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 [m0088_Input_Impedance_for_Open_and_Short_Circuit_Terminations], 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 $Z_0 = \sqrt{ Z_{in}^{(SC)} \cdot Z_{in}^{(OC)} }$ If we instead divide these measurements, we find $\frac{ Z_{in}^{(SC)} }{ Z_{in}^{(OC)} } = -\tan^2\beta l \nonumber$ Therefore: $\tan\beta l = \left[ - \frac{ Z_{in}^{(SC)} }{ Z_{in}^{(OC)} } \right]^{1/2}$ 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 $$v_p = \omega/\beta$$.