This article is demonstration of measurement of Transmission Loss in a section of coaxial transmission line. The scenario is chosen to expose the experiment to some of the things that complicate such measurements.

The very popular nanoVNA-H will be used to make the measurements.

The scenario:

- nanoVNA fully calibrated from 1-5MHz using a 200mm length coax lead on Port 2 (nanoVNA CH1);
- 35m of CCS RG6/U (close to an electrical quarter wavelength);
- three 50Ω terminations in shunt with VNA Port 2; and
- f=1.65MHz (close to a quarter wavelength.

The transmission line load is four 50Ω loads in parallel, one of them being VNA Port 2. Only one quarter of the output power is captured by the VNA, so there is effectively a loss of 6.02dB in that configuration. It also delivers a 12.5+j0Ω load the the transmission lines, VSWR is about 6. Note this power division is based on the assumption that Zin of Port 2 is 50+j0Ω, and error in Zin flows into the result. A 10dB attenuator is fitted to Port 2 prior to calibration to improve accuracy of Zin.

Above is a block diagram of the test configuration. nanoVNA measurements are wrt 50Ω, so \(P=\frac{V^2}{50}\) and \(V=\sqrt{50P}\).

1650000 0.710496228 1.456050447 0.255798404 -84.390357361 1.0 0.0 1.0 0.0

Above is a record from the .s2p file in MA format.

Above is calculation of Zin=293.8+j211.42Ω from s11 from the .s2p file. Also calculated is MismatchLoss=3.052dB. This method of calculating MismatchLoss is only correct if either source or load impedance is purely real, which is true in this case.

Converting the magnitude of s21 from the .s2p file, we get |s21|=-11.84dB.

## So, a bit of accounting is in order

Let's review some meanings of terms (in the 50Ω VNA context):

- \(TransmissionLoss=\frac{PowerIn}{PowerOut}=\frac{P_1}{P_2}\);
- \(s11=\frac{V_{1r}}{V_{1i}}\);
- \(InputMismatchLoss=\frac{P_{1i}}{P_{1}}=\frac1{(1-|s11|^2)}\); and
- \(s21=\frac{V_{2i}}{V_{1i}}\).

It is assumed that Zin of VNA Port 2 is 50+j0Ω, and that therefore P2r=0. Error in Zin of VNA Port 2 flows into the results. A 10dB attenuator is fitted to Port 2 prior to calibration to improve accuracy of Zin.

With the quantities expressed in dB, we can derive that \(TransmissionLoss=-|s21|-InputMismatchLoss\).

In the example given above, \(TransmissionLoss=-|s21|-InputMismatchLoss \\=11.84-3.052=8.79\: dB\).

As explained, the 6.02dB loss of the power divider in the load needs to be deducted to obtain the coax TransmissionLoss of 2.77dB.

You might be tempted to apply (Smith 1995)'s expression for loss due to standing waves \(\frac{Loss_{mismatched}}{Loss_{matched}}=\frac{1+S^2}{2S}\) but this scenario does not satisfy his conditions for it to apply, a much misused expression.

## References

- Smith, P. 1995. Electronic applications of the Smith chart 2nd ed. Noble Publishing Tucker.

## Conclusions

The Transmission Loss of a line section may not be directly given by any measures displayed by a VNA, it may take some interpretation and some accounting for elements that can be measured.