Table Of ContentCHAPTER 1
INTRODUCTION
BY E. M. PURCELL
1.1. Microwaves.—The microwave region of the electromagnetic
spectrum iscommonly taken toinclude frequencies of 109cycles persecond
and higher. The upper frequency limit exists only as an active frontier
lying, at the time of writing, not far below 10” cycles per second. It is
not necessary here to attempt to- fix the boundaries more precisely.
Instead, it maybe asked why it is profitable and proper to single out for
special treatment this section of the r-f spectrum. The answer to this
question will provide a broad of “microwaves” and may serve
to indicate the scope and purpose of this book.
The distinguishing features of this region of the spectrum are most
striking from the point of view of the electrical engineer. Indeed, from
thepoint of view of the physicist concerned with the properties of matter
and radiation, boundaries such as those suggested above would appear
wholly arbitrary. If he were obliged to choose aname for the region, his
choice might suggest very long, rather than very short, wavelengths. In
thecentimeter wavelength range, the interaction of radiation with matter
appears toafford,with certain notable exceptions, lessabundant evidence
of the structure of molecules and atoms than the spectroscopist finds at
much shorter wavelengths. Onthe other hand, inthe study of the dielec-
tricconstants and magnetic permeabilities of matter inthebulk no special
distinction ismade microwave frequencies and much lower radio
frequencies. This is not to suggest that the microwave region is unin-
teresting to the physicist but rather that its aspect and extent arevarious,
depending onthe nature ofthe problem athand. The engineer, however,
is concerned with the techniques of producing, controlling, transmitting,
and detecting electromagnetic energy, and in the microwave region these
techniques take on a novel and characteristic form.
The low-frequency end of the microwave spectrum marks roughly
thepoint at which many of the familiar techniques of the radio-frequency
artbecome difficult orineffective. Perhaps fortuitously, perhaps inevita-
bly, approximately at this point those methods and devices which exploit
theshortness ofthe wavelength become practical and effective. A simple
resonant circuit, for example, for afrequency of 30 Me/see might consist
of a coil and condenser, as in Fig. l.la.
1
2 I.Y1’ROII1’CTIO.V [SEC.11
It is not hard to SI1OWthat if all linear dimensions of this circuit were
reduced by a factor of 100, the resonant frequency lvould be increased by
the same factor, to 3000 31c, sec. This method of scaling isnot practical
for reasons other than the ridiculously small size of the resulting object.
In the circuit shown in Fig. 1lb, the resistance of the wire forming the
coil is ten times as effecti~e in damping the oscillations of the circuit;
that is, the Q of circuit b is one-tenth that of circuit a. l~oreover, the
amount of energy that can be stored in the circuit without dielectric
breakdown-often an important consideration—is smaller for circuit b
(a) (e)
.
(b) (f)
I’IG,1,1.–-Resonantcircuitsatlowfrequenciesandatmicrowavefrequencies.
by a factor of about 1000. It would be better, of course, to reduce the
number of turns in the coil, as shown in Fig. 1.lCwhere the inductance of
a single turn is combined with the capacitance of a gap of reasonable
size. One further step leads to the reduction of the area of the single
turn and to the widening ofthe condenser gap, asshown in Fig. 1.ld.
Thus the disadvantages of circuit b have been overcome to a con-
siderable degree. The circuit ddiffers from b,however, in one important
respect: its physical dimensions are notsmall compared with its resonant
wavelength. Consequently, the circuit, if excited, will lose energy by
radiation; the condenser now acts as an antenna, and the circuit behaves
as if a series resistance had been inserted to absorb energy and damp the
oscillations. This loss of energy can be avoided by enclosing the entire
circuit within conducting walls, which might be done in the manner
shown in Fig. 1.le where the two conductors of circuit d have become
coaxial cylinders which, provided certain requirements on the thickness
and conductivityy of the walls are met, confine all electric and magnetic
fields to the region between the two cylinders. Another possible solution
is shown in Fig. 1.Ij. This resonant circuit is simply a hollow metal
SEC.1.21 ,llIC’ROII”.II-R CIR(; [’1’l’,i 3
cylinder. It will merely be asserted here that if the dimensions of the
cylinder are correctly chosen, the cylindrical cavity will display an elec-
trical resonance at the desired frequency.
The forms of the resonant circuits r and ,fare characteristic of micro-
wave devices, which are usually distinguished by two features: (1) the
physical size of the circuit elements is comparable with the wavelength
involved, and (2) the electromagnetic fields are totally confined within
conducting walk, except where it is desired that energy be radiated into
space. As has already been s~~ggested,the second of these features is an
~mavoidable consequence of the first. It is v-orth remarking that the
features of the circuit of Fig. 1.lb which made it impractical in the micro-
wave region and which led to devices comparable to a wavelength in size
will eventually be encountered again as higher and higher frequencies are
reached. It will no longer be possible, even by the use of microwave
techniques, to design circuits of convenient physical size that permit
transmission of high power or storage of much energy or that have
reasonably low loss. These difficulties are not equally fundomenlal, nor
will they necessarily become acute at the same point in the spectrum.
Nevertheless, considerations of this sort will ultimately determine the
practical upper frequency limit of what is here called the microwave
region and will stimulate the search for other methods of handling electro-
magnetic energy.
In this book, then, the word “microwave” will be used to imply, not
necessarily a particular range of frequencies, but a characteristic tech-
nique and a point of view. A resonator of the form of Fig. 1~If comes
within the scope of this book, even if it is 20 ft in diameter, with a cor-
responding fundamental resonant frequency of 40 hfc/sec. It may be
regarded as more or less accidental that the methods of analysis and
measurement to be discussed find their widest application at centimeter
and millimeter wavelengths.
102. Microwave Circuits.—The example of the resonant circuit, dis-
cussed above in rather oversimplified language, will already have sug-
gested to the reader that the application of the ordinary low-frequency
twminology of induchwe and capacitance (and even the word circuit
i,self) to the objects of Fig. 1.ld, e, and f is of uncertain validity. In
the progression from circuits a and bthrough to j, the concepts of capac-
itance and inductance lose their identity. In the hairpin-shaped reso-
nator d, for instance, although it is both useful and meaningful to regard
one end of the structure as a condenser and the other as an inductance,
these concepts cannot be made quantitative except by an arbitrary and
artificial convention. Inductance and capacitance are thoroughly dis-
guised in the cylindrical resonator j. To a varying extent, the notions of
current, voltage, and resistance have likewise lost their uniqueness.
4 INTROI)lJCTION [sm. 12
Evidently, if theelectrical properties of such structures aretobe investi-
gated, adirect analysis of the electromagnetic field must be made.
In the study of low-frequency circuits, it is possible to avoid an
analysis of the field only because all relevant properties of a circuit ele-
ment, such as a coil or condenser, can be described by one or two num-
bers, and different elements can be combined in various ways without
impairing the validity of the description. . A simple coil ischaracterized
by two quantities: induct ante and resistante. These two quantities
suffice to determine, at any frequency, the relation between the current
flowing at the terminals of the coil and the voltage between these ter-
minals. Since current and voltage have here a perfectly definite mean-
ing, and since the relation of one to the other at any frequency isall that
need be known, it is not necessary to inquire into the structure of the
complicated magnetic field surrounding the coil unless, on occasion, it is
desired to calculate, rather than to measure, the inductance of the coil.
To be able in this manner to avoid solving Maxwell’s equations for each
new structure saves an enormous amount of work and makes feasible
the analysis of quite complicated circuits. Examples in which this
abbreviated description in terms of inductance, capacitance, and resist-
ance is inadequate are, however, not hard to find, even at ordinary radio
frequencies. The apparent inductance of the coil of Fig. 1~la would, in
fact, be found to vary noticeably with the frequency. To attribute this
variation, as is the custom, to the effect of the “distributed capacitance”
of the windings is to acknowledge that the behavior of the coil and the
associated electromagnetic field cannot be exactly described by a single
number, except at asingle frequency. In this special case, for purposes of
circuit analysis, the statement that the coil is a linear device with two
terminals presenting a certain impedance Z is a complete and accurate
description. At microwave frequencies the problem is to analyze an
electrical structure that cannot be broken down into such simple elements
and for which it is not apparent that voltage and current have a unique
meaning.
There is at least one way out of this difficulty. Renouncing all
attempts to describe circuits in terms of voltage, current, and impedance
and restricting the emphasis to those quantities which appear explicitly
in the equations of the electromagnetic field, the generality of Maxwell’s
equations can be utilized. Each new problem could then be stated as a
boundary-value problem; that is, a solution of Maxwell’s equations satis-
fying certain prescribed conditions appropriate to the particular circuit
at hand would be sought. Although it would be easy to state the prob-
lem in this way, in all but the simplest cases the actual solution would be
hopelessly difficult. Furthermore, any modification of the original cir-
cuit would usually lead to an entirely new problem. It would not be
SEC.13] MICROWAVE MEASUIMMEN1’S 5
possible to combine several elements in a new w-ayand to predict and
understand the behavior of the combination without re-examining in
detail the behavior of each element. If no course other than this were
open, it is likely that knowledge of this art would consist of a meager
collection of completely solved special cases, together with an extensive
assortment of practical devices, designed by trial and error and imper-
fectly understood.
Fortunately, the situation is not nearly so hopeless. In the first
place, the foregoing discussion has overemphasized the complexity of
microwave circuits. Ifelectronic devices andantennas areexcluded, the
components of microwave circuits consist chiefly of cavity resonators
and transmission lines of various types. The analysis of the electrical
properties of cavities of various shapes has received much attention in
recent years, especially in connection with the development of velocity-
modulation tubes. A cavity resonator by itself is a rather simple device,
in the sense that the properties that are usually of interest can be
described by a very few parameters, much as the properties of a low-
frequency resonant circuit are summarized in the statement of the reso-
nant frequency, the value of Q, and the impedance at resonance. The
uniform transmission line isperhaps even more familiar to most engineers.
It will be part of the purpose of this book to show how the standard meth-
ods for the analysis of uniform transmission lines can be applied to the
propagation of energy through hollow pipes of various sorts—in other
words, to generalize the notion of a transmission line. This is not diffi-
cult. It can be said that certain isolated elements of the microwave cir-
cuit problem arerather easily handled. The main problem isencountered
when things are connected together or when nonuniformities are intro-
duced into apreviously uniform line. In the following chapters methods
for attacking this problem will be worked out.
The direction that the development of microwave circuit analysis
and design techniques takes is strongly influenced by (1) the kind of
measurements that can be made at microwave frequencies, (2) the nature
of the question to be answered, and (3) the existing well-tried and power-
fulmethods for the solution oflow-frequency circuit problems, from which
it is, naturally, expedient to borrow as often as possible.
1.3. Microwave Measurements.—At microwave frequencies neither
the circuit elements to be examined nor the measuring apparatus itself
can convenient y be made small compared with a wavelength. This
circumstance calls for new experimental techniques and, more important
for this discussion, shifts attention from the conventional circuit quan-
tities of voltage, current, and resistance to other quantities more directly
accessible to measurement.
At very low frequencies, the voltage between the two conductors of a
6 IN1’RODUCTZON [SEC.13
two-wire transmission line might be measured at some point along the
line by connecting an a-c voltmeter, of either the dynamometer or the
iron-vane type, between the two conductors. This becomes impractical
even in the audio-frequency range because of the inductance of the coil
through which the current must flow. It is then necessary to resort to
instruments in which the high-frequency circuit more nearly approaches a
simple resistance element, the current through which ismeasured by indi-
rect means—perhaps by a thermal effect or by the use of a rectifier.
For example, a fine wire of suitably high resistance might be stretched
between the conductors, itsriseintemperature asaresult ofthe Joule heat
developed determined by measuring its d-c resistance, and the result com-
pared with the temperature rise produced in the same environment by a
known direct current passing through thewire. With suitable corrections
for the difference between the d-c resistance of the wire and its resistance
at the frequency in question, this method is sound and reliable up to
frequencies for which the length of the wire isan appreciable fraction of a
wavelength. At frequencies higher than this the inductance of the wire
cannot be neglected, which isanother way of saying that the wire itself is
becoming a transmission line. The current that flows in the wire is not
the same at different points along the wire, and it isnot surprising to find
that the average temperature rise of the wire isnot related inasimple way
to the voltage between the two terminals. Since the wire must neces-
sarily span the distance between the two conductors of the main trans-
mission line, it is clear that afundamental difficulty stands in the way of
any attempt to devise an instrument which may be legitimately called a
“voltmeter” when applied to circuits whose dimensions are comparable
with a wavelength.
The root of the difficulty goes deeper than the foregoing remarks
suggest. The potential difference between two points ordinarily means
the line integral of the electric field strength, JE. A, taken at one instant
of time, along some path joining the two points. This concept is unique
and useful only ifthe value of the line integral isindependent of the path.
When the path necessarily extends over a distance not small compared
with awavelength, the line integral isnot, in general, independent of the
path, and the significance of the term “voltage” is lost.
This suggests that attention be directed to the electric field. A1-
though it is not common engineering pyactice to measure the electric
field strength at a point in absolute terms, it isnot hard to think of ways
in which it might be done. For instance, a tiny dielectric rod might be
suspended in the field in such a way that the torque tending to line it up
with- the field could be directly measured. From a knowledge of the
shape of the rod and the dielectric constant of the material of which it is
made, the value of the field strength involts per meter could be computed.
Usually it is much more convenient and just w useful to mewwe the
.
SEC.13] JfICROW.4 VE MEAS UIiE.Wh’NTS 7
ratio of the electric field strengths at two points in a system. A familiar
example of this procedure is the use of a “probe” to examine the varia-
tion of the field strength along a coaxial transmission line. This probe
usually consists of ashort length of auxiliary line, the central conductor of
which projects through a small hole into a slot in the outer conductor of
the transmission line under examination. The other end of the auxiliary
line terminates in some detecting device of a rectifying or of a thermal
type. If precautions are taken to ensure that the probe itself causes
only a negligible disturbance in the line under test, the excitation of the
auxiliary line will be proportional to the field in the main line, which
allows comparison of the field at one location in the main line with the
field at some other point.
In much the same way, relative measurements of the magnetic field
strength can be made, and these, in turn, may be considered to replace
the measurement of current. In microwave circuits, current usually
appears asavolume orsurface current density, and whereas these quanti-
ties have a perfectly unique meaning, it is more natural to associate with
the field the related quantities that can be measured.
The usual measurements of field quantities just described are relative
ones. The quantities that are customarily measured in absolute terms
are frequency or wavelength, and power. The r-f power delivered to a
load can be measured directly by calorimetric means; the rate of evolution
of heat in some object is inferred from its temperature rise and may be
indic~,ted in various ways. To be sure, absolute m~~surements could be
made, ii necessary, of other dectr;V1 quantities. A method .of measur-
ing the electric fieid smwngt~ has already been suggested; it would also
be possible in principle and, in certain instances, in practice to measure
the radiation pressure associated with the electromagnetic field at a
boundary. However, it is approximately correct to say that the wattis
the only electrical unit that is of direct importance here.
In the microwave region both frequency and wavelength can be
measured with great precision. The determination of wavelength
involves the accurate measurement of one ormore lengths associated with
a simple resonant circuit such as a cylindrical cavity or a transmission
line. Microwave frequencies, on the other hand, can be compared
directly with lower frequencies by standard methods of frequency multi-
plication. This rather cumbersome and inflexible method is preferred
where extreme absolute accuracy is required, as in the establishment of
frequency standards. . For most work, the measurement of wavelength is
much more convenient. Moreover, the physical quantity wavelength is
directly -and intimately associated with the dimensions of the microwave
circuit, and it becomes natural to think in terms of wavelength rather
than frequency.
Although this book is-not directly concerned with the extensive sub-
8 INTRODUCTION [sm. 14
ject of microwave measurement techniques-to which Vol. llofthe Series
is devoted—the reader will more readily understand the emphasis and
approach adopted in the following chapters if the situation just described
iskept in mind. For example, the term impedance will be used again and
again to denote adimensionless ratio (the normalized impedance) and only
rarely to denote the ratio of a potential difference in volts to a current in
amperes.
1.4. The Aims of Microwave Circuit Analysis.—A microwave circuit
is a region enclosed by metallic walls of any shape and communicating
with the exterior only by way of a number of transmission lines or wave-
guides, which may be called the terminals of the circuit. This definition
is not so narrow as it might at first appear, for it is easy to extend the
notion of transmission lines as terminals to include waveguicies carrying
many modes, and even antennas.
The final object of microwave circuit analysis, as of low-frequency
circuit analysis, is to provide a complete description of what goes on at
the terminals of the circuit. Suppose that a given circuit has certain
arbitrarily selected impedances connected to all pairs of terminals but
one. If we are able to predict correctly the impedance that will be
measured at the remaining terminal paifino matter which terminal
this happens to be and no matter what the frequency—our description of
the circuit can be said to be complete. The methods by which such a
prediction can be made, once the properties of the component parts of
the circuit are given, and the various ways in which the results of the
analysis can be expressed are the main topics of this book.
The properties of the circuit elementswill not be derived ab initio, for
the most part. Such problems involve extensive theoretical calculations.
The results, however, of such calculations, as well as numerous experi-
mental results, are summarized in Vol. 10, the Waveguide Handbook.
The limitation in scope of the present volume is thus characteristic of a
book on network analysis in which one would not expect to find a deriva-
tion of the inductance of a coil of a certain shape. This illustration fails
to suggest, however, the variety and novelty of the problems that can be
solved and the degree to which the usefulness of the methods to be
described here is enhanced by the availability of solutions to those
problems.
One often has to deal with a circuit whose complete description can-
not easily be deduced. Nevertheless it may be possible to make certain
restricted statements, based on very general considerations, about the
behavior of the circuit. Such observations prove very useful, not only
in reducing the number of parameters that have to be determined experi-
mentally to complete the description of the circuit, but in disclosing %sic
similarities between circuits superficially different and in avoiding vain
SEC.1.7] SYMMETRY 9
efforts to contrive circuits having properties prohibited by one of these
general principles. For these reasons much attention will be devoted in
this book to those general theorems which can be shown to apply to
certain classes of circuits. One topic, for example, will be the properties
of aIossless three-terminal-pair junction, by which is meant an enclosure
of arbitrary shape, with perfectly conducting walls, provided with three
outlets.
1.5. Liieari&.-It is probably. obvious that the program outlined
does not include nonlinear circuit elements in its scope. The situation in
thk respect is similar to that which prevails at lower frequencies, except
that, at least until now, the need for general techniques of nonlinear
circuit analysis has been less acute in the microwave region. The micro-
wave oscillator, the gas switch, and the crystal detector are the only
nonlinear microwave devices of present importance, and each of these
must be treated in a special way and is so treated elsewhere in thisseries.
It is, of course, just this restriction to linear circuit elements which makes
possible the sort of analysis with which we shall be concerned.
1.6. Dissipation.—The reader will find that much more attention is
devoted to di.wipationless networks than would be justified in a study of
low-frequency circuits. It is one of the attractive features of the micro-
wave region that most microwave circuit elements show such a small loss
that the error made in assuming that all conductors are perfect conduc-
tors isusually negligible-except, of course, in the case of highly resonant
circuits. The reason for this, broadly speaking, is that low-frequency
circuit elements and microwave circuit elements do not differ much in
physical size, and the Qof elements of the same size varies directly asthe
square root of the frequency, if skin resistance only is concerned. Ferro-
magnetic materials arenot employed asarule, and the dielectric materials
are used sparingly.
1.7. Symmetry .-One other conspicuous feature of microwave cir-
cuits is the symmetry properties possessed by many widely used circuit
elements. By simply joining a number of similar waveguides together
in a geometrically symmetrical way, it is relatively easy to construct a
circuit whose low-frequency equivalent would consist of a complicated
network of accurately matched elements. In other words, geometrical
and electrical symmetry are here naturally and closely allied. The
result is to stimulate the investigation of symmetrical networks of
many types, including those with as many as four or even six terminal
pairs, and to simplify the analysis of many useful devices.
CHAPTER 2
ELECTROMAGNETIC WAVES
BY C. G. MONTGOMERY
THE FIELDRELATIONS
2.1. Maxwell’s Equations.—In electromagnetic theory, the inter-
action between charges and their mutual energy is expressed in terms of
four field vectors E and B, D and H. - The first pair define the force on a
charge density pmoving with avelocity vby the equation
F = p[E + (V X B)].
The vectors E and B satisfy the field equations
dB
div B = O. (1)
curl E = – z’
The second pair of field vectors are determined by the charges and
currents present and satisfy the equations
curl H= J+~, div D = p, (3
where J is the current density and p the charge density. The set of
Eqs. (1) and (2) is known asMaxwell’s equations of the electromagnetic
field. Sometimes theforce equation isincluded asamember of the set.
Tke connection between D and E, and B and H, depends upon the
properties of the medium in which the fields exist. For free space, the
connection is given by the simple relations
D = eoE, B = ~,H.
For material mediums of the simplest type the relations are of the same
form but with other characteristic parameters
D = cE, B = ~H.
The symbol edenotes the permittivity of the medium, and ~ the perme-
ability. For crystalline mediums which are anisotropic, the scalars cand
pmust be generalized to dyadic quantities. Then D and Enolonger have
the same direction in space. This more complex relationship will ?ot
be dealt with here. If the medium is a conductor, then ohm’s law is
10
