Notes on waveguides

I mainly follow Pozar’s book


A waveguide is one or more prismatic conductors used to transfer power in the form of time-varying electromagnetic fields.

Different wave modes can propagate in a waveguide, depending on the frequency and the propagation characteristics (dependent on the geometry and the materials used).

Field model

Again, I use $\exp (-j\omega t)$ in the expansion rather than $\exp (+j\omega t)$

For a waveguide along $z$ we write the complex electric and magnetic fields of the forward wave at angular frequency $\omega$ as:

\(\bm E = (E_x(x,y)\bm 1_x + E_y(x,y)\bm 1_y + E_z(x,y)\bm 1_z) \exp (-j\omega t + jkz - \alpha z)\) \(\bm H = (H_x(x,y)\bm 1_x + H_y(x,y)\bm 1_y + H_z(x,y)\bm 1_z) \exp (-j\omega t + jkz - \alpha z)\)

TEM waves

Imposing $E_z=0$, $H_z=0$ and entering with these expressions into the sourceless Maxwell equations, \(\nabla\times\bm E = j\omega \mu H,\) \(\nabla\times\bm H = -j\omega \varepsilon E,\) we find that for a non-trivial solution we need: $k^2 = \omega^2\mu\varepsilon$.

We also find that the traverse electric and magnetic fields $E_x\bm 1_x +E_y\bm 1_y$ and $H_x\bm 1_x + H_y\bm 1_y$ are solenoidal and irrotational.

The wave impedance of the TEM modes, i.e. the ratio between electric and magnetic fields at each point, is: \(Z_{TEM} = \frac{E}{H} = \sqrt{\frac{\mu}{\varepsilon}}\)

TE waves

We now take $E_z=0$ but $H_z\neq 0$. After some algebra, the sourceless Maxwell equations yield the transverse fields $E_x$, $E_y$, $H_x$ and $H_y$ as a function of the transverse derivatives of $H_z$ and the square of the cutoff wavenumber, \(k_c^2 = \omega^2 \mu \varepsilon - k^2\)

\[Z_{TE} = \frac{E}{H} = \mu\frac{\omega}{k}\]

TM waves

The same as TE waves, but we now take $E_z\neq 0$ but $H_z=0$. The same applies (although boundary conditions for the electric and magnetic fields on a perfect conductor are different, and thus the resulting cutoffs are also different)

\[Z_{TM} = \frac{E}{H} = \frac{1}{\varepsilon}\frac{k}{\omega}\]

Network analysis

Two-conductor TEM lines have well-defined voltage, current, characteristic impedance. This is not so in single-conductor waveguides.

For each TE and for each TM mode, we can define equivalent voltages and currents (a pair for the forward wave and another for the backward wave), following these guidelines

Multiport networks

We can treat any microwave component as a black box with $N$ ports. We want to figure out the (linear) relation between any incident wave on one port and the outgoing wave through all ports (including the one)

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Scattering matrix

This linear relation can be expressed as a matrix that gives the $V^-$ amplitudes on each port as a function of the $V^+$ amplitudes on each port

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Further study

Pozar’s chapter 4 introduces many other topics of interest: