Transmission Lines and Matching

Syllabus: Transmission Lines: Equations, characteristic impedance, impedance matching, impedance transformation, S-parameters, Smith chart.

14 hand-picked visual resources on this topic, 6 of them little-known gems — under each card, look for See it. Every link is checked against the live site before it ships.

Line parameters

A line matters once it is an appreciable fraction of a wavelength long. $Z_0$ is set by the geometry alone; for a lossless line it is real and frequency-independent.

$Z_0=\sqrt{\dfrac{R+j\omega L}{G+j\omega C}}\ \xrightarrow{\text{lossless}}\ \sqrt{\dfrac{L}{C}}$
$\gamma=\alpha+j\beta=\sqrt{(R+j\omega L)(G+j\omega C)}$
$v_p=\dfrac{1}{\sqrt{LC}}=\dfrac{\omega}{\beta},\qquad \lambda=\dfrac{v_p}{f}$
$\text{distortionless: }\dfrac{R}{L}=\dfrac{G}{C}\Rightarrow\alpha=\sqrt{RG},\ \beta=\omega\sqrt{LC}$

See it

Impedance transformation and matching

A line transforms whatever is on its end, periodically in $\lambda/2$ . The quarter-wave transformer and the shorted/open stub follow from one formula, and almost every matching question is one of those three.

$Z_{in}=Z_0\dfrac{Z_L+jZ_0\tan\beta l}{Z_0+jZ_L\tan\beta l}$

Reflection, VSWR and the Smith chart

$\Gamma$ and VSWR are two readings of the same mismatch. On the Smith chart, moving toward the generator is clockwise, and a $\lambda/4$ move takes you to the diametrically opposite point — which is why the chart converts impedance to admittance by inspection.

$\Gamma_L=\dfrac{Z_L-Z_0}{Z_L+Z_0},\qquad \Gamma(l)=\Gamma_Le^{-2\gamma l}$

S-parameters

Defined with every other port terminated in $Z_0$ , so $S_{11}$ is simply the input reflection coefficient with a matched load. Reciprocity and losslessness each impose one testable condition.

$b_i=\sum_jS_{ij}a_j,\qquad S_{11}=\dfrac{b_1}{a_1}\Big|_{a_2=0}$

← Plane Waves, Media and Interfaces Waveguides, Fibres and Antennas →

Loading revision…