3.18: Measurement of Transmission Line Characteristics

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 $v_p = \omega/\beta$.