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.