=mV~=~(~--m)(~ The electromagnetic

idW

~gdi

2

_.,

force and the dynamic efficiency are expressed as '

~t.,

-2-

o.,

- - d.~'

:'

Z

The important feature of this mode is the possibility of a considerable tion force. Indeed, subject to the constraint

increase of the trac-

i d q r = qr di

(9)

there is no increment of the magnetic field energy (dW M = 0). The electromagnetic force according to (5) is determined entirely by the complete differential of the electromagnetic energy Fz-.~- i Oqr di oi d*

The dynamic

efficiency

can attain

the

value

Oqr . OW ] i o.--7= ~-SI"

of

,~ =

l

n = 1, -

because

(/%/1'~.

In the fourth mode, it is impossible to estimate the dynamic efficiency because there is no consumption of electromagnetic energy (dWem = 0, because dP = 0). The traction force in this mode should be computed on the basis of the magnetic field differential 4

1 ,) qrdi .

d[l,,~ = All",, = -.~- air (i -- Ai)

Its value is

dWM4

1 "tit di 2 d.r

de"

F.,

The motion is maintained at the expense of the decreasing magnetic field energy. For the fifth mode, a reduction of the flux linkage is observed, i.e., the electromotive force changes sign. A deceleration of the motor operation is introduced. Considering the change of the magnetic field differential

aw.,,, = :,)I~,, = ~ ( ~ - A;)(,),- A,i,)- ~ , ) ~ we write the expression for the electromagnetic t F5 - -

The dynamic

efficiency

.iiq' di

t

2 ~ oid.,

corresponding

"iPlr

+ "2-:

this

~

I = -- ~

operation I

lit'

2

~'d;,

force ~ llr di

2 ~ -o., 7

to

.,~ ~ v - - s -

cdq r " aL -7 + 7

~

di qt T "

mode is

Po -- P~,

2P Z

The sixth mode can be regarded as a special c a s e of the fifth mode. we have

For the sixth mode

335

diI'M~ -- \L;~, = -~i(T-- A T ) - - - ~ _

tv --

.1

. il~|

_

t

.i

Ti

=

- - - - i d T

=

d i - - 4 i ~ , r rj~Fdx ,

".'1 i

t di

I

. o~1r

I

. dl[ r

d.c

."

t o.r

_"

g ~.r '

~ q6

-"

PQ . 2P z

CONCLUSIONS The foregoing analysis indicates that, depending on the operational modes of an electromagnetic motor, the magnetic field energy differential and, therefore, the absolute value and the direction of the electromagnetic force can vary substantially. For example, at a given electromagnetic energy differential, the electromagnetic force in the third mode is twice as large as in this first one. The expression for the traction force in the second mode coincides with the established formula, confirming that this formula is applicable only for constant current. In addition, for a given magnetic flux linkage this expression yields a medium-level electromagnetic force relative to the other modes. The maximum traction force is observed with the third operational model, subject to condition (9). In energy terms the mode is recommended where work is performed at the expense of the electromagnetic energy with a constant magnetic field energy; the constancy is ensured by condition (9) by automatic regulation of the current and the flux linkage. For a practical implementation of this mode, magnetic field energy should be stored with the armature braked. In the course of motion, the proper regulation formula should be implemented. An electromagnet with such regulation is equivalent to an active direct current motor of sequential excitation (where the excitation winding is placed on the stator and the armature winding on the rotor). This raises a question: Why does the second mode of electromagnetic force relationship yield generally satisfactory results if it represents a single specific case? We explain this universally by the following factors. First of all, traction characteristics are observed at a constant current, i.e., the experimental conditions coincide with the situation where the formula is applicable. Secondly, as mentioned before, the expression estimates the average force relative to other modes. Considering that in dynamic processes condition (9) for maximum force is not satisfied unless special techniques are employed and that the energetically preferred mode corresponds to high-motion speeds; the duration of this operational mode in a working cycle of the electric magnet is short. In that case, the average dynamic force in an electromagnetic motor without automatic regulation of ~ and i is approximately equal to the force calculated from the usual expression for the second mode. The study has demonstrated that the maximum dynamic efficiency can reach 100%, provided an appropriate regulation formula between the current and magnetic flux linkage and the elimination of thermal loss. LITERATURE CITED I. 2.

3. 4.

336

N. P. Ryashentsev, "Vibration pulse systems in mining," Fiz.-Tekhn. Probl. Rzrab. Polezn. Iskop., No. 6 (1987). V. M. Kazanskii, L. I. Malinin, and V. I. Malinin, "Dynamic model of the electromechanical module of industrial robots and analytic methods," in: Electromechanical Component of Automatic Systems [in Russian], N~TI, Novosibirsk (1977). A. V. Gordon and A. G. Slivinskaya, Direct Current Electromagnets [in Russian], Gosenergoizdat, Moscow-Leningrad (1960). N. P. Ryashentsev and V. N. Ryashentsev, Electromagnetic Drive for Linear Machines [in Russian], Nauka, Novosibirsk (1989).