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

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5.5 Waveguides and Cavities in Spherical Coordinates 291

Figure 5.36: Fields of the TM

101

and TE

101

modes in a spherical cavity.

∂

∂θ ∂r

= k

sin kr

−

cos kr

−

sin kr

sin θ, (5.274)

jωµ

∂V

∂θ

= −jηk

sin kr

−

cos kr

sin θ, (5.275)

where

k =

4.493

. (5.276)

We can see that the TM

101

mode is the lowest mode in spherical cavity.

The ﬁeld maps of the TM

101

and TE

101

modes in a a spherical cavity are

shown in Fig. 5.36.

5.5.2 Biconical Lines and Biconical Cavities

Biconical lines, shown in Fig. 5.37(a), (b), are spherical radial waveguides.

In a biconical line, the p olar axes, θ = 0 and θ = π, are excluded from

the ﬁeld region, so the solutions must include two independent Legendre

functions, in which m is an integer because the ﬁeld region includes the whole

circumference in φ, but ν is not necessarily an integer. So the two independent

solutions can be P

(cos θ) and Q

(cos θ) or P

(cos θ) and P

(−cos θ). The

orientation of φ = 0 is chosen such that Borgnis’ function becomes an even

function respect to φ.

(1) TM and TE Modes in Biconical Lines

The solution U for the TM mode in a biconical line becomes

U =

√

(1)

ν+1/2

(kr) + b

√

(2)

ν+1/2

(kr)

[CP

(cos θ)+DQ

(cos θ)]cos(mφ).

Applying the short-circuit boundary condition U |

= 0, we have

D = −C

(cos θ

)

(cos θ

)

292 5. Metallic Waveguides and Resonant Cavities

Figure 5.37: Biconical lines and biconical cavity.

and then

U =

√

(1)

ν+1/2

(kr) + B

√

(2)

ν+1/2

(kr)

¤£

(cos θ

(cos θ)

−P

(cos θ

(cos θ)

cos(mφ), (5.277)

where

A = a

(cos θ

)

, B =

(cos θ

)

Applying the short-circuit boundary condition U |

= 0, we have

(cos θ

) − P

(cos θ

) = 0. (5.278)

This is the eigenvalue equation for the TM mode in a biconical line. The pth

root of this equation of mth order is denoted by ν

The solution V for the TE mode in a biconical line becomes

V =

√

(1)

ν+1/2

(kr) + b

√

(2)

ν+1/2

(kr)

[CP

(cos θ)+DQ

(cos θ)]cos(mφ).

(5.279)

Applying the short-circuit boundary condition at θ = θ

and θ = θ

dθ

= 0 and

dθ

= 0,

we have

(cos θ)

dθ

+ D

(cos θ)

dθ

= 0

and

(cos θ)

dθ

+ D

(cos θ)

dθ

= 0.

5.5 Waveguides and Cavities in Spherical Coordinates 293

These homogeneous equations are satisﬁed by nontrivial solutions only when

their determinants vanish, that is

(cos θ)

dθ

(cos θ)

dθ

−

(cos θ)

dθ

(cos θ)

dθ

= 0.

(5.280)

This is the eigenvalue equation for the TE mode in a biconical line. The pth

root of this equation of mth order is denoted by ν

. Generally, ν

is not an integer, it is an integer only when θ

and θ

are some special

angles.

The expressions for the ﬁeld component in biconical line can be obtained

by substituting the above U or V into the ﬁeld component formulas respect

to Borgnis’ potentials. They are spherical traveling waves in the +r and −r

directions.

(2) TEM Mode in Biconical Lines

The lowest-order mode in the biconical line is the one for which m = 0 and ν

takes the ﬁrst root of (5.278) and (5.280), i.e., ν = 0. For m = 0 and ν = 0,

all the ﬁeld components of the TE mode are zero and the function U for the

TM mode is given by

U =

√

(1)

1/2

(kr)+B

√

(2)

1/2

(kr)

(cos θ

(cos θ)−P

(cos θ

(cos θ)].

(5.281)

Considering

(cos θ) = 1, and Q

(cos θ) = ln cot

and applying the expressions for the half-order Bessel functions in Appendix

C.5.1:

(1)

1/2

(x) = −j

πx

, H

(2)

1/2

(x) = j

πx

−jx

yields

U =

ln cot

− ln cot

−jkr

− U

−

jkr

, (5.282)

where

−

= −jA

πk

, U

= −jB

πk

Substituting (5.282) into (4.233) to (4.238), and using

dθ

ln cot

= −

sin θ

we have the ﬁeld components

r sin θ

−jkr

+ U

−

jkr

r sin θ

−jkr

+ E

−

jkr

, (5.283)

294 5. Metallic Waveguides and Resonant Cavities

jω²

r sin θ

−jkr

− U

−

jkr

r sin θ

−jkr

−

jkr

, (5.284)

= 0, E

= 0, H

= 0,

where

= jkU

, E

−

= jkU

−

This is the TM

mode in a biconical line, in which only the transverse com-

ponent of the electric ﬁeld E

and the transverse component of the magnetic

ﬁeld H

exist; they form two spherical TEM waves in the +r and −r direc-

tions. So the TM

mode in a biconical line is also known as the spherical

TEM mode, denoted by TEM

(r)

According to (3.70), the characteristic impedance of the spherical TEM

mode in biconical line becomes

)rdθ

2π

±H

r sin θdφ

2π

tan(θ

/2)

tan(θ

/2)

. (5.285)

(3) Biconical Cavities

If the biconical line is closed by a short-circuit surface at r = a and the two

conical tips separated by an inﬁnitesimal gap it becomes a biconical cavity,

shown in Fig. 5.37(c). Applying the boundary condition at r = a, we have

the expressions for the ﬁeld components of the dominant TEM modes in a

biconical cavity as follows

= E

sin k(a − r)

r sin θ

, E

jη

cos k(a − r)

r sin θ

. (5.286)

The input impedance seen from the tip is

= jZ

tan ka,

which is the same as the formula for a uniform transmission line. The reso-

nant condition for a biconical cavity is

ka =

pπ

, ω

pπ

√

µ²

, p = 1, 2, 3 ···. (5.287)

The capacity-loaded biconical cavity or reentrant spherical cavity is used

in some active devices and gas-discharge microwave duplexers, refer to Prob-

lem 5.15.

5.6 Reentrant Cavities 295

Figure 5.38: Reentrant cavities.

5.6 Reentrant Cavities

In microwave active devices, such as klystrons, microwave triodes and

tetrodes, Gunn diode oscillators, varactor parametric ampliﬁers, microwave

duplexers, and particle accelerators, it is essential for eﬃcient energy transfer

between the carriers, i.e., electrons, protons, or holes, and the ﬁelds that the

electric ﬁeld in the interactive region be strong enough and the transit time

of carriers across the ﬁeld region b e small enough. Special shapes are em-

ployed which have a small gap in the interactive region and are known as the

small-gap cavities or reentrant cavities. Some examples of such cavities are

capacitance-loaded coaxial lines, capacitance-loaded radial lines, capacitance-

loaded biconical lines, and reentrant cylindrical cavities, shown in Fig. 5.38.

The reentrant cavity shown in Fig. 5.38(a) or (a’) is a circular cylindri-

cal cavity with a small gap at the central part of the cavity. The boundary

surface of z = constant is no longer uniform in the ρ direction, and the prob-

lem becomes a boundary-value problem with complicated boundaries. This

problem can be solved by means of the method given in Section 4.11. The

complicated boundary conditions for a reentrant cavity can not be satisﬁed

by the ﬁelds with simple sine or cosine functions, i.e., single harmonics in z.

The functions of the ﬁelds must be a series with inﬁnite terms, or so called

inﬁnite space harmonics.

The interesting mode in the circular cylindrical reentrant cavity is the

circumferential uniform TM mode, in which V = 0 and U is an even function

with respect to z, because the geometry of the cavity is symmetric with

respect to z = 0, refer to Fig. 5.38(a). The function U can then be expressed

by a series of space harmonics with even symmetrical functions with respect

296 5. Metallic Waveguides and Resonant Cavities

to z and each term of the series is given by the following general form

U = [a

(T ρ) + a

(T ρ)] cos βz. (5.288)

The ﬁeld region of the reentrant cavity is divided into two sub-regions:

region 1, the gap region, ρ ≤ b and region 2, the cavity region, b ≤ r ≤ a.

Region 1. Region 1 includes the axis ρ = 0, so the coeﬃcient of N

(T ρ) in

(5.288) is zero, a

= 0. Applying the short-circuit boundary conditions on

the surfaces of the gap z = ±d, we have

∂U

∂z

z=±d

= 0, i.e., sin βd = 0, β

mπ

, m = 0, 1, 2, 3, ···. (5.289)

Borgnis’ function in region 1 is then written as the following series:

∞

m=0

ρ) cos β

z, (5.290)

where

− β

µ² −

mπ

. (5.291)

The expressions for the ﬁeld components in region 1 are then given by

∞

m=0

ρ) cos β

z, (5.292)

ρ1

∞

m=0

ρ) sin β

z, (5.293)

φ1

∞

m=0

jω²T

ρ) cos β

z, (5.294)

φ1

= 0, H

ρ1

= 0.

Region 2. Region 2 does not include the axis ρ = 0, so neither the coeﬃcient

of J

(tρ) nor the co eﬃcient of N

(tρ) in (5.288) are zero. Applying the short-

circuit boundary conditions on the end surfaces of the cavity z = ±l gives

∂U

∂z

z=±l

= 0, i.e., sin βl = 0, β

nπ

, n = 0, 1, 2, 3, ···. (5.295)

Borgnis’ function in region 2 is then written as

∞

m=0

ρ) + a

ρ)] cos β

z, (5.296)

5.6 Reentrant Cavities 297

where

− β

µ² −

nπ

. (5.297)

In region 2, the function U

must satisfy the short-circuit boundary con-

dition on the cylindrical surface ρ = a, that is

ρ=a

= 0, i.e., a

a) + a

a) = 0,

so that

= −

= b

. (5.298)

Then the expression for U

becomes

∞

m=0

a)J

ρ) − J

a)N

ρ)] cos β

z. (5.299)

The expressions for the ﬁeld components in region 2 are then given by

∞

n=0

a)J

ρ) − J

a)N

ρ)] cos β

z, (5.300)

ρ2

∞

n=0

a)J

ρ) − J

a)N

ρ)] sin β

z, (5.301)

φ2

∞

m=0

jω²T

a)J

ρ) − J

a)N

ρ)] cos β

z, (5.302)

φ2

= 0, H

ρ2

= 0.

The next step is to match the ﬁelds of the two regions at the boundary

ρ = b. The methods used to ﬁnd a strict solution and approximate solutions

will be introduced in the next subsections.

5.6.1 Exact Solution for the Reentrant Cavity

According to the theory given in Section 4.1.2, for obtaining the exact so-

lution, both the tangential electric ﬁeld and the tangential magnetic ﬁeld of

the two regions must be continuous at the boundary ρ = b, i.e.,

ρ=b

= E

ρ=b

, H

φ1

ρ=b

= H

φ2

ρ=b

. (5.303)

The z component of the electric ﬁeld in region 1 at ρ = b is obtained from

(5.292):

(b) =

∞

m=0

cos

mπz

, where A

= T

b). (5.304)

298 5. Metallic Waveguides and Resonant Cavities

The φ component of the magnetic ﬁeld (5.294) in region 1 at ρ = b becomes

φ1

(b) =

∞

m=0

cos

mπz

, where Y

jω²

. (5.305)

The z component of the electric ﬁeld in region 2 at ρ = b is obtained from

(5.300):

(b) =

∞

n=0

cos

nπz

, (5.306)

where

= T

a)J

b) − J

a)N

b)]. (5.307)

The φ component of the magnetic ﬁeld (5.294) in region 2 at ρ = b becomes

φ2

(b) =

∞

n=0

cos

nπz

, (5.308)

where

jω²

a)J

b) − J

a)N

a)J

b) − J

a)N

. (5.309)

The ﬁeld-matching conditions at the cylindrical boundary ρ = b are

(b)=E

(b) →

∞

n=0

cos

nπz

∞

m=0

cos

mπz

, |z| ≤ d, (5.310)

(b)=0 →

∞

n=0

cos

nπz

=0, d ≤ |z| ≤ l, (5.311)

φ2

(b)=H

φ1

(b) →

∞

n=0

cos

nπz

∞

m=0

cos

mπz

, |z| ≤ d. (5.312)

Considering the right-hand sides of (5.310) and (5.311) as the given func-

tion, we can ﬁnd the coeﬃcient of the Fourier series of the left-hand side, B

∞

m=0

cos

mπz

cos

nπz

dz =

∞

m=0

. (5.313)

Then, considering the left-hand side of (5.312) as a given function, we can

ﬁnd the coeﬃcient of the Fourier series of the right-hand side, A

, as

∞

m=0

cos

nπz

cos

mπz

dz =

∞

n=0

. (5.314)

5.6 Reentrant Cavities 299

In the above two expressions,

, δ

1 n, m = 0,

2 n, m 6= 0,







1 n = 0, m = 0,

0 n = 0, m 6= 0,

cos

mπz

cos

nπz

dz n 6= 0.

Substituting the expression for B

(5.313) into (5.314), and using p in-

stead of m in (5.313), we have

∞

n=0

∞

p=0

. (5.315)

Let

∞

n=0

. (5.316)

Then (5.315) becomes

∞

p=0

− A

= 0,

i.e.,

∞

p=06=m

+ (Q

− Y

= 0. (5.317)

This is a set of homogeneous linear equations of inﬁnite order, and the co-

eﬃcients are an inﬁnite series. The homogeneous equations are satisﬁed by

nontrivial solutions only when the determinant of the coeﬃcients vanishes,

that is

− Y

··· Q

···

− Y

··· Q

···

− Y

··· Q

···

··· ··· ··· ··· ··· ···

··· Q

− Y

···

··· ··· ··· ··· ··· ···

= 0. (5.318)

This is the eigenvalue equation of the eigenvalue problem, which is an inﬁnite

algebraic equation and all the coeﬃcients are inﬁnite series. The eigenvalue

equation can be solved by numerical method. The roots of this equation are

the natural frequencies of the cavity, which is involved in equations (5.305),

(5.309), (5.316), and (5.318).

The coeﬃcients of the ﬁeld components A

and B

are then found by

solving the linear equations (5.317) and using the relations of (5.313). One

of the coeﬃcients cannot be determined, which is determined by the strength

of the excitation.

This is an exact solution of the reentrant cavity, the accuracy of the

solution depends upon how many terms are in the series and how many

equations are taken into account in the calculation. [83]

300 5. Metallic Waveguides and Resonant Cavities

5.6.2 Approximate Solution for the Reentrant Cavity

To obtain the exact solution given in the last subsection is an onerous task.

We try to ﬁnd the approximate solution by means of the method given in

Section 4.11.

(1) Single-Term Approximation in the Gap Region

In region 1, the gap region, the width of the gap is much smaller than the

wavelength, 2d ¿ λ, so we may neglect the high-order space harmonics in

the series of the solutions (5.292)–(5.294), and take the term with m = 0 as

the trial functions,

m = 0, β

mπ

= 0, T

− β

= k.

The solutions (5.290)–(5.294) become

= a

(kρ), (5.319)

= k

(kρ), (5.320)

φ1

= jω²ka

(kρ), (5.321)

ρ1

= 0, E

φ1

= 0, H

ρ1

= 0.

In region 2, the cavity region, the length of the cavity is relatively large,

so we take the series solutions (5.299)–(5.302) as the trial functions.

The electric ﬁeld and the magnetic ﬁeld in region 1 at the boundary

between regions 1 and 2, ρ = b, are given by

(b) = k

(kb) = A

, (5.322)

φ1

(b) = jω²ka

(kb) = A

, where Y

jω²

(kb)

. (5.323)

The electric ﬁeld and the magnetic ﬁeld in region 2 at the boundary ρ = b

are still given by expressions (5.306)–(5.309).

The ﬁeld-matching condition of the electric ﬁeld E

at the cylindrical

boundary ρ = b is that (5.306) equals (5.322):

∞

n=0

cos

nπz

, |z| ≤ d,

0, d ≤ |z| ≤ l.

(5.324)

The coeﬃcient of the above Fourier series, B

, is

cos

nπz

dz, where δ

1, n = 0,

2, n 6= 0.

(5.325)

The integral in the above expression is

cos

nπz

dz = d

sin(nπd/l)

nπd/l

= sinc

nπd

. (5.326)