[ 2018-12-18 ]

## VNA OSL calibration

Categories: Electronics

*A simple application of Mason's gain formula can be used to derive the VNA OSL calibration routine.*

When making measurements with a VNA, it is important to calibrate the test equipment before the measurements in order to remove sources of error arising from non-ideal directional couplers, test cables and connectors. This is done using a so called cal-kit which contains devices whose S-parameters are well known (i.e. they have been accurately characterized for example by using a more accurate VNA than the one that is being calibrated).

In the case of a one-port measurement the calibration is often done using three calibration standards: open, short and load. This calibration routine is thus referred to as OSL calibration.

## Error model

The error model for a one-port VNA can be represented by the following signal flow graph (SFG).

```
S21
b0 ------> a1 ------> b2 ------> a3
| ^ |
S11 | | S22 | SDUT
| | |
v | v
a0 <------ b1 <------ a2 <------ b3
S12
```

where S_{11}, S_{21}, S_{12} and S_{22} are the S-parameters of the error network that sits between
the ideal VNA and the device under test (DUT) S_{DUT}. The error network thus represents all
sources of measurement error within the VNA itself and in the connection to the DUT.

In order to find the reflection coefficient measured by the VNA

from the SFG, Mason's gain formula (MGF) can be used.

## Masons's gain formula

The following section has been directly copied from Wikipedia:

### Definitions

- Path: a continuous set of branches traversed in the direction that they indicate.
- Forward path: A path from an input node to an output node in which no node is touched more than once.
- Loop: A path that originates and ends on the same node in which no node is touched more than once.
- Path gain: the product of the gains of all the branches in the path.
- Loop gain: the product of the gains of all the branches in the loop.

### Steps to find MGF solution

- Make a list of all forward paths and label them G
_{k} - Make a list of all loops in the SFG and label these L
_{i}. Make a list of all pairwise non-touching loops and calculate their gains L_{i}L_{j}etc. Continue until there are no more loops left. - Compute the determinant and the cofactors.
- Subsitute into the equation for G.

### Solution using MGF

According to the steps outlined above and noting that the input is b_{0}
and the output is a_{0}, the value of S_{M} can be found as follows:

With regards to the VNA error model, the terms are usually referred to as follows

This equation can be rearranged such that it is linear in three unknowns:

Using the three calibration standards a system of three linear equations can be formulated.

where

Once all three measurements have been made the system of equations can be solved
for the parameters S_{11}, S_{22} and S'.

S_{21}S_{12} can be calculated as follows:

Now all unknown parameters of the error network have been determined.

## Correcting DUT measurements

The reflection coefficient of an arbitrary DUT S_{DUT} using
the measurement S_{M} is now given by