Electromagnetics

V ELECTROMAGNETICS

Hung-Yu David Yang Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, Illinois, USA

Electromagnetics is fundamental in electrical and electronic engineering. Electromagnetic theory based on Maxwell's equations establishes the basic principle of electrical and electronic circuits over the entire frequency spectrum from dc to optics. It is the basis of Kirchhoff's current and voltage laws for lowfrequency circuits and Shell's law of reflection in optics. For low-frequency applications, the physics of electricity and magnetism are uncoupled. Coulomb's law for electric field and potential and Ampere's law for magnetic field govern the physical principles. Infrared and optical applications are usually described in the content of photonics or optics as separate subjects. This section emphasizes the engineering applications of electromagnetic field theory that relate directly to the coupling of space and time-dependent vector electric and magnetic fields, and, therefore, most of the subjects focus on microwave and millimeter-wave regimes. The eleven chapters in this section cover a broad area of applied electromagnetics, including fundamental electromagnetic field theory, guided waves, antennas and radiation, microwave components, numerical methods, and radar and inverse scattering. Chapter 1 discusses the basic theory of magnetostatics. Magnetic field and energy due to a direct current is defined based on Amp~re's law and the Biot-Savart law. Macroscopic properties of magnetic material are described. In addition, domains and hysteresis are introduced. Inductance relating the magnetic flux to the current is defined. The concept of a magnetic circuit, which finds important applications in power transformers, is also introduced. Chapter 2 is devoted to the fundamental theory of electrostatics. The concept of electric field and potential based on

Coulomb's law and Gauss's law is introduced. Electric energy and force based on the field and potential are described. Boundary value problem based on the Poisson's equation and Laplace's equation for electric potential is formulated and canonical examples are given. Waves propagating in a homogeneous isotropic region are usually in the form of plane-waves. Chapter 3 describes the basic properties of plane waves for both lossless and lossy media. These basic properties include the nature of the electric and magnetic fields, the properties of the wave number vector, and the power flow of the plane wave. Special attention is given to the specific case of a homogenous (uniform) plane wave, i.e., one having real direction angles because this case is most often met in practice. Properties such as wavelength, phase and group velocity, penetration depth, and polarization are discussed for these plane waves. Chapter 4 describes the theory of transmission lines. Transmission equations for voltage and current are derived based on lumped-element circuit models. The propagation characteristics of both lossless and lossy lines are discussed with the latter emphasized on low-loss cases and cases lacking distortion. Useful parameters of a terminated transmission line, including impedance, reflection coefficient, voltage, and current, at various locations on the line are discussed in detail. The basic operation of the Smith chart to relate the reflection coefficient to the input impedance at the transmission line is explained and examples are given. Distributing electromagnetic power from one point to another in a prescribed way usually requires transmission lines or waveguides. Chapter 5 discusses the properties of a

478 class of guided wave structures. The emphasis is on non-TEM structures where the guided waves are dispersive. The mode characteristics of rectangular metallic waveguides, circular metallic waveguides, microstrip lines, slot lines, coplanar waveguides, and the circular dielectric waveguides are summarized. In wireless communication systems, it is necessary to send signals in the form of electromagnetic waves through air, such as in radio or television broadcasting, or via point-to-point microwave links. An antenna is a device for transmission or reception of electromagnetic signals. Chapter 6 describes the basic theory of antennas and their arrays. This chapter presents the fundamental properties of electromagnetic waves emanating from any antenna as well as the antenna parameters, including polarization, radiation patterns, beam width, side lobe level, efficiency, gain, bandwidth, input impedance, directivity, and receiving cross section. Chapter 6 discusses the radiation/reception properties of selected antenna structures, including a dipole, a monopole, a wire-loop, a slot, and a microstrip. The theory of antenna arrays made of a number of individual antenna elements at different locations is also discussed. Active and passive components are essential building blocks of microwave circuits and systems that have become increasingly important due to the booming of next-generation wireless communications. Chapter 7 describes the basic characteristics of a class of microwave passive components, including tuning stubs, lumped elements, impedance transformers and matching network, couplers, power dividers/combiners, resonators, and filters. Scattering parameters are usually used to characterize the frequency-dependent components. Examples are given mostly for rectangular metallic waveguides and microstrip circuits. Active microwave components will also be discussed in this chapter. The engineering applications of electromagnetic fields and waves usually require accurate solutions of Maxwell's equations subject to proper boundary conditions. Due to the increasing capability of computers, many of the complicated electromagnetic problems are now becoming solvable. Numerical computation has become an indispensable subject in electromagnetics. The two most widely applied numerical

Hung-Yu David Yang methods are discussed in Chapter 8. Chapter 8 also describes a frequency-domain integral-equation based approach known as the method of moments. This method is particularly useful when the Green's function (the kernel) can be found analytically. This section also discusses the recent progress of using a fast algorithm to deal with large electromagnetic systems. Chapter 9 describes a time-domain differential equation-based approach known as the finite-difference time-domain (FDTD) method. This method is particularly useful for complicate noncanonical or nonlinear structures with impulsive sources. The volume of the structure usually dictates computer time and memory required. Perfect absorbing boundary facilitates fast computation with minimum required memory. An early application of electromagnetics is on radio detection and ranging known as radar, which is an interdisciplinary subject involving communications, signal processing, and propagation. Chapter 10 discusses the principle of both pulsed and CW radar systems and radar parameters. Specialized radars for various applications are also addressed, including MTI radar, Doppler radar, tracking radar, high cross-range resolution radar, and synthetic aperture radar. An inverse scattering problem is to reconstruct or recover physical or geometric properties of an object from measured electromagnetic fields. The basic principles of inverse scattering are also discussed in this chapter. The approach based on an integral equation formulation in conjunction with an iterative scheme to solve the inverse scattering is also discussed. Trends in compact communication systems to involve the integration of antenna (including matching network) and active circuits (such amplifiers) together as one component. Chapter 11 discusses the basic principles of active integrated circuit antennas. The basic operation of microwave transistors (both BJT and FET) is discussed. Active circuits of amplifiers, oscillators, and detectors/mixers are described. Active antennas using microstrip patches or printed slots are also described. The section 5 editor would like to thank all the authors and reviewers for their volunteer effort and cooperation to make this chapter possible. It is our hope that this chapter will be valuable to electrical and electronic engineers who are interested in the subject of electromagnetics.

1 Magnetostatics Keith W. Whites Department of Electrical and Computer Engineering, South Dakota School of Mines and Technology, Rapid City, South Dakota, USA

1.1 1.2

Introduction ....................................................................................... Direct Current .....................................................................................

479 479

1.2.1 Current and Current Density • 1 . 2 . 2 0 h m ' s Law • 1.2.3 Resistance • 1.2.4 Power and Joule's Law • 1.2.5 Conservation of Charge and Kirchhoff's Current Law

1.3

Governing Equations of Magnetostatics ...................................................

482

1.3.1 Postulates of Magnetostatics • 1.3.2 Biot-Savart Law and Vector Magnetic Potential A • 1.3.3 Boundary Conditions for B, H, and ]

1.4

Magnetic Force and Torque ...................................................................

485

1.4.1 Ampbre's Force Law • 1.4.2 Lorentz Force Equation • 1.4.3 Torque and Magnetic Dipole Moment

1.5

Magnetic Materials ...............................................................................

487

1.5.1 Magnetization Vector and Permeability • 1.5.2 Magnetic Materials • 1.5.3 Domains and Hysteresis • 1.5.4 Permanent Magnets

1.6

Inductance ..........................................................................................

491

1.6.1 Magnetic Flux and Flux Linkage • 1.6.2 Definition of Inductance

1.7

Stored Energy .....................................................................................

494

1.7.1 Energy Stored in a Magnetic Field • 1.7.2 Energy Stored in an Inductor

1.8

Magnetic Circuits ................................................................................ References ..........................................................................................

1.1 I n t r o d u c t i o n

1.2 Direct Current

Magnetostatics involves the computation of magnetic forces and fields produced by direct (i.e., time-stationary) currents and from materials with permanent magnetization (magnets). Only magnetic forces and fields that do not change with time are magnetostatic. There are many applications of magnetostatics and even a few industries that are almost wholly based upon it. The magnetic recording and electric power industries both apply principles from magnetostatics. Other applications include magnetic resonance imaging (MRI) (Inan and Inan, 1999), magnetic brush applicators in electrophotography (laser printers) (Schein, 1992), and aurora in the earth's atmosphere (Paul et al., 1998), to name a few.

1.2.1 Current and Current Density

Copyright© 2005by AcademicPress. Allrightsof reproductionin any form reserved.

495 497

Current is the flow of charge. By convention, the direction of this flow is with the movement of positive charge. The amount of charge ~Q flowing through (i.e., perpendicular) to a surface in time 8t is defined as ~Q = ISt, where I is the current. In the limit of infinitesimally small time increments, the current I through the surface can be defined as:

I =~[A],

(1.1)

where the units are coulombs per second [C/s] or amperes [A]. 479

480

K e i t h W. W h i t e s

Magnetostatics is a field theory and, consequently, the quantities of interest are usually distributed throughout space. As such, the v o l u m e c u r r e n t density, ] ( x , y, z) is often employed. In terms o f the charge carriers, the current density is given by (1.2)

] = Nqv[A/m2],

where N is the n u m b e r o f charge carriers per unit volume (i.e., the n u m b e r density), q is the charge, and v is the average (or drift) velocity. In addition, a current density through an open surface S is related to the current as:

I = l l . ds[A].

l = a E [A/m2],

through the electrical c o n d u c t i v i t y o" of the material. The units o f a are siemens per meter IS/m]. The conductivities for various materials are listed in Table 1.1. It is apparent that a varies enormously for different materials. The materials near the top of the table are called conductors, whereas those near the b o t t o m are called insulators. The electrical conductivity o" o f metals varies with temperature. As a simple estimate of this variation, the conductivity can be assumed to change linearly with temperature (Halliday et al., 2002):

(1.3)

$

A surface c u r r e n t d e n s i t y , / , [ A / m ] , is an approximation for 1 in a very thin layer.

Example 1.1 One end of a copper wire (diameter = 2 mm) is attached to one end of an a l u m i n u m wire (diameter = 4 mm). A direct current I = 10 mA is passing through the wires. To determine the current density in each wire, equation 1.3 is applied assuming the current density is uniformly distributed over the cross-section: I

0.01A

]cu - - Areac~ -- ~(0.002)2m 2 -- 795"8A/m2"

(1.4)

I 0.01A Areaa~ -- g(0.004)Zm 2 -- 198"9A/m2'

(1.5)

Jal

--

The much smaller current density in A1 is due solely to the larger diameter and not because of the material type. The drift speed of the conduction electrons in each wire can be determined using equation 1.2 and by knowing that there are approximately 8.49 x 1028 conduction charges/m 3 in Cu and 6.02 x 1028 conduction charges/m 3 in A1 (Halliday et al., 2002). Therefore, the following two equations result: _ Vcu

Jcu 795.8 A/m 2 Ncu~e-- (8.49 x 1028/m3)(1.6022 x 10 19C) = 5.850 X 10 8m/s

(1.6)

]AI 198.9A/m2 VAI = Ncu~e -- (6.02 x 1028/m3)(1.6022

x

10-19C)

=

2.062 x 10-8m/s. (1.7)

1.2.20hm's

Law

At each point in an ohmic material, such as in a conductor, the volume current density I and electric field E are related by O h m ' s law:

(1.8)

1

1

O"

¢70

-

c~ ¢70

( T - To).

(1.9)

In this linear equation, a0 is the conductivity at temperature To, c~ is the temperature coefficient o f the conductor (see Table 1.1), and a is the conductivity at temperature T. For metals with a positive c~, the conductivity decreases with increasing T (or resistivity _= 1 / a increases with increasing T). Example 1.2 From Table 1.1, the conductivity of copper at 20°C is o"0 = 5.8 X 107. We can use equation 1.9 to determine the temperature at which the conductivity is half that at 20°C. Solving for T in equation 1.9 using To = 20°C, a = 0-o/2, and = 0.00393/°C gives the following: T = To + -

1

= 274.5°C.

(1.10)

TABLE 1.1 Electrical Conductivity a and Temperature Coefficient (Near 20°C) for Selected Materials at dc. Material Silver1 Copper (annealed)1 Gold1 Aluminum1 Tungsten Iron (99.98% pure) 1 Tin1 Constantan I Nichrome1 Carbon (graphite) Seawater Silicon (pure) Distilled water Glass Polystyrene Hard rubber Quartz (fused) 1Weast (1984).

¢(S/m) (20°C)

~x(per degree Celsius)

6.29 X 1 0 7 5.8001 × 1 0 7 4.10 x 107 3.541 x 107 1.90 × 1 0 7 1.0 X 1 0 7 8.70 X 1 0 6 2.0 × 106 1.0 x 106 7.1 × 104 4 4 x 10 4

0.0038 0.00393 0.0034 0.0039 0.0045 0.005 0.0042 0.00001 0.0004 -0.0005 --0.07

~

--

10 -4

~ 10 lO_ > 10 14 ~ 10 15 ~

10 -16

10-14

__

__ __ --

1

481

Magnetostatics

1.2.3 Resistance The ratio of the potential difference along a conductor to the current through the conductor is called the resistance R with units of ohms (f~). Referring to an arbitrary conductor as in Figure 1.1, this ratio of voltage to current can be expressed as:

v

f

.al

c

rE.a1 c

R - - I -- I J" d~s - I a E . d~s [a]. 5

(1.11)

274.5°C. At 20°C, using equation 1.12 and a from Table 1.1, you get: R=

1

5.8001 x 107 S/m(~ • 0.0010252 m 2)

As mentioned previously, the conductivity of copper decreases with increasing temperature. At T = 274.5°C, ¢ = 2.9001x 107 S/m from equation 1.9. Hence, the following results

S

R= To conform to the convention that R > 0 for passive conductors, the path of integration c is from the surfaces of higher to lower potential through the conductor, and ds is in the direction of current, as shown in Figure 1.1. If the conductor in this Figure is homogeneous with a cross-sectional area A, then from equation 1.11: 1 V L R -- aA"1~(L) = aA [ ~ ] "

(1.12)

Equation 1.12 can be used to compute R for any straight, homogeneous conductor with a uniform cross-sectional area A at zero frequency. Conversely, if the conductor is inhomogeneous or has a nonuniform cross section, R must be computed using equation 1.11. The resistance R and capacitance C of two perfect conductors (or, simply, two constant potential surfaces) at zero frequency are related as (Cheng, 1981): RC

= -r7,

t~.,J)

where e (permittivity) and a are the material parameters of the otherwise homogeneous space between the perfect conductors.

= 5.224mfL (1.14)

1

2.9001 x 107 S / m ( g . 0.0010252m 2)

= 10.45mfl. (1.15)

This resistance is twice that at 20°C as expected, because the conductivity has decreased by half.

1.2.4 Power and Joule's Law Ohm's law in equation 1.8 relates the conduction current ] to the electric field E at every point in conductive material. Because of collisions between the charge carriers (electrons) comprising the current with the lattice of atoms forming the conductive material, there will be a loss of electrical energy. The power P delivered to electrical charges in a volume v is given by Joule's law:

P = J E. l d v [ W ] ,

(1.16)

v

where P has units of joules per second [J/s] or watts [W]. This power is dissipated as heat in the conductive material through an irreversible process since P is unchanged when the direction of E in equation 1.16 is reversed with ] given in equation 1.8. Considering a conductor with a uniform cross section and a length L, if both E and J are directed along the conductor's length at all points, then from equation 1.16:

Example 1.3

Suppose you wish to determine the resistance of a 1-m length of 12-gauge copper wire at 20°C and at a temperature of

P=J'EdlIJds L

s

(1.17)

=vi[w]. L

I, O', E

.."'"

.......J l

.

""~

i

/

i

,I ,I

;'

(os i ',, s i --,......~ ~ ~

A

This familiar expression for power in electrical circuits can be expressed in two alternative forms using Ohm's law for resistors (in equation 1.11) as:

-

V 2

'~-t" /

P --

".., )

+l'l'v

FIGURE 1.1 Conventions Used in the Computation of Resistance Using Equation 1.11

R

-- I2R[W].

(1.18)

This power is dissipated in the resistor and transferred to its surroundings through joule heating. Example 1.4 If a 75-W incandescent light bulb is connected to a 120-V electrical outlet, equation 1.18 can be used to compute the

482

Keith W. Whites

resistance of the light bulb provided Vand ! are root-meansquare values. In this case: R-

V ~ s -- 1202 _ _ 19a, P 75

(1.19)

which is the resistance when the light bulb is turned on. If the cost of electricity is 5 ¢(kW.h), it would cost $2.70 (= 75 x 24 × 30 x 0.05/1000) to light this bulb continuously for 1 month. The resistance in equation 1.19 will likely be very different than when the light bulb is off and cold. From equation 1.9 using equation 1.12: R - R0 = ~Ro(T - To)[~]

(1.20)

Assuming the tungsten filament is approximately 3000°C when lit and using a = 0.0045/° C from Table 1.1, then using this last equation gives R0 ~ 13 1) for the resistance of a roomtemperature 75-W light bulb. This agrees closely with a measured value of 13.1 ~ even though using a at such a large temperature was not justified.

1.2.5 Conservation of Charge and Kirchhoff's Current Law

Z/j

= 0,

(1.24)

J which is the circuit form of KCL. Example 1.5 Conductors are intrinsically charge neutral even if a current exists in the conductor. If an excess charge is somehow introduced into the conductor, it will redistribute itself until the electric field due to the excess charge is zero. For example, suppose an excess charge is introduced into an isolated conductor. Beginning with equation 1.21, substituting equation 1.8, and using Gauss's law V • (eE) = p, we can determine that the free volume charge p must satisfy the equation (Cheng, 1989): Op a O~-+ 7 p = 0,

(1.25)

with solution p = po e-t#

[C/m31,

(1.26)

where z =-

[s].

(1.27)

O"

A basic postulate of physics is that electrical charge can neither be created nor destroyed. This fact is manifested in electromagnetics through the continuity equation: Op V .1 = -~-~.

(1.21)

This equation relates the net outward flux of J per unit volume to the time rate of change of the volume electric charge density p at every point. When there is no time variation (which is the situation for magnetostatics), the conservation of charge equation 1.21 becomes:

In these results, P0 is the charge density distribution at an initial time and r is the relaxation time constant. For all practical purposes, the charge is redistributed in a time equal to 5z. This time is very brief for metals and much longer for good dielectrics. At these two extremes, copper has a relaxation time z ~ 1.5 x 10 -19 S, while for fused quartz, r ~ 4 days.

1.3 Governing Equations of Magnetostatics 1.3.1 Postulates of Magnetostatics

V . I = 0.

(1.22)

Physically, this equation tells us that the net outward flux of ] per unit volume at every point must vanish. In other words, the electric current density ] acts like an incompressible fluid. Applying the divergence theorem (Paul et al., 1998) to equation 1.22 gives the integral form of the static continuity equation as:

~

l.

The natural phenomenon of magnetostatics is governed by a short and succinct set of equations. The circulation of the magnetic field intensity H [A/m] is governed by Ampbre's law"

V x H = ] (point form). ~ H . dl

=

Ine t

(integral form).

(1.28) (1.29)

c

d s = O.

(1.23)

5

This result is Kirchhoff's current law (KCL) expressed in integral form. Using equation 1.23 at a junction of N conducting wires in a nonconducting space (such as air), the currents /j in all wires satisfy:

The Ine t is the net current passing through the open surface bounded by the closed contour c (Cheng, 1989). Furthermore, the net outward flux of the magnetic flux density B [Wb/m 2 or T] is governed by Gauss's law for magnetic fields:

V . B = 0 (point form).

(1.30)

1

483

Magnetostatics

~

B. ds = 0 (integral form).

I~ [ l(r') x R B(r) = ~ j - -~3 dr' [T].

(1.31)

5

(1.45)

rI

The B and H vector fields are related through the constitutive equation: B=~H[T],

The vector R = r - r' points from the source point to the observation point. For a surface current density ],, the BiotSavart law reads:

(1.32)

where # is the permeability [H/m] of the material. All magnetostatic fields must satisfy the equations 1.28 through 1.31. Very few magnetostatic problems, however, have simple and analytical solutions for the vector fields B and H. Table 1.2 contains a representative list of problems that do have simple analytical solutions. These problems all contain much symmetry, which is the key to arriving at these simple solutions.

B(r) = I~ ~j

fls(r')~3 × R ds'

[T],

(1.46)

{

whereas for a filamentary current I (pointing in the direction of dl' at r'), the Biot-Savart law is written as: bt ~ I ( r ' ) d l ' × R [T]. R3

(1.47)

B(r) = 4 ~ J

cI

Example 1.6 In rare instances, the integral forms of Amp6re's law equation 1.29 and Gauss's law equation 1.31 can be used to solve for H and B. This occurs when the problem contains sufficient symmetry so that H and B can be factored out of the integrals in these two equations (Paul et al., 1998). As an example consider the toroid shown in Table 1.2. Assuming that the wire (carrying current /) is "tightly wound" on the toroid, then using Ampere's law around a circular contour inside the toroid gives:

~

H . dl = NI,

The B field can also be computed from the magnetic vector potential A as: B=g7x

A [T]

(1.48)

where for a volume current density J at source coordinates r':

A(r) = 4~ I ](r~ff) dv' [ W b / m ]. (1.42)

c

because the current pierces the open surface (bounded by contour c) N times. By symmetry and using a cylindrical coordinate system with dl = (#rd(~, in this case, gives: 2"rrr H¢ = 741,

(1.43)

#NI B4~ = ~ IT].

(1.44)

Equation 1.44 is the solution given in Table 1.2. Outside the toroid, Inet = 0, and consequently Be = 0.

1.3.2 B i o t - S a v a r t L a w a n d V e c t o r M a g n e t i c Potential A Amp&e's law (equations 1.28 and 1.29) and Gauss's law (equations 1.30 and 1.31) are the rules that all magnetostatic fields must obey. Except in very limited situations, as discussed in the previous section, these laws cannot be used to directly compute B or H. Instead, if a given current density I is prescribed at source coordinates r', the B field can be directly computed at any observation coordinate r using the Biot-Savart law (Paul et al., 1998):

(1.49)

v'

Expressions for A produced by ]s and Iare similar to equations 1.46 and 1.47, respectively (Paul et al., 1998). In rare instances, computation of A first and then B is simpler than computing B directly using the Biot-Savart law. Example 1.7 A direct current/exists in the circular loop shown in Table 1.2. You can use the Biot-Savart law (equation 1.47) to find a simple analytical solution for B along the z-axis. (This is not true for observation points off the z-axis. Instead, numerical integration of (equation 1.47) would be required) (Whites, 1998, Example 4.10). For this geometry, referring to equation 1.47 and the circular loop figure in Table 1.2, dl' = (o'ad(J and R = r r' = ~:z Fa. Consequently, from equation 1.47: -

-

2x

2~r

~(a ~ z2)3~/7 o

_

[2f

#I 4,rr(a2 + z2)3/2 ~az

4~r(a2~

Z2)3/2 J o

cos4'd4' + ~,az

(~"az +

za 2) d~b !

sin~o'd4' + ~a 2

2;1

d~' .

0

(1.50)

Keitk W. Whites

484 TABLE1.2 Examplesof Problemswith SimpleAnalyticalSolutionsfor B [T] Name

Geometry

Solution

Z

Infinitesheet1

x~

y

(1.33)

B = ±~.~y<>0

Z

L

Straightwire2

''al

Finite:

--k .....

B = ~/~4~/r( cos o{2 -- cos 51)

~

i'~ ' Y

(1.34)

Infinite:

(1.35) -L

O)

(1.36)

B ~ (i'2cos0 + 0 sin0)

(1.37)

B = z^ ~~;~

Circularloop (alongz axis) 3

(x --- y =

Y x

I Z

Magneticdipole1

0 ~ r m=2m

,t ~ Longsolenoid(L >> a) 1

I Z

:Y

~

L

Ntums k/z/

B .~ ~ # ( ~ ) I

inside

(1.38)

Z Toroid'

h

#

~

x~ r - ~ .

~ y Nturns

Coaxialcable1

B= Z p

"~~i Helmholtzcoils(alongz axis)4

0

outside

inside

(1.39)

4~~/~

rw < r < r~

(1.40)

0

r>rs

B= {~Nr r

Nturns each

B= ~.PN2Ia2{ [a2 q- (z + k2)2]-3/2 q_ [a2 q_ ( z , ~)2] 3/2}

(alongz axis)

1Paulet al. (1998).

2Inanand Inan (1999).

3Cheng(1989).

4Whites(1998),Sec4.3, Prob.4.3.4.

(1.41)

1 Magnetostatics

485

1.4 Magnetic Force and Torque

Therefore,

B(z)

#Ia 2 = ~

2(a 2 + Z2)3/2

[T],

(1.51)

which is the result of equation 1.36 listed in Table 1.2.

1.3.3 B o u n d a r y C o n d i t i o n s for B, H, and ] The vector field quantities B and H behave in a prescribed manner at the interface between two different magnetic materials. The component of B perpendicular to the interface (in the direction of the unit vector n) is continuous across the interface. That is: h. (B2 - B1) = 0,

=

(1.53)

Bnl.

The B1 (or B2) is B at the interface but just inside material 1 (or 2) and h is a unit vector pointing from region 1 toward region 2. Conversely, the tangential components of H are discontinuous across a boundary that has a surface current density Js as:

h × (Hi - H1) =

]s.

One of the most fundamental principles in magnetostatics is that a current immersed in a magnetic field experiences a force. This magnetic force behaves very differently than the electrostatic force. Considering two closed loops of current shown in Figure 1.2, the total net magnetic force on the current in loop 1 due to the current in loop 2 is given by Amp6re's force law (Paul et al., 1998): F12 = ~ # I l I 2 ~ dll x(dl2xR12)R~ 2 [N].

This force law can also be expressed in the slightly different form:

,uI, I2 ~R12(d~3.dl2) F12-

4"rr J J Cl C2

F21 =

Hi2

--

Ht~ = Is,,.

(1.55)

A surface current density Is exists at an interface only in certain situations such as an impressed source layer, on the surface of superconductors, and, for time-varying fields, on the surface of perfect electrical conductors (a ---+oo) (Paul etal., 1998). When Is does not exist at the interface, then from equation 1.55:

Ht2 = Htl,

(1.56)

so that the tangential components of H are continuous across the interface. At the interface between two materials with conductivities al and o'2, the components o f ] that are normal (i.e., perpendicular) to the interface are continuous across the interface: ]~2 = J,1,

(1.57)

while the tangential components are discontinuous according to the relationship:

Jt2 ]tl -- = --. O"2

O"

(1.58)

R12

[N].

(1.60)

By interchanging indices 1 and 2 in equation 1.60, it is easy to see that:

(1.54)

In scalar form, using the right-handed coordinate system {n, t, u}, the tangential components of H(Ht2, and Hti) are discontinuous across the material interface by an amount equal to Is,,, which is the component of ]s perpendicular to Ht2 and Htl. That is:

(1.59)

cl c2

(1.52)

or equivalently,

Bn2

1.4.1 A m p 6 r e ' s Force Law

-F12,

(1.61)

as required by Newton's third law (Whites, 1998). Substituting equation 1.47 into equation 1.59, the net force on loop 1 can be expressed as: F12 =

~Ildll

x B2(r~) [N].

(1.62)

Cl It is very clear from this result that a current immersed in a magnetic field experiences a force. Specifically, the magnetic field is produced by the current in loop 2, and the force F12 in equation 1.62 is that experienced by the current in loop 1. Example 1.8 Using equation 1.62 it can be shown that the net magnetic force on any closed loop of current immersed in a uniform B

Loop

1~] ~ d~l

Loop 2 H12

FIGURE 1.2 Differential Current Elements in Two Current Loops

11dll and I2dl2

Located

486

K e i t h W. W h i t e s

C~~~

Because of this second fact, a magnetic force can only change the direction of a moving charge and not its speed. This can be easily understood by considering the differential work d W performed by the magnetic field when the particle is displaced a small distance d l = v d t in time d t (Ulaby, 2001):

X

d W = Fro" d l = ( F m . v ) d t = 0,

B = ,~B o

FIGURE 1.3 Closed Loop of Current I Immersed in a Uniform B Field

(1.66)

since from equation 1.65, F m • v = O.

Example 1.9

field is zero (Ulaby, 2001). When B is uniform, then from equation 1.62:

The Lorentz force equation can be applied to charged particle motion through solids and gases. An example of the former is the Hall effect in a conductor as illustrated in Figure 1.4. In steady-state, the force of equation 1.65 on an electron qe passing through this material is zero such that:

(1.63) E = -v =0

This result is true regardless of the shape of the current loop so long as the B field is uniform. As an example, the net magnetic force on the loop in Figure 1.3 can be computed using equation 1.62 as:

x B = -j'vBo

[V/m].

(1.67)

The difference in potentials at the y = w and y = 0 faces of the material is as follows: w

VI4 = - I E r d y = vwBo [V],

(1.68)

0

F = I~dl

x B = I

(3¢dx) x (3¢Bo) +

c 7~

adqb) x (3¢Bo 0

= I J ( - 3¢sin q~+ j, cos q~)ad(a x (dcBo) = - £IaBo sin ~b]0 = 0, 0

(1.64) which is zero, as expected in light of equation 1.63.

1.4.2 Lorentz Force Equation Moving charges in a magnetic field also experience a magnetic force, similar to current in the previous section. The total force experienced by a charge q moving with velocity v is given by the celebrated Lorentz force equation: F=

q ( E ÷ v x B ) [N],

because of the charge accumulation on these two faces. This VH is called the Hall voltage. From the sign of V,, the polarity of the charge can be determined; from the magnitude of V,, the number density of charge carriers can be computed. A Hall device can also be used to measure conductivities of materials and to measure magnetic field strengths (Inan and Inan, 1999; Halliday, 1992). Imagine that a copper foil of thickness d = 100~m with current I = 20 A is placed in magnetic field with B0 = 0.5 T. Substituting equation 1.2 in equation 1.68 and rearranging gives the following: VH-

//30 dNe

(20A)(0.5 T) (100x 10 6 m)(8.49x 1028 m-3)(1.6022 × 10

19

C)

= 7.35/zV. (1.69)

(1.65)

which is the sum of an electric force ( F e = qE) and a magnetic force (Fro = q v x B). There are two important distinctions concerning the behavior of the electric and magnetic forces in the Lorentz force equation. First, it is apparent from equation 1.65 that the electric force F e acts on both moving and stationary charges, whereas the magnetic force Fm acts only on moving charges. Second, energy is transferred from the electric field to the charged particle, whereas no energy is transferred from the magnetic field to a moving (or stationary) charged particle.

Y B=/Bo÷

/ ++++++++++++

FIGURE 1.4 Hall Effect and a Conducting Material Placed in a Uniform B Field

1 Magnetostatics

487

1.4.3 Torque and Magnetic Dipole Moment In example 1.8, it was mentioned that the net magnetic force on a closed loop of current in a uniform B field is zero. However, this closed loop of current does experience a torque and will rotate if it is free to do so. The torque T exerted on any planar loop of current immersed in a uniform B field is given as (Cheng, 1989):

microscopic magnetic dipole moments m. These magnetic dipole moments can be used to develop a phenomenological model for the classical effects and many quantum mechanical aspects of magnetization (Paul et al., 1998; Plonus, 1978). The macroscopic effects of magnetization are described by a magnetization vector field M as a vector sum of mi in a small volume Av': N

T = m x B [N-m],

~, mi

(1.70)

M = lim i=1 where the magnetic dipole moment m of a planar current loop of area A is as follows:

m = nm = hIA [A-m2].

(1.71)

The unit normal vector h is determined using the right-hand rule. That is, with the fingers pointing in the direction of the current, the thumb points in the direction of n. The magnetic dipole moment m of any planar loop can be computed using equation 1.71. (Note, however, that the B field from equation 1.37 produced by m is valid only at distances "far" from the loop [White, 1998, Example 4.10].)

Av'~0~ Av

(1.74)

[A/m],

where N moments are assumed contained in Avq By definition, the magnetic field intensity H is then given as:

B

H = - - - M [A/m]. /.to

(1.75)

The permeability of free space, ~0, is identically equal to 4w x 10 7 [H/m]. From experimentation, it has been found that for many materials, M and H are simply related as:

M = Zm H [A/m].

Example 1.10 The object shown in Figure 1.3 is an example of a planar current loop in a uniform B field that experiences a torque. The magnetic dipole moment of this loop can be found using equation 1.71:

m= ~a 2

2 [A-m2],

In this expression, Zm is the magnetic susceptibility of the material and is a dimensionless quantity. The values of Zm are typically found through experimental measurement. Substituting equation 1.76 into 1.75 and rearranging gives the following equations:

(1.72)

and the torque can be determined from equation 1.70 as:

T---- (z~'rra 2) × (~¢Bo) = j ~ Bo"rra2 [N-m].

(1.76)

(1.73)

B=P0(1 + Zm)H [T]

(1.77)

B = ~ H [T],

(1.78)

or

where /'/ = # 0 # r = ]A0(1 q-

Xm) [H/m]

(1.79)

The directions of T and the loop rotation--if it were free to move--are both indicated in Figure 1.3. The direction of rotation is determined from T using the right-hand rule: With the thumb in the direction of T, the fingers give the sense of rotation. The loop will not experience a torque when the unit normal vector h is pointing in the direction of B.

is the permeability and #r is the relative permeability of the material. For free space, #r = 1. Equation 1.78 is the constitutive relationship for magnetic fields.

1.5 Magnetic Materials

A magnetic sphere of radius a and permeability # is placed in a uniform incident field Hinc = ~H0. It can be shown that the secondary field inside and outside the sphere is the following (Plonus, 1978; Ramo et al., 1999):

1.5.1 Magnetization Vector and Permeability When magnetic material is placed in an incident (or external) magnetic field, a secondary magnetic field is produced by the material. Similar to a dielectric material, the magnetic material is said to have become polarized by the alignment of

Example 1.11

H sec°ndary =

{ ~-~-~0.( --~-o lio p - p o r d a3

uGSG0~0 V ~'2 cos 0 + 0sin 0

)

r < a

r>a

[A/m]. (1.80)

488

Keith W. Whites TABLE 1.3

z

Material

H

/

0

FIGURE 1.5 Magnetic Sphere of Permeability Ix Immersed in a Uniform H Field. The field lines become strictly vertical when the sphere is removed.

The total H at any point is the sum H = H inc q- H sec°ndary. A plot of the H field lines is shown in Figure 1.5. The magnetic sphere causes H to be disturbed from its original uniform nature. Inside the sphere, however, H is uniform and pointed along the same axis as the incident field, though its amplitude is now less than H0 (and the amplitude of B is greater than/30) when # > #0- This disturbance in H is due to the secondary H produced by the induced magnetization M of the sphere. Using equation 1.76: M =

( Z 3Zm#° H 0 ~- Z 3(~t-/~°) Hd

~

Relative Permeability Ix of Selected Materials

at Zero Frequencyand at Room Temperature1

~

0

o

r < a [A/m], r>a

(1.81)

which is zero outside the sphere because there is nothing to magnetize in free space.

1.5.2 Magnetic Materials There are five major classifications for magnetic materials as briefly discussed in this subsection (Inan and Inan, 1999; Paul et al., 1998; Halliday et al., 1992). Diamagnetic materials have Zm < 1, and the remaining four types all have Zm > 1. Because of this, diamagnetic materials are repelled by a strong magnet, whereas specimens of the other four categories are attracted to a strong magnet. • Diamagnetic materials have a small negative Zm so that #r < 1 as shown in Table 1.3. M1 materials are diamagnetic to some extent although this behavior may be superceded by a more dominant effect, such as ferromagnetism. Diamagnetism is a classical (versus quantum mechanical) effect produced by moving charges. The induced magnetization M is opposed to the applied B, thus reducing the

#r

Diamagnetic (~r ~ 1) Water Copper Silver Gold Bismuth

0.99999 0.99999 0.99998 0.99996 0.99983

Paramagnetic (P'r "~ 1) Air Magnesium Aluminum Titanium FeO2

1.000004 1.000012 1.000021 1.00018 1.0014

Ferromagnetic (~ . . . . . ) Cobalt Nickel Mild steel Iron Mumetal Supermalloy

250 600 2,000 5,000 100,000 1,000,000

1Paul et al. (1998).

total B in such a material sample. This effect is directly analogous to the polarization effects in ordinary dielectrics. • Paramagnetic materials have a small positive )~,~ so that #r > 1 as shown in Table 1.3. Paramagnetism is a quantum mechanical effect largely due to the spin magnetic moment of the electron. While these permanent magnetic moments are usually randomly oriented (so that M = 0), they become partially aligned in an applied B. In this latter state, the magnetization M is aligned with the applied B, thus increasing the total B in the sample (and, consequently, decreasing H). • Ferromagnetic materials also have a positive Zm, but the resulting #r is usually much greater than 1. There is a much stronger quantum mechanical interaction between neighboring spin moments than with paramagnetic materials that can also lead to magnetization M without an applied B. Ferromagnetism is strongly temperature dependent. Only the elements iron, nickel, and cobalt are ferromagnetic at room temperature. Above the Curie temperature To a ferromagnetic material becomes paramagnetic, and the magnetic susceptibility decreases with increasing temperature. The Tc of the room temperature ferromagnetic materials are 770°C for iron, 354°C for nickel, and 1115°C for cobalt. • Antiferromagnetic materials are quite similar to ferromagnetic materials in that there is a strong quantum mechanical interaction between neighboring atomic molecules. However, this strong interaction causes an antiparallel alignment of magnetic moments yielding a zero

1 Magnetostatics

489

magnetization M. The elements chromium and manganese are examples of antiferromagnetic materials. This effect is highly temperature dependent. Below the Nfel temperature TN, the magnetic susceptibility increases with increasing temperature but decreases for temperature greater than TN (Kittel, 1996). • Ferrimagnetic materials possess characteristics between those of ferromagnetic and antiferromagnetic materials. There is an incomplete antiparallel alignment of magnetic moments as in the antiferromagnetic materials, and, consequently, there is a net magnetization M, though typically less than ferromagnetic materials. Ferrites are a special class of ferrimagnetic material that have a low conductivity at high frequencies. These ceramic-like materials are extremely useful in high-frequency applications due to their small conduction losses. Examples include MnZn with/~ . . . . . ~ 2000 and NiZn with #r, max ~ 100.

1.5.3 D o m a i n s a n d Hysteresis The origin of the extremely large permeabilities of ferromagnetic materials is the existence of magnetized domains in these materials (Plonus, 1978; Kittel, 1996). A magnetized domain is a microscopic region (on the order of 0.1 to 1 m m 3) where the magnetization is uniform and saturated. Between adjacent domains are domain walls. Without an external field (at point O in Figure 1.6), these domains are randomly orientated so that the net magnetization of the material is zero. In the presence of external magnetic rid&, these domains align and produce an enormous secondary B in the same direction as the applied field, thus giving a possibly enormous permeability. If the applied fields become strong enough, the domains rotate and the material leaves the initial linear region of operation and enters the nonlinear and multivalued region indicated by the hysteresis curve in Figure 1.6. The ratio B/H at any point on the curves is the permeability #. Around point O, the slope is nearly constant, whereas on the hysteresis curve it is not; this indicates that the ferromagnetic material has become nonlinear.

B[T] ff

B s (Saturation)

(Remanen

(Coercivity)S/ .~~ Hysteresis

H [A/m]

'

FIGURE 1.6 Magnetization Curves for a Typical Ferromagnetic Material. The initial magnetization curve is indicated by the dashed line, while the hysteresis curve is indicated by the solid line. The m a x i m u m B that the material attains for any H is called the saturated B, Bs in Figure 1.6, corresponding to total alignment of domains. When H is then reduced from this point to zero, B reduces to Br, which is the remanent B field or remanance. The negative (or antiparallel) H required to further reduce B to zero is Hc, the coercivity. Materials with a large Hc (called hard ferromagnetic materials) expend more energy per complete transversal of the hysteresis curve and consequently find applications as permanent magnets (see the next section). Conversely, materials with small ~ (called soft ferromagnetic materials) expend much less energy per hysteresiscurve cycle and find uses in transformers, relays, and generators. The properties of a few selected soft ferromagnetic materials are listed in Table 1.4, including tx along the initial magnetization curve in Figure 1.6 and the m a x i m u m Ix found anywhere along the hysteresis curve.

1.5.4 P e r m a n e n t M a g n e t s Permanent magnets are materials that possess a magnetization M in the absence of an applied magnetic field. These magnets

TABLE 1.4 Properties of Selected Soft Ferromagnetic Materials 1 Material (% by weight; remainder is Fe)

Initial Pr

Max/#

Bs [T]2

B~ IT]2

Commercial iron (0.2 impurities) Purified iron (0.05 impurities) Silicon-iron (4 Si) Silicon-iron (3 Si) Mumetal (5 Cu, 2 Cr, 77 Ni) 78 Permalloy (78.5 Ni) Supermalloy (79Ni, 5Mo)

250 10,000 1,500 7,500 20,000 8,000 100,000

9,000 200,000 7,000 55,000 100,000 100,000 1,000,000

2.15 2.15 1.95 2 0.65 1.08 0.79

0.77 -0.5 0.95 0.23 0.6 0.5

1plonus (1978). 2Multiply by 10,000 for cgs unit of gauss [G]. 3Multiply by 47 x 10 3 for cgs unit of oersted JOe].

Hc [A/m]3 ~ 80 4 20 8 4 4 0.16

490

Keith W. Whites B [T]

-H(A/m) • O (ell)max BH (Jim3) Demagnetization-- ] -~ Energyproduct Hc

FIGURE 1.7 Demagnetization and Energy Product Curves for a Permanent Magnet can be constructed from ferromagnetic or other materials that are nonlinear with magnetization curves containing hysteresis, as shown in Figure 1.6. If the material has been properly magnetized, then a remanent B field Br exists in the material even when H = 0. Consequently, this material acts as a per-

manent magnet. In many applications of permanent magnets--such as in electrical motors, generators, and relays--the permanent magnet is part of a magnetic circuit containing an air gap. In these situations, as derived in example, 1.12, the permanent magnet generally operates in the second (or fourth) quadrant of the hysteresis curve (Bozorth, 1978). This second-quadrant portion of the hysteresis curve is called the demagnetization curve and is illustrated in Figure 1.7. One quality measure of a permanent magnet is the "size" of this demagnetization curve. A large Br and/4~ in Figure 1.7 ensures that the intrinsic magnetization of the magnet is large and remains "permanent" even for large H. Another selection criterion used for permanent magnets is based on the energy product BH. This quantity is appropriately named since energy density is proportional to this product (see subsection 1.7.2), which is also plotted in Figure 1.7 by multiplying B and H at each point along the demagnetization curve. The maximum energy product(BH)ma~ is yet another measure of the quality of a permanent magnet since H in the air gap produced by the permanent magnet shown in Figure 1.8 is maximum (with I = 0) when the energy product is maximum. This maximum energy product (BH)max for a selected number of steels, alloys, and other materials is listed in Table 1.5 together with the r e m a n a n c e Br and coercivity Hc.

Hm~oermanent /

Example 1.12 The geometry shown in Figure 1.8 is an excellent canonical problem to illustrate the salient features of permanent magnet circuits. Assume a small cross-section and ignore all leakage flux. The first topic is to show the steps necessary to determine B in this problem, which is the same in the gap and the magnet because of the boundary condition shown in equation 1.52. To do this, two equations will be solved simultaneously. The first equation is the hysteresis curve for the permanent magnet, which is shown in Figure 1.9. The second equation comes from an application of Ampbre's law of equation 1.29 around the permanent magnet circuit in Figure 1.8:

~ H . dl = 741.

(1.82)

¢

Assuming no "flux leakage" from the permanent magnet or "flux fringing" in the air gap (see Section 1.8), then: Hmlm + Hglg = NI,

(1.83)

where Hm and Hg are the magnetic fields in the magnet and air gap, respectively. In the air gap, Hg = B/p, o. Substituting this into equation 1.82 and rearranging gives:

B=-#o

NI Hm+#o~-g [T].

l•g

(1.84)

This straight-line equation 1.84 is also drawn in Figure 1.9. The intersections of this "load line" (Bozorth, 1978, Ch. 9) with the hysteresis curve give the two possible solutions for B in the permanent-magnet circuit of Figure 1.8. The actual operating point depends on the previous time history because this is a nonlinear circuit. Nevertheless, note from this example that the second quadrant of the hysteresis curve (which is symmetrical with the fourth) is the important quadrant. This illustrates why the second quadrant is used as a measure of comparison for magnets, as shown in Figure 1.7. The second topic considered in this example is the relationship between Hg in the air gap and the maximum energy product (BHm)max of the permanent magnet in Figure 1.8. With 1 = 0 in equation 1.82, then:

Magnet Hg = - ~gg H m [A/m].

(1.85)

Multiplying this result by Hg and using the constitutive relationship B = polls yields (Bozorth, 1978, Ch. 9): FIGURE 1.8 Ring of Permanent Magnet Material and Air Gap Excited by a Coil of Wire with N Turns and Current I

i_12 _

lm B H m

Ig

Po

(1.86)

1

491

Magnetostatics TABLE 1.5 Properties of Selected Permanent Magnet Materials 1 Br [T]2

Hc [A/m]3

0.2 0.57

120,000 61,000

8,000 12,800

1 0.95 0.95 0.95

4,000 5,200 5,900 13,500

1,600 2,200 2,600 5,200

Alnico (% by weight; remainder is Fe): I (12 AI, 20 Ni, 5 Co) II (10 A1, 17 Ni, 12.5 Co, 6 Cu) III (12 A1, 25 Ni) IV (12 A1, 28 Ni, 5 Co) V (8 A1, 14 Ni, 24 Co, 3 Cu) VI (8 AI, 15 Ni, 24 Co, 3 Cu, 1.25 Ti) VIII (7 A1, 15 Ni, 35 Co, 4 Cu, 5 Ti)

0.71 0.72 0.68 0.55 1.25 1.03 1.04

33,800 43,400 36,600 55,700 47,700 59,700 126,000

10,700 13,100 10,700 10,300 39,800 29,000 44,000

Samarium Cobalt, SmCO (rare earth): 18 22 26 H 27 H 32 H ~

0.86 0.985 1.06 1.1 1.16

573,000 696,000 736,000 820,000 756,000

140,000 180,000 210,000 220,000 250,000

Neodymium Iron Boron, NdFeB (rare earth): 27 30 H 35 39 H 45 48

1.085 1.12 1.23 1.28 1.355 1.41

768,000 851,000 899,000 979,000 935,000 1,030,000

210,000 240,000 280,000 320,000 350,000 380,000

Material Barium ferrite (Ferroxdure) Iron powder (100% Fe) Steel (% by weight; remainder is Fe): Carbon steel (0.9 C, 1 Mn) Chromium steel (1 C, 0.5 Mn, 3.5 Cr) Tungsten steel (0.7 C, 0.5 Mn, 0.5 Cr, 6 W) Cobalt steel (0.7 C, 0.35 Mn, 2.5 Cr, 8.25 W, 17 Co)

(BH)max[J/m3] 4

lPlonus ( 1978); Pollock ( 1993); Magnet Sales + Manufacturing (1995). 2Multiply by 10,000 for cgs unit of gauss [G]. 3Multiply by 4w x 10-3 for cgs unit of oersted [Oe]. 4Multiply by 4"rr x 10 5 for cgs unit of mega-gauss-oersted [MG-Oe].

1.6 Inductance

B [T]

1.6.1 Magnetic Flux and Flux Linkage

Loadli~?

N~l\.x,.J [A/m] ,,

Hm

HysteresiHsC/ curve ~

Magnetic flux a n d flux linkage are two qualities of a spatially distributed magnetic field that are closely related to inductance. The flux of the B field, ~bm, t h r o u g h a loop is the integral of the c o m p o n e n t of B n o r m a l to the loop over the loop crosssection. Mathematically, the m a g n e t i c flux t h r o u g h loop i p r o d u c e d by c u r r e n t in loop j is defined as (Paul et al., 1998): = [Bj-dsi

[Wb].

(1.87)

5i

FIGURE 1.9 Figure 1.8

Solution for B in the Permanent-Magnet Circuit of

That is, the square of the m a g n e t i c field in the air gap is directly p r o p o r t i o n a l to the m a x i m u m energy p r o d u c t of the magnet. In other words, choosing a m a g n e t with a larger m a x i m u m energy p r o d u c t will p r o d u c e a larger magnetic field in the air gap s h o w n in Figure 1.8.

In the case that i = 1 a n d j = 2, for example, t h e n @m,12 is the magnetic flux t h r o u g h loop 1 p r o d u c e d b y the c u r r e n t in loop 2. Alternatively, magnetic flux can also be c o m p u t e d as (Paul et al., 1998):

Cm,;j

~ a j . dli [Wb], 6i

(1.88)

Keith W. Whites

492 where Aj is the magnetic vector potential of current loop j. Inductors are commonly constructed by wrapping many turns of wire around a core. This is an efficient method of increasing the inductance since for the same amount of current, the flux through (or "linking") the open surface bounded by the contour is approximately proportional to the number of wire turns. Accordingly, for loop i with Ni identical and "tightly wound" turns, the flux linkage is defined as:

Aij = NitP m, ij[Wb ] .

(1.89)

This flux linkage represents the total magnetic flux that passes through (or is "linked" by) the open surface bounded by the multiturn contour. Example 1.13 The toroid in Figure 1.10 will be used to illustrate the computation of magnetic flux and flux linkage. Assume that p is so large that all of the magnetic field is trapped in the toroid. In other words, ignore all "flux leakage." With a current I1 in coil 1, applying Amp~re's law of equation 1.29 along a closed contour in the toroid yields:

Nx Ix

B,b,1 = U-~7-r [T].

(1.90)

Using the definition in equation 1.87, the magnetic flux through coil 1 due to 11 (the "self flux") is as follows:

h/2 b ~m, ll = jB1 dsl = I I *" #N~II"* ~ sl -h/2 a

#Nl2"rrll h ln (\a,/ b~ [Wb], (1.91)

which is also equal to t~m,21 (the "mutual flux") since the cross sections of the two coils are identical. The flux linkage through coil 1 is from equations 1.89 and 1.91:

AH :

g@m,

I l h In (b'~ 11 -- # g (2~ \ a J [Wb],

Z /t0 . a ~ . . ~

N1 turns /1

FIGURE 1.10 Two Wire Coils Wrapped Around a Toroid with Large #

[Wb].

(1.93)

1.6.2 D e f i n i t i o n o f I n d u c t a n c e An inductor is a device that can store energy in a magnetic field. Coils, solenoids, and toroids are all examples of inductors. For an ideal two-terminal inductor with inductance L, the voltage-current relationship is:

VL(t) = L

[V].

(1.94)

For practical inductors, however, nonideal effects may alter this voltage-current relationship through an additional series resistance and possible lead capacitance and inductance (Paul, 1992, Ch. 6). Assuming an ideal inductor, the computation of inductance becomes a strictly magnetostatic problem. Considering one or more loops of current in space, it can be shown that the magnetic flux through a loop is proportional to the current that produces the magnetic flux (Paul et al., 1998). (This behavior is evident in equation 1.91 for the toroid.) The constant of proportionality is called inductance with units of henry (H). Specifically, inductance is defined in terms of the flux linkage and current as: L;j =

flux linkage through ith coil due to current in jth coil [H] current in jth coil (1.95)

or

Lij - Aij _ Ni~m,O [H],

(1.96)

assuming from equation 1.89 that all Ni turns of the inductor are identical. The Neumann formula (Cheng, 1989):

Lij = ~ j j ~ - - / ) (1.92)

whereas the flux linkage through coil 2 is as written here:

N2 t u r n s T ~ ' ~

a21 = S2~//rn,21 -- pN2NIllhln(!)2~

[HI

(1.97)

ci cj is a useful alternative form to equation 1.96, especially for filamentary currents. If i = j, Lii ( = L) is called a self-inductance, whereas if i ¢ j, Lij is called a mutual inductance. A list of self-inductances for a few selected geometries is shown in Table 1.6. Inductance will not depend on current if the inductor is constructed from linear materials. Conversely, if ferromagnetic materials are used to fabricate the inductor, the inductance may be dependent on current since, as discussed previously, ferromagnetic materials can behave nonlinearly. Other than for very simple and ideal geometries, neither equations 1.96 and 1.97 can be analytically evaluated to give

493

1 Magnetostatics TABLE 1.6 Self-Inductancesfor a Selected Set of Geometries Name

Geometry

Inductance [H]

z

T°r°idt

~

(

~

h ..~..~ x

,,y

~ V x "

L :#N2hln(b'~ 2~ \aJ

(1.98)

N turns

z

Solenoid (radius = a)

h

~

~

T L/

#N2~a2

Ntums

(1.99)

h >> a]3: L ~ - - h

lOflN2,a2

h ~- 0.4a (Wheeler formula)3: L ~ - 10h + 9a

(1.100)

L ~ -~.ln(f~'~ +~9.h

(1.1Ol)

L,,~#hln(d) n krw/

(1.102)

2a

Coaxial cablel(h >) r,)

g

Two-wireline2

"~ )~2rw

T0

O J" o 1paul et al. (1998).

)

2Plonus(1978).

3Ramoet al. (1994).

simple formulas for inductance. This is only possible for inductors with high degrees of symmetry, such as the toroid or infinitely long coaxial cable shown in Table 1.6. Mternatively, numerical integration can be used instead to evaluate equations 1.96 and 1.97; the latter is particularly suited for coil-type inductors (Whites, 1998, Example 4.15).

Example 1.14 The self- and mutual inductances of the wire coils on the toroid in Example 1.13 can be computed using equation 1.96 together with the flux linkages in equations 1.92 and 1.93. For coil 1, the self-inductance is the following:

Ln-

An _#N21hln(b_~ [H], 11

2¢r

\aJ

A21

L21- I ~ -

t~N2Nlhln(b" ~ 2w kay [H].

(1.1o4)

In both cases, the inductance is proportional to the square of the number of wire turns. This behavior is common to all coiltype inductors. The self-inductance for coil 2 and the mutual inductance L12 can also be computed using equation 1.96. A current 12 is assumed to exist in coil 2 for the purposes of these inductance calculations. Since the inductance of inductors formed from linear materials does not depend on the current in the wire, assuming this current 12 is only a construction step. From equation 1.96:

(1.103)

which is the first entry in Table 1.6, while the mutual inductance is the following:

A22 _ #N2hln(_b'~ [H], L 2 2 - 12 2"rr \a/ and the mutual inductance is

(1.105)

Keith W. Whites

494

L12-

#N~NZhln(b"]

A12 I2 --

k, aJ

2n

[H].

r~

(1.106)

Comparing equations 104 and 106, it is apparent that L12 = L2> This is a general result. In particular, by interchanging indices i and j in equation 1.97 it can be shown that:

Lq=Lji

i¢j

(1.107)

1.7 Stored Energy

FIGURE 1.11 Coaxial Cable with a Nonmagnetic Center Conductor. All current I in the center conductor is assumed to return on the outer conductor.

and I Hour = ~b~wr rw < r < rs [A/m].

1.7.1 Energy Stored in a Magnetic Field As described by Amp&e's force law discussed in Section 1.4.1, a magnetic force exists between loops of constant current. Were the loops flee to move, they would migrate toward or away from each other depending on their orientations. The work done to keep the current loops stationary is stored as energy in the magnetic field around the loops (Halliday et al., 2002). Energy is also stored in the magnetic field of a single current loop as well as other situations where a magnetic field exists. Magnetic energy Wm can be computed by integrating the dot product of B and H through space as (Cheng, 1989):

Wm= ~

B. Hdv

[]].

(1.108)

From equations 1.108 and 1.78: z0+l 2n rs

Win=~ J l~lHI2dv=~J J Jt~H~rdrdOdz coax

zo

1 win(r) : ~ B ( r ) . H ( r )

[j/m3].

(1.109)

[

wm(r)dv

[J].

(1.110)

,3

all space

Example 1.15 We will compute the magnetic energy stored in a 1-m section of a long coaxial cable shown in Figure 1.11. Using Amp6re's law, the magnetic field inside and outside the center conductor are, respectively, (Whites, 1998, Example 4.2):

Ir

Hin = ~bZ--5- r < rw [A/m] L~r;

(1.111)

(1.113)

i }

Substituting He inside and outside the center conductor from equations 1.111 and 1.112, respectively, and integrating gives

ffln(rs'~]\rwjl

W~ : i2[ 1v~ +#4 0

[J]'

(1.114)

for the 1-m section of coax.

1.7.2 Energy Stored in an Inductor Inductors are the primary circuit elements for storage of magnetic energy. Ideally, they are the only circuit elements with this property. The magnetic energy Wm stored by an inductor with current I is as follows:

The total stored magnetic energy can be computed by integrating Wm throughout space as:

Win=

0 0

=IZ P~o H~,inrdr+D H~,outrdr [J]" 0 rw

all space

Any change to this stored magnetic energy occurs when the magnetic field changes with time according to Faraday's law. Once B and H have reached a steady-state, equation 1.108 provides a method to compute the time-stationary stored magnetic energy. A magnetic energy density Wm can be defined at every point r in space from the integrand of equation 1.108 as:

(1.112)

W m

= ~ZI 2 [J].

(1.115)

In this expression, L is the self-inductance of the inductor as discussed in Subsection 1.6.2. With coupled inductors--such as when two or more coils are wrapped around a common high permeability core--the energy expression becomes more complicated. In the case of two coupled inductors with currents I1 and I2 in coils 1 and 2, respectively, the energy stored in the magnetic field is (Paul et al., 1998):

Wm=

LHI (

1 2 +L1211/2 [J], +~L22I~

(1.116)

1 Magnetostatics

495

w h e r e Lll and L22 are the self-inductances of coils 1 and 2, respectively, and L12( = L21 ) is the mutual inductance between

the two coils. As expected, the first two terms in equation 1.116 represent the magnetic energies stored in coils 1 and 2, respectively, while the last term is the energy stored in the mutual inductances of the two-coil system. One potential subtlety is that these self-inductances in equation 1.116 are those computed for a coil with the other one present but having a current equal to zero (Paul et aL, 1998). Example 1.16 The energy stored in a 1-m section of the long coaxial cable in Figure 1.11 was computed to be equation 1.114 in example 1.15. This magnetic energy calculation will be repeated here but using the inductance expression of equation 1.115. The inductance for a 1-m section of coaxial cable is given from the third entry in Table 1.6 as:

mation used to solve for magnetic fields and magnetic fluxes (Cheng, 1989; Paul et al., 1998; Planus, 1978). Applications for this method include the design of electrical generators, motors, transformers, and actuators. Magnetic circuit analysis is quite similar to electrical circuit analysis. Quantities that serve analogous purposes in both types of circuit analyses are listed side by side in Table 1.7. Magnetic circuit analysis is illustrated with the specific example of the toroid shown in Figure 1.12(A). There are two primary assumptions made in magnetic circuit analysis (Paul et al., 1998). First, it is assumed that the permeability of the structure is very large (# >> #0). Second, since # >> #0, it is further assumed that all of the magnetic field lines flow along the magnetic core and do not deviate outside of the structure; that is, it is assumed there is no flux leakage out of the toroid in Figure 1.12(A). Applying Amp~re's law of equation 1.29 gives:

~

H.

L= #in(r') 2~r

#o

77w + ~

(1.117)

[H].

dl

=

NI,

(1.119)

C

such that inside a linear magnetic core: Using this inductance in equation 1.115 with a current I in the coaxial cable gives:

Wm

=

1 [# ln(r'~

#0112

2 [2-rr krw/ + ~

(1.118)

[J]'

~NI B + = ~ - ~ - [T].

(1.120)

Another common assumption in magnetic circuit analysis though not required - is that the cross-sectional dimensions

which is identical to equation 1.114 as expected. In some situations, applying this procedure in reverse can be a convenient method for computing inductance. That is, the stored magnetic energy is computed first, and from this, the inductance is determined using equation 1.115.

If/m

1.8 Magnetic Circuits Magnetic circuit analysis is a technique that can be applied to certain magnetic field problems to greatly simplify the solution. In short, this technique is a lumped-element approxi-. TABLE 1.7

(A) Physical Geometry

(B) Equivalent Magnetic Circuit

F I G U R E 1.12 Toroid with High Permeability Core of Circular Cross-Sectional Area A

Analogous Quantities in Electrical and Magnetic Circuit Analysis I

Electrical circuits

Magnetic circuits

Conductivity [S/m]

Permeability [H/m]

//

g~

EMF [V]

V

MMF [A]

Current [A]

I

Magnetic flux [Wb]

Resistance [1)]

Reluctance [H 1]

Ohm's law

R = ±aA V = IR

KVL (around a loop)

Y~ Vi = ~ RjIi

KCL (at a junction)

~ Ij = 0

i J

1paul et al., (1998); Cheng (1989).

Vm = t)mR

j

i

Vm,i = ~ RjOm,j J

E ~bmj = 0 J

496

Keith W. Whites

are small with respect to the length (the "large aspect ratio" assumption). In these situations, B will be approximately uniform over the cross section. Assuming that is the case here, then from equation 1.120: B4 ~ p N I = B [T].

(1.121)

2wa

Consequently, the magnetic flux @m in a core with crosssectional area A will be approximately I~m ,'~ B A --

pANI

2"rra

[Wb],

(1.122)

and the flux will have this value all around the toroid. The equivalent magnetic circuit for the toroid in Fig. 1.12(A) is developed by expressing equation 1.122 as the ratio of two quantities: NI ~Jm-

Vm

[Wb].

(2rraX~ - - R k PA/I

(1.123)

In the numerator, Vm=NI

[A].

(1.124)

Equation 1.124 shows what is called the magnetomotive force (MMF) that serves an analogous function in this circuit as a voltage source (EMF) in an electrical circuit. In the denominator of equation 1.123:

ent magnetic flux paths, as appropriate, then computing the reluctances of the paths using equation 1.125. The last two entries in Table 1.7 list the laws used to solve magnetic circuit problems. In particular, the sum of mmfs around a closed loop is equal to the sum of"reluctance drops" (KVL analogy), while the sum of magnetic fluxes into a junction equals zero (KCL analogy) (Cheng, 1989; Paul et al., 1998) Example 1.17

The geometry shown in Figure 1.13(A) is used to illustrate the solution of magnetic field problems using magnetic circuit analysis. The structure is assumed to have a square cross section of area 10 6 m 2, a core with #r = 1,000, and dimensions 11 = 1 cm,/3 = 3 cm, and 14 = 2 cm. One distinguishing characteristic of this structure is the air gap at the bottom of the center section shown in Figure 1.13(A). If the length of the air gap is small with respect to the cross-sectional dimensions, we can ignore flux fringing (or "spreading out" of the B field lines) in the air gap (Plonus, 1978). Consequently, this gap can be simply modeled as another reluctance as indicated by Rg in the equivalent magnetic circuit of Figure 1.13(B). The goal here will be to solve for the magnetic flux density in the air gap. Using equation 1.125 the four reluctances in Figure 1.13(B) can be calculated as: R] -

211 + 14 _ 31.83 x 106 1000PoA

[H-l].

R 2 = 14 - lg - 0.001/2 = 15.44 × 1 0 6

1000#oA R =--

1

pA

[H-l].

(1.125)

Equation 1.125 shows the reluctance of the core (with mean length 1), which is analogous to resistance as shown in Table 1.7. The equivalent magnetic circuit for the toroid is shown in Figure 1.12(B). Other magnetic field problems can be solved using this magnetic circuit analysis by dividing the geometry into differ-

I1

,

R3 -

213 + 14 _ 63.66 × 106 1000PoA

lg _ 79.58 × 106 R g - P° A

[H-l].

[H-l].

6

10 mA

II /2

turns

II

~m,1

0.1 ~'

mm I/fm, 3

1 mm

(A) PhysicalGeometry

[H-l].

(1.127) (1.128) (1.129)

The magnetic flux through the source coil, I~m,1 , c a n be calculated as the source mmf divided by the total reluctance seen by the source as:

Iffm, 1 ~1

2000

(1.126)

(B) Equivalent Magnetic Circuit

FIGURE 1.13 MagneticCircuit Geometry with a Small Air Gap

1 Magnetostatics i)m, 1 --.

V m

Rtotal

--

497 NI

R1 + (R2 + Rg)IIR3 = 0.286 x 10-6[¥Vb]. (1.130)

Using "flux division" (which is analogous to current division in electrical circuits), the magnetic flux through the air gap is then:

R3 Ore.1 = 0.115 × 10-6[Wb]. @m,2 = ~P~3-[- ~PL2-[- ~ g

(1.131)

References Bozorth, R.M. (1978). Ferromagnetism. New York: IEEE Press. Cheng, D.K. (1989). Field and wave electromagnetics. (2d ed.). Reading, MA: Addison-Wesley. Inan, U.S., and Inan, A.S. (1999). Engineering electromagnetics. Menlo Park, CA: Addison Wesley. Halliday, D., Resnick, R., and Krane, K.S. (2002). Physics. (5th ed.). New York: John Wiley & Sons.

Kittel, C. (1996). Introduction to solid state physics. (7th ed.). New York: John Wiley & Sons. Magnet Sales & Manufacturing. (1995). Catalog 7, High-performance

permanent magnets. Paul, C.R., Whites, W., and Nasar, S.A. (1998). Introduction to electromagnetic fields. (3d ed.). New York: McGraw-Hill. Paul, C.R. (1992). Introduction to electromagnetic compatibility. New York: John Wiley & Sons. Plonus, M.A. (1978). Applied electromagnetics. New York: McGrawHill. Pollock, D.D. (1993). Physical properties of materials for engineers. (2d ed.). Boca Raton: CRC Press. Ramo, S., Whinnery, ].R., and Van Duzer, T. (1994). Fields and waves in communication electronics, (3d ed.). New York: John Wiley & Sons. Schein, L.B. (1992). Electrophotography and development physics. (2d ed.). Berlin: Springer-Verlag. Ulaby, ET. (2001). Fundamentals of applied electromagnetics. Upper Saddle River, NJ: Prentice Hall. Weast, R.C. (Ed.). (1984). CRC handbook of chemistry and physics. (65th ed.). Boca Raton, FL: CRC Press. Whites, K.W. (1998). Visual electromagneticsfor mathcad. New York: McGraw-Hill.