Transmission lines

KVL and KCL on a section $\Delta z$ as $\Delta z\to 0$; valid for any waveform.

Sinusoidal steady state with $e^{j\omega t}$; the third equation follows by differentiating the first and substituting the second.

$V_0^{+}$ is the wave travelling toward the load and $V_0^{-}$ the reflected wave. The minus sign in $I(z)$ matters: the reflected wave's current flows back toward the generator.

$\alpha$ in Np/m (1 Np = 8.686 dB), $\beta$ in rad/m. $Z_0=V^{+}/I^{+}$ for ONE travelling wave; it is not $V/I$ at a point on a mismatched line. Inverting, $R+j\omega L=\gamma Z_0$ and $G+j\omega C=\gamma /Z_0$ recover the line constants from measured $\gamma$ and $Z_0$.

$Z_0$ is real and, with $L$ and $C$ constant, independent of frequency. On any TEM line whose dielectric is homogeneous (coaxial, two-wire, parallel-plate) $LC=\mu \epsilon$; for a non-magnetic dielectric of relative permittivity ${\epsilon }_r$ this gives $v_p=c/\sqrt{{\epsilon }_r}$, so the wavelength on the line is shorter than in free space by the factor $\sqrt{{\epsilon }_r}$.

The line is lossy, but (taking $R$, $L$, $G$, $C$ as constants) $\alpha$ and $v_p$ do not depend on frequency and $Z_0$ is real, so a signal is attenuated and delayed without changing shape.

The first term of $\alpha$ is conductor loss, the second dielectric loss.

Inner conductor radius $a$, inner radius of the outer conductor $b$, lossless dielectric with $\mu ={\mu }_0$ and $\epsilon ={\epsilon }_0{\epsilon }_r$. $L$ is the external inductance only (internal inductance neglected, as at high frequency where the current flows in a thin skin); the last form uses $\sqrt{{\mu }_0/{\epsilon }_0}\approx 120\pi \Omega$.

Two-wire: conductor radius $a$, centre-to-centre spacing $D$; for $D\gg a$, ${\cosh }^{-1}(D/2a)\approx \ln (D/a)$, so $Z_0\approx (120/\sqrt{{\epsilon }_r})\ln (D/a)\Omega$ for a non-magnetic dielectric. Parallel-plate: strip width $w$ and separation $h$, with $w\gg h$ so that fringing can be neglected. Both, like the coaxial line, satisfy $LC=\mu \epsilon$ (external inductance, lossless dielectric).

Voltage reflection coefficient; $|{\Gamma }_L|\le 1$ for any passive load ($\mathrm{Re}{Z}_L\ge 0$) on a line with real $Z_0$. The current reflection coefficient is $-{\Gamma }_L$.

Voltage is continuous at the load or junction, so transmitted voltage = incident + reflected. For a junction the 'load' is the second line's $Z_{02}$ (assumed matched beyond). $\tau$ can exceed 1, but power still balances: for real $Z_{01}$, $Z_{02}$ the fraction of incident power transmitted is $|\tau {|}^2{Z}_{01}/Z_{02}=1-|\Gamma {|}^2$.

${\theta }_L=\angle {\Gamma }_L$ and $d$ is measured from the load toward the generator. On a lossy line $|\Gamma |$ shrinks as $e^{-2\alpha d}$ toward the generator.

$|V|$ is largest where $\Gamma (d)$ is real and positive, and smallest where it is real and negative; $|I|$ does the opposite.

Every input-impedance result in the next section is a special case of this formula. $Z_{in}$ repeats every $\lambda /2$, because $\tan \beta d$ has period $\pi$.

Here $Z_0$ and $\gamma$ are in general complex. With $\alpha =0$ it reduces to the lossless form, because $\tanh (j\beta d)=j\tan \beta d$.

Lossless line: $|V{|}_{max}=|V_0^{+}|(1+|\Gamma |)$ and $|V{|}_{min}=|V_0^{+}|(1-|\Gamma |)$. The current maxima sit at the voltage minima.

Lossless line, $V_0^{+}$ a peak phasor. $|\Gamma {|}^2$ is the fraction of incident power reflected. The same delivered power is $P_L=|V{|}_{max}|V{|}_{min}/(2Z_0)$. Return loss is quoted here as a positive number for a passive load (a matched load has $RL=\infty$); some texts attach a minus sign, so read the question's convention.

$V_g$ is the source's open-circuit voltage (peak phasor) and $Z_g$ its internal impedance; $Z_{in}$ is the input impedance of the line together with its load. On a lossless line all of $P_{in}$ reaches the load. For a given source, $P_{in}$ is largest when $Z_{in}=Z_g^{\ast }$ (conjugate matching); when $Z_g=Z_0$ is real, that happens exactly when the load is matched, $Z_L=Z_0$.

A step of height $V_g$ is applied at $t=0$ through a source resistance $R_g$ to a line of one-way delay $T=\ell /v_p$. The line first looks like $Z_0$, so it launches $V_1^{+}$. At $t=T$ this reaches the load, which reflects ${\Gamma }_L{V}_1^{+}$, so the load voltage jumps to $V_1^{+}(1+{\Gamma }_L)$; at $t=2T$ the reflection reaches the source and is reflected again with ${\Gamma }_g$, and so on. The voltage at any point is the sum of every wave that has passed it so far (a bounce diagram). The final value is the DC voltage divider, because at DC a lossless line is just a pair of wires.

Lossless line, ${\theta }_L$ in radians, $n$ an integer; the first maximum (or minimum) is the smallest $d\ge 0$ the formula gives. Successive maxima (or minima) are $\lambda /2$ apart and a maximum is $\lambda /4$ from the next minimum. The line impedance is purely resistive at both.

The slotted-line measurement on a lossless line: $d_{min}$ is the distance from the load to any voltage minimum; $|{\Gamma }_L|$ follows from $S$.

Purely reactive. Inductive for $0<\ell <\lambda /4$ (with $Z_{sc}=j{Z}_0$ at $\ell =\lambda /8$); an open circuit at $\ell =\lambda /4$, near which it behaves like a parallel resonant circuit; capacitive for $\lambda /4<\ell <\lambda /2$; a short circuit again at $\ell =\lambda /2$.

Purely reactive. Capacitive for $0<\ell <\lambda /4$ (with $Z_{oc}=-j{Z}_0$ at $\ell =\lambda /8$); a short circuit at $\ell =\lambda /4$, near which it behaves like a series resonant circuit; inductive for $\lambda /4<\ell <\lambda /2$; an open circuit again at $\ell =\lambda /2$.

Valid for lossy lines too, since $Z_{sc}=Z_0\tanh \gamma \ell$ and $Z_{oc}=Z_0\coth \gamma \ell$; take the root of $Z_0$ with $\mathrm{Re}{Z}_0>0$. For a lossless line write $Z_{sc}=j{X}_{sc}$ and $Z_{oc}=j{X}_{oc}$: then $Z_0=\sqrt{-X_{sc}{X}_{oc}}$ and $\tan \beta \ell =X_{sc}/Z_0$, which fixes $\ell$ to within a multiple of $\lambda /2$. (Taking $\tan \beta \ell =\sqrt{-Z_{sc}/Z_{oc}}$ loses the sign: a $3\lambda /8$ line would be read as $\lambda /8$.)

A short becomes an open, an open becomes a short, and an inductive load looks capacitive (and vice versa).

For any $Z_0$: at a frequency where the section is a whole number of half-wavelengths long, it repeats the load impedance.

No reflected wave; $|V|$ and $|I|$ are the same everywhere and the VSWR is 1.

A $\lambda /4$ section of characteristic impedance $Z_T$ makes a real load $R_L$ look like $Z_0$. Exact only at the design frequency (where the section is $\lambda /4$ long) and at odd multiples of it.

First move from the load to a point where the line impedance is real ($S{Z}_0$ at $d_{max}$, $Z_0/S$ at $d_{min}$), cut the line there and insert the $\lambda /4$ section: $Z_T=\sqrt{Z_0\cdot S{Z}_0}$ or $\sqrt{Z_0\cdot Z_0/S}$.

Same form as the impedance formula, written in normalised admittances. Use it whenever elements are connected in shunt.

Lossless line; the upper signs go together. Obtained by setting $\mathrm{Re}y(d)=1$ with $y_L=1/r$. Take $\beta d$ in $(0,\pi )$, adding $\pi$ to a negative angle.

The centre of the chart is $\Gamma =0$, the matched point $z=1$.

Centre $\left(\frac{r}{1+r},0\right)$, radius $\frac{1}{1+r}$; every one passes through $\Gamma =1$. $r=0$ is the outer rim and $r=1$ passes through the centre.

Centre $(1,1/x)$, radius $1/|x|$; only the arcs inside the unit circle are drawn. $x>0$ arcs lie in the upper half, $x<0$ in the lower half, and $x=0$ is the horizontal axis.

With the $e^{j\omega t}$ convention: clockwise when moving toward the generator, counter-clockwise toward the load. One full turn is $\lambda /2$, so the rim scales run from 0 to 0.5 wavelengths.

$a_n$ enters port $n$, $b_n$ leaves it. The net time-average power into port $n$ is $\frac{1}{2}\left(|a_n{|}^2-|b_n{|}^2\right)$.

$S_{11}$ is the reflection coefficient at port 1 with port 2 terminated in $Z_0$ (matched, not shorted); $S_{21}$ is the transmission coefficient from port 1 to port 2 under the same condition.

${\Gamma }_L$ is the reflection coefficient of the load on port 2, referred to $Z_0$. ${\Gamma }_{in}=S_{11}$ only when ${\Gamma }_L=0$. Looking back into port 2 with a source of reflection coefficient ${\Gamma }_s$ on port 1, ${\Gamma }_{out}=S_{22}+S_{12}{S}_{21}{\Gamma }_s/(1-S_{11}{\Gamma }_s)$.

All with port 2 matched. $|S_{21}{|}^2$ is the fraction of the incident power delivered to the matched load on port 2.

Holds for linear networks built only from reciprocal elements (line sections, R, L, C, coupled coils and transformers), with the same real reference impedance at both ports. Fails for magnetically biased ferrite devices (isolators, circulators) and for active circuits such as amplifiers.

Each column carries unit power and different columns are orthogonal (equivalently, $[S][S{]}^{\ast T}=[I]$, so the rows are too). For a lossless two-port this forces $|S_{11}|=|S_{22}|$ and $|S_{12}|=|S_{21}|$. Any passive network has $|S_{11}{|}^2+|S_{21}{|}^2\le 1$; the shortfall is the fraction of the power incident on port 1 that is dissipated inside the network (port 2 matched).

Port 1 sees $Z+Z_0$ when port 2 is matched.

Here $S_{21}=1+S_{11}$ because the voltage is the same on both sides of a shunt element.

A line of the reference impedance $Z_0$ and length $\ell$. Moving the reference plane at port $n$ outward by a length ${\ell }_n$ of matched line changes only phases: $S_{11}^{\prime }=S_{11}{e}^{-j2\beta {\ell }_1}$.