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

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5.6 Reentrant Cavities 301

Then we have the ﬁnal expression for the coeﬃcient B

sinc

nπd

. (5.327)

The exact matching conditions for the electric ﬁeld E

and the magnetic

ﬁeld H

at the cylindrical boundary ρ = b cannot be satisﬁed simultaneously,

for they are both trial functions but are not true ﬁelds in the gap region. Once

the electric ﬁeld E

is matched exactly at ρ = b, the magnetic ﬁeld H

can

only be matched approximately by applying the average matching condition

given in Section 4.11, which is

2π

φ1

(b)dz dφ =

2π

φ2

(b)dz dφ. (5.328)

Applying (5.323) and (5.308), we have

dz =

∞

n=0

cos

nπz

dz.

Substituting (5.327) into the above expression gives

d =

∞

n=0

sinc

nπd

cos

nπz

dz.

Applying the integral formula (5.326), we have

∞

n=0

sinc

nπd

. (5.329)

Substituting the expressions for Y

and Y

, (5.323) and (5.309), into the

above expression yields

(kb)

kbJ

(kb)

∞

n=0

a)J

b) − J

a)N

a)J

b) − J

a)N

sinc

nπd

(5.330)

where

− β

−

nπ

, k = ω

√

µ².

Equation (5.330) is the approximate eigenvalue equation of the cylindrical

reentrant cavity. The pth root of (5.330) is the eigenvalue, i.e., the natural

angular wave number of the TM

0p0

mode of the cavity, denoted by k

0p0

The coeﬃcients of the ﬁeld components in region 2, b

, in terms of a

may be obtained from (5.307), (5.322), (5.327), and (5.330). Then we have

all the ﬁeld components in the two regions with one unknown coeﬃcient a

to be determined by the strength of the excitation of the cavity.

302 5. Metallic Waveguides and Resonant Cavities

The other approximate matching condition is the speciﬁc-point matching

condition. Instead of the average matching condition (5.328), the magnetic

ﬁeld in the two sides are matched at a particular point on the boundary ρ = b,

z = 0:

φ1

(b, 0) = H

φ2

(b, 0). (5.331)

Applying (5.323) and (5.308) in the above equation and substituting (5.327)

into it, we have

∞

n=0

. and Y

∞

n=0

sinc

nπd

Substituting (5.323) and (5.309) into the above expression yields

(kb)

kbJ

(kb)

∞

n=0

a)J

b) − J

a)N

a)J

b) − J

a)N

sinc

nπd

, (5.332)

The only diﬀerence between (5.330) and (5.332) is a factor sinc(nπd/l). Its

inﬂuence is small when d is not too large.

The ﬁeld map in the reentrant cavities is shown in Fig. 5.39(a).

(2) Single-Term Approximation in Both Regions

If the length of the cavity region is also small, 2l ¿ λ, see Fig. 5.39(b),

we may neglect the high-order space harmonics in both the region 1 and

the region 2 and take the terms with n = 0 in (5.299)–(5.302) as the trial

functions. Then we have

n = 0, β

nπ

= 0, T

− β

= k,

= b

(ka)J

(kρ) − J

(ka)N

(kρ)], (5.333)

= k

(ka)J

(kρ) − J

(ka)N

(kρ)], (5.334)

φ2

= jω²kb

(ka)J

(kρ) − J

(ka)N

(kρ)]. (5.335)

The ﬁeld components at the boundary ρ = b become

(b) = k

(ka)J

(kb) − J

(ka)N

(kb)] = B

, (5.336)

φ2

(b) = jω²kb

(ka)J

(kb) − J

(ka)N

(kb)] = B

, (5.337)

where

jω²

(ka)J

(kb) − J

(ka)N

(kb)

(ka)J

(kb) − J

(ka)N

(kb)

. (5.338)

In region 1, the gap region, d < l ¿ λ, the trial functions are still

the single-term expressions (5.319)–(5.321), and the ﬁeld components at the

boundary ρ = b are still (5.322) and (5.323).

5.6 Reentrant Cavities 303

Figure 5.39: Field maps in reentrant cavities of diﬀerent approximations.

The ﬁeld-matching condition for the electric ﬁeld E

at the cylindrical

boundary ρ = b is (5.324) and (5.325) and the resulting relation between B

and A

is (5.327) with n = 0 is

l = A

The physical meaning of this relation is that the potential diﬀerences at ρ = b

in regions 1 and 2 are equal.

The ﬁeld-matching condition for the magnetic ﬁeld H

at ρ = b is that

(5.323) equals (5.337), which gives

= B

, i.e., Y

l = Y

Substituting the expressions for Y

and Y

, (5.323) and (5.338), into the

above equation yields

(kb)

(ka)J

(kb) − J

(ka)N

(kb)

(ka)J

(kb) − J

(ka)N

(kb)

. (5.339)

This approximate eigenvalue equation is just the single-term formulation of

(5.330) and (5.332).

(3) Uniform Field Approximation in the Gap Region,

Capacitance-Loaded Radial Lines

In the above two approaches, the ﬁeld distributions in the gap region are

Bessel functions in the ρ direction, which is the ﬁeld of the radial TEM

mode. If the radius of the gap region is small enough, then the electric ﬁeld

in the gap will be approximately uniform and the magnetic ﬁeld in the gap

will be inﬁnitesimally small. In this case, the gap region can be considered as

304 5. Metallic Waveguides and Resonant Cavities

a parallel-plate capacitor and the cavity becomes a capacitance-loaded radial

line as shown in Fig. 5.39(c).

The approximate eigenvalue equation of a capacitance-loaded radial line

cavity can be obtained by means of admittance matching as follows. The

admittance of the capacitor is

= jωC = jω

²πb

. (5.340)

The voltage and the current in the shorted radial line at ρ = b are given

V = 2l E

(b) = 2l B

, I = 2πbH

φ2

(b) = 2πbB

and the input admittance of the shorted radial line at ρ = b is

πb

jω²

(ka)J

(kb) − J

(ka)N

(kb)

(ka)J

(kb) − J

(ka)N

(kb)

. (5.341)

Let the two admittances be equal to each other. This gives

(ka)J

(kb) − J

(ka)N

(kb)

(ka)J

(kb) − J

(ka)N

(kb)

. (5.342)

This is the eigenvalue equation of the capacitance-loaded radial line cavity.

If we consider the following approximate formulas for the Bessel functions,

lim

x→0

(x) = 1, lim

x→0

(x) =

then (5.339) reduces to (5.342).

(4) Capacitance-Loaded Coaxial Line Cavities and

Quasi-Lumped-Element Cavities

If the outer radius a is small and the length of the cavity l is large, the

cavity becomes a capacitance-loaded coaxial line as shown in Fig. 5.39(d).

Its eigenvalue equation is

jωC = j

tan kl

ln(a/b) tan kl

If all of a, l, and d are small, it becomes a resonant circuit with lumped

elements L and C, which is known as a quasi-lumped-element cavity, refer to

Fig. 5.39(e).

The capacitance-loaded biconical cavity or reentrant spherical cavity and

the ridge waveguide can also be analyzed by means of the above methods.

They are given as problems 5.15 and 5.16.

5.7 Principle of Perturbation 305

Figure 5.40: Cavity wall perturbations.

5.7 Principle of Perturbation

When some parameters such as the conﬁguration of the boundary, the ma-

terial in the volume or the material of the boundary change slightly, the

electromagnetic system is said to be perturbed. If the solution of an unper-

turbed problem is known, then the solution of the perturbed problem, which

is slightly diﬀerent from the unperturbed one, can be obtained by means of

the principle of perturbation.

We have already used the perturbational method for the calculation of

waveguide attenuation coeﬃcients and resonator Q factors, in which the per-

turbation of the wall material is considered. In this section, the general

formulations of the principle of perturbation will be given. [37, 91]

5.7.1 Cavity Wall Perturbations

An ideal resonant cavity formed by a perfect-conductor surface S and enclos-

ing a lossless region V is shown in Fig. 5.40(a). The cavity wall perturbation

or the conductor perturbation of the cavity is to introduce a small deforma-

tion in the wall, shown in Fig. 5.40(b) or to introduce a small conductive

perturbing object into the cavity, shown in Fig. 5.40(c). Deformation of the

wall may also be considered as a conductive perturbing object stuck on the

wall.

Suppose the volume of the perturbing object is ∆V , the surface enclosing

the perturbing object is ∆S. The positive direction of ∆S is the outward

direction of the volume ∆V . The volume of the perturbed cavity is V

and

the surface enclosing it is S

. The positive direction of S

and S is the

outward direction of the cavity volume V

and V . So we have S

= S − ∆S,

306 5. Metallic Waveguides and Resonant Cavities

= V − ∆V .

Let ω

, E

, and H

represent the natural angular frequency and the ﬁelds

of the unperturbed cavity, and ω, E, and H represent the corresponding

quantities of the perturbed cavity. In both cases Maxwell’s equations must

be satisﬁed, that is

∇ × E

= −jω

µH

, (5.343)

∇ × H

= jω

²E

, (5.344)

∇ × E = −jωµH, (5.345)

∇ × H = jω²E. (5.346)

Scalar multiplying the equation (5.346) by E

∗

and the conjugate of the

equation (5.343) by H, we have

∗

· ∇ × H = jω²E · E

∗

, H · ∇ × E

∗

= jω

µH

∗

· H.

Subtracting these two equations and applying the vector identity (B.38),

∇ · (A × B) = B · ∇ × A − A · ∇ × B,

we have

∇ · (H × E

∗

) = jω²E · E

∗

− jω

µH · H

∗

. (5.347)

By the similar operations on the equation (5.344) and the equation (5.345),

we obtain

∇ · (H

∗

× E) = jωµH · H

∗

− jω

²E · E

∗

. (5.348)

These two equations, (5.347) and (5.348), are added and the sum is in-

tegrated throughout the volume of the perturbed cavity V

, and then the

divergence theorem is applied to the left-hand side. The resulting equation

(H ×E

∗

×E)·dS = j(ω−ω

)

(²E ·E

∗

+µH ·H

∗

)dV. (5.349)

The second term of the surface integral vanishes, because the perturbed elec-

tric ﬁeld E satisﬁes the short-circuit boundary condition on S

, n×E|

= 0.

Then the above equation becomes

H × E

∗

· dS = j(ω − ω

)

(²E · E

∗

+ µH · H

∗

)dV. (5.350)

Since S

= S − ∆S, the left-hand side of the equation becomes

H × E

∗

· dS =

H × E

∗

· dS −

∆S

H × E

∗

· dS.

5.7 Principle of Perturbation 307

The ﬁrst term of the right-hand side vanishes, because the unperturbed elec-

tric ﬁeld E

and its conjugate E

∗

satisﬁes the short-circuit boundary condi-

tion on S, n × E

∗

= 0. Then we can rewrite (5.350) as

∆ω = ω − ω

∆S

H × E

∗

· dS

(²E · E

∗

+ µH · H

∗

)dV

. (5.351)

This is an exact formula for the change in the natural frequency due to the

perturbation of the cavity wall.

For practical application of the formula, we must replace the unknown

perturbed ﬁelds E and H by the unperturbed ﬁelds E

and H

. For small

perturbations this is certainly reasonable in the denominator, i.e.,

(²E · E

∗

+ µH · H

∗

)dV ≈

²E

+ µH

dV.

In the numerator, the tangential component of the perturbed magnetic ﬁeld

is approximately equal to the unperturbed value when the deformation of the

wall is small, shallow, and smooth. With this approximation and applying

the complex Poynting theorem in the loss-less source-free volume ∆V , we can

rewrite the numerator of (5.351) as

∆S

H × E

∗

· dS ≈

∆S

× E

∗

· dS = jω

∆V

²E

− µH

dV.

Substituting these two approximate expressions into (5.351), we have

∆ω

ω − ω

≈

∆V

µH

− ²E

²E

+ µH

. (5.352)

This is the perturbation formula for conductor perturbation or wall pertur-

bation of a cavity. Note that the denominator is proportional to the total

energy stored in the cavity, whereas the terms in the numerator are propor-

tional to the electric and magnetic energies removed by the perturbation.

Hence, (5.352) can be rewritten as

∆ω

≈

∆W

− ∆W

, (5.353)

where W denotes the total energy stored in the original cavity, and ∆W

and

∆W

denote the time average magnetic energy and electric energy, respec-

tively, originally stored in the small volume ∆V . The perturbation formula

shows that an inward perturbation of the wall will raise the natural frequency

if it is made at a point with large magnetic ﬁeld (high w

) and small electric

ﬁeld (low w

), and will lower the natural frequency if it is made at a point

with large electric ﬁeld (high w

) and small magnetic ﬁeld (low w

The perturbation formula (5.352) or (5.353) is valid only when the in-

troduction of the perturbing object does not inﬂuence the ﬁelds outside the

308 5. Metallic Waveguides and Resonant Cavities

Figure 5.41: Material perturbation of a cavity.

perturbing body. This means that the perfect-conductor surface of the per-

turbing object is p erpendicular to the original electric ﬁeld and parallel to

the original magnetic ﬁeld. Otherwise, the perturbation formula must be

modiﬁed as follows:

∆ω

= K

∆V

µH

− ²E

²E

+ µH

= K

∆W

− ∆W

, (5.354)

where K is a constant and depends upon the shape of the perturbing object

and the orientation of it in the ﬁelds. The constant K may be obtained by

means of experiment or quasi-static approach.

5.7.2 Material Perturbation of a Cavity

If an insulating object is introduced into the cavity, the natural frequency of

the cavity will change. This phenomena is known as material perturbation

of the cavity. [37]

Suppose the permittivity and permeability of the lossless material ﬁlling

an unperturbed cavity of volume V are ² and µ, and the ﬁelds and the natural

frequency of the cavity are E

, H

, and ω

, respectively. See Fig. 5.41(a).

The perturbed cavity is one in which a material object with volume ∆V and

medium constants ² + ∆² and µ + ∆µ is introduced in the cavity, shown

in Fig. 5.41(b), (c). Note that µ and ² in V and ∆µ and ∆² in ∆V are

not necessarily uniform. The volume of the perturbed cavity outside the

perturbation object is V

= V − ∆V . The ﬁelds and the natural frequency

of the perturbed cavity are E, H, and ω.

Fields E

and H

satisfy Maxwell’s equations in the volume V :

∇ × E

= −jω

µH

, ∇ × H

= jω

²E

. (5.355)

5.7 Principle of Perturbation 309

Fields E and H satisfy the following Maxwell equations in the volume V

∇ × E = −jωµH, ∇ × H = jω² E. (5.356)

But in the volume ∆V , ﬁelds E and H satisfy the the perturbed Maxwell

equations:

∇ × E = −jω(µ + ∆µ)H, ∇ × H = jω(² + ∆²)E. (5.357)

In the volume V

, we have

∗

· (∇ × H) = jω²E · E

∗

. (5.358)

In the volume ∆V , we have

∗

· (∇ × H) = jω(² + ∆²)E · E

∗

. (5.359)

In the volume V , i.e., in b oth V

and ∆V , we have

−H · (∇ × E

∗

) = −jω

²H

∗

· H. (5.360)

Adding equations (5.358) and (5.360), then applying the vector identity

(B.38), we have

∇ · (H × E

∗

) = jω²E · E

∗

− jω

µH · H

∗

, in V

. (5.361)

Adding equations (5.359) and (5.360), then applying the vector identity

(B.38), we have

∇ · (H × E

∗

) = jω(² + ∆²)E · E

∗

− jω

µH · H

∗

, in ∆V. (5.362)

By similar operations we obtain

∇ · (H

∗

× E) = jωµH · H

∗

− jω

²E · E

∗

, in V

, (5.363)

and

∇ · (H

∗

× E) = jω(µ + ∆µ)H · H

∗

− jω

²E · E

∗

, in ∆V. (5.364)

Adding the two equations (5.361) and (5.363), and adding the two equa-

tions (5.362) and (5.364), then integrating the sums throughout the volume

V and applying the divergence theorem to the left-hand side, we have

(H × E

∗

+ H

∗

× E) · dS = j(ω − ω

)

(²E · E

∗

+ µH · H

∗

)dV

+ jω

∆V

(∆²E · E

∗

+ ∆µH · H

∗

)dV. (5.365)

The surface integral on the left-hand side of the above equation vanishes,

because both the unperturbed and the perturbed electric ﬁelds E

and E

310 5. Metallic Waveguides and Resonant Cavities

satisfy the short-circuit boundary condition on the cavity boundary S, i.e.,

n × E

= 0 and n × E|

= 0:

(H × E

∗

+ H

∗

× E) · dS = 0.

Then (5.365) becomes

∆ω

ω − ω

= −

∆V

(∆²E · E

∗

+ ∆µH · H

∗

)dV

(²E · E

∗

+ µH · H

∗

)dV

. (5.366)

This is an exact formula for the change in the natural frequency due to the

material perturbation of the cavity.

For small perturbation, i.e., ∆² and ∆µ are small, we may replace E, H

by E

, H

, respectively, and replace ω by ω

except for the factor ω − ω

the perturbation formula becomes

∆ω

ω − ω

≈ −

∆V

∆²E

+ ∆µH

²E

+ µH

. (5.367)

The perturbation formula shows that any increase in µ and ² can only

decrease the natural frequency of a cavity, no matter whether the electric ﬁeld

or the magnetic ﬁeld is perturbated. In the above mathematical treatment

the volume ∆V has not been supposed to b e small. If the changes in the

medium constants extend all over the cavity, ∆V → V , the p erturbation

formula becomes

∆ω

ω − ω

≈ −

∆²E

+ ∆µH

²E

+ µH

. (5.368)

If ∆² or ∆µ is not small enough, but ∆V is small as compared with V ,

in the perturbation formula (5.366), the perturbed ﬁelds E and H in the

volume integral over ∆V may not be replaced by E

and H

, but E and H

in the volume integral over V may be replaced by E

and H

, because the

perturbation of the ﬁelds is limited to a small region inside and around ∆V .

The perturbation formula (5.366) becomes

∆ω

≈ −

∆V

(∆²E · E

∗

+ ∆µH · H

∗

)dV

²E

+ µH

≈ −K

∆V

∆²E

+ ∆µH

²E

+ µH

(5.369)

The ratio of the perturbed ﬁelds E and H inside the perturbation object to

the unperturbed ﬁelds E

and H

and the constant K can be approximately

obtained by using a quasi-static approximation and supposing that the un-

perturbed ﬁelds are uniform. This assumes that the time-varying ﬁelds inside

∆V are related to those outside ∆V in the same manner as the static ﬁelds,

because, in a region small compared to the wavelength, Helmholtz’s equation

can be approximated by Laplace’s equation.