2 waveguides and equivalent transmission linesmilton/p5573/chap2.pdf16 2 waveguides and equivalent...

2

Waveguides and Equivalent Transmission Lines

A waveguide is a device for transferring electromagnetic energy from one pointto another without appreciable loss. In its simplest form it consists of a hollowmetallic tube of rectangular or circular cross section, within which electromag-netic waves can propagate. The two conductor transmission line discussed inChapter 1 is a particular type of waveguide, with special properties. Thesimple physical concept implied by these examples may be extended to in-clude any region within which one-dimensional propagation of electromagneticwaves can occur. It is the purpose of this chapter to establish the theory ofvarious types of simple waveguides, expressed in the general transmission-linenomenclature developed in the preceding chapter.

2.1 Uniform Waveguides

The major portion of this chapter will be devoted to the theory of uniformwaveguides – metallic tubes which have the same cross-section in any planeperpendicular to the axis of the guide. Initially, the simplifying assumptionwill be made that the metallic walls of the waveguide are perfectly conducting.Since the field is then entirely confined to the interior of the waveguide, theguide is completely described by specifying the curve C which defines a cross-section σ of the inner waveguide surface S. The curve C may be a simple closedcurve, corresponding to a hollow waveguide, or two unconnected curves, as ina coaxial line.

2.1.1 Transmission-Line Formulation

We first consider the problem of finding the possible fields that can exist withina waveguide, in the absence of any impressed currents. This is equivalent toseeking the solutions of the Maxwell equations

∇× E = ikζH , (2.1a)

∇×H = −ikηE , (2.1b)

14 2 Waveguides and Equivalent Transmission Lines

where we have defined [SI units]

k = ω√εµ =

ω

c, (2.2a)

c being the speed of light in the medium inside the guide, and introduced theabbreviations

ζ =

√

µ

ε, η =

√

ε

µ= ζ−1 . (2.2b)

These equations are to be solved subject to the boundary condition

n×E = 0 on S . (2.3)

Recall that the other two Maxwell equations, in charge-free regions,

∇ ·D = 0 , ∇ ·B = 0 , (2.4)

are contained within these equations, as is the boundary condition n ·B =0. The medium filling the waveguide is assumed to be uniform and non-dissipative. In view of the cylindrical nature of the boundary surface, it isconvenient to separate the field equations into components parallel to theaxis of the guide, which we take as the z axis, and components trasverse tothe guide axis. This we achieve by scalar and vector multiplication with e, aunit vector in the z direction, thus obtaining

∇ · e×E = −ikζHz , (2.5a)

∇ · e×H = ikηEz , (2.5b)

and

∇Ez −∂

∂zE = ikζe×H , (2.6a)

∇Hz −∂

∂zH = −ikηe×E . (2.6b)

On substituting (2.5b) [(2.5a)] into (2.6a) [(2.6b)], one recasts the latter intothe form

∂

∂zE = ikζ

(

1 +1

k2∇∇

)

·H× e , (2.7a)

∂

∂zH = ikη

(

1 +1

k2∇∇

)

· e× E , (2.7b)

in which 1 denotes the unit dyadic. This set of equations is fully equivalentto the original field equations, for it still contains (2.5a), (2.5b) as its z com-ponent. The transverse components of (2.7a), (2.7b) constitute a system ofdifferential equations to determine the transverse components of the electricand magnetic fields. Thes equations are in transmission line form, but with

2.1 Uniform Waveguides 15

the series impedance and the shunt admittance per unit length appearing asdyadic differential operators. The subsequent analysis has for its aim the re-placement of the operator transmission line equations by an infinite set ofordinary differential equations. This is performed by successively suppressingthe vectorial aspect of the equations and the explicit dependence on x and y,the coordinates in a transverse plane.

Any two-component vector field, such as the transverse part of the electricfield Et, can be represented as a linear combination of two vectors derivedfrom a potential function and a stream function, respectively. Thus

Et = −∇tV′ + e×∇V ′′ , (2.8)

where V ′(r) amd V ′′(r) are two arbitrary scalar functions and ∇t indicatesthe transverse part of the gradient operator. In a similar was, we write

Ht = −e×∇I ′ −∇tI′′ , (2.9a)

orH× e = −∇tI

′ + e×∇I ′′ , (2.9b)

with I ′(r) and I ′′(r) two new arbitrary scalar functions. This general repre-sentation can be obtained by constructing the two-component characteristicvectors (eigenvectors) of the operator 1 + 1

k2 ∇∇. Such vectors must satisfythe eigenvector equation in the form

∇t∇ ·At = γAt . (2.10)

Hence, either ∇ ·At = 0 and γ = 0, implying that At is the curl of a vectordirected along the z axis; or ∇ ·At 6= 0, and At is the gradient of a scalarfunction. The most general two-component vector At is a linear combinationof these two types, and e×A is still of the same form, as it must be. In con-sequence of these observations, the substitution of the representation (2.8),(2.9a) into the differential equations (2.7a), (2.7b) will produce a set of equa-tions in which every term has one or the other of these forms. This yields asystem of four scalar differential equations, which are grouped into two pairs,

∂

∂zI ′ = ikηV ′ ,

∂

∂zV ′ = ikζ

(

1 +1

k2∇2

t

)

I ′ , (2.11a)

∂

∂zI ′′ = ikη

(

1 +1

k2∇2

t

)

V ′′ ,∂

∂zV ′′ = ikζI ′′ , (2.11b)

where ∇2t is the Laplacian for the transverse coordinates x and y. (Any con-

stant annihilated by ∇t is excluded because it would not contribute to theelectric and magnetic fields.) The longitudinal field components may now bewritten as

ikηEz = ∇2t I

′ , (2.12a)

ikζHz = ∇2tV

′′ . (2.12b)


The net effect of these operations is the decomposition of the field intotwo independent parts derived, respectively, from the scalar functions V ′, I ′,and V ′′, I ′′. Note that the first type of field in general possesses a longitudi-nal component of electic field, but no longitudinal magnetic field, while thesituation is reversed with the second type of field. For this reason, the variousfield configurations derived from V ′ and I ′ are designated as E modes, whilethose obtained from V ′′ and I ′′ are called H modes; the nomenclature in eachcase specifies the non-vanishing z component of the field.1

The scalar quantities involved in (2.11a), (2.11b) are functions of x, y, andz. The final step in the reduction to one-dimensional equations consists inrepresenting the x, y dependence of these functions by an expansion in thecomplete set of functions forming the eigenfunctions of ∇2

t . For the E mode,let these functions be ϕa(x, y), satisfying

(∇2t + γ′2a )ϕa(x, y) = 0 , (2.13)

and subject to boundary conditions, which we shall shortly determine. Onsubstituting the expansion

V ′(x, y, z) =∑

a

ϕa(x, y)V ′a(z) , (2.14a)

I ′(x, y, z) =∑

a

ϕa(x, y)I ′a(z) , (2.14b)

into (2.11a), we immediately obtain the transmission line equations

d

dzI ′a(z) = ikηV ′

a(z) , (2.15a)

d

dzV ′

a(z) = ikζ

(

1− γ′2ak2

)

I ′a(z) . (2.15b)

In a similar way, we introduce another set of eigenfunctions for ∇2t .

(∇2t + γ′′2a )ψa(x, y) = 0 . (2.16)

and expand the H mode quantities in terms of them:

V ′′(x, y, z) =∑

a

ψa(x, y)V ′′a (z) , (2.17a)

I ′′(x, y, z) =∑

a

ψa(x, y)I ′′a (z) . (2.17b)

The corresponding differential equations are

1 A more common terminology for E modes are TM modes, meaning “transverse

magnetic”; and for H modes, TE modes, for “transverse electric.” Still another

notation is ⊥ for “perpendicular,” refering to H modes, and ‖ for “parallel,”

referring to E modes.


d

dzI ′′a (z) = ikη

(

1− γ′′2a

k2

)

V ′′a (z) , (2.18a)

d

dzV ′′

a (z) = ikζI ′′a (z) . (2.18b)

The boundary conditions on the electric field require that

Ez = 0, Es = 0 on S , (2.19)

where Es is the component of the electric field tangential to the boundarycurve C. These conditions imply that

∇2t I

′ = 0,∂

∂sV ′ =

∂

∂nV ′′ on S , (2.20)

where ∂∂n is the derivative normal to the surface of the waveguide S, and

∂∂s is the circumferential derivative, tangential to the curve C. Since theseequations must be satisfied for all z, they impose the following requirementson the functions ϕa and ψa:

γ′2a ϕa = 0 ,∂

∂sϕa = 0 ,

∂

∂nψa = 0 on C . (2.21)

If we temporarily exclude the possibility γ ′a = 0, the second E mode boundarycondition is automatically included in the first statement, that ϕa = 0 on theboundary curve C. Hence, E modes are derived from scalar functions definedby

(

∇2t + γ′2a

)

ϕa(x, y) = 0 , (2.22a)

ϕa(x, y) = 0 on C , (2.22b)

while H modes are derived from functions satisfying

(

∇2t + γ′′2a

)

ψa(x, y) = 0 , (2.23a)

∂

∂nψa(x, y) = 0 on C , (2.23b)

These equations are often encountered in physics. For example, they describethe vibrations of a membrane bounded by the curve C, which is either rigidlyclamped at the boundary [(2.22b)], or completely free [(2.23b)]. Each equa-tion defines an infinite set of eigenfunctions and eigenvalues ϕa, γ′a and ψa,γ′′a . Hence, a waveguide possesses a two-fold infinity of possible modes of elec-tromagnetic oscillation, each completely characterized by one of these scalarfunctions and its attendant eigenvalue.

We shall now show that the discarded possibility, γ ′a = 0, cannot occurfor hollow waveguides, but does correspond to an actual field configuration intwo conductor lines, being in fact the T mode discussed in the first chapter.The scalar function ϕ associated with γ ′a = 0 satisfies Laplace’s equation


∇2tϕ(x, y) = 0 , (2.24)

and is restricted by the second boundary condition, ∂∂sϕ(x, y) = 0 on C, or

ϕ(x, y) = constant on C . (2.25)

Since ϕ satisfies Laplace’s equation, we deduce that

∫

C

ds ϕ∂

∂nϕ =

∫

σ

dS (∇tϕ)2 , (2.26)

in which the line integral is taken around the curve C and the surface integralis extended over the guide cross-section σ. For a hollow waveguide with across-section bounded by a single closed curve on which

ϕ = constant = ϕ0 , (2.27)

we conclude∫

C

ds ϕ∂

∂nϕ = ϕ0

∫

C

ds∂

∂nϕ = ϕ0

∫

σ

dS∇2tϕ = 0 , (2.28)

and therefore from (2.26) ∇tϕ = 0 everywhere within the guide, which impliesthat all field components vanish, effectively denying the existence of such amode. If, however, the contour C consists of two unconnected curves C1 andC2, as in a coaxial line, the boundary condition, ∂

∂sϕ = 0 on C, requires thatϕ be constant on each contour

ϕ = ϕ1 on C1, ϕ = ϕ2 on C2 , (2.29)

but does not demand that ϕ1 = ϕ2. Hence

∫

C

ds ϕ∂

∂nϕ = ϕ1

∫

C1

ds∂

∂nϕ+ ϕ2

∫

C2

ds∂

∂nϕ = (ϕ1 − ϕ2)

∫

C1

ds∂

∂nϕ ,

(2.30)since

0 =

∫

C

ds∂

∂nϕ =

∫

C1

ds∂

∂nϕ+

∫

C2

ds∂

∂nϕ , (2.31)

and the preceding proof fails if ϕ1 6= ϕ2. The identification with the T modeis completed by noting [(2.12a] that Ez = Hz = 0.

The preceding discussion has shown that the electromagnetic field withina waveguide consists of a linear superposition of an infinite number of com-pletely independent field configurations, or modes. Each mode has a character-istic field pattern across any section of the guide, and the amplitude variationsof the fields along the guide are specified by “currents” and “voltages” whichsatisfy transmission-line equations. We shall summarize our results by collect-ing together the fundamental equations describing a typical E mode and Hmode (omitting distiguishing indices for simplicity).


• E mode:

Et = −∇tϕ(x, y)V (z) , (2.32a)

Ht = −e×∇ϕ(x, y)I(z) , (2.32b)

Ez = iζγ2

kϕ(x, y)I(z) , (2.32c)

Hz = 0 , (2.32d)

(∇2t + γ2)ϕ(x, y) = 0, ϕ(x, y) = 0 on C , (2.32e)

d

dzI(z) = ikηV (z) , (2.32f)

d

dzV (z) = ikζ

(

1− γ2

k2

)

I(z) . (2.32g)

• H mode:

Et = e×∇ψ(x, y)V (z) , (2.33a)

Ht = −∇tψ(x, y)I(z) , (2.33b)

Ez = 0 , (2.33c)

Hz = iηγ2

kψ(x, y)V (z) , (2.33d)

(∇2t + γ2)ψ(x, y) = 0,

∂

∂nψ(x, y) = 0 on C , (2.33e)

d

dzI(z) = ikη

(

1− γ2

k2

)

V (z) , (2.33f)

d

dzV (z) = ikζI(z) . (2.33g)

The T mode in a two-conductor line is to be regarded as an E mode withγ = 0, and the boundary condition replaced by ∂

∂sϕ = 0. It may also beconsidered an H mode with γ = 0.

The transmission line equations for the two mode types, written as

• E mode:

d

dzI(z) = i

ωε

cV (z) , (2.34a)

d

dzV (z) =

(

iωµ

c+γ2c

iωε

)

I(z) , (2.34b)

• H mode:

d

dzI(z) =

(

iωε

c+γ2c

iωµ

)

V (z) , (2.35a)

d

dzV (z) =

iωµ

cI(z) , (2.35b)


are immediately recognized as the equations of the E amd H type for thedistributed parameter circuits discussed in Chapter 1. The E mode equiv-alent transmission line has distributed parameters per unit length specifiedby a shunt capacitance C = ε, and series inductance L = µ, and a seriescapacitance C ′ = ε/γ2. The H mode line distributed parameters are a se-ries inductance L = µ, a shunt capacitance C = ε, and a shunt inductanceL′′ = µ/γ2, all per unit length. Thus, if we consider a plane wave, I ∝ eiκz,with V = ZI , the propagation constant κ and characteristic impedence Zassociated with the two types of lines are

• E mode:

κ =√

k2 − γ2 , (2.36a)

Z = ζκ

k, (2.36b)

• H mode:

κ =√

k2 − γ2 , (2.36c)

Y = ηκ

k. (2.36d)

We may again remark on the filter property of these transmission lines,which has been discussed in Chapter 1. Actual transport of energy along awaveguide in a particular mode can only occur if the wave number k exceedsthe quantity γ associated with the mode. The eigenvalue γ is therefore referredto as the cutoff or critical wavenumber for the mode. Other quantities relatedto the cutoff wavenumber are the cutoff wavelength,

λc =2π

γ, (2.37)

and the cutoff (angular) frequency

ωc = cγ(εµ)−1/2 . (2.38)

When the frequency exceeds the cutoff frequency for a particular mode, thewave motion on the transmission line, indicating the field variation along theguide, is described by an associated wavelength

λg =2π

κ, (2.39)

which is called the guide wavelength. The relation between the guide wave-length, intrinsic wavelength, and cutoff wavelength for a particular mode is,according to (2.36a),

1

λg=

√

1

λ2− 1

λ2c

, (2.40a)


or

λg =λ

√

1−(

λλc

)2. (2.40b)

Thus, at cutoff (λ = λc), the guide wavelength is infinite and becomes imagi-nary at longer wavelengths, indicating attenuation, while at very short wave-lengths (λ � λc), the guide wavelength is substantially equal to the in-trinsic wavelength of the guide medium. Correspondingly, the characteristicimpedance for a E (H) mode is zero (infinite) at the cutoff frequency and isimaginary at lower frequencies in the manner typical of a capacitance (induc-tance). The characteristic impedance approaches the intrinsic impedance ofthe medium for very short wavelengths.

The existence of a cutoff frequency for each mode involves the implicitstatement that γ2 is real and positive; γ is positive by definition. A proof iseasily supplied for both E and H modes with the aid of the identity

∫

C

ds f∗∂

∂nf =

∫

σ

dS |∇f |2 − γ2

∫

σ

dS |f |2 , (2.41)

where f stands for either an E-mode function ϕ or an H-mode function ψ. Ineither event, the line integral vanishes and

γ2 =

∫

σ dS |∇f |2∫

σdS |f |2 , (2.42)

which establishes the theorem. It may be noted that we have admitted, in allgenerality, that f may be complex. However, with the knowledge that γ2 isreal, it is evident from the form of the defining wave equation and boundaryconditions that real mode functions can always be chosen.

The impedance (admittance) at a given point on the transmission linedescribing a particular mode,

Z(z) =1

Y (z)=V (z)

I(z), (2.43)

determines the ratio of the transverse electric and magnetic field componentsat that point. According to (2.32a) and (2.32b), an E mode magnetic field isrelated to the transverse electric field by

E mode: H = Y (z)e×E , (2.44a)

which is a general vector relation since it correctly predicts that Hz = 0. Theanalogous H-mode relation is

H mode: E = −Z(z)e×H , (2.44b)

For either type of mode, the connections between the rectangular componentsof the transverse fields are


Ex = Z(z)Hy, Ey = −Z(z)Hx , (2.45a)

Hx = −Y (z)Ey, Hy = Y (z)Ex . (2.45b)

In the particular case of a progressive wave propagating (or attenuating) inthe positive z direction, the impedance at every point equals the characteristicimpedance of the line, Z(z) = Z. The analogous relation Z(z) = −Z describesa wave progressing in the negative direction.

2.1.2 Hertz vectors

The reduction of the vector field equations to a set of transmission-line equa-tions, as set forth in Sec. 2.1.1, requires four scalar functions of z for its properpresentation. However, it is often convenient to eliminate two of these func-tions and exhibit the general electromagnetic field as derived from two scalarfunctions of poistion, which appear in the role of single component Hertz vec-tors. On eliminating the functions V ′(r) amd I ′′(r) with the aid of (2.15a)and (2.18b), the transverse components of E and H, (2.8), (2.9a), become

Et =i

kζ∇t

∂

∂zI ′ + e×∇V ′′ , (2.46a)

Ht = −e×∇I ′ +i

kη∇t

∂

∂zV ′′ , (2.46b)

which can be combined with the expressions for the longitudinal field compo-nents, (2.12a), (2.12b), into general vector equations

E = ∇× (∇×Π′) + ikζ∇×Π′′ , (2.47a)

H = −ikη∇×Π′ + ∇× (∇×Π′′) , (2.47b)

The electric and magnetic Hertz vectors that appear in this formulation onlyposses z components, which are given by

Π′z =

i

kζI ′ , Π′′

z =i

kηV ′′ . (2.48)

The Maxwell equations are completely satisfied if the Hertz vector componentssatisfy the scalar wave equation:

(∇2 + k2)Π′z = 0 , (∇2 + k2)Π′′

z = 0 , (2.49)

which is verified by eliminating V ′ and I ′′ from (2.11a), (2.11b). For a partic-ular E mode, the scalar function I ′ is proportional to the longitudinal electricfield and

E mode: Π′z =

1

γ2Ez . (2.50a)

Similarly,

H mode: Π′′z =

1

γ2Hz . (2.50b)

Hence the field structure of an E or H mode can be completely derived fromthe corresponding longitudinal field component.


2.1.3 Orthonormality Relations

We turn to an examination of the fundamental physical quantities associatedwith the electromagnetic field in a waveguide – energy density and energyflux. In the course of the investigation we shall also derive certain orthogonalproperties possessed by the electric and magnetic field components of thevarious modes. Inasmuch as these relations are based on similar orthogonalproperties of the scalar functions ϕa and ψa, we preface the discussion by aderivation of the necessary theorems. Let us consider two E-mode functionsϕa and ϕb, and construct the identity

∫

C

ds ϕa∂

∂nϕb =

∫

σ

dS∇ϕa ·∇ϕb − γ′2b

∫

σ

dS ϕaϕb . (2.51)

If we temporarily exclude the T mode of a two-conductor guide, the lineintegral vanishes by virtue of the boundary condition. On interchanging ϕa

and ϕb, and subtracting the resulting equation, we obtain

(γ′2a − γ′2b )

∫

σ

dS ϕaϕb = 0 , (2.52)

which demonstrates the orthogonality of two mode functions with differenteigenvalues. In consequence of the vanishing of the surface integral in (2.51),we may write this orthogonal relation as

∫

σ

dS∇ϕa ·∇ϕb = 0 , γ′a 6= γ′b . (2.53)

If more than one linearly independent mode function is associated with aparticular eigenvalue – a situation which is referred to as “degeneracy” –no guarantee of orthogonality for these eigenfunctions is supplied by (2.52).However, a linear combination of degenerate eigenfunctions is again an eigen-function, and such linear combinations can always be arranged to have theorthogonal property. In this sense, the orthogonality theorem (2.53) is validfor all pairs of different eigenfunctions. The theorem is also valid for the Tmode of a two-conductor system. To prove this, we return to (2.51) and choosethe mode a an an ordinary E mode (ϕa = 0 on C), and the mode b as the Tmode (γ′b = 0); the desired relation follows immediately. Note, however, thatin this situation orthogonality in the form

∫

dS ϕaϕb = 0 is not obtained.Finally, then, the orthogonal relation, applicable to all E modes, is

∫

dS∇ϕa ·∇ϕb = δab , (2.54)

which also contains a convention regarding the normalization of the E-modefunctions:

∫

dS (∇ϕ)2 = 1 , (2.55)


a convenient choice for the subsequent discussion. With the exception of theT mode, the normalization condition can also be written

γ′2a

∫

dS ϕ2a = 1 . (2.56)

The corresponding derivation for H modes proceeds on identical lines, withresults expressed by

∫

dS∇ψa ·∇ψb = δab , (2.57)

which contains the normalization convention∫

dS (∇ψa)2 = γ′′2a

∫

dS ψ2a = 1 . (2.58)

2.1.4 Energy Density and Flux

The energy quantities with which we shall be concerned are the linear energydensities (that is, the energy densities per unit length) obtained by integrat-ing the volume densities across a section of the guide. It is convenient toconsider separately the linear densities associated with the various electricand magnetic components of the field. Thus the linear electric energy densityconnected with the longitundinal electric field is

UEz=ε

4

∫

dS |Ez |2 . (2.59)

On inserting the general superposition of individual E-mode fields [cf. (2.32c)],

Ez =iζ

k

∑

a

γ′2a ϕa(x, y)I ′a(z) , (2.60)

we find

UEz=ε

4

ζ2

k2

∑

a

γ′2a |I ′a(z)|2 , (2.61)

in which the orthogonality and normalization of the E-mode functions hasbeen used. The orthogonality of the longitudinal electric fields possessed bydifferent E modes is thus a trivial consequence of the corresponding propertyof the scalar functions ϕa. The longitudinal electric field energy density canalso be written

UEz=

1

4

∑

a

1

ω2C ′a

|I ′a(z)|2 , (2.62)

by introducing the distributed series capacitance, C ′a = ε/γ′2a , associated with

the transmission line that describes the ath E mode. In a similar way, thelinear energy density


UHz=µ

4

∫

dS |Hz|2 (2.63)

derived from the longitudinal magnetic field [(2.33d)]

Hz =iη

k

∑

a

γ′′2a ψa(x, y)V ′′z (z) (2.64)

reads

UHz=µ

4

η2

k2

∑

a

γ′′2a |V ′′a (z)|2 , (2.65)

in consequence of the normalization condition for ψa and the orthogonalityof the longitudinal magnetic fields of different H modes. The insertion of thedistributed shunt inductance characteristic of the ath H-mode transmissionline, L′′a = µ/γ′′2a , transforms this energy density expression into

UHz=

1

4

∑

a

1

ω2L′′a|V ′′

a (z)|2 . (2.66)

To evaluate the linear energy density associated with the transverse electricfield

UEt=ε

4

∫

dS |Et|2 , (2.67)

it is convenient to first insert the general representation (2.8), thus obtaining

UEt=ε

4

[∫

dS |∇tV′|2 +

∫

dS |e×∇V ′′|2 − 2Re

∫

dS∇V ′ · e×∇V ′′∗

]

.

(2.68)The last terms of this expression, representing the mutual energy of the E andH modes, may be proved to vanish by the following sequence of equations:

∫

σ

dS∇V ′ · e×∇V ′′∗ = −∫

σ

dS∇V ′′∗ · e×∇V ′

= −∫

σ

dS∇ · (V ′′∗e×∇V ′)

= −∫

C

ds V ′′∗n · e×∇V ′ =

∫

C

ds V ′′∗ ∂

∂sV ′ = 0 ,

(2.69)

in which the last step involves the generally valid boundary condition, ∂∂sV

′ =0 on C, see (2.21). (A proof employing the boundary condition V ′ = 0 on Cwould not apply to the T mode.) It has thus been shown that the transverseelectric field of an E mode is orthogonal to the transverse electric field of anH mode. For the transverse electric field energy density of the E mode, wehave from (2.14a)


ε

4

∫

dS |∇tV′|2 =

ε

4

∑

a

|V ′a(z)|2 , (2.70)

as an immediate consequence of the orthonormality condition (2.54), whichdemonstrates the orthogonality of the transverse electric fields of different Emodes. Similarly, from (2.17a) the transverse electric field energy density ofthe H modes:

ε

4

∫

dS |e×∇V ′′|2 =ε

4

∫

dS |∇tV′′|2 =

ε

4

∑

a

|V ′′(z)|2 , (2.71)

is a sum of individual mode contributions, indicating the orthogonality of thetransverse electric fields of different H modes. Finally, the transverse electricfield energy density is

UEt=

1

4

∑

a

C|V ′a(z)|2 +

1

4

∑

a

C|V ′′a (z)|2 , (2.72)

where C = ε is the distributed shunt capacitance common to all E- and H-mode transmission lines.

The discussion of the transverse magnetic field energy density,

UHt=µ

4

∫

dS |Ht|2 =µ

4

∫

dS |H× e|2 , (2.73)

is precisely analogous and requires no detailed treatment, for in virtue of (2.8)and (2.9a) it is merely necessary to make the substitutions V ′ → I ′, V ′′ → I ′′

(and ε → µ, of course) to obtain the desired result. The boundary conditionupon which the analogue of (2.69) depends now reads ∂

∂sI′ = 0 on C, which

is again an expression of the E-mode boundary condition. Hence

UHt=

1

4

∑

a

L|I ′a(z)|2 +1

4

∑

a

L|I ′′a (z)|2 , (2.74)

where L = µ is the distributed series inductance characteristic of all modetransmission lines. The orthogonality of the transverse magnetic fields associ-ated with two different modes, which is contained in the result, may also bederived from the previously established transverse electric field orthogonal-ity with the aid of the relations between transverse field components that isexhibited in (2.44a), (2.44b).

The complex power flowing along the waveguide is obtained from the lon-gitudinal component of the complex Poynting vector by integration across aguide section:

P =1

2

∫

dS E×H∗ · e =1

2

∫

dSE · (H× e)∗ , (2.75)

whence from (2.8) and (2.9b)


P =1

2

[

∫

dS∇tV′ ·∇tI

′∗ +

∫

dS e×∇V ′′ · e×∇I ′′∗

−∫

dS∇V ′ · e×∇I ′′∗ −∫

dS e×∇V ′′ ·∇I ′∗

]

=1

2

∑

a

V ′a(z)I ′a(z)∗ +

1

2

∑

a

V ′′a (z)I ′′a (z)∗ , (2.76)

which uses (2.14a), (2.14b) and (2.17a), (2.17b), and the analogue of (2.69).Hence the complex power flow is a sum of individual mode contributions,each having the proper transmission-line form; it will now be evident thatthe normalization condtions (2.55) and (2.58) were adopted in anticipation ofthis result. The orthogonality that is implied by expression (2.76) is a simpleconsequence of the orthogonality property of transverse electric fields, sinceH× e for an individual mode is proportional to the corresponding transverseelectric field.

It will have been noticed that the linear energy densities associated withthe different field components are in full agreement with the energies storedper unit length in the various elements of the distributed parameter circuits.Thus the E- and H-type circuits give a complete pictorial description of theelectromagnetic properties of E and H modes in the usual sense: Capacitanceand inductance represents electric and magnetic energy; series elements areassociated with longitudinal electric and transverse magnetic fields (longitudi-nal displacement and conduction currents); shunt elements describe transverseelectric and longitudinal magnetic fields (transverse displacement and conduc-tion currents). The air of precise definition attached to the line parameters,however, is spurious. We are at liberty to multiply a transmission line voltageby a constant and divide the associated current by the same constant with-out violating the requirement that the complex power have the transmissionline form. Thus, let a mode voltage and current be replaced by N−1/2V (z)and N1/2I(z), respectively, implying that the new voltage and current areobtained from the old definitions through multiplication by N 1/2 and N−1/2,respectively. In order the preserve the form of the energy expressions, theinductance parameters must be multiplied by N , and the capacitance param-eters dividied by N . It follows from these statements that the characteristicimpdeance must by multiplied by N , in agreement with its significance asa voltage-current ratio. The propagation constant is unaffected by this al-teration, of course. We may conclude that one of the basic quantities thatspecifies the transmission line, characteristic impdedance, remains essentiallyundefined by any considerations thus far introduced. The same situation arosein the field analysis of the two-conductor transmission line and it was shownthat a natural definition for the characteristic impedance could be obtainedby ascribing the customary physical meaning to either the current or voltage,the same result being obtained in either event. This somewhat artificial pro-cedure was employed in order to emphasize the rather different character of


waveguide fields for, as we shall now show, a precise definition of character-istic impedance can be obtained by ascribing a physical significance to eitherthe current or the voltage, depending on the type of mode, but not to bothsimultaneously.

An E mode is essentially characterized by Ez, from which all other fieldcomponents can be derived. Associated with the longitudinal electric field isan electric displacement current, the current density being [(2.32c)]

−iωεEz = γ2ϕ(x, y)N1/2I(z) . (2.77)

In addition, there is a longitudinal electric conduction current on the metalwalls, with the surface density, according to (2.32b)

−(n×H)z =∂

∂nϕ(x, y)N1/2I(z) , (x, y) ∈ S . (2.78)

The total longitudinal electric current is zero,2 and it is natural to identifythe total current flowing in the positive direction with I(z), which leads tothe following equation for N :

N1/2

[

γ2

∫

+

dS ϕ+

∫

+

ds∂

∂nϕ

]

= 1 , (2.79)

where the surface and line integrals, denoted by∫

+, are to be conducted over

those regions where ϕ and ∂∂nϕ are positive. This equation is particularly

simple for the lowest E mode in any hollow waveguide, that is, the modeof minimum cutoff frequency, for this mode has the property, to be estab-lished in Sec. ??, that the scalar function ϕ is nowhere negative, and vanishesonly on the boundary. It follows that the normal derivative on the boundarycannot be positive. Hence the displacement current flows entirely in the pos-itive direction, and the conduction current entirely in the negative direction.Consequently,

N =1

γ4(∫

dS ϕ)2

=1

γ2

∫

dS ϕ2

(∫

dS ϕ)2

, (2.80)

on employing the normalization condition for ϕ, (2.56), to expressN in a formthat is independent of the absolute scale of the function ϕ, Therefore, for thelowest E mode in any guide, a natural choice of characteristic impedance is,from (2.36b),

Z = ζκ

k

1

γ2

∫

dS ϕ2

(∫

dS ϕ)2

. (2.81)

For the other E modes, the φ normalization condition can be used in ananalogous way to obtain

2 This follows from the fact that H = 0 in the conductor, so that∫

Cds ·H =

∫

σdS · ∇×H = 0.


Z = ζκ

k

1

γ2

∫

dS ϕ2

(

∫

+dS ϕ+ 1

γ2

∫

+ds ∂

∂nϕ)2

. (2.82)

It may appear more natural to deal with the voltage rather than the cur-rent in the search for a proper characteristic impedance definition, since thetransverse electric field of an E mode is derived from a potential [(2.32a)].The voltage could then be defined as the potential of some fixed point withrespect to the wall in a given cross section, thus determining N . In a guideof symmetrical cross section the only material reference point is the center,which entails the difficulty that there exists an infinite class of modes forwhich ϕ = 0 at the center, and the definition fails. In addition, when thisdoes not occur, as in the lowest E mode, the potential of the center point doesnot necessarily equal the voltage, if the characteristic impedance is definedon a current basis as we have done. Hence, while significance can always beattached to the E-mode current, no generally valid voltage definition can beoffered.

By analogy with the E-mode discussion, we shall base a characteristicadmittance definition for H modes on the properties of Hz, which can be saidto define a longitudinal magnetic displacement current density, from (2.33d)

−iωµHz = γ2ψ(x, y)N−1/2V (z) . (2.83)

The total longitudinal magnetic displacement current is zero3 and we shallidentify the total magnetic current flowing in the positive direction with V (z).The voltage thus defined equals the line integral of the electric field intensitytaken clockwise around all regions through which positive magnetic displace-ment flows. Accordingly,

N = γ4

(∫

+

dS ψ

)2

= γ2

(

∫

+dS ψ

)2

∫

dS ψ2(2.84)

and [cf. (2.36d)]

Y = ηκ

k

1

γ2

∫

dS ψ2

(∫

dS ψ)2

. (2.85)

It would also be possible to base an admittance definition on the identificationof the transmission-line current with the total longitudinal electric conductioncurrent flowing in the positive direction on the metal walls. However, thecharacteristic admittance so obtained will not agree in general with that justobtained.

Although we have advanced rather reasonable definitions of characteristicimpedance and admittance, it is clear that these choices possess arbitrary fea-tures and in no sense can be considered inevitable. This statement may convey

3 Because E = 0 in the conductor, so is∫

Cds · E =

∫

σdS ·∇×E.


the impression that the theory under development is essentially vague and ill-defined, which would be a misunderstanding. Physically observable quantitiescan in no way depend on the precise definition of a characteristic impedance,but this does not detract from its appearance in a theory which seeks toexpress its results in conventional circuit language. Indeed, the arbitrarinessin definition is a direct expression of the greater complexity of waveguidesystems compared with low-frequency transmission lines. For example, in thejunction of two low-frequency transmission lines with different dimensions, theconventional transmission-line currents and voltages are continuous to a highdegree of approximation and hence the reflection properties of the junctionare completely specified by the quantities which relate the current and voltagein each line. In a corresponding waveguide situation, however, physical quan-tities with such simple continuity properties do not exist in general, and it istherefore not possible to describe the properties of the junction in terms of twoquantities which are each characteristic of an individual guide. Armed withthis knowledge, which anticipates the results to be obtained in subsequentchapters, we are forced to the position that the characteristic impdedance isbest regarded as a quantity chosen to simplify the electrical representationof particular situation, and that different defintions may be advantageouslyemployed in different circumstances. In particular, the impedance definitionimplicitly adopted at the beginning of the chapter, corresponding to N = 1,is most convenient for general theoretical discussion since it directly relatesthe transvers electric and magnetic fields. This choice, which may be termedthe field impedance (admittance), will be adhered to in the remainder of thischapter.

Before turning to the discussion of particular types of guides, we shallderive a few simple properties of the electric and magnetic energies associatedwith propagating and nonpropagating modes. The tools for the purpose areprovided by the complex Poynting vector theorem

∇ · (E×H∗) = iω(

µ|H|2 − ε|E|2)

, (2.86)

and the energy theorem (ignoring any dependence of ε and µ on the frequency)

∇ ·(

∂E

∂ω×H∗ + E∗ × ∂H

∂ω

)

= i(

ε|E|2 + µ|H|2)

. (2.87)

[Proofs of these theorems are given in the Problems at the end of this chapter.]If (2.86) is integrated over a cross section of the guide, only the longitudinalcomponent of the Poynting vector survives, and we obtain the transmission-line form of the complex Poynting vector theorem, as applied to a single mode[cf. (2.76)]:

d

dz

[

1

2V (z)I(z)∗

]

=d

dzP = 2iω(UH − UE) , (2.88)

where UE and UH are the electric and magnetic linear energy densities. Asimilar operation on (2.87) yields


d

dz

{

1

2

[

∂V (z)

∂ωI(z)∗ + V ∗(z)

∂I(z)

∂ω

]}

= 2i(UE + UH) = 2iU , (2.89)

since ∂Et/∂ω, for example, involves ∂V (z)/∂ω in the same way that Et con-tains V (z), for the scalar mode functions do not depend upon the frequency.As a first application of these equations we consider a propagating wave pro-gressing in the positive direction, i.e.,

V (z) = V eiκz ,I(z) = Ieiκz ,

V = ZI . (2.90)

The complex power is real and independent of z:

P =1

2V I∗ =

1

2Z|I |2 , (2.91)

whence we deduce from (2.88) that UE = UH ; the electric and magnetic linearenergy densities are equal in a progressive wave. To apply the energy theoremwe observe that

∂V (z)

∂ω=∂V

∂ωeiκz + i

dκ

dωzV eiκz , (2.92a)

∂I(z)

∂ω=∂I

∂ωeiκz + i

dκ

dωzIeiκz , (2.92b)

and that

1

2

[

∂V (z)

∂ωI(z)∗ + V ∗(z)

∂I(z)

∂ω

]

=1

2

(

∂V

∂ωI∗ + V ∗ ∂I

∂ω

)

+ idκ

dωz1

2(V I∗ + V ∗I) . (2.93)

Therefore, (2.89) implies

dκ

dω

1

4(V I∗ + V ∗I) =

dκ

dωP = U , (2.94)

orP = vU , (2.95)

where

v =dω

dκ= c

dk

dκ= c

κ

k. (2.96)

The relation thus obtained expresses a proportionality between the powertransported by a progressive wave and the linear energy density. The coef-ficient v must then be interpreted as the velocity of energy transport. It isconsistent with this interpretation that v is always less than c, and vanishesat the cutoff frequency. At frequencies large in comparison with the cutoff fre-quency, v approaches the intrinsic velocity of the medium. It is interesting tocompare this velocity with the two velocities already introduced in discussing


one-dimensional propagation in a dispersive medium – the phase and groupvelocities. The phase velocity equals the ratio of the angular frequency andthe propagation constant:

u =ω

κ= c

k

κ, (2.97)

while the group velocity is the derivative of the angular frequency with respectto the propagation constant:

v =dω

dκ. (2.98)

That the group velocity and energy transport velocity are equal is not unex-pected. We notice that the phase velocity always exceeds the intrinsic velocityof the medium, and indeed is infinite at the cutoff frequency of the mode. Thetwo velocities are related by

uv = c2 (2.99)

which is reminiscent of (??). A simple physical picture for the phase and groupvelocities will be offered in Chapter 3.

Another derivation of the energy transport velocity, which makes moreexplicit use of the waveguide fields, is suggested by the defining equation:

v =P

U=

∫

dS e · E×H∗

1

2

∫

dS (ε|E|2 + µ|H|2) . (2.100)

In virtue of the equality of electric and magnetic linear energy densities, andthe relation e×E = ZH, which is valid for an E-mode field peopagating inthe positive direction [(2.44a)], we find using (2.36b)

v =Z

µ

∫

dS |Ht|2∫

dS |H|2 = cκ

k, (2.101)

since H has no longitudinal component. Similarly, the energy transport ve-locity for an H mode is from (2.44b) and (2.36d)

v =Y

ε

∫

dS |Et|2∫

dS |E|2 = cκ

k. (2.102)

When the wave motion on the transmission line is not that of a simpleprogressive wave, but the general superposition of standing waves (or runningwaves) described by

V (z) = V cosκz + iZI sinκz , (2.103a)

I(z) = I cosκz + iY V sinκz , (2.103b)

or

V (z) = (2Z)1/2(

Aeiκz +Be−iκz)

, (2.104a)

I(z) = (2Y )1/2(

Aeiκz −Be−iκz)

, (2.104b)


the electric and magnetic linear energy densities are not equal, in general,since the complex power is a function of position on the line:

P (z) =1

2

[

V I∗ cos2 κz + V ∗I sin2 κz + i sinκz cosκz(

Z|I |2 − Y |V |2)]

,

(2.105)or

P (z) = |A|2 − |B|2 −AB∗e2iκz +A∗Be−2iκz . (2.106)

However, equality is obtained for the total electric and magnetic energiesstored in any length of line that is an integral multiple of 1

2λg . To prove this,

we observe that by integrating (2.88)

P (z2)− P (z1) = 2iω(WH −WE) , (2.107)

where WE and WH are the total electric and magnetic energies stored in thelength of transmission line between the points z1 and z2. Now, the complexpower is a periodic function of z with the periodicity interval π/κ = 1

2λg

[(2.38], from which we conclude that P (z2) = P (z1) if the two points areseparated by an integral number of half-guide wavelengths, which verifies thestatement. An equivalent form of this result is that the average electric andmagnetic energy densities are equal, providing the averaging process is ex-tended over an integral number of half-guide wavelengths, or over a distancelarge in comparison with 1

2λg .

An explicit expression for the average energy density can be obtainedfrom the energy theorem. The total energy W , stored in the guide betweenthe planes z = z1 and z = z2, is given by the integral of (2.89), or

W =1

4i

[

∂V (z)

∂ωI(z)∗ + V (z)∗

∂I(z)

∂ω

]z2

z1

. (2.108)

On differentiating the voltage and current expressions (2.103a), (2.103b) withrespect to the frequency, we find

∂V (z)

∂ω= iz

dκ

dωZI(z) +

[

cosκz∂V

∂ω+ i sinκz

∂ZI

∂ω

]

, (2.109a)

∂I(z)

∂ω= iz

dκ

dωY V (z) +

[

cosκz∂I

∂ω+ i sinκz

∂Y V

∂ω

]

. (2.109b)

Hence

∂V (z)

∂ωI(z)∗ + V (z)∗

∂I(z)

∂ω= iz

dκ

dω

[

Z|I(z)|2 + Y |V (z)|2]

+ . . . , (2.110)

where the unwritten part of this equation consists of those terms, arising fromthe bracketed expressions in (2.109a) and (2.109b), which are periodic func-tions of z with the period 1

2λg . Thus, if the points z1 and z2 are separated by


a distance that is an integral multiple of 1

2λg , these terms make no contribu-

tion to the total energy. We also note that the quantity Z|I(z)|2 + Y |V (z)|2is independent of z:

Z|I(z)|2 + Y |V (z)|2 = Z|I |2 + Y |V |2 = 4(|A|2 + |B|2) , (2.111)

from which we conclude that the total energy stored in a length of guide,l, which is an integral number of half-guide wavelengths, is, in terms of theenergy velocity (2.96)

W = ldκ

dω

1

4

(

Z|I |2 + Y |V |2)

=l

v

1

4

(

Z|I |2 + Y |V |2)

=l

v

(

|A|2 + |B|2)

.

(2.112)The average total energy density is W/l, which has a simple physical signif-icance in terms of running waves, being just the sum of the energy densitiesassociated with each progressive wave component if it alone existed on thetransmission line.

The energy relations for a nonpropagating mode are rather different; thereis a definite excess of electric or magnetic energy, depending on the type ofmode. The propagation constant for a nonpropagating (below cutoff) mode isimaginary:

κ = i√

γ2 − k2 = i|κ| , (2.113)

and a field that is attenuating in the positive direction is described by

V (z) = V e−|κ|z

I(z) = Ie−|κ|zV = ZI . (2.114)

The imaginary characteristic impedance (admittance) of an E(H) mode isgiven by

E mode: Z = iζ|κ|k

= i|Z| , (2.115a)

H mode: Y = iη|κ|k

= i|Y | . (2.115b)

The energy quantities of interest are the total electric and magnetic energystored in the positive half of the guide (z > 0). The difference of these energiesis given by (2.107), where z1 = 0 and z2 → ∞. Since all field quantitiesapproach zero exponentially for increasing z, P (z2) → 0, and

WE −WH =1

2iωP (0) =

1

4iωV I∗ . (2.116)

For an E modeV I∗ = Z|I |2 = i|Z||I |2 , (2.117)

and so

E mode: WE −WH =1

4ω|Z||I |2 , (2.118)


which is positive. Hence an E mode below cutoff has an excess of electricenergy, in agreement with the capacitive reactance form of the characteristicimpedance. Similarly, for an H mode,

V I∗ = Y ∗|V |2 = −i|Y ||V |2 , (2.119)

whence

H mode: WH −WE =1

4ω|Y ||V |2 , (2.120)

implying that an H mode below cutoff perponderantly stores magnetic en-ergy, as the inductive susceptance form of its characteristic admittance wouldsuggest.

To obtain the total energy stored in a nonpropagating mode, we employ(2.108), again with z1 = 0 and z2 →∞:

W =i

4

(

∂V

∂ωI∗ + V ∗ ∂I

∂ω

)

. (2.121)

On differentiating the relation V = ZI with respect to ω, and making appro-priate substitutions in (2.121), we find

W =i

4

[

dZ

dω|I |2 + (Z + Z∗)

∂I

∂ωI∗

]

. (2.122)

The imaginary form of Z (= i|Z| for an E mode) then implies that

W = −1

4

d|Z|dω

|I |2 . (2.123)

We may note in passing that the positive nature of the total energy demandsthat |Z| be a decreasing function of frequency, or better, that the reactancecharacterizing Z (= −iX) be an increasing function of frequency. This gener-alization of the low-frequency theorems discussed in Chapter 1 will be furtherdeveloped in Chapter X. The requirement is verified by direct differentiation:

−d|Z|dω

=1

ω

γ2

γ2 − k2|Z| , (2.124)

and

E mode: W =1

4ω

γ2

γ2 − k2|Z||I |2 . (2.125)

A comparison of this result with (2.118) shows that

2WH

W=k2

γ2. (2.126)

Thus, the electric and magnetic energies are equal just at the cutoff frequency(k = γ), and as the frequency diminishes, the magnetic energy steadily de-creases in comparison with the electric energy. In conformity with the latter


remark, the E-mode characteristic impedance approaches iζγ/k = iγ/ωε whenk/γ � 1, which implies that a transmission line describing an attenuated Emode at a frequency considerably below the cutoff frequency behaves like alumped capacitance C = ε/γ = ελc/(2π).

The H-mode discussion is completely analogous, with the roles of electricand magnetic fields interchanged. Thus the total stored energy is

W = −1

4

d|Y |dω

|V |2 , (2.127)

implying that the susceptance characterizing Y (= −iB) must be an increasingfunction of frequency. Explicitly,

H mode: W =1

4ω

γ2

γ2 − ω2|Y ||V |2 , (2.128)

and2WE

W=k2

γ2. (2.129)

At frequencies well below the cutoff frequency, the electric energy is negligiblein comparison with the magnetic energy, and the characteristic admittancebecomes iηγ/k = iγ/ωµ. Thus, an H-mode transmission line under these cir-cumstances behaves like a lumped inductance L = µ/γ = µλc/(2π).

2.1.5 Problems for Chapter 2

1. Prove the complex Poynting vector theorem, (2.86), and the energy the-orem, (2.87), starting from the definitions of the Fourier transforms intime:

E(ω) =

∫ ∞

−∞

dt eiωtE(t) , (2.130a)

H∗(ω) =

∫ ∞

−∞

dt e−iωtH(t) . (2.130b)

2 waveguides and equivalent transmission linesmilton/p5573/chap2.pdf16 2 waveguides and equivalent...

Documents