3.18: Measurement of Transmission Line Characteristics
selected template will load here
This action is not available.
This section presents a simple technique for measuring the characteristic impedance \(Z_0\), electrical length \(\beta l\), and phase velocity \(v_p\) of a lossless transmission line. This technique requires two measurements: the input impedance \(Z_{in}\) when the transmission line is short-circuited and \(Z_{in}\) when the transmission line is open-circuited.
In Section 3.16, it is shown that the input impedance \(Z_{in}\) of a short-circuited transmission line is
\[Z_{in}^{(SC)} = +jZ_0 \tan\beta l \nonumber \]
and when a transmission line is terminated in an open circuit, the input impedance is \[Z_{in}^{(OC)} = -jZ_0 \cot\beta l \nonumber \] Observe what happens when we multiply these results together: \[Z_{in}^{(SC)} \cdot Z_{in}^{(OC)} = Z_0^2 \nonumber \] that is, the product of the measurements \(Z_{in}^{(OC)}\) and \(Z_{in}^{(SC)}\) is simply the square of the characteristic impedance. Therefore \[Z_0 = \sqrt{ Z_{in}^{(SC)} \cdot Z_{in}^{(OC)} } \nonumber \] If we instead divide these measurements, we find \[\frac{ Z_{in}^{(SC)} }{ Z_{in}^{(OC)} } = -\tan^2\beta l \nonumber \] Therefore: \[\tan\beta l = \left[ - \frac{ Z_{in}^{(SC)} }{ Z_{in}^{(OC)} } \right]^{1/2} \nonumber \] If \(l\) is known in advance to be less than \(\lambda/2\), then the electrical length \(\beta l\) can be determined by taking the inverse tangent. If \(l\) is of unknown length and longer than \(\lambda/2\), one must take care to account for the periodicity of tangent function; in this case, it may not be possible to unambiguously determine \(\beta l\). Although we shall not present the method here, it is possible to resolve this ambiguity by making multiple measurements over a range of frequencies.
Once \(\beta l\) is determined, it is simple to determine \(l\) given \(\beta\), \(\beta\) given \(l\), and then \(v_p\). For example, the phase velocity may be determined by first finding \(\beta l\) for a known length using the above procedure, calculating \(\beta = \left(\beta l\right)/l\), and then \(v_p = \omega/\beta\).