Question 13.3.Q3: Electric field E and magnetic field B in a uniform EM wavegu......

(a) Solutions to wave equations (13.64) and (13.65) depend on boundary conditions and, in general, uniform EM waveguides are governed by boundary conditions on the tangential component of electric field {E} and normal component of magnetic field \mathcal{B}, respectively, as

\boldsymbol{E} \times\left.\hat{\mathbf{n}}\right|_S= {E}_{\text {tang }} \mid_S=0 \quad \text { and }\left.\quad \boldsymbol{B} \cdot \hat{\mathbf{n}}\right|_S=\mathcal{B}_{\text {norm }} \mid_S=0               (13.66)

For a circular waveguide of radius a boundary conditions (13.66) can be expressed as follows

(1) Dirichlet-type boundary condition on tangential component of electric field: {E}_{\text {tang }} \mid_S=0.

\left. {E}_z\right|_S=0 \quad \text { or }\left.\quad  {E}_z\right|_{r=a}=0 .           (13.67)

(2) Neumann-type boundary condition on normal component of magnetic field: \mathcal{B}_{\text {norm }} \mid_S=0.

\left.\frac{\mathrm{d} \mathcal{B}_z}{\mathrm{~d} n}\right|_S=0 \quad \text { or }\left.\quad \frac{\mathrm{d} \mathcal{B}_z}{\mathrm{~d} r}\right|_{r=a}=0 .         (13.68)

(b) Since the problem deals with a circular waveguide, we will seek solutions to wave equations (13.64) and (13.65) in the cylindrical coordinate system, as indicated in Fig. 13.5(A). The standard relationship between the Cartesian and cylindrical coordinate system with a common z axis is presented in Fig. 13.5(B) and is given as

x=r \cos \theta, \quad y=r \sin \theta, \quad \text { and } \quad z=z \text {. }           (13.69)

The electric field {E} of (13.64) has three components in cylindrical coordinates and each component is a function of spatial coordinates r, θ , and z as well as of the temporal coordinate t

{E}=\left[ {E}_r(r, \theta, z, t),  {E}_\theta(r, \theta, z, t),  {E}_z(r, \theta, z, t)\right]              (13.70)

Similarly, the magnetic field \mathcal{B} of (13.65) has three components in cylindrical coordinates and each of them depends on spatial coordinates r, θ , and z as well as the temporal coordinate t

\boldsymbol{B}=\left[\mathcal{B}_r(r, \theta, z, t), \mathcal{B}_\theta(r, \theta, z, t), \mathcal{B}_z(r, \theta, z, t)\right]            (13.71)

Wave equations (13.64) for E and (13.65) for B contain the vector Laplacian operator ∇² which, when applied to an arbitrary vector field A (such as {E}\ and\ \mathcal{B}) with components A_r,\ A_θ ,\ and\ A_z, generates another vector field. The generated vector field is equal to the vector field of the scalar Laplacian operator applied to the individual components of the vector field.
Expressed in cylindrical coordinates, (13.64) and (13.65) are given in the following format

\nabla^2 \mathbf{A}=\left|\begin{array}{l} \frac{\partial^2 A_r}{\partial r^2}+\frac{1}{r} \frac{\partial A_r}{\partial r}+\frac{1}{r^2} \frac{\partial^2 A_r}{\partial^2 \theta^2}+\frac{\partial^2 A_r}{\partial z^2}-\frac{2}{r^2} \frac{\partial A_\theta}{\partial \theta}-\frac{A_r}{r^2} \\ \frac{\partial^2 A_\theta}{\partial r^2}+\frac{1}{r} \frac{\partial A_\theta}{\partial r}+\frac{1}{r^2} \frac{\partial^2 A_\theta}{\partial^2 \theta^2}+\frac{\partial^2 A_\theta}{\partial z^2}+\frac{2}{r^2} \frac{\partial A_r}{\partial \theta}-\frac{A_\theta}{r^2} \\ \frac{\partial^2 A_z}{\partial r^2}+\frac{1}{r} \frac{\partial A_z}{\partial r}+\frac{1}{r^2} \frac{\partial^2 A_z}{\partial^2 \theta^2}+\frac{\partial^2 A_z}{\partial z^2}+0+0 \end{array}\right|=\frac{1}{c^2} \frac{\partial^2 \mathbf{A}}{\partial t^2}=\frac{1}{c^2}\left|\begin{array}{l} \frac{\partial^2 A_r}{\partial t^2} \\ \frac{\partial^2 A_\theta}{\partial t^2} \\ \frac{\partial^2 A_z}{\partial t^2} \end{array}\right| .           (13.72)

As evident from (13.72), the individual relationships for the r and θ components of vector field A are quite complicated; however, the relationship for the z component of A retains the original form of the wave equation, expressed by the scalar Laplacian operator in the cylindrical coordinate system as

\frac{\partial^2 A_z}{\partial r^2}+\frac{1}{r} \frac{\partial A_z}{\partial r}+\frac{1}{r^2} \frac{\partial^2 A_z}{\partial \theta^2}+\frac{\partial^2 A_z}{\partial z^2} \equiv \frac{1}{r} \frac{\partial}{\partial r}\left(\frac{\partial A_z}{\partial r}\right)+\frac{1}{r^2} \frac{\partial^2 A_z}{\partial \theta^2}+\frac{\partial^2 A_z}{\partial z^2}=\frac{1}{c^2} \frac{\partial^2 A_z}{\partial t^2} .             (13.73)

Using the scalar Laplacian operator in cylindrical coordinates of (13.73), we now express the wave equations for {E}_z\ and\ \mathcal{B}_z as follows

\nabla^2  {E}_z \equiv \frac{\partial^2  {E}_z}{\partial r^2}+\frac{1}{r} \frac{\partial  {E}_z}{\partial r}+\frac{1}{r^2} \frac{\partial^2  {E}_z}{\partial \theta^2}+\frac{\partial^2  {E}_z}{\partial z^2}=\frac{1}{c^2} \frac{\partial^2  {E}_z}{\partial t^2}           (13.74)

and

\nabla^2 \mathcal{B}_z \equiv \frac{\partial^2 \mathcal{B}_z}{\partial r^2}+\frac{1}{r} \frac{\partial \mathcal{B}_z}{\partial r}+\frac{1}{r^2} \frac{\partial^2 \mathcal{B}_z}{\partial \theta^2}+\frac{\partial^2 \mathcal{B}_z}{\partial z^2}=\frac{1}{c^2} \frac{\partial^2 \mathcal{B}_z}{\partial t^2}           (13.75)

Equations (13.74) and (13.75) for {E}_z\ and\ \mathcal{B}_z, respectively, are known as 3- dimensional wave equations in cylindrical coordinates; they are linear partial differential equations of the second order in four variables (three spatial variables: r, θ , and z, and one temporal variable: t) with constant coefficients. The two equations have identical form and can in general be written as follows

\nabla^2 \eta \equiv \frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial \eta}{\partial r}\right)+\frac{1}{r^2} \frac{\partial^2 \eta}{\partial \theta^2}+\frac{\partial^2 \eta}{\partial z^2}=\frac{\partial^2 \eta}{\partial r^2}+\frac{1}{r} \frac{\partial \eta}{\partial r}+\frac{1}{r^2} \frac{\partial^2 \eta}{\partial \theta^2}+\frac{\partial^2 \eta}{\partial z^2}=\frac{1}{c^2} \frac{\partial^2 \eta}{\partial t^2}           (13.76)
with η a function of r,θ,z, and t representing both {E}_z (r,θ,z,t) and \mathcal{B}_z(r,θ,z,t). The conditions imposed on η(r,θ,z,t) fall into two categories:

(1) Those involving spatial coordinates r, θ , and z and governed by boundary conditions, given in (13.67) for Dirichlet-type boundary condition and in (13.68) for Neumann-type boundary condition.
(2) Those involving the temporal coordinate t and governed by initial conditions.

The most common approach to solving the 3-dimensional wave equation (13.76) is to apply the method of separation of variables. This method usually provides a solution to a partial differential equation in the form of an infinite series, such as a Fourier series, for example. We first separate out the time factor by defining η(r,θ,z,t) as a product of two functions: \phi and T

\eta(r, \theta, z, t)=\phi(r, \theta, z) T(t)           (13.77)

where \phi is a function of spatial coordinates r, θ , and z only and T is a function of time t only.
Inserting (13.77) into (13.76) and dividing by \phi(r,θ,z)T (t) gives

\frac{\nabla^2 \phi}{\phi} \equiv \frac{1}{c^2} \frac{1}{T} \frac{\partial^2 T}{\partial t^2},            (13.78)

with the left hand side of (13.78) depending on spatial coordinates r, θ , and z only, and the right hand side depending on time t only. If (13.78) is to hold for all r, θ , z, and t, it is evident that each side must be equal to a constant. This constant is identical for both sides of (13.78) and usually referred to as the separation constant Λ. From (13.78) we thus get two equations

\nabla^2 \phi=\Lambda \phi         (13.79)

and

\frac{\partial^2 T}{\partial t^2}=\Lambda c^2 T            (13.80)

Equation (13.79) is referred to as the Helmholtz partial differential equation representing an eigenvalue problem in three dimensions with \phi the eigenfunction, Λ the eigenvalue, and ∇² the scalar Laplacian operator in cylindrical coordinates [see (13.76)]. The Helmholtz equation (13.79) results in three different types of solution, depending on the value of the separation constant Λ:

(1) For Λ > 0 the solutions are exponential functions.
(2) For Λ = 0 the solution is a linear function.
(3) For Λ < 0 the solutions are trigonometric functions.

The Dirichlet boundary condition of (13.67) can be satisfied only for Λ < 0 and this will result in trigonometric solutions for function η. We now concentrate on finding solutions to the wave equation and set Λ = −k² to satisfy the usual periodicity requirement. Parameter k is called the free space wave number or free space propagation coefficient and is related to angular frequency ω through the standard relationship

k=\frac{\omega}{c},                (13.81)

with c the speed of light in vacuum. Incorporating Λ = −k² into (13.79) and (13.80) yields the following equations for \phi(r,θ,z) and T (t), respectively

\nabla^2 \phi=k^2 \phi=0           (13.82)

and

\frac{\partial^2 T}{\partial t^2}+k^2 c^2 T=\frac{\partial^2 T}{\partial t^2}+\omega^2 T=0 .           (13.83)

The solutions for T (t) of (13.83) are either trigonometric or exponential functions but we reject the latter on physical grounds. Instead of using real trigonometric functions we express T (t) as

T(t) \propto e^{-i \omega t}            (13.84)

and assume that ω may be either positive or negative.
For the Helmholtz equation, given in (13.82), we again use the method of separation of variables and express \phi(r,θ,z) as a product of three functions: R(r), Θ(θ), and Z(z) to get

\phi(r, \theta, z)=R(r) \Theta(\theta) Z(z)           (13.85)

We now insert (13.85) into (13.82), divide the result by R(r)Θ(θ)Z(z), and get the following

\frac{1}{R} \frac{\partial^2 R}{\partial r^2}+\frac{1}{r R} \frac{\partial R}{\partial r}+\frac{1}{r^2} \frac{1}{\Theta} \frac{\partial^2 \Theta}{\partial \theta^2}+\frac{1}{Z} \frac{\partial^2 Z}{\partial z^2}+k^2=0           (13.86)

Since in our waveguide geometry the RF wave is propagating in the positive z direction as a plane wave, we rewrite (13.86) as

-\left[\frac{1}{R} \frac{\partial^2 R}{\partial r^2}+\frac{1}{r R} \frac{\partial R}{\partial r}+\frac{1}{r^2} \frac{1}{\Theta} \frac{\partial^2 \Theta}{\partial \theta^2}\right]=\frac{1}{Z} \frac{\partial^2 Z}{\partial z^2}+k^2            (13.87)

and note that the left hand side of (13.87) is a function of r and θ only and the right hand side of (13.87) is a function of z only. This can hold only when the two sides are equal to a constant that we designate as γ^2 _n . The right hand side of (13.87) now gives

\frac{1}{Z} \frac{\partial^2 Z}{\partial z^2}+k^2=\gamma_n^2 \quad \text { or } \quad \frac{1}{Z} \frac{\partial^2 Z}{\partial z^2}+k_{\mathrm{g}}^2=0           (13.88)

and results in the following trigonometric solution for propagation in the positive z direction

Z(z) \propto e^{i k_z z}         (13.89)

where k_g is referred to as the waveguide wave number or waveguide propagation coefficient and defined as

k_{\mathrm{g}}^2=k^2-\gamma_n^2 .           (13.90)

Equation (13.90) is a general form of the dispersion relationship for uniform circular EM waveguide dependent on constant γ_n. Once γ_n is determined through boundary conditions on z components {E}_z\ and\ \mathcal{B}_z of the electric field {E} and magnetic field \mathcal{B}, the dispersion relationship is expressed in the form ω = f (k_g) and used in calculation of cutoff frequency ω_c, phase velocity υ_{ph} and group velocity υ_{gr} for a given transmission mode.
We now address the left hand side of (13.87) which, after insertion of (13.88), reads

-\left[\frac{r^2}{R} \frac{\partial^2 R}{\partial r^2}+\frac{r}{R} \frac{\partial R}{\partial r}+\gamma_n^2 r^2\right]=\frac{1}{\Theta} \frac{\partial^2 \Theta}{\partial \theta^2} .           (13.91)

The left hand side of (13.91) depends on r alone and the right hand side on θ alone and again, as seen above, this can hold in general if both sides are equal to a constant that must be negative to provide physically relevant solutions. We therefore set the constant equal to −m² and get the following expression for the right hand side of (13.91)

\frac{\partial^2 \Theta}{\partial \theta^2}+m^2 \Theta=0 \quad(m=0,1,2,3, \ldots)            (13.92)

Equation (13.92) has the following standard general trigonometric solution leading to trigonometric or complex exponential functions that serve as eigenfunctions

\Theta(\theta)=A \cos m \theta+B \sin m \theta            (13.93)

Inserting (13.92) into (13.91) and multiplying the result with R/r² gives the following expression for R(r)

\frac{\partial^2 R}{\partial r^2}+\frac{1}{r} \frac{\partial R}{\partial r}+\left(\gamma_n^2-\frac{m^2}{r^2}\right) R=0           (13.94)

representing the Bessel differential equation of order m or an eigenvalue equation with eigenvalue γ^2 _n when boundary conditions are imposed on R(r). The physical conditions imposed on {E}_z\ and\ \mathcal{B}_z, and thus on R(r) as well, stipulate that:

(1) R(r) must be finite at r = 0.
(2) R(r = a) must satisfy either the Dirichlet boundary condition \left.R(r)\right|_{r=a}=0 of (13.67) or the Neumann boundary condition \mathrm{d} R /\left.\mathrm{d} r\right|_{r=a}=0 \text { of (13.68). }

The general solution to the Bessel equation (13.94) of order m consists of cylindrical functions; among these, given for non-negative integer values of m, the best known are the Bessel functions of the first kind J_m\left(γ_nr\right) and Bessel functions of the second kind N_m\left(γ_nr\right) (also known as Neumann functions). A few important features of J_m(x)\ and\ N_m(x) are apparent:

(1) With increasing x, functions J_m(x)\ and\ N_m(x) oscillate about zero with a slowly diminishing amplitude and a decrease in separation between successive roots (zeros).
(2) The two Bessel functions J_m(x)\ and\ N_m(x) possess an infinite number of roots, usually designated as x_{mn} and defined as those values of x at which the Bessel functions cross zero, i.e., where J_m(x) = 0\ or\ N_m(x) = 0.
(3) For x = 0, the Bessel functions of the first kind are finite; for integer m > 0 all Bessel functions of the first kind are equal to zero, i.e., \left.J_{m>0}(x)\right|_{x=0}=0 and for m = 0 the zero order Bessel function of the first kind equals to 1, i.e., \left.J_0(x)\right|_{x=0}=1.
(4) For x = 0, the Bessel functions of the second kind (Neumann functions) exhibit a singularity, i.e., \lim _{x \rightarrow 0} N_m(x)=-\infty.

The general solution to the Bessel differential equation (13.94) is given as

R(r)=C J_m\left(\gamma_n r\right)+D N_m\left(\gamma_n r\right),            (13.95)

where C and D are coefficients determined from the initial conditions. Since the Neumann functions are singular at r = 0, to obtain a physically relevant solution to (13.94) we set D = 0 in (13.95) to get the following general solution for R(r)

R(r)=C J_m\left(\gamma_n r\right) .           (13.96)

Combining solutions for R(r), Θ(θ), Z(z), and T (t) given in (13.96), (13.94), (13.89), and (13.84), respectively, we get the following general solution of the wave equation (13.76) for η(r,θ,z,t) representing the electric field component {E}_z and the magnetic field component \mathcal{B}_z. The general solution is written in the form of a double series with A_{mn}\ and\ B_{mn} that can be determined with the help of initial conditions

In (13.97) m is the order of the Bessel function, n the rank order number of the given root of the Bessel function, and \left(k_{\mathrm{g}} z-\omega_{m n} t\right) is usually referred to as the phase of the wave φ. Each pair of integers (m,n) corresponds to a particular characteristic mode of RF propagation through the uniform waveguide. The general solution (13.97) to the wave equation (13.76) is given as a linear superposition of all allowed modes for m = 0, 1, 2,… and n = 1, 2, 3,… . The value of γn is determined using the boundary condition (13.67) for electric field {E} or (13.68) for magnetic field B in conjunction with the general solution (13.97) for {E}_z\ or\ \mathcal{B}_z.

(c) The z components of the electric field {E} and magnetic field \mathcal{B} are written, respectively, in general form as double series with both {E}_z\ and\ \mathcal{B}_z different from zero

and

where A_{m n}, B_{m n}, C_{m n}, \text { and } D_{m n} are coefficients that can be determined from initial conditions. On the other hand, parameter γ_n in the argument of the Bessel function of (13.98) and (13.99) is determined from the boundary conditions on {E}_z\ and\ \mathcal{B}_z. Since these are generally different, they cannot be applied simultaneously and the fields are split into two special categories: transverse magnetic (TM) modes and transverse electric (TE) modes, characterized as follows:

(1) For the \mathrm{TM}_{m n} \text { modes, } \mathcal{B}_z=0 everywhere in the waveguide core and {E}_z is governed by the Dirichlet-type boundary condition \left. {E}_z\right|_{r=a}=0 which specifies that {E}_z = 0 at the boundary between waveguide core and conducting wall of the waveguide. The \left. {E}_z\right|_{r=a}=0 boundary condition results in the following solution for γ_n of (13.98)

\left. {E}_z\right|_{r=a}=\left.J_m\left(\gamma_n r\right)\right|_{r=a}=J_m\left(\gamma_n a\right)=0 \quad \rightarrow \quad \gamma_n=\frac{x_{m n}}{a}          (13.100)

where x_{mn} is the n-th zero (root) of the m-th order Bessel function. Roots of Bessel functions for 0 ≤ m ≤ 2 and 1 ≤ n ≤ 3 are listed in Table 13.2. The lowest TM_{mn} mode will be for m = 0 and n = 1, giving the following expression for {E}_z.

{E}_z(r, \theta, z, t)= {E}_{z 0} J_0\left(\frac{x_{01}}{a} r\right) e^{i \varphi}= {E}_{z 0} J_0\left(\frac{2.405}{a} r\right) e^{i \varphi}          (13.101)

where {E}_{z0} is the electric field amplitude, φ is the phase of the wave, and x_{01} = 2.405 is the first zero (root) of the J_0(z) Bessel function, as found in standard tables of Bessel functions and listed in Table 13.2.

(2) For the \mathrm{TE}_{m n} \text { modes, }  {E}_z=0 everywhere in the waveguide core and \mathcal{B}_z is governed by the Neumann-type boundary condition \mathrm{d} \mathcal{B}_z /\left.\mathrm{d} r\right|_{r=a}=0, that results in the following solution for γ_n of (13.99)

\left.\frac{\mathrm{d} \mathcal{B}_z}{\mathrm{~d} r}\right|_{r=a}=\left.\frac{\mathrm{d} J_m\left(\gamma_n r\right)}{\mathrm{d} r}\right|_{r=a}=\frac{\mathrm{d} J_m\left(\gamma_n a\right)}{\mathrm{d} r}=0 \quad \rightarrow \quad \gamma_n=\frac{y_{m n}}{a},          (13.102)

where y_{mn} is the n-th zero (root) of the first derivative of the m-th order Bessel function. Roots of the first derivative of Bessel functions for 0 ≤ m ≤ 3 and 1 ≤ n ≤ 3 are listed in Table 13.2.

As shown in Table 13.2, the lowest non-trivial TE_{mn} mode will be for m = 1 and n = 1, resulting in the following expression for \mathcal{B}_z.

\mathcal{B}_z(r, \theta, z, t)=\mathcal{B}_{z 0} J_1\left(\frac{y_{11}}{a} r\right) e^{i \varphi}=\mathcal{B}_{z 0} J_1\left(\frac{1.841}{a} r\right) e^{i \varphi},             (13.303)

where \mathcal{B}_{z0} is the magnetic field amplitude, φ is the phase of the wave, and y_{11} = 1.841 is the first zero (root) of the derivative of the J_1(z) Bessel function, as found in standard tables of Bessel functions and listed in Table 13.2.