[ 2018-12-18 ]
VNA OSL calibration
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.
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 S11, S21, S12 and S22 are the S-parameters of the error network that sits between the ideal VNA and the device under test (DUT) SDUT. 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:
- 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 Gk
- Make a list of all loops in the SFG and label these Li. Make a list of all pairwise non-touching loops and calculate their gains LiLj 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 b0 and the output is a0, the value of SM 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.
Once all three measurements have been made the system of equations can be solved for the parameters S11, S22 and S'.
S21S12 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 SDUT using the measurement SM is now given by