cs_2013070111220490

8/10/2019 CS_2013070111220490

1/7

Circuits and Systems, 2013, 4, 316-322http://dx.doi.org/10.4236/cs.2013.43043 Published Online July 2013 (http://www.scirp.org/journal/cs)

The Magnetic Field Study of a Finite Solenoid

A. K. Al-Shaikhli, H. Fatima, K. Omar, A. Fadhil, J. LinaElectrical Engineering Department, University of Technology, Baghdad, Iraq

Email: [email protected]

Received April 15,2013; revised May 16, 2013; accepted May 24, 2013

Copyright 2013 A. K. Al-Shaikhli et al. This is an open access article distributed under the Creative Commons Attribution License,

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

This paper presents the axial and radial magnetic fields strength equations at any point inside or outside a finite solenoid

with infinitely thin walls. Solution of the equation has been obtained in terms of tabulated complete elliptic integrals.

For the axial field, an accurate approximation is given in terms of elementary function. Internal and external magneticfields to the solenoid are presented in graphical form with normalized values for a wide variety of solenoid dimensions.A comparison with previous studies has been done, a good correlation between the two results is achieved and the ratio

of error is calculated.

Keywords:Solenoid Magnetic Field; Elliptic Integrals

1. Introduction

Magnetic fields have long been the interest of scientists.

Electric currents produce magnetic fields. As with the

electric field, magnetic fields have magnitude and direc-

tion. The direction is determined by the direction that a

compass will point. The magnitude is determined by thesize and location of the electric currents that produce the

magnetic field. The simplest way to create a uniform

magnetic field is to run a current through a solenoid of

wire [1,2]. Consider a solenoid of Nturns and length L,

carrying a current I; the number of turns/length is n=N/L,

as shown in Figure 1.

For a long solenoid of many turns, the magnetic flux

density B within the coil is nearly uniform, while the

field outside is close to zero. The B-field within the coil

can be calculated from Amperes Law. This law may be

derived from the Biot-Savart Law [1,3]. There are many

papers published in the field of electromagnetic technol-

ogy. Since the discovery of magnetism, engineers and

scientists alike have struggled continuously to devise

models and techniques to aid in the understanding and

prediction of the electromagnetic field in different situa-

tions. Along the way, many useful methods have been

developed and refined, for example, the analytical

method, the method of images, the magnetic circuit ap-

proach, the use of conformal transformation and the nu-

merical methods, which are complex and need special

packages for execution [4].

The aim of this paper is modeling and programming

the axial and radial magnetic field strength at any point

within or outside a finite solenoid in terms of standard

tabulated functions, in order to wide study the effect of

various parameters in system response to which solenoid

are connected.

2. Mathematical RepresentationThe calculation of the fields generated by various elec-

tromagnetic configurations such as loops, finite helical

solenoids, and infinite solenoids has been treated by the

early classical physicists, but only the simplest cases

such as the single loop have been calculated for the entire

field both inside and outside the loop [5,6]. In other cases

such as helical solenoid, or finite solenoid the calculation

have been limited to the axis [1,7]. Derivations of the

off-axis positions have been done by Foelsch [8], and the

solutions are obtainable by means of the large number of

approximate expressions which are valid over restricted

ranges of size or position. The principal difficulty in the

calculation of the fields of nearly all configurations has

resulted from the fact that the integral solution cannot be

achieved without the use of various elliptic integrals.

3. Derivation of Equations

Consider a solenoid as shown in Figure 2. The magnetic

field due to this coil is given in terms of the magnetic

vector potentialAby:

B AX (1)

where, for the geometry assumed, only the A compo-

ent can be nonzero. Then Equation (1) yields simplyn

Copyright 2013 SciRes. CS

8/10/2019 CS_2013070111220490

2/7

A. K. AL-SHAIKHLI ET AL. 317

Figure 1. Solenoid with N turns.

r i

L/2 L/2

a z, l

Figure 2. Schematic diagram of a solenoid.

1r z rAz r r

,AB B

(2)

For a single circular filament, one has

cos d

i a

R4

A (3)

where R is the distance from the local point on the

filament to the field point. For a solenoid made up of a

series of n filaments per unit length, we have then

If

2

2 2 22 0

cos dd

2

L

L

a niA l

z l r a

2 cosar

(4)

, 2 z l z L and l is the axial distancefrom the origin to the filament [1]. Then Equation (4)

becomes

2 2 20

cos dd

2 2 cos

ainA

r a ar

2 2 2 2 cos

(5)

Let c r a ar tan, and c2d sec d

Then

c

2

2 2 20

sec dcos d

2 tan

ani cA

c c

2 2tan 1 sec

(6)

For Then

0

sec d cos d2

aniA

(7)

, this becomesOn integrating with respect to

0

cos ln sec tan d2

aniA

tan

(8)

From c

2 2

0

cos ln d2

caniA

c c

(9)

2 2

0

cos ln ln d2

aniA c c

(10)

ln zeroc

2 2 2

0

cos ln 2 cos d2

aniA r a ar

(11)

by using integration by parts to Equation (11)

2 2 2ln 2 cosu r a ar ,

sind cos d ,v v

2 2 2

0

2

2 2 2 2 2 20

sin 2 cos

sin d

2 2 cos 2 cos

2

aniA In r a ar

ani ar

r a ar r a ar

(12)

The first term vanishes [1].

2

2 2 2 2 2 20

sin d

2 cos 2 cos

ar

r a ar r a ar

2

aniA (13)

On multiplying integrand by


8/10/2019 CS_2013070111220490

3/7

8/10/2019 CS_2013070111220490

4/7

A. K. AL-SHAIKHLI ET AL. 319

2k

22

4ar

a r cos (24)

In a similar manner, Bz Equation (20) can be re-

duced to standard elliptic integrals. First change the

variable of integration to t . There follows suc-cessively

1

2 2 2 2 21 2

d

4 2z

at t

ni rB

ar r a r at1

2 2t t

ar ar

(25)

2

202

21

d42

1z

rsn uK

ni a a rB k uara r r

sn ua r

(26)

, k4

za rni kB K ka rar

(27)

1tana r

, k

0r

(28)

As before, kis given by Equation (24). The human

lambda function is tabulated in references

[1,9-11]. For many purposes, it is convenient simply to

know the variation of the fields near the axis. As ,

Equations (23) and (27) reduce to the following ex-

pressions:

2

3 22 24r

ni a r B

a

(29)

2 22z

niB

a

(30)

A convenient approximation for Bz, valid whenever

r a and accurate to 1 percent in this range, is

2 12

1 2 2 sin4 4

z

m mniB m k k

(31)

where2

1 1 , 1 m k k k k .Equation (31) reduces exactly to Equation (30) atthe axis. Equations (23) and (27) are readily written in

dimensionless form with the distances given in units of

the solenoid radius. Then Equations (23)-(30) still hold

but with 1, r a , a replacing a, r, , throughout.

4. Computer Implementation

Computer programs with the sequence of flow chart

which is shown in Figure 3have been written in MAT-

LAB package to implement the estimation of radial and

axial magnetic fields from Equations (23) and (27). Set

of normalized curves for a wide variety of solenoid pa-

rameters are obtained.

5. Comparison

The same data used by reference [12] are considered to

estimate the magnetic field strength for different number

of turns of solenoid. The filament carries current I at

distance h from the surface of slab. The magnetic field

strength at any point P along the surface of the slab can

be calculated. The conductor will produce a magnetic

field H on the surface, according to Amperes Law,

equal to

2 2

ep

e

hIH

h z

(32)

where z is the distance of point P along the load

measured from a point directly beneath the conductor

and, ho the effective height for a circular conductor

equal to:

2 2

oh h a (33)

Choice the values of distance z, the length of coil L,

the conductor radius a, the distance from surface to

slab h, the number of turns n and the current per unit

length I to estimate the magnetic field strength H,

from this calculation limited the set of curves which

compared with it. The calculated results are plotted and

compared with that obtained by this reference. A good

correlation between the two results is achieved, as can be

seen in Figures 4 and 5. The maximum deviation is

about 5%, for a coil of 30 turns and 4% of 40 turns.

This deviation is reduced when the number of turns

n of the coil is increased, and approaches to zero when

n equal to (2500) or more turns. The ratio of error is

shown in Figure 6.


8/10/2019 CS_2013070111220490

5/7

A. K. AL-SHAIKHLI ET AL.320

start

Input (L & a)Solenoid

of

i = 1

ZL = 0

),

)

)

k

r

r

(

(

)(

4 a

a

kK

ar

kni

zB

k

kE

k

)(

)

K

k

k

r

ani

rB (

2

22

If

i = 100i=i+1

Zl = Zl + H

Plot Bz , r

Br, r / a

/ a

END

I = 100

i = i + 1

Zl = Zl + H

No

Yes

),k

)(

)(

)(

4 ra

ra

kK

ar

kni

zB

22

2

+ni a - k E(k)

B K(k) -r

r k k-

Calculate Magn

Bz & B

etic Field

r

Figure 3. Flow chart of the computer program.

6. The Use of Normalized Curve

The normalized data of a solenoid leads to estimate the

magnetic field strength, so the system behavior can be

obtained. Such result is very useful to designers and us-

ers in determining the best combination to be used in a

particular problem. The use of normalized curves is very

straightforward. To clarify the extraction of the required

information from such curves, the following may be con-

sidered as a helpful procedure:

1) Take the distance from origin z, the length of the

coil L, the radius of the coil a, number of turns n

and current per unit length I.

2) Calculate the ratio L/a, then use this value to as-

sign the normalized figure associated with them.

3) Calculate the ratio 2z/L and use this value to as-

sign the necessary curve. If there is no curve having ex-

act value of 2z/L, interpolation may be needed by

imagining a curve lies between the nearest lower and

Figure 4. Comparison of magnetic field strength H.

Figure 5. Comparison of magnetic field strength H.

Figure 6. Error variation with number of turns.

upper values of 2z/L.

As an example, consider the same data used by refer-

ence [13] having dimensions L = 5 cm, a = 1 cm, z = 3

cm, r = 0.5 cm and 0.0001 wb/m2 whereB B

nI . By following the previously mentioned procedure;


8/10/2019 CS_2013070111220490

6/7

8/10/2019 CS_2013070111220490

7/7

A. K. AL-SHAIKHLI ET AL.


322

Figure 11. Dimensionless radial field of finite solenoid for

(L/a= 25).

Figure 12. Dimensionless axial field of finite solenoid for

(L/a= 25).

3) The calculated results of magnetic field strength are

plotted and compared with a reliable reference and a

good correlation was achieved, the difference between

two results is reduced when the number of turns n of

the coil is increased i.e. the conductor radius decreases

for constant radius value of a solenoid.4) The general form of result leads to a quick study of

various parameters which determines the system behav-

ior containing a solenoid, uniform fields with total varia-

tion of 1% which can be achieved over as much as 60%

of the internal volume of the solenoid if the length is 25

radii or greater.

REFERENCES[1] F. H. Faris, The Magnetic Field Study of a Finite Sole-

noid, M.S. Thesis, University of Technology, 2005.

[2]

D. E. Mapother, The Axial Variation of the Magnetic

Field in a Solenoid of Finite Thickness, University of Il-linois, Engineering Experiment Station Circular, Urbana,

1955.

[3]

B. Mason, Designs for Transverse, Super Conducting

MRI Magnets, Department of Electrical and ComputerEngineering, University of Queensland, Brisbane, 2000.

[4] F. Kh. Abood, Design and Implementation of an Ex-

perimental Coil Gun System, M.S. Thesis, University ofTechnology, 2002.

[5]

J. Novickij, S. Balevieius, et al., Axial Magnetic FieldMeasurment of Pulsed Solenoids, Elektronika IR Elek-

trotechnika, 2004, pp. 1392-1215.http://www.ee.ktu.lt/journal/2004/2/Novickij.pdf

[6] W. T. Scott, The Physics of Electricity and Magnetism,John Wiley & Sons, Inc., New York, 1959.

[7] D. E. Mapother and J. N. Snyder, The Axial Variation of

the Magnetic Field in Solenoids of Fini Thickness, Tech.Rep.5 University I11., 1954.

[8] K. Foelsch, Magnetic Field and Induktivitaet Einer Zyl-indrischen Spule, Archiv fr Elektrotechnik, Vol. 30, No.

3, 1936, pp. 139-157.

[9] P. F. Byrd and M. D. Friedman, Handbook of Elliptic

Integrals for Engineers and Physicists, Springer-Verlag,Berlin, 1954.

[10] C. Heuman, Tables of Complete Elliptic Integrals,Journal of Mathematical Physics, Vol. 20, 1941, pp. 127-

207.

[11] M. Abramowitz and I. A. Stegun, Handbook of Mathe-matical Functions with Formulas, Graphs and Mathe-matical Tables, Dover Publications, New York, 1964.

[12]

A. K. M. Al-Shaikhli, A Novel Method for the Design of

Induction Heating Work Coils, Ph.D. Thesis, Lounghbor-ough University of Technology, Lounghborough, 1985.

[13] C. Thoma, Test of L.S.P. offAxis Magnetic Field Ex-pansion in Cylindrical Coordinates, 2002.

http://www.helsinki.fi/~karimaki/swph/07/ProjectPal070522/FastParametrization/thoma.pdf

cs_2013070111220490

Documents