Examples

We now give examples of very important, cross-sections for which we can derive explicit expressions of the vector basis functions.

5.5.1 Planar waveguide

We start with the simplest case, which is the planar waveguide. The geometry of this waveguide is depicted in figure 5.8.

The solution is obtained from the following one-dimensional eigenvalue problems:







d²Y(y)

dy² + γY (y) = 0, 0≤ y ≤ b Y(y) = 0, y= 0, b

and 









d²Y(y)

dy² + γY (y) = 0, 0≤ y ≤ b dY

dy(y) = 0, y= 0, b

Examples 87

Eigenfunctions vn, wn, ψ Eigenvaluesk²_{t n} TM_n v_n=

r2

b sinnπy b

 π²n²

b² TEn wn=

r2

b cosnπy b



π²n² b²

TEM ∇^Tψ =

r1

byˆ 0

Table 5.1: A table of the normalized eigenfunctions to equations (5.4), (5.5) and (5.18) for the planar waveguide, c.f., figure 5.8 for the definition of the geometry. The integer n has the values n = 1, 2, 3, . . ..

The solutions to these problems are given by Y_n(y) = sinnπy

b



, n= 1, 2, 3, . . . and

Y_n(y) = cosnπy b



, n= 0, 1, 2, 3, . . .

respectively. The eigenvalues are in both cases γ = n²π²/b². These sets of functions are complete. From the functions we construct the functions v_nand w_n, that in this case only depend on the coordinate y. The normalized TM-case basis functions are

v_n(y) = r2

bsinnπy b



, n= 1, 2, 3, . . . and the TE-case basis functions are

w_n(y) = r2

bcosnπy b



, n= 1, 2, 3, . . .

The index n = 0 does nor correspond to a TM-mode, as seen from page 78.

The planar waveguide has two separated surfaces which means that a TEM-mode exists. We use (5.18) to determine the TEM-mode basis functions











d²ψ(y)

dy² = 0, 0≤ y ≤ b ψ(y) =

(C₁, y= 0 C2, y= b The solution is

ψ(y) = C₁+C₂− C1

b y

and the TEM-mode has the normalized basis function (see (5.16))

∇Tψ^±(y) = r1

byˆ The results are collected in table 5.1.

x y

a b

Figure 5.9: The geometry for a waveguide with rectangular cross-section.

5.5.2 Waveguide with rectangular cross-section

We continue and determine the eigenfunctions for the rectangular waveguide. This is the most common type of hollow waveguide. The geometry is depicted in figure 5.9. The surface is simply connected and hence no TEM-mode exists. The convention is to let the longest side of the rectangle be along the x-axis.

The eigenvalues that are to be solved are







∂²v(ρ)

∂x² +∂²v(ρ)

∂y² + k²_tv(ρ) = 0 v(ρ) = 0 ρ on Γ

(TM-case)

and 









∂²w(ρ)

∂x² + ∂²w(ρ)

∂y² + k_t²w(ρ) = 0

∂w

∂n(ρ) = 0 ρ on Γ

(TE-case)

The solution is based on the following one-dimensional eigenvalue problems:





d²X(x)

dx² + γX(x) = 0, 0≤ x ≤ a X(x) = 0, x= 0, a

and 









d²X(x)˜

dx² + γ ˜X(x) = 0, 0≤ x ≤ a d ˜X

dx(x) = 0, x= 0, a The solutions to these two problems are

X_m(x) = sinmπx a



, m= 1, 2, 3, . . . and

X˜_m(x) = cosmπx a



, m= 0, 1, 2, 3, . . . ,

Examples 89

Eigenfunctions vmn, wmn Eigenvalues k_{t mn}² TM_mn v_mn= 2

√absinmπx a

sinnπy b

 π²

m² a² + n²

b²



TEmn wmn =

rεmεn

ab cosmπx a



cosnπy b

 π²

m² a² + n²

b²



Table 5.2: Table of normalized eigenfunctions to equations (5.4) and (5.5) for rectangular waveguides, see figure 5.9. The integers m and n can have values m, n = 0, 1, 2, 3, . . ., with the exception that m and n are not zero for TM-modes, and m and n cannot both be zero for the TE-modes (ε_m= 2− δm,0), see page 78. The convention in this book is always to have the long side of the rectangle along the x-axis, i.e., a > b. The mode with the lowest cut-off frequency is then the TE₁₀ mode. This mode is called the fundamental mode and is very important.

respectively. These sets of functions are orthogonal and complete on the interval x∈ [0, a].

The solution to the two-dimensional eigenvalue problems for the rectangular waveguide are obtained as a product of these sets of one-dimensional eigenfunctions ⁴, i.e.,









sinmπx a



sinnπy b



, TM-case

cosmπx a



cosnπy b



, TE-case

The eigenvalues in the two cases are the same k_t² = π² m²/a²+ n²/b²

. The normalized functions are 













v_mn= 2

√absinmπx a

sinnπy b



, TM-case

w_mn=

rε_mε_n

ab cosmπx a

cosnπy b



, TE-case

where the Neumann-factor is εm= 2− δm,0. The results are collected in table 5.2.

Example 5.3

The fundamental mode of a rectangular waveguide with a > b is the TE10 mode. It has the cut-off frequency f_c10 = c₀

2a and w₁₀ = r 2

abcosπx a

. The normalized electric field is

E_10TE(x, ω) = ˆyiωµ₀ π

r2a

b sinπx a

 (5.29)

4A common method to create complete sets of functions in two dimensions is to take the product of one-dimensional systems, i.e., if{f^m(x)}^∞m=1and{gⁿ(y)}^∞n=1are complete systems on the intervals x∈ [a, b] and y ∈ [c, d], respectively, then

{f^m(x)gn(y)}^∞m,n=1

is a complete set of functions in the rectangle [a, b]× [c, d].

m 1 2 0 1 2 3 3 4 0 1

n 0 0 1 1 1 0 1 0 2 2

f_cmn (GHz) 3.19 6.38 6.81 7.52 9.33 9.57 11.7 12.8 13.6 14.0 k_{z mn} (m⁻¹)^a 43.3 107i 119i 136i 179i 184i 233i 255i 274i 282i k_{z mn} (m⁻¹)^b 144.6 86.6 70.6 22.6 114i 122i 188i 215i 237i 246i

aThe frequency is f = 3.8 GHz.

bThe frequency is f = 7.6 GHz.

Table 5.3: Table of the lowest cut-off frequencies f_cmnand the longitudinal wavenumber k_{z mn} for a rectangular waveguide with dimensions 4.7 cm × 2.2 cm. Only TE-modes can have m- or n-values that are zero. For frequencies below the cut-off frequency the longi-tudinal wavenumber k_{z mn} is imaginary and the corresponding mode is non-propagating.

The attenuation of that mode is exp(−Im{kz mn}z).

If a > 2b then the second mode is TE20that has cut-off frequency f_c20 = c0

a. If b < a < 2b then TE01 is the second mode with cut-off frequency f_c01= c₀

2b. In order to maximize the bandwidth it is common to have rectangular waveguides with a > 2b. Then the bandwidth is BW = c₀

2a and the fractional bandwidth is b_f = 2(c₀/2a)/(3c₀/2a) = 2/3 = 0.67.

Example 5.4

A rectangular waveguide has dimensions 4.7 cm × 2.2 cm. The cut-off frequencies fcmn

for the different modes are easy to calculate from (5.12) and table 5.2. The longitudinal wavenumbers k_{z mn}, given by (5.9), are related to the frequency f and the cut-off frequency f_cmn in the following way

k_{z mn} = 2π c₀

õ q

f²− f²cmn

The results are given in table 5.3. The bandwidth is BW = 3.19 Ghz and the fractional bandwidth is b_f = 1.

5.5.3 Waveguide with circular cross-section

The geometry of the circular waveguide with radius a is depicted in figure 5.10. The geometry has only one simply connected surface and hence there is no TEM-mode. It is best to solve the eigenvalue problem in cylindrical-(polar)coordinates. The eigenvalue problems are given by







∇²Tv(ρ) + k²_tv(ρ) = 1 ρ

∂

∂ρ



ρ∂v(ρ)

∂ρ

 + 1

ρ²

∂²v(ρ)

∂φ² + k_t²v(ρ) = 0 v(a, φ) = 0

(TM-case)

and









∇²Tw(ρ) + k_t²w(ρ) = 1 ρ

∂

∂ρ



ρ∂w(ρ)

∂ρ

 + 1

ρ²

∂²w(ρ)

∂φ² + k²_tw(ρ) = 0

∂w

∂n(a, φ) = 0

(TE-case)

Examples 91

ρ φ a

x y

Figure 5.10: Geometry for waveguide with circular cross-section.

We solve these eigenvalue problems by the method of separation of variables. We make the ansatz v(ρ, φ) = f (ρ)g(φ) and insert this into the differential equation. After division with f (ρ)g(φ)/ρ² we get

ρ f(ρ)

∂

∂ρ



ρ∂f(ρ)

∂ρ



+ k²_tρ²=− 1 g(φ)

∂²g(φ)

∂φ²

The right hand side depends only on φ and the left hand side depends only on ρ. That means that they both have to be equal to a constant and we denote this constant γ. We

get 







 ρ ∂

∂ρ



ρ∂f(ρ)

∂ρ



+ k²_tρ²− γ

f(ρ) = 0

∂²g(φ)

∂φ² + γg(φ) = 0

The solution to the eigenvalue problem in the variable φ is g(φ) =

cos mφ sin mφ



, m= 0, 1, 2, 3, . . .

Only integer values of m are allowed since the function must be periodic in φ with period 2π, i.e., only γ = m², m = 0, 1, 2, 3, . . . are possible values. The corresponding set of functions is complete on the interval φ ∈ [0, 2π). The solution to the equation in the ρ-variable is a Bessel function, see appendix A. Only solutions that are regular in ρ = 0 are valid, i.e.,

f(ρ) = Jm(ktρ)

The boundary conditions vm(a, φ) = 0 and ^dw_dρ^m(a, φ) = 0 for the TM- and TE-cases, respectively, add extra conditions. For these boundary conditions to be satisfied, the transverse wavenumber has to satisfy

k_ta=

(ξmn, (TM-case) η_mn, (TE-case)

Eigenfunctionsvmn, wmn Eigenvalues k_{t mn}² TM_mn v_mn =

√εmJm(ξmnρ/a)

√πaJ_m⁰ (ξmn)

cos mφ sin mφ

! ξ²_mn

a² TEmn wmn =

√εmηmnJm(ηmnρ/a) pπ (η_mn² − m²)aJm(ηmn)

cos mφ sin mφ

! η_mn²

a²

Table 5.4: Table of the normalized eigenfunctions to equations (5.4) and (5.5) for waveguides with circular cross-section, see figure 5.10 for definition of geometry. (εm = 2− δm,0). The first values of the positive zeros ξ_mn to J_m(x) and the positive zeros η_mn to J_m⁰ (x), i.e., J_m(ξ_mn) = 0 and J_m⁰ (η_mn) = 0, m = 0, 1, 2, 3, . . ., n = 1, 2, 3, . . . are listed in tables A.1 and A.2 in appendix A. The mode with the lowest cut-off frequency is the TE₁₁ mode.

where ξ_mn and η_mn, n = 1, 2, 3, . . ., are zeros to the Bessel function J_m(x) and to the derivative of the Bessel function, respectively, i.e., J_m(ξ_mn) = 0 and J_m⁰ (η_mn) = 0. Nu-merical values of the first of these zeros are given in appendix A.

The sets of functions {Jm(ξ_mnρ/a)}^∞_n=1, {Jm⁰ (η_mnρ/a)}^∞_n=1 are both complete on the interval ρ ∈ [0, a] for every value of m. The complete set of functions in the circle is, in analogy with the rectangular waveguide, given by the product of the sets of basis functions. The results of the normalized functions (the normalization integrals are given in appendix A) are











 v_mn =

√εmJm(ξmnρ/a)

√πaJ_m⁰ (ξmn)

cos mφ sin mφ

!

, TM-case

w_mn=

√ε_mη_mnJ_m(η_mnρ/a) pπ(η_mn² − m²)aJ_m(η_mn)

cos mφ sin mφ

!

, TE-case

where ε_m = 2− δm,0. The results are collected in table 5.4.

Example 5.5

The fundamental mode is the TE₁₁ mode. The cut-off frequency is given by f_c11 = c₀η₁₁ where a is the radius of the cylinder and η11= 1.841 is the first zero of J₁⁰(x). The second2πa mode is the TM₀₁ mode with cut-off frequency f_c01 = c₀ξ₀₁

2πa where ξ₀₁= 2.405 is the first zero of J₀(x). The bandwidth is BW = f_c01 − fc11 = c₀

2πa(2.405− 1.841) = 0.564 c₀ 2πa. The fractional bandwidth is b_f = 2 0.564

1.841 + 2.405 = 0.265.

In document Microwave theory Karlsson, Anders; Kristensson, Gerhard (Page 100-106)