A VNA is usually calibrated by the user at some chosen reference plane using standard parts, commonly an open circuit, short circuit, and nominal (50Ω) load. As a result of this OSL calibration, the VNA is able to correct measured s11 to that reference plane, and display its results wrt that reference plane.

There are occasions where it is not possible, or not convenient to locate the DUT at the reference plane. This article discusses the problem created, and some solutions that might give acceptable accuracy for the application at hand.

The discussion assumes the VNA is calibrated for nominal 50+j0Ω.

Above is a diagram of a configuration where the unknown Zl is not located exactly at the reference plane, but at some extension.

The problem created is that Vr50’/Vi50′ (s11) is not the same as if Zl was attached directly at the reference plane.

Let’s look at some possible solutions.

## Extension is a known lossy uniform transmission line

If the extension can be well characterised as a uniform transmission line, we can estimate the effect of the extension by calculation and approximately correct it.

The impedance transformation due to a section of uniform transmission line is \(Z_{in}=Z_0 \frac{Z_l+Z_0 tanh(\gamma l)}{Z_0+Z_l tanh(\gamma l)}\). If the transmission line parameters are known, the the impedance ‘measured’ at the reference plane can be corrected to the actual DUT location.

This is not a common feature, but it appears in Rigexpert’s Antscope (1) as the add/subtract cable feature.

The accuracy depends on the accuracy of the transmission line model and the line characterisation, it needs to be verified, it is not an insignificant issue.

## Extension is a uniform 50Ω transmission line of negligible loss

If we can assume that port extension conductors form a uniform transmission line with negligible loss, we can use the relationship that \(Z_{in}=Z_0 \frac{Z_l+\jmath Z_0 tan(\beta l)}{Z_0+\jmath Z_l tan(\beta l)}\).

We can correct the measurement by adjusting for the extension round trip phase delay due to βl, so at each measurement frequency we add that phase delay back into the phase of ‘measured’ s11.

Accuracy depends on Zo=50+j0Ω and negligible loss, though acceptable results might be obtained for small departures. Note that negligible loss infers a short line section.

This facility is often provided in VNAs and PC client programs.

## Extension is a short uniform transmission line other than 50Ω and of negligible loss

This can also be used with good effect to approximately compensate for a test fixture that does not exactly meet the conditions specified previously. The error may be small if:

- the line section is electrically very short; and
- Zl >>Zo; or
- Zl<<Zo.

If we assume that port extension conductors form a uniform transmission line with negligible loss, we can use the relationship that \(Z_{in}=Z_0 \frac{Z_l+\jmath Z_0 tan(\beta l)}{Z_0+\jmath Z_l tan(\beta l)}\).

Let’s look in more detail at the two cases.

### Case: \(Z_l>>Z_0\)

\(Z_{in}=Z_0 \frac{Z_l+\jmath Z_0 tan(\beta l)}{Z_0+\jmath Z_l tan(\beta l)}\). \(For Z_l>>Z_0, Z_{in} \to Z_0 \frac{1}{\jmath tan(\beta l)}\) so we can derive an equivalent length at Zo: \(Z_{0_1} \frac{1}{tan(\beta_1 l_1)} \approx Z_{0_2} \frac{1}{tan(\beta_2 l_2)}\).

For small values of \(\beta l, tan(\beta l)=\beta l\), and noting that \(\frac{\beta_1}{\beta_2}=\frac{v_{f2}}{v_{f1}}\) (where vf is the velocity factor), \(l_2 \approx \frac{Z_{0_2}v_{f2}}{Z_{0_1}v_{f1}}l_1\).

To calculate e-delay, Z_{02} and v_{2} will be 50 and 1 respectively, and \(edelay=\frac{l}{c_0 v_f}=\frac{l \cdot \text{1e12}}{299792458} \text{ps}\).

So, it turns out that when Zl dominates the expression for Zin, that an electrically short uniform transmission line port extension of some Zo other than 50Ω can be offset as in the expression above.

The appropriate e-delay (as it is often known) can be discovered by measurement of an OC at the end of the extension, and adjusting it until the phase of s11 is 0° independent of frequency.

Whilst the assumptions might seem quite restrictive, this technique can be used with good utility for suitable DUT and fixture, eg finding the high impedance self resonant frequency of an inductor / resonator.

### Case: \(Z_l<<Z_0\)

\(Z_{in}=Z_0 \frac{Z_l+\jmath Z_0 tan(\beta l)}{Z_0+\jmath Z_l tan(\beta l)}\). \(For Z_l<<Z_0, Z_{in} \to Z_0 \jmath tan(\beta l)\) so we can derive an equivalent length at Zo: \(Z_{0_1} tan(\beta_1 l_1) \approx Z_{0_2} tan(\beta_2 l_2)\).

For small values of \(\beta l, tan(\beta l)=\beta l\), and noting that \(\frac{\beta_1}{\beta_2}=\frac{v_{f2}}{v_{f1}}\) (where vf is the velocity factor), \(l_2 \approx \frac{Z_{0_1}v_{f2}}{Z_{0_2}v_{f1}}l_1\).

To calculate e-delay, Z_{02} and v_{2} will be 50 and 1 respectively, and \(edelay=\frac{l}{c_0 v_f}=\frac{l \cdot \text{1e12}}{299792458} \text{ps}\).

So, it turns out that when Zo dominates the expression for Zin, that an electrically short uniform transmission line port extension of some Zo other than 50Ω can be offset as in the expression above.

The appropriate e-delay (as it is often known) can be discovered by measurement of an SC at the end of the extension, and adjusting it until the phase of s11 is 180° independent of frequency (easiest with a wrapped phase plot if available).

Whilst the assumptions might seem quite restrictive, this technique can be used with good utility for suitable DUT and fixture, eg finding the low impedance resonant frequency and input impedance of a transmission line section.