Loss, Insertion Loss, and Mismatch Loss terms pre-date VNAs and S parameters, but a VNA can be a very productive way of measuring / calculating these quantities for two port networks.

This article explains the basic S parameters and their use to measure and calculate Loss, Insertion Loss, and Mismatch Loss.

## S parameters

Review of s parameters of a two port network.

Above, a two-port network showing incident waves (a1, a2) and reflected waves (b1, b2) used in s-parameter definitions. (“Waves' means these are voltages, not power.)

The linear equations describing the two-port network are then:

\(b_1=s_{11}a_1+s_{12}a_2\\\) \(b_2=s_{21}a_1+s_{22}a_2\)## Applied VNA

In a VNA test of a two port network, the DUT, Zl=Zs=Zc, the nominal calibration impedance, often 50+j0 ohms; The ongoing discussion assumes that the calibration impedance is purely real (which is usually the case). Note that the VNA correction processes can correct for various errors, including in Zs and Zl. Lets assume it does and discuss the exceptions later.

### Without DUT in place

Without DUT in place, b1=0, a2=0, b2=a1, s11=0, s21=1.

### DUT in place

With DUT in place, a2=0, b2=s21 a1, s11 and s21 as measured.

## Insertion Loss

Insertion Loss is the ratio of power into a matched load (to mean that the load impedance is the complex conjugate of the Thevenin equivalent source impedance) to the power in the load with the subject network / device inserted.

In terms of the network shown:

- the power into the matched load without DUT \(P_{fwd}=\frac{{|a_1|}^2}{Z_c}\); and
- the power into the load the DUT in place \(P_{out}=\frac{{|b_2|}^2}{Z_c}=\frac{{|s_{21}|}^2 {|a_1|}^2}{Z_c}\).

Insertion Loss can be calculated as \(InsertionLoss=\frac{P_{fwd}}{P_{out}}=\frac1{{|s_{21}|}^2}\) (after simplification).

InsertionLoss can be expressed in dB, \(InsertionLoss_{dB}=10 \log InsertionLoss\).

## Mismatch Loss

Mismatch Loss is the ratio of output power of a source into a matched load to the power into the DUT under a given mismatch.

In terms of the network shown:

- the power into the matched load without DUT \(P_{fwd}=\frac{{|a_1|}^2}{Z_c}\); and
- the power into the DUT \(P_{in}=P_{fwd}-P_{ref}\) (valid only when Zc is real).
- since \(P_{ref}=P_{fwd}\frac{{|b_1|}^2}{{|a_1|}^2}\), \( P_{in}=P_{fwd} (1-({\frac{|b_1|}{|a_1|}})^2)=P_{fwd} (1-{|s_{11}|}^2)\)

So \(MismatchLoss=\frac{P_{fwd}}{P_{in}} = \frac{1}{(1-{|s_{11}|}^2)}\)

MismatchLoss can be expressed in dB, \(MismatchLoss_{dB}=10 \log MismatchLoss\).

This measurement is also valid for a one port network… but note the caveats here, Zs=Zc and they are real.

### Loss

Loss (or Transmission Loss if you must qualify it for clarity) is simply \(Loss=\frac{P_{in}}{P_{out}}\).

In terms of the network shown:

as derived above \(P_{out}=\frac{{|s_{21}|}^2 {|a_1|}^2}{Z_c}\) and since \(\frac{{|a_1|}^2}{Z_c}=P_{fwd}\), \(P_{out}={|s_{21}|}^2 P_{fwd}\)

as derived above, \( P_{in}=P_{fwd} (1-{|s_{11}|}^2)\)

We can calculate \(Loss=\frac{P_{in}}{P_{out}}=\frac{P_{fwd} (1-{|s_{11}|}^2)}{ P_{fwd}{|s_{21}|}^2}=\frac{(1-{|s_{11}|}^2)}{{|s_{21}|}^2}=\frac{InsertionLoss}{MismatchLoss}\)

Loss can be expressed in dB, \(Loss_{dB}=10 \log Loss\).

Of course if InsertionLoss and MismatchLoss are expressed in dB, we can calculate \(Loss_{dB}=InsertionLoss_{dB}-MismatchLoss_{dB}\)

## Errors

VNAs are usually used with a calibration / correction process, and 12 term correction deals with errors in source and load impedance.

Lesser correction schemes, such as the internal correction in a NanoVNA-H4, do not implement 12 term correction, and they do not correct for load impedance, ie the input impedance of Port 2.

There will be some residual error / noise in measurement and calculation.

Error caused by the test fixture should not be overlooked.