TEM mode

A TEM mode (transverse electromagnetic) is a propagating electromagnetic wave in which both $\mathbf{E}$ and $\mathbf{H}$ are entirely perpendicular to the direction of propagation — no field components along the travel direction.

For a wave moving in $+\hat{\mathbf{z}}$:

$E_z=0,H_z=0.$

The transverse fields ${\mathbf{E}}_t$ and ${\mathbf{H}}_t$ lie in the $xy$-plane, with $({\mathbf{E}}_t,{\mathbf{H}}_t,\hat{\mathbf{z}})$ forming a right-handed orthogonal triad. This is the simplest possible field configuration that satisfies Maxwell’s equations in a guided or unbounded geometry.

Where TEM modes occur

Free-space plane waves are always TEM. The wave equation in unbounded space admits a transverse-field solution with $\mathbf{E}$ and $\mathbf{B}$ perpendicular to $\hat{\mathbf{k}}$, the propagation direction. Any plane EM wave you encounter in vacuum is automatically TEM.

Two-conductor transmission lines support TEM: coax, parallel-wire, stripline, parallel-plate (ideally). The defining feature: there are two separate conductors, so a DC voltage can exist between them, and the static-like field pattern in the line’s cross-section “advances” along the line as a TEM wave.

Single-conductor waveguides (rectangular, circular, ridge) do not support TEM. They support only TE (transverse electric, $E_z=0$) and TM (transverse magnetic, $H_z=0$) modes, each with a cutoff frequency below which the wave cannot propagate. This is why hollow waveguides have a minimum operating frequency — the lowest mode (TE₁₀ in rectangular guide) has $f_c=c/(2a)$ where $a$ is the wide dimension.

The rule of thumb: TEM requires two conductors. With one conductor (or none), only higher-order modes exist.

Why two conductors are needed

In a TEM mode, the transverse field pattern ${\mathbf{E}}_t(x,y)$ at any cross-section is identical to the static electrostatic field that would exist between two conductors held at fixed potentials. It’s just a “frozen-in” static-like pattern that propagates at the phase velocity.

For a static electric field to exist between conductors, you need at least two of them — one at high potential, one at low. The voltage difference drives the field pattern. A single conductor with no return path has $\mathbf{E}=0$ outside (in the static limit), so no TEM mode is possible.

This is the physical content of the “two-conductor requirement”: TEM = a propagating version of a static voltage pattern, which requires a voltage difference to define.

Properties of TEM modes

No cutoff frequency. TEM modes propagate at all frequencies, down to DC. This is what makes coax suitable for everything from DC power feeds to GHz signals on the same line.

Phase velocity is medium-dependent only: $u_p=\frac{1}{\sqrt{\mu ϵ}}=\frac{c}{\sqrt{{\mu }_r{ϵ}_r}},$ independent of line geometry and frequency. This means TEM lines are non-dispersive in the ideal lossless case — a pulse keeps its shape as it propagates.

Voltage and current are well-defined. Because the transverse field is static-like, you can integrate $\mathbf{E}$ between the two conductors to get a unique voltage $V(z,t)$, and integrate $\mathbf{H}$ around one conductor to get a unique current $I(z,t)$. These are the quantities the Telegrapher’s equations talk about. In TE/TM waveguides, there is no unique definition of “voltage” — it depends on the path chosen.

Characteristic impedance $Z_0=V/I$ for a forward wave is real and frequency-independent (for a lossless TEM line).

Versus TE and TM

Mode	$E_z$	$H_z$	Where it lives
TEM	0	0	Two-conductor lines, plane waves
TE	0	≠ 0	Hollow waveguides (e.g., TE₁₀ in rectangular)
TM	≠ 0	0	Hollow waveguides, some dielectric guides
Hybrid (HE, EH)	≠ 0	≠ 0	Optical fibers, dielectric waveguides

Each of these has different cutoff behavior, different dispersion, different field structure. Electromagnetics focuses on TEM because two-conductor lines are the dominant technology up through low-GHz frequencies. Microwave and millimeter-wave engineering increasingly relies on TE/TM modes in waveguides and on hybrid modes in fibers and microstrip (which is “quasi-TEM” with a small longitudinal field component at higher frequencies).

Quasi-TEM

Real microstrip and similar planar lines aren’t exactly TEM because the wave travels partly in the dielectric and partly in air, with different propagation speeds. The mode is almost TEM with a small longitudinal component appearing at high frequency — called quasi-TEM. For most practical purposes below a few GHz, the TEM approximation is fine; above that, full-wave simulation (HFSS, CST) is needed.

Why this matters for the Smith chart and matching

The whole transmission-line formalism — characteristic impedance, reflection coefficient, SWR, Smith chart — assumes a TEM (or quasi-TEM) line where $V$ and $I$ are well-defined. In a hollow waveguide you can still talk about modal amplitudes and wave impedance, but the “circuit-element” picture of two-port networks with a characteristic impedance and a reflection coefficient is fundamentally a TEM-mode concept.