# 4.4: Extraction of Transmission Line Parameters

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This section describes approaches for extracting the per unit length $$RLGC$$ parameters of a transmission line, and the effective permittivity and permeability of transmission line mediums, from measurements. The measurements could be made experimentally or through simulation.

### Ideal Calibration

The procedure described here assumes ideal calibration to the ports of a uniform transmission line that will yield symmetrical transmission line $$S$$ parameters with $$S_{11} = S_{22}$$ and $$S_{12} = S_{21}$$ for the reciprocal structure. The propagation constant of a transmission line is

$\label{eq:1}\gamma=\sqrt{(R+\jmath\omega L)(G+\jmath\omega C)}$

and the characteristic impedance is

$\label{eq:2}Z_{0}=\sqrt{\frac{(R+\jmath\omega L)}{(G+\jmath\omega C)}}$

Then

$\label{eq:3}R=\Re\{\gamma Z_{0}\},\quad L=\Im\left\{\frac{\gamma Z_{0}}{\omega}\right\},\quad G=\Re\{\gamma /Z_{0}\},\quad C=\Im\left\{\frac{\gamma /Z_{0}}{\omega}\right\}$

These parameters can be extracted from the measured $$S$$ parameters of a transmission line provided that sufficiently accurate calibration can be achieved. This extraction was described in [22, 23, 24], yielding the line’s characteristic impedance

$\label{eq:4}Z_{0}=Z_{\text{REF}}\sqrt{\frac{(1+S_{11})^{2}-S_{21}^{2}}{(1-S_{11})^{2}-S_{21}^{2}}}$

where $$Z_{\text{REF}}$$ is the reference impedance of the $$S$$ parameters. The line’s propagation constant is obtained from

$\label{eq:5}\text{e}^{-\gamma\ell}=\left(\frac{1-S_{11}^{2}+S_{21}^{2}}{2S_{21}}\pm K\right)^{-1}$

where

$\label{eq:6}L=\left[\frac{(S_{11}^{2}-S_{21}^{2}+1)^{2}-(2S_{11})^{2}}{(2S_{21})^{2}}\right]^{\frac{1}{2}}$

It is also possible to use specialized test structures to obtain the transmission line parameters. In [25] measurements of the input impedances of short- and open-circuited stubs are used to derive the transmission line parameters.

### Nonideal Calibration

The measurement of the $$S$$ parameters of a transmission line described above assumes that the error model of the fixtures can be accurately calibrated. Sometimes it can be difficult to achieve a sufficiently accurate calibration because of the difficulty of inserting a known resistive impedance standard. In the through-line calibration methods described in Section 4.3.5, measurement of a fixtured transmission line and measurement of back-to-back fixtures (a through connection) yields the propagation constant $$\gamma$$ of the transmission line, even if an impedance standard is not available. One way of extracting the characteristic impedance, $$Z_{0}$$, of a line from measurements in this situation was described in Section 4.5. The key idea was varying the electrical length of a transmission line to trace out a circle on the Smith chart. The frequency-dependent characteristic impedance can be obtained by using a number of transmission lines of different length. So with $$\gamma$$ and $$Z_{0}$$ known, the $$RLGC$$ parameters can be obtained using Equation $$\eqref{eq:3}$$.

## 4.4.1 Summary

The complexity and expense of equipment involved in measuring $$S$$ parameters depends greatly on the frequency range with vector network analyzers able to measure $$S$$ parameters above a hundred gigahertz costing many $$100\text{s}$$ of thousands of dollars. Also the time involved in calibration can be significant taking many hours to days at the very highest of microwave frequencies but only a few minutes at single-gigahertz frequencies. Microwave measurement requires the development of considerable expertise.

This page titled 4.4: Extraction of Transmission Line Parameters is shared under a CC BY-NC license and was authored, remixed, and/or curated by Michael Steer.