ELEMENTS
OF MICROWAVE

NETWORKS
Basics of Microwave Engineering

CARMINE VITTORIA
Nor-/lwcls/rmUniv., USA

Y

$World Scientific
Singapore*New
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Vinoria C.
Wemenu of microwave networks : basics of microwave engineering /
Carmine Vittoria.
p. cm.
Includes bibliographical references(p. ) and index.

ISBN 9810234244
I. Microwave circuits -- Mathematical models. 2. Microwaves - Mathematical models. 3. Microwave transmission lines -- Mathematical
models. 1. Title.
TK7876.VS8 1998
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Preface
The purpose of this book is to bridge the knowledge gap between fundamental
textbooks used in required courses and the rapid advances made in the

microwave technology. Today, microwave systems and/or requirements are
too complex or advanced to be analyzed by simple graphical approaches as
the Smith chart, for example. As such, this book is to be used in an elective
course at the undergraduate level. In putting together this book, I chose topics
which (1) did not duplicate topics covered in textbooks used in required
courses, (2) were a logical extension of required courses, and (3) reached out
to a larger audience of students majoring in science and engineering.
Using criteria (1) as a basis I have purposely left out all formal
derivations of Maxwell's equations and subsequent material related to it. I
introduced Maxwell's equations as a given fact only to make some
connections in later chapters. From criteria (2) the following topics were
chosen: (a) lossy medium, (b) scattering S parameters, (c) matrix
representation of wave propagation, and (d) microwave properties of ferrite
networks. All microwave systems contain some element of losses. As such,
lossy medium as well as absorbing networks were extensively covered. The
S matrix and its properties is introduced for the sole purposes of preparing
students for the demands of the microwave industry. The matrix
representation of wave propagation has been used for many years in other
fields of science. We have re-introduced it here in relation to microwave
networks. Finally, ferrite usage represents the largest industry in technology.
We should introduce topics in ferrite networks in a way that is complementary
to the core material in undergraduate education. We need to go beyond the
description of a scalar permeability at the undergraduate level.
The book consists of eight chapters. The first two chapters are
introductory material for students who do not have an adequate background
in electromagnetic theory, such as students in other engineering disciplines
and science majors. These two chapters are not to be covered in a regular
course. A semester course covers six chapters starting with chapter 3. It is not
necessary to cover every single topic in each chapter. It is possible to adapt
or re-arrange the topics in a way to meet the needs of students depending on

the makeup of the class. For example, as an elective course we covered
chapters 3-8 minus topics in ferrite networks and specialized topics in a
quarter which is shorter than a semester. Although it was an electrive course
V


vi

Elemcnb o j Micrmwve Nctmonkr

for juniors/seniors, the book could be used as a springboard to a more
advanced course at the graduate level bu including advanced topics in ferrite
networks and matrix representation of microwave networks in general.

I wish to thank Professor George Alexandrakis for his hospitality while
I was preparing the manuscript at the University of Miami. Also, the able
assistance of Drs. H. How and W.Hu, and Y. Biaz is acknowledged.


Contents
Preface

V

Review of Maxwell Equations
Chapter I
A. Maxwell's Equations in MKS System of Units
B. General Constitutive Relations
C. External, Surface and Internal Electromagnetic Fields
D. Practical Example

E. Electric Polarization of Microwave Signal
F. Microwave Response and Polarization
References
Examples

i
1

2
5
6
10
13
14
14

Common Waveguide Structures
Chapter II
A. Parallel - Plate Waveguides
B. Coaxial Line
C. Rectangular Waveguide
References
Examples

21
21
26
29
33
34


Chapter 111 Telegraph's Equations
A. Types of Transmission Lines
B. Wave Equation
C. Connection to Circuit Parameters
D. Formal Solution
E. Electrical Quantities
F. Bounce Diagrams
References
Examples

39
39
39

Chapter IV Analytical Solution
A. Lossy Case
B. Real Time Solutions
C. Lossless Case

48

52
57
68
79
79
90
91
96

97

vii


viii

Elements of Micrwwovc Networks

D. Determination of I, and I,
E. Coupling Between Source and Transmission Line
References
Examples

102

Chapter V
Graphical Solution
A. Mathematical Terminologies
B. Plot of r ( - I )in Complex Plane
C. Projection of Z(-I) onto the r(-I)Complex Plane-Smith Chart
D. Projection of V(4) on Smith Chart
E. Projection of I(-() on Smith Chart
F. Graphical Methods for Lossy Lines
References
Examples

127
127
128

131

109
112
113

141
146
147
150
151

Chapter V1 Special Topics in 'hansmission Lines
A. Stub Tuners
B. Graphical Solution
C. Quarter and Half Wavelength Transmission Lines
D. Microwave Absorbers
References
Examples

160
1 60

Chapter VII Electromagnetic Scattering Parameters
A. Definitions of S-Parameters
A. 1 . Algebraic Properties of S Matrix
B. Relationship Between Measurements and Network
Electrical Parameters
C. Example of Inverse Scattering Problem
D. General Comments

References
Examples

199
199
199

Chapter VIlI Matrix Representation of Microwave Networks
A. Transfer Function Matrix of Two Port Network

163
169
176
188

189

210
214

215
216
217
224
224


B. Transmission Line Analysis Using Matrix Representation
C. Connection Between Scattering Parameters and Matrix
Representation

D. General Properties of Matrix Representation
D. 1. Single Networks
D.2. Cascaded Networks
D.3. Periodic Network Systems
D.4. Application
E. Some Special Properties of Ferrite Networks
F. Relationship Between Scattering S-Parameters and
Matrix Elements
G. Example of Inverse Scattering
H. Special Applications of Matrix Representation
1. Three and Four Port Networks
I. 1. Three Port Network
1.2. Four Port Network
J. Equivalent Circuit of Ferrite Circulator
References
Examples
Index

226
229
232
232
236
237
24 1
244
247
252
254
256

257
26 1
262
264
265
283


Chapter I
Review of Maxwell Equations
The foundations of electromagnetic wave propagation in a medium are
Maxwell’s equations. In this book we introduce measurement quantities such
as voltage, V, and current, I, to solve for electromagnetic wave propagation
characteristics of specialized guided structures. As a result of introducing
measurement variables V and I. it is mathematically convenient to introduce
electrical circuit parameters, such as R, L, C and G. Before we go much into
details of the analysis of wave propagation let’s re-acquaint ourselves with
Maxwell equations written in terms of field variables E and f i , the electric
and magnetic field variables.

A. Maxwell’s Equations in MKS System of Units
Maxwell’s equations in MKS maybe written in the differential form as
follows:

where

j = d

The two constitutive equations are usually written as
B=h(fi+@=pfi


Gi=&$+jLEE

mi

,

The units of each field quantity are defined as follows

J = Current Density (amplm’)

B = Magnetic Flux Density (webedm’)
H = Magnetic Field Intensity (amp/m)
1


2

Elmentr of Macroawvc Networh

D = Electric Displacement (coulomb/m’)
p = Charge Density (coulomb/m’)

u

= Conductivity (mhoslm)

M = Magnetization (amp/m2)
P = Electric Polarization (coulomb/m*)
p,, = Permeability of free space (henrys/m)


q,= Dielectric constant of free space (faradsh)
E and p may be defined in terms of their respective susceptibilities x

P=N(l+X,)
E=Eo(l+X,)

or

.

where x, = magnetic susceptibility and x, = electric susceptibility.

B. General Constitutive Relations
We have assumed in the above relations that both p and E are scalar
quantities. This implies for example that the differential permeability is
isotropic.

and that

Similarly,

ax 6Dy a!
4 4- 4

&=-=---

and



Review of M w w c l l Equotioru 3

For most practical microwave materials p and E are not scalar, but
tensors. The above relations state that E and p are proportional to induced
incremental polarizations (upon application of external E or g ) . The
polarization may be in the form of induced magnetization or electric
polarization. The more one is able to polarize a material the higher the values
of p or E . However, if we allow the possibility that the induced polarization
direction (magnetic or electric) may be orthogonal (in direction) to the
application of an external field, we find that we can no longer describe p and
E in terms of a scalar quantity. In general a change in polarization along a
given direction may be due to fields applied in an arbitrary direction.
Mathematically, an incremental 6B along a given direction may be expressed
as follows

We have implied in the above equation that a change in 6B,, can be
induced by a small change in field along the x, y and z directions. In general,
one may write

c

In matrix form one may represent eq.(8) symbolically as follows

=z
MI

We define a differential susceptibility tensor element X,
, where
i, j = x, y, z. is the unit matrix. Hence, pii = (1 +x,J p,, and pIj’, for i j.
Finally, a tensor for [p] may be defined as follows


m


4

Elements of Micnwow Ncturwlks

or equivalently

The tensor for [E] can be written in a similar fashion.

[&I= 4 ~ l + [ r , l ]
or equivalently

Here now

(&)q

1B:

=--

for i, j = x, y and z.

4 a,

In summary, the value of each matrix element of [E] and [p] is
proportional to the induced electric or magnetic polarization. The existence
of the off-diagonal matrix elements of [E] and [p] is due to anisotropic

inductions of polarization in a magneto-dielectric medium. If changes in
external fields are defined with respect to a zero field reference, we may write
the constitutive equations simply as follows


Review of Manuel1 Equotiona 5

where 5, is a unit vector in the i direction (x, y,

2.).

C. External, Surface and Internal Electromagnetic Fields
Maxwell's equations represent a set of equations which relate the internal
electromagnetic fields in a magneto-dielectric medium to the polarization
fields of that medium. It establishes fundamental relationships between these
set of fields. Polarization fields are the result of local interactions between
the electromagnetic fields and the medium. A convenient way to relate
internal fields with surface electromagnetic fields is via the so-called Poynting
relation or integral

where

cf?

is a vector normal to a surface containing the medium of interest

and is in units of m2. The left hand side of the equation is a surface integral,
where the surface is enclosed and the fields are determined on the surface.
The right hand side of the equation is a volume integral in which the fields are
determined within the enclosed surface. They are internal and polarizing

fields. The minus sign is important, if we want to conserve energy as the
electromagnetic wave propagates in and out of the enclosed surface. In Eq.
(16) p and E may be either scalars or tensors. The total stored energy density
is usually written as

where U, and U, are the stored electric and magnetic energies, respectively.
The Poynting relation may be written in terms of U

The above equation states that the net time rate of change of
electromagnetic energy within an enclosed volume is equal to the negative of
the total work done by the fields on the medium. It is understood that the left
hand side of the above equation contains fields only at the surface of the
medium. The beauty of the above relationship is that any changes in the
stored energy or the potential energy within a medium manifest itself as a
change in the surface fields. In fact it is exactly this principle that allows


6

Elements of M i c m w v c Networks

experimentalists to be able to characterize the properties of microwave
materials. Interaction between internal fields and the medium are included in
U. Finally, surface fields may be related to fields far removed from the
medium by simple application of electromagnetic boundary conditions at the
surface, see figure 1.1.

p

External


I

or Incident
Wave

dg

Fig. 1 . 1 Internal, external and surface
elctromagnetic fields.

D. Practical Example
If there is no absorption of microwave energy within the enclosed surface, the
steady state power entering and exiting the enclosed surface is the same.
Mathematically, it means that if we integrate over the enclosed volume and
average the power over one period of time, the average power enclosed is
zero, assuming the dissipative term
is zero. Let's consider conceptually
the application of eq.(16) by assuming wave propagation in one direction, zaxis, and l? and fi perpendicular to each other but transverse to the z-axis.
Thus, surface integration over the x-z and y-z planes is zero, since E and A
are in the x-y plane, see figure 1.2.
The two surfaces of integration are the surfaces in which
electromagnetic energy is impinged upon and in which the energy is exiting
the volume in question. At the input surface one can always adjust the
relative phase of E and fi to be zero so that one can write at the input
surface the following fields


Revicw of M o w e l l Equatim 7


(19)

Fig. I .2 Field configuration relative to enclosed
volume.

At the other surface E, and fi, are again assumed to be in the x-y plane but
have now some phase relative to the input fields at surface (1). Write at the
output surface the following
<”(t) = E2

H2

+ k)ix

(20)

* HI .

The phases 6. and 4+, may be a result of wave propagation from one surface
to the other of the enclosed volume.
The superscripts (1) and (2) designate the input and output surfaces,
respectively. Q may be less then E,,if there are dissipations in the enclosed
medium. According to the left hand side of eq.( la), the surface integral yields
at the input surface

j(E!l)(~)xel)(t)).di=
E, H,~

a


~ = pw
0 .r

(22)


8

Elemmb of MicrOuMw Netwoda

This is the instantaneous power applied at surface (1). The units in the above
expression are as follows
volts amps
---.m2
m
m

e watts

The input power averaged over time is

(P,)

= E~HI ~ / d a ~ d

T o
( P N ) = E l H , A / 2,

where A is the cross area and T the period.


The above result can be deduced conveniently by making use of the
complex number representation for I? and A . Thus, for example, introduce
the complex number notation
= El ii,
= Hliiy

,

p(,)
=

C

and
where

~

I

I
,

~

for
] example.

In the above complex notation or representation E, and H,may be
complex numbers. In order to compare with previous results we will assume

both to be real. (PN)may now be calculated from
( P , , , )1= ~ R + ~ x~ ( ~ ~ i i ~ ) I’ . Hd , -A~. ~ ,

Let’s now calculate the average output power, (P-,),
)‘f(

=

21 Re/

x

,!?!j2)

(q2’).
.&,

where in the phasor representation we have

q’= E28 4 (3,
and

R2)= H~eJAii,

Thus,

(p,)=jE2H2cos(4-4)A

I


.

-


Let's now formally perform the contour integral over the whole surface and
average over time.

The minus sign in the above is the result of d? pointing in -z direction
at the input surface. The ratio (E,H, / El HI) is always less than 1, if there
is dissipation in the enclosed medium. However, if the ratio is exactly equal
to I , + . = &. Since the system is conservative, the average power out must
be equal to the average input power. This means that 6 and fi must
necessarily have gained the same phase at the output of the enclosed medium
for a lossless medium.
We may express above results in terms of measurable quantities, such
as voltage, V, and current, I, at the two surfaces in question. Thus,
let's write the following

Clearly, V is related to E, and some length factor and I to H, and a length
factor. V and I may be complex. The asterisk implies complex conjugate. We
can now re-write the Poynting vector in the following form

Clearly, the left hand side is the average dissipated power enclosed in
the volume or - ( p o r n ) , if the right hand side is non zero. Write

In general the scalar permeability and permittivity may be written as
p =p'- j p"

and


E

=

E'-~E".


10 Elemenls of Micmwavc Netmorka

Let's draw the analogy from circuit analysis in which power dissipated in a
circuit may be written as

(Pms) = -21 I 'R + -2I GV'

*

Thus, we can identify G with E" and R with p" and
in eq. (28).
-.-.-.m3
hv

sec

-'

I

m
m'

md fanads volrs'

-.-.-.
s8c
and

m
mhos d r s '
-.-.m3
m
m'

oh.anlpF'rwms

m3I mhm.wirs'

m'

1
;
. Let's check the units

I watts

=mhos.vdrs'rwafis

In some sense it is plausible to describe wave propagation in terms of
voltage, current and circuit elements, since there is an exact accountability
of energy flow.
E. Electric Polarization of Microwave Signal

We need to distinguish between the polarization of a material and wave
polarization. By material's polarization we mean either the material's
magnetization field, Q , or electric polarization field, p . The polarization
of the wave is described by the locus of the tip of the E vector as time
increases.
In general the electric field is a vector with components in the x, y and
z- directions. For simplicity, consider a microwave electric field represented
by complex numbers in the x and y directions.
(30)

= E,,e-lhix -jEoe-Jkiy*

In real time the electric field may be obtained as follows.
E(z,r) =

&.r)

= Re(ED+'-'')b,

h

+ R{EOe"*'-'*-*'')

-

-

= E, C O ~ W
- k)ii,+ E,, S ~ ( Wk)Z,


If we were to plot E(2.r) at
circle, see figure 1.3.

Z=O

for increasing time,

9

(31)

E(w) traces out a


Revim of MorwcIl Equatiom 11

Fig. 1.3 Locus of E(0,t) as t increases.

The locus of E(0.r) for increasing time can be shown to be a circle by
recognizing that

Ex([) = Eocos WJ
and

EJr) = Eo sin

WJ

The amplitude of E(0J) is then


lE
Jq
=J E ~
cos*

wl

+ E; sin’d

This is the equation of a circle in the x-y plane.
For z + 0, we have a wave traveling in the +z direction. However, at
given z a circle is traced out whose rotation sense is shown in figure 1.3 .
This sense of rotation is referred to as a right hand circular polarization. This
convention may have been adopted From the notion that if your thumb in your
right hand is pointing toward the propagation direction, the fingers will point
toward the sense of rotation. Another convention is to denote such a wave
with the superscript (+) or write


12

Elements of Micnnvaue Network

A?? = E,,e-"ii, -jEoe-lbZy

(33)

*

For a left hand circular polarization, one would write
?l = Eoe-J"is,+jEoe-*iY

(34)

*

For this polarization the sense of rotation is clockwise, see figure 1.3.
Sometimes, it is convenient to combine the two types of polarization and write

Linear polarization is a special case of the circular polarization. The electric
field is only along one direction. For example, for polarization in the y and
x directions one writes the following.
Ex = 0,Ey # 0
and

E,

* 0, Ey = 0 .

We can decompose a linear polarization into two circular polarizations. For
example, a linear polarized electric field along the xdirection may be decomposed as follows:

and

(36)

In real time the linear polarizations become
~,(z.r)=~ee
( ~, ., 1 ] = ~ e ( ~ , d ( " ' - ~ . ) I = ~ , c o d ~ - k )

and

~ , ( t , r )= Re[E,

~ , c o ~ ( O- rk)1

~ J Y ' ] =

for example.

Clearly, if the amplitudes of E'and E- are not equal, the resulting
polarizations from the sum and differences result into elliptical polarizations.
Common usage is to designate an E field by superscripts in order to represent
the sense of circular polarization instead of explicitly writing out the E field
dependence on direction, for example.


M e w of Mamuell EqwHons 13

F. Microwave Response and Polarization
The microwave response of a material depends on the polarization of the
wave of the exciting microwave or the input microwave stimulus. For
example, for dielectric and metallic mediums one expects that if the material
is excited by a linearly polarized wave, the output will also be linearly
polarized wave. Also,if the input is circularly or elliptically polarized wave,
the output would have the same wave polarization. Thus, polarization is
conserved. Generally, this axiom does not hold for ferrite materials,
especially ferrites biased by a DC external magnetic field. Let me now
explain this basic difference between dielectric and ferrite materials.

Let's consider two special cases: magnetically (1) biased and (2)
unbiased ferrite materials. By magnetically biased we mean that a DC
external field is applied to the ferrite material while a microwave signal
excites the ferrite. For case (l), wave polarization is not conserved. This
means, for example, if a ferrite is excited by a microwave signal which is
linearly wave polarized, the output or the scattered microwave signal is not
linearly wave polarized. The output signal or response may be either
elliptically or circularly polarized. There are special situations in which the
output may be linearly polarized, but we do not wish to dwell on this
exception. For most practical applications the non-conservation of wave
polarization is a desirable property of biased ferrites, as in applications of
non-reciprocal microwave devices.
The difference in the microwave response between dielectric and ferrite
materials is based upon fundamental laws of physics. The material's electric
polarization, p , of a dielectric material consists of electric dipole moments
and is a result of a linear displacement of atoms or ions. Thus, the response
of a dielectric material mimics that of the excitation of an input microwave
signal. In a ferrite material the magnetic polarization, fi,or magnetization
consists of magnetic dipole moments, ti, whose origin is due to circular
motion of currents. Femtes do not respond to electric field excitations but do
respond to microwave magnetic field excitations. Application of magnetic
fields induces a circular motion of fi or a precessional motion of fi,
irrespective of the wave polarization of the input signal. Thus, the normal
modes or the natural motion of the magnetization is circular. It is not
surprising to observe that the microwave response of a ferrite material is a
wave which is circularly polarized, although the input microwave signal may
be linearly polarized wave, for example.


14

Elements of Macmwave Network

For case (2) in which the ferrite is unbiased by an external DC
magnetic field, wave polarization is conserved. In such cases the ferrite
consist of small ferrites randomly positioned with respect to each other.
These small ferrites are sometimes called crystallites and they are too many
to count. In this limit microwave wave polarization other than the input
polarizations cancel each other out within the composite ferrite, if the
crystallites are randomly positioned in the ferrite. The net effect of this
randomness in magnetization direction gives rise to reciprocal response of a
ferrite material.

SUGGESTED REFERENCES
1. S. Ramo, J.R. Whinnery and T. Van Duzer, Fields and Waves in
Communications and Electronics, Wiley, New York, 1965.
2. B.I. Bleaney and B. Bleaney, Electricity and Magnetism, Clarendon Press,
Oxford, 1965.
3. Leo Young, Systems of Units in Electricity and Magnetism, Oliver and
Boyd, Edinburgh, 1969.
4. C. Vittoria, Microwave Properties of Magnetic Films, World Scientific,
Singapore, 1993.
5 . B. Lax and K.J. Button, Microwave Ferrites and Ferrimagnetics,
McGraw-Hill, New York, 1962.
6. 0. Dunlap, The Story of Radio, Dial Press, New York, 1935.
7. H.A. Meyer, A History of Electricity and Magnetism, MIT Press,
Cambridge, MA, 1971.
8. E. Larsen, Telecommunications: A History, F. Muller, Ltd., London,
1977.
9. G.C. Southworth, Forty Years of Radio Research, Gordon & Breach, New

York,1962.

Example 1
An incident wave is incident upon a thin layer of dimensions of
1 o o X 1 ~ X 1 0 4d .
The incident power is 1 mW, the transmitted power

through the thin layer is 0.8 mW and the reflected power is 0.1 mW.


Review of Matwell Equationt 15

Calculate the internal electric field if the absorption is only due to the
conduction of the layer and is equal to 2 mhos/m.
Let’s designate the power absorbed as,,,P
transmitted power as PT.Then,
pDISS =PINC

where

,P

reflected power as PR,and

- (PR + pT>,

= incident power.
1
2

P , , ~= ~ o - ~ w = - ~ ~ ’ Y ,
where

V volume of layer

= 10%).

IEl= 10 volts/m.

Example 2
A circularly polarized magnetic field, A , is applied to a magnetic sample.

f i = ~ ~ +i wi :r i i X
The induced magnetization,

a=AQz

+sinoriiy) .

fi,of the sample takes the following form.

+ X h ( a o r i j , +dnwrii,,)

.

a) What is the susceptibility tensor for DC and ac fields?
b) What is the corresponding permeability tensor?
Let’s defke
j =x, y and 2.
Thus,

ml
x, = for ac fields and xDC=

h,
It is convenient to put

M = M, ii, + xh& -jxhii,

H,

for DC fields, where i,

and A into phasor representation.
;

p=4iiz
+hiix-jhiiy.
It is now simple to apply above definitions of susceptibility. Thus,


16

Elements oJ Microwave Network

x,, = x
xu = X z y = x,

=0

*

The ac susceptibility tensor is then summarized in this matrix form.

Another definition of the ac susceptibility tensor is possible, if the microwave
signal is circularly polarized. A? and A can also be written as follows

and

Kf= M,ii: +Xhi= 4 6 : +dip.
fi= HaZz +hfii,,

where

hf =I(qq z y ) .

z,

is a unit vector in the radial direction. The superscript indicates the sense
of rotation. The (+) sign indicates CCW rotation. The time dependence of
hi is implied, when we require hi to be a real quantity. It is meaningful to
introduce a susceptibility in terms of m' and h' . Specifically,
Xf =-mi
hi '
The ac susceptibility tensor then becomes (for a circular polarized wave)

3

It is simple to show that the ac permeability tensor becomes

(p)=&(~ll+M)=A[;

;;


R m c w of Mtuwnll Equations 17

for linear polarized ac fields and

for left(-) and right (+) circularly polarized ac fields. The DC susceptibility
tensor reduces to a single scalar number, since the induced polarization or
magnetization is along the applied DC field direction, the z-axis. Thus,
xm = M , / H , .

The DC permeability is then
hC=PO61+X03.

Example 3
Calculate the characteristic impedance or internal impedance of a
~ mhos/m. The operating frequency
conductor with conductivity o=0 . 5 lo*’
is 9 Ghz. The characteristic impedance is defined as

= free space permeability,

where

pen =

and

cCIf = a/jo for a metal.

2 may also be written as

where

6 =J+

I

skin depth.

Substituting above values of a and f, we obtain
s=237x i04mr237p ,

lp=lOdm ,
and

2, = 0.084 (I +j) .


18

Elements of Miemwave Networks

Table 1.1
Conversion from MKS to CGS System of Units

Ouantitv

Length
Mass
Time
Density
Force

MI(s
1 meter
1 kilogram

1o2

centimeters
grams
second
g/cm3
dynes
ergs
statcoulombs
statvolts
statamperes
statcoulombs/crn2
statvolts/cm
statcoulombs/cm2
statohms
sec'l
stathenrys
statfarads
gauss
Oersteds

maxwells
gauss
gauss cm3
stathenryskm
statfaradskm

Id

1 second
1 kg/rn3
1 newton
Work
1 joule
Charge
1 coulomb
Potential
1 volt
Current
1 ampere
Displacement, D 1 coulomb/m2
Electric Field, E 1 volt/m
Polarization, P
1 coulomb/m*
Resistance, R
1 ohm
Conductivity, u 1 mho/m
Inductance, L
1 henry
Capacitance, C 1 farad
Magnetic Density, B 1 weber/m2

Magnetic Field, H 1 ampereh
Magnetic Flux, @ 1 weber
Magnetization, M 1 ampere/m
Magnetic moment, m 1 ampm2
Permeability, p 1 henryh
Dielectric Constant, E 1 farad/m

1
103

Id
107
3 x 10'
11300
3 x 10'
12n x Id
10'13

3xld
lO'l/9
9 x 10'
10"/9
9 x 10"
1o4
4n x 1 0 3
10'
104/(4n)
3 x 1013
107/(4x)
36x x lo9

Table 1.2
Conversion Table for Various Units of Energy
ey
1

mi!

"I(

1

1. I6049X 10'
1.4387
1
0.73219 x IF
3.03071 x IOU

fal

0.80657 X lo'

1.60210 x
1.98630~
IOU
1.3853 x I O U
1
4.1840

3.8291 x 10'

4.74~
10"
3.2995 x 10"
2.3995 x 10'
1

1.23981x lo4 I
1 . 8 6 1 7 0 ~ 1 0 ~0.69511
0.62418~lo'* 0.50344~1OZ1
2.61 158x l o t 9 2.10642 x lo"