Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

For the TE case $E_{z} = 0$ but $H_{z} \neq = 0$ , and $H_{z}$ obeys the wave equation

Δ H_{z} (x, y, z) + k^{2} H_{z} (x, y, z) = 0. (7.2.2)

With $E_{z} = 0$ and $\partial / \partial x = 0$ the four formulas, Eq. (7.1.11), read

H_{x} H_{y} E_{x} E_{y} = - \frac{1}{h ^{2}} j τ \frac{\partial H _{z}}{\partial x} = 0, = - \frac{1}{h ^{2}} j τ \frac{\partial H _{z}}{\partial y}, = - \frac{1}{h ^{2}} j ω μ \frac{\partial H _{z}}{\partial y}, = \frac{1}{h ^{2}} j ω μ \frac{\partial H _{z}}{\partial x} = 0. (7.2.3)

The boundary conditions are

E_{x} (x, 0, z) = 0, E_{x} (x, b, z) = 0 ⟺ \frac{\partial H _{z}}{\partial y} (x, 0, z) = 0, ⟺ \frac{\partial H _{z}}{\partial y} (x, b, z) = 0. (7.2.4)

The condition is on the slope of $H_{z}$ because $E_{x}$ is proportional to $\partial H_{z} / \partial y$ , so $E_{x} = 0$ on a plate means $\partial H_{z} / \partial y = 0$ there. The other tangential component, $E_{z}$ , is zero everywhere for a TE mode.

The ordinary differential equation

Write out Eq. (7.2.2),

\frac{\partial ^{2} H _{z}}{\partial x ^{2}} + \frac{\partial ^{2} H _{z}}{\partial y ^{2}} + \frac{\partial ^{2} H _{z}}{\partial z ^{2}} + k^{2} H_{z} = 0.

With $\partial / \partial x = 0$ and $\partial^{2} / \partial z^{2} \to - τ^{2}$ ,

\frac{\partial ^{2} H _{z}}{\partial y ^{2}} + (k^{2} - τ^{2}) H_{z} = 0 ⟹ \frac{\partial ^{2} H _{z}}{\partial y ^{2}} + h^{2} H_{z} = 0. (7.2.5)

Separating the travelling factor, $H_{z} (y, z) = Y (y) e^{- j τ z}$ , leaves an ordinary differential equation for the profile across the gap,

\frac{d ^{2} Y}{d y ^{2}} + h^{2} Y = 0 ⟹ Y (y) = C_{1} cos (h y) + C_{2} sin (h y) . (7.2.6)

It is the equation of a mass on a spring: harmonic in $y$ , with "wavenumber" $h$ . Its two constants are fixed by the two plates.

Applying the boundary conditions

With $H_{z} (y, z) = [C_{1} cos (h y) + C_{2} sin (h y)] e^{- j τ z}$ the first condition gives

\frac{\partial H _{z}}{\partial y} (x, 0, z) = h C_{2} e^{- j τ z} = 0 ⟹ C_{2} = 0, H_{z} (y, z) = C_{1} cos (h y) e^{- j τ z},

and the second

\frac{\partial H _{z}}{\partial y} (x, b, z) = - h C_{1} sin (hb) e^{- j τ z} = 0 ⟹ sin (hb) = 0 ⟹ h_{n} b = nπ, n = 1, 2, \dots

Each allowed value $h_{n} = nπ / b$ gives one solution,

H_{z n} (y, z) = C_{1 n} cos (\frac{nπ}{b} y) e^{- j τ_{n} z} . (7.2.7)

The condition $h_{n} b = nπ$ says that $H_{z}$ fits $n$ half-periods of a cosine across the gap, with its slope zero on both plates.

Why not

n = 0

$h = 0$ gives a constant $H_{z}$ with no transverse field at all (and the formulas (7.2.3) divide by $h^{2}$ ): not a wave. The TE modes start at $n = 1$ .

The general TE field is the sum of all these solutions. With Eq. (7.2.3) and

\frac{\partial H _{z n}}{\partial y} = - \frac{nπ}{b} C_{1 n} sin (\frac{nπ}{b} y) e^{- j τ_{n} z}

the transverse components follow:

H_{z} (y, z) = n = 1 \sum \infty C_{1 n} cos (\frac{nπ}{b} y) e^{- j τ_{n} z}, (7.2.8)

H_{x} (y, z) = 0, E_{y} (y, z) = 0, (7.2.9)

E_{x} (y, z) = j ω μ n = 1 \sum \infty (\frac{nπ}{b h _{n}^{2}}) C_{1 n} \times sin (\frac{nπ}{b} y) e^{- j τ_{n} z}, (7.2.10)

H_{y} (y, z) = j n = 1 \sum \infty (\frac{nπ τ _{n}}{b h _{n}^{2}}) C_{1 n} \times sin (\frac{nπ}{b} y) e^{- j τ_{n} z} . (7.2.11)

Each term in the series is called a mode. The $n$ th term is the TE $_{n}$ mode. Since $h_{n} = nπ / b$ , the factor $nπ / (b h_{n}^{2})$ is simply $1/ h_{n}$ , and a single mode reads

TE$_n$ mode of the parallel-plate guide

H_{z} E_{x} H_{y} = C cos (h_{n} y) e^{- j τ_{n} z}, h_{n} = \frac{nπ}{b}, = \frac{j ω μ}{h _{n}} C sin (h_{n} y) e^{- j τ_{n} z}, = \frac{j τ _{n}}{h _{n}} C sin (h_{n} y) e^{- j τ_{n} z}, (7.2.12)

with $\frac{E _{x}}{H _{y}} = \frac{ω μ}{τ _{n}} = Z_{T E}$ and $H_{x} = E_{y} = E_{z} = 0$ .

In plain words

The electric field of a TE $_{n}$ mode points along the plates ( $x$ ) and follows $sin (nπ y / b)$ , zero on both plates. The magnetic field lies in the $y z$ plane and closes on itself in loops. The ratio of the transverse fields is the TE wave impedance of Section 7.1.7, as it must be.

Cutoff of the TE modes

The phase constant of TE $_{n}$ follows from $h_{n}^{2} = k^{2} - τ_{n}^{2}$ :

τ_{n} = k^{2} - h_{n}^{2} = k^{2} - (\frac{nπ}{b})^{2} . (7.2.13)

In order to have a propagating (not evanescent) TE $_{n}$ mode, $τ_{n}$ must be real:

k > \frac{nπ}{b} ⟹ f > f_{co, n} = \frac{c}{ε _{r} μ _{r}} \frac{n}{2 b} = \frac{n v}{2 b} . (7.2.14)

$f_{co, n}$ is called the cutoff frequency of the TE $_{n}$ mode. If $f < f_{co, n}$ , the TE $_{n}$ mode cannot propagate. It is the general result, Eq. (7.1.14) in Section 7.1, with $h_{n} = nπ / b$ . The lowest TE mode is TE $_{1}$ , with $f_{co, 1} = v /2 b$ : at cutoff the gap is half a wavelength, $b = λ /2$ . The dispersion diagram of the modes is Figure 7.1.6 in Section 7.1.

The TE_1 mode, side view, real fields at t=0 (f=1.5f_{co,1}). The magnetic field lines (blue, dashed) are closed loops in the yz plane, the contours of \sin(\pi y/b)\cos\tau z. The electric field is along x. It is largest midway between the plates and between the loops, where it points into the page (\otimes, +x), it points out of the page half a guide wavelength further on, at the two edges of the picture, and it is zero on the plates. — **Figure 7.2.2.** The TE $_{1}$ mode, side view, real fields at $t = 0$ ( $f = 1.5 f_{co, 1}$ ). The magnetic field lines (blue, dashed) are closed loops in the $y z$ plane, the contours of $sin (π y / b) cos τ z$ . The electric field is along $x$ . It is largest midway between the plates and between the loops, where it points into the page ( $\otimes$ , $+ x$ ), it points out of the page half a guide wavelength further on, at the two edges of the picture, and it is zero on the plates.

Common mistake · Calling TE

_{1}

the dominant mode

TE $_{1}$ is the lowest TE mode, but it is not the dominant mode of the guide. TM $_{1}$ has the same cutoff, and the TEM mode (Section 7.2.4) propagates at every frequency: the dominant mode of the parallel-plate guide is TEM.

7.2.2 TE modes

The ordinary differential equation

Applying the boundary conditions

Cutoff of the TE modes