Zhang K., Li D. Electromagnetic Theory for Microwaves and Optoelectronics

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5.1 General Characteristics of Metallic Waveguides 241

For TE modes, it becomes

2 η

+ E

dS =

2 η

dS =

dS, (5.11)

and for TM modes

+ H

dS =

2 η

dS. (5.12)

If we use (4.63)–(4.68), applying Helmholtz’s equations and boundary

conditions, the above expressions, (5.11) and (5.12), become functions of U

and V :

dS =

dS, (5.13)

2 η

dS =

dS. (5.14)

5.1.6 Attenuation

(1) Volume Loss

If the volume loss of the material ﬁlling the metallic waveguide is considered,

the angular wave number of the medium becomes complex and the phase

coeﬃcient of the waveguide is also complex. It becomes

β =

1 −

= β − jα, (5.15)

where

k denotes the complex angular wave number of the lossy medium,

given in Section 2.1.3. The imaginary part α is the attenuation coeﬃcient.

The wave will attenuate exponentially along the waveguide.

(2) Surface Loss

The wall of a practical waveguide is made of high-conductivity metal instead

of a perfect conductor. There are power losses on the surface of the wall

and the wave in the waveguide will be attenuated along the direction of

propagation.

The attenuation of the waveguide due to surface loss can be obtained by

solving Helmholtz’s equations with non-perfect conductor boundary condi-

tions, which gives the accurate ﬁeld solution of the problem, but it is an

onerous task. In practice, for low-loss waveguides made with good conductor

walls, the perturbation technique given as follows is suitable.

In practical lossy waveguides, the tangential comp onent of the electric

ﬁeld at the boundary becomes nonzero. The tangential electric ﬁeld accom-

panied by the tangential magnetic ﬁeld forms the Poynting vector pointing

242 5. Metallic Waveguides and Resonant Cavities

Figure 5.3: Fields and power ﬂows in an ideal waveguide and a lossy waveg-

uide.

normally into the wall and gives rise to the attenuation. But in ideal waveg-

uides, the tangential component of the electric ﬁeld at the boundary is always

zero. We recognize that this is the only diﬀerence that must be considered,

because it is the diﬀerence of a ﬁnite value from zero. However, the other

components of the ﬁelds in most regions of a lossy waveguide are also diﬀer-

ent from those in an ideal waveguide, but the diﬀerences are small enough

for a low-loss waveguide and can be neglected. See Fig. 5.3. Therefore, the

tangential component of the electric ﬁeld at the boundary of a low-loss waveg-

uide can be obtained by means of the expression for the surface impedance

of a good conductor given in Section 2.1.3 and the value obtained in the

perfect waveguide for the tangential component of the magnetic ﬁeld. Here,

the plane-wave approximation is used because the penetration depth for a

good conductor is very small compared with the radius of curvature of the

wall, and the Poynting vector is always approximately normal to the good

conductor surface, refer to Section 2.4.9.

The z dependence of the ﬁelds in a lossy waveguide with propagation

coeﬃcient γ = α + jβ is e

−jβz

−αz

, and the power ﬂow along z is P (z) =

P e

−2αz

. Then we have

α =

dP/dz

[Np/m] = 4.343

dP/dz

[dB/m]. (5.16)

The average power ﬂow density normal to the conducting wall is obtained in

Section 2.1.3 as

S =

σδ

, (5.17)

where H

denotes the tangential component of the magnetic ﬁeld at the

surface of the wall, which is obtained approximately by the solution for a

perfect waveguide. The average power ﬂow entering a wall of length dz is

given by

dP = dz

Sdl, and

2σδ

5.2 General Characteristics of Resonant Cavities 243

Substituting it into (5.16) yields

α =

4σδ

[Np/m], (5.18)

where l denotes the enclosed boundary curve of the cross section of the inner

wall of the waveguide, P denotes the average power ﬂow along the waveguide

given in the last subsection, σ denotes the conductivity of the wall material,

and δ denotes the skin depth.

Finally, considering (5.11) and (5.12), we have

2σδβ

, (5.19)

2σδβ

. (5.20)

5.2 General Characteristics of Resonant

Cavities

A lossless electromagnetic system completely enclosed by a short-circuit or

open-circuit surface forms an adiabatic system, which is known as an ideal

resonant cavity or ideal resonator. Practically, a resonant cavity can be a

metallic box with an arbitrary geometry in which the short-circuit boundary

is approximately realized by means of a high-conductivity metal wall.

The electromagnetic problem of an ideal resonant cavity is a typical 3-

dimensional eigenvalue problem. The electromagnetic ﬁelds can exist in an

ideal cavity only when the frequency is equal to one of the discrete natural

frequencies or resonant frequencies, which is known as the oscillation mode

of the cavity. In the resonant state, in an ideal resonator, the maximum

electric energy is equal to the maximum magnetic energy stored in the res-

onator. They convert to each other periodically and become electromagnetic

oscillations. No energy is needed to sustain the oscillation because an ideal

resonator is a lossless system.

If the cavity is ﬁlled with lossy dielectric material or the loss on the

metallic wall of the cavity is no longer negligible, the oscillation in a source-

free resonator will damp out with respect to time. A source must exist to

compensate the power loss and sustain oscillations, and the discrete natural

frequencies will expand into frequency bands. The frequency response of a

mode of a resonant cavity is the same as that of an L-C resonant circuit.

5.2.1 Modes and Natural Frequencies of the Resonant

Cavity

The natural angular wave number of the mth mode of an ideal resonant

cavity is equal to the mth eigenvalue of the enclosed electromagnetic system

244 5. Metallic Waveguides and Resonant Cavities

given in Section 4.10.2:

|∇ × E

. (5.21)

The corresponding natural angular frequency is

√

µ²

. (5.22)

The ﬁelds E

and H

are the eigenfunctions of the boundary value problem

and satisfy the Maxwell equations. So, (5.21) becomes

= ω

µ²

µH

²E

= k

We have

= W

This is consistent with the result of Poynting’s theorem given in Section 1.4.2.

5.2.2 Losses in a Resonant Cavity, the Q Factor

An important parameter specifying the performances of a resonant circuit is

the quality factor or Q. The general deﬁnition of Q that is applicable to all

resonant systems is

Q =

time−average energy stored in the system

energy loss per radian of oscillation in the system

For the mth mode of a resonant cavity,

= ω

, (5.23)

where ω

denotes the natural angular frequency of the mth mode, W denotes

the time-average energy stored in the cavity, and P denotes the power loss,

i.e., the energy loss per unit time interval in the cavity.

The time-average energy density stored in the cavity at the mth mode,

w, is

w =

µH

²E

µH

²E

The Joule-loss and polarization-loss densities inside the cavity give

σE

ω²

The Q of a cavity due to the volume loss is then given by

= ω

wdV

ω²

σ + ω²

tan δ

. (5.24)

5.3 Waveguides and Cavities in Rectangular Coordinates 245

Applying the perturbational technique, we ﬁnd that the average power

ﬂow density normal to the cavity wall is given in (5.17) as

σδ

where H

denotes the tangential component of the magnetic ﬁeld at the

surface of the cavity wall, which is obtained approximately by the solution

for an ideal cavity. The Q of a cavity due to the surface loss is then given by

= ω

. (5.25)

The Q of a cavity due to the total loss becomes

+ P

, (5.26)

where Q

denotes the Q of an unloaded cavity and is known as the unloaded

quality factor.

When a cavity is coupled to an external load, we deﬁne the external

quality factor Q

as follows

time−average energy stored in the cavity

energy loss per radian of oscillation in the load

= ω

. (5.27)

The Q factor of the system is known as the loaded quality factor, denoted by

, and is given by

time−average energy stored in the cavity

energy loss per radian of oscillation in the system

=ω

sys

. (5.28)

sys

+ P

. (5.29)

5.3 Waveguides and Cavities in Rectangular

Coordinates

The waveguides and cavities in rectangular coordinates include the rectangu-

lar waveguide, parallel plate transmission line, and rectangular cavity. Their

boundaries coincide with the coordinate planes of the rectangular coordinate

system.

5.3.1 Rectangular Waveguides

The rectangular waveguide is a metallic pipe with a rectangular cross section.

See Fig. 5.4. The transverse dimensions of the waveguide are a and b in the

x and y directions, respectively. It is uniform and inﬁnitely long in the

longitudinal direction z.

246 5. Metallic Waveguides and Resonant Cavities

Figure 5.4: Rectangular waveguide.

(1) TE Modes

For the TE mode, U = 0. The function V (x, y, z) was given in (4.132)–

(4.134) or (4.138)–(4.140). As the ﬁelds in the waveguide are standing waves

along x, y and traveling waves along +z, the functions X(x) and Y (y) must be

sinusoidal functions and Z(z) must be an exponential function with imaginary

argument.

V (x, y, z) = X(x)Y (y)Z(z)

= (A sin k

x + B cos k

x)(C sin k

y + D cos k

y)e

−jβz

, (5.30)

where β = k

denotes the longitudinal phase coeﬃcient, and

β = k

− T

, T =

+ k

. (5.31)

The above V (x, y, z) satisﬁes Helmholtz’s equation, and the next step is to

satisfy the boundary conditions.

The boundary conditions of function V on the boundary of the waveguide

is given from Section 4.4 as

∂V

∂n

= 0, i.e.,

∂V

∂x

x=0,a

= 0 and

∂V

∂y

y =0,b

= 0. (5.32)

Taking the derivatives of (5.30), we have

∂V

∂x

= k

(A cos k

x − B sin k

x)(C sin k

y + D cos k

y)e

−jβz

∂V

∂y

= k

(A sin k

x + B cos k

x)(C cos k

y − D sin k

y)e

−jβz

Substituting them into the boundary equations (5.32) yields

∂V

∂x

x=0

= 0 −→ A = 0, (5.33)

5.3 Waveguides and Cavities in Rectangular Coordinates 247

∂V

∂x

x=a

= 0 −→ sin k

a = 0, k

mπ

, m is an integer, (5.34)

∂V

∂y

y =0

= 0 −→ C = 0, (5.35)

∂V

∂y

y =b

= 0 −→ sin k

b = 0, k

nπ

, n is an integer. (5.36)

The expressions for T , ω

, and λ

become

T =

mπ

nπ

, ω

mπ

nπ

µ²

, λ

√

(5.37)

The expression for β becomes

β = k

µ² −

mπ

nπ

. (5.38)

Substituting (5.33)–(5.36) into (5.30) gives

V (x, y, z) = V

cos(k

x) cos(k

y)e

−jβz

, (5.39)

where V

= BD.

Substituting (5.39) into the expressions for the transverse ﬁeld compo-

nents, (4.147)–(4.152), and considering U = 0 yields

= −jωµ

∂V

∂y

= jωµk

cos(k

x) sin(k

y) e

−jβz

, (5.40)

= jωµ

∂V

∂x

= −jωµk

sin(k

x) cos(k

y) e

−jβz

, (5.41)

= T

U = 0, (5.42)

= −jβ

∂V

∂x

= jβk

sin(k

x) cos(k

y) e

−jβz

, (5.43)

= −jβ

∂V

∂y

= jβk

cos(k

x) sin(k

y) e

−jβz

, (5.44)

= T

V =

+ k

cos(k

x) cos(k

y) e

−jβz

. (5.45)

In the above expressions, m and n are arbitrary integers. A set of speciﬁc m

and n represents a mode denoted by TE

. It can be seen from (5.40)–(5.45)

that one of m or n can be zero, to form the TE

and TE

modes, but m

and n cannot both be zero, if so, all ﬁeld comp onents would be zero.

Field maps of some low-order TE modes are given in Figure 5.5.

248 5. Metallic Waveguides and Resonant Cavities

Figure 5.5: Field maps of TE modes in a rectangular waveguide.

(2) TM Modes

For the TM mode, V = 0. The function U (x, y, z) is given by

U(x, y, z) = X(x)Y (y)Z(z)

= (A sin k

x + B cos k

x)(C sin k

y + D cos k

y)e

−jβz

. (5.46)

The boundary conditions for function U on the boundary of the transverse

cross section are given from Section 4.4 as follows:

x=0

= 0 −→ B = 0, (5.47)

x=a

= 0 −→ sin k

a = 0, k

mπ

, m is an integer, (5.48)

y =0

= 0 −→ D = 0, (5.49)

y =b

= 0 −→ sin k

b = 0, k

nπ

, n is an integer. (5.50)

The expressions for T , ω

, λ

and β become

T =

mπ

nπ

, ω

mπ

nπ

µ²

, λ

√

(5.51)

5.3 Waveguides and Cavities in Rectangular Coordinates 249

Figure 5.6: Field maps of TM modes in a rectangular waveguide.

The expression for β becomes

β = k

µ² −

mπ

nπ

. (5.52)

The above expressions for cutoﬀ condition and dispersion are the same as

those for TE modes given in (5.37) and (5.38).

Substituting (5.47)–(5.50) into (5.46) gives

U(x, y, z) = U

sin(k

x) sin(k

y)e

−jβz

, (5.53)

where U

= AC.

Substituting (5.53) into the expressions for the transverse ﬁeld compo-

nents, (4.147)–(4.152), and considering V = 0 yields

= −jβ

∂U

∂x

= −jβk

cos(k

x) sin(k

y) e

−jβz

, (5.54)

= −jβ

∂U

∂y

= −jβk

sin(k

x) cos(k

y) e

−jβz

, (5.55)

= T

U =

+ k

sin(k

x) sin(k

y) e

−jβz

, (5.56)

= jω²

∂U

∂y

= jω²k

sin(k

x) cos(k

y) e

−jβz

, (5.57)

= −jω²

∂U

∂x

= −jω²k

cos(k

x) sin(k

y) e

−jβz

, (5.58)

= T

V = 0. (5.59)

The ﬁeld maps of some low-order TM modes are given in Fig. 5.6.

A set of sp eciﬁc m and n represents a TM

mode. It can be seen from

(5.54)–(5.59) that all ﬁeld components will be zero when m or n is zero. So m

250 5. Metallic Waveguides and Resonant Cavities

0.0 .5 1.0 1.5 2.0 2.5 3.0

0.0

1.0

1.5

2.0

2.5

3.0

TE10

TE01

TE20

TE02

TE30

TE03

TE11 TM11

TE21

TM21

TE12 TM12

TE31

TM31

TE13 TM13

TE33

TM33

Figure 5.7: Mode distribution in a rectangular waveguide.

and n are both nonzero integers for TM modes, and TM

and TM

modes

cannot exist in rectangular waveguides. The cutoﬀ angular wave number

= T and the phase coeﬃcient β are the same for TE

modes and TM

modes, so the TE and TM modes with the same m, n are degenerate modes.

(3) Propagation Characteristics

Substituting the formula of T in (5.37) or (5.51) into (5.1), (5.3), (5.4), (5.5),

(5.6), and (5.8), we have the cutoﬀ angular frequency ω

, cutoﬀ wavelength

, phase velocity v

, guided wavelength λ

, group velocity v

, and the cutoﬀ

attenuation coeﬃcient α of a sp eciﬁc mode in the rectangular waveguide.

The distribution of the cutoﬀ wavelengths of some lower-order modes in

rectangular waveguides is given in Fig. 5.7, and the dispersion curves (f - β

diagram) are given in Fig. 5.8.

If the width in the x direction is larger than the height in the y direction of

a rectangular waveguide, a > b, the cutoﬀ frequency of the TE

mode is the

lowest one, namely the lowest-order mode. When the height in the y direction

is less than the half-width in the x direction, b < a/2, the TE

mode is the

next lower-order mode, and when the height in the y direction is larger than

the half-width in the x direction, b > a/2, the TE

mode becomes the next

lower-order mode. When the frequency is higher than the cutoﬀ frequency of

the lowest-order mode, and lower than the cutoﬀ frequency of the next lower-

order mode, the lowest-order mode becomes the only propagating mode. This

is known as the single-mode state of a transmission system. In a rectangular

waveguide, single mode propagation can be realized only with the TE

mode,

which is known as the dominant mode or principal mode.