Tech

Go Techie: 1-Port SOL Calibration (S-parameter)

6th April 2018

Now we’re going techie to talk about ‘1-Port SOL Calibration’!

If you have dabbled in electromagnetics and done some microwave experiments, you might have already used vector network analysers (VNAs) several times. Before use, however, you have to calibrate the analyser.

Ever wondered how calibration works? Since this is the first official techie post, we are only going to talk about 1-port calibration. To understand the calculation process, let us first take a look at our measurement setup in Fig. 1.

Fig. 1. 1-port measurement setup.

Without calibration, the VNA sees not only the device under test (DUT), but also the error box, and hence gives us distorted S-parameters we do not want. Now, to consider the error, we are going to introduce the ‘signal flow graph’ (Fig. 2). A signal flow graph is, literally, a linear model that we use to describe the flow of signals. The value on each branch indicate the gain of the signal through that route, and the value on a node indicate the sum of all signals entering that node.

Fig. 2. Signal flow graph representation of a two-port network.

We take the error box in Fig. 1 and consider it as the 2-port device in Fig. 2. See the similarities? σ denotes the theoretic S11 of the DUT while ρ denotes the measured S11 (reflection coefficient). Also, e00, e01, e10, e11 are the S-parameters of the error box. After a quick calculation, we have the relation equation as:

\rho=e_{00}+\frac{e_{10}e_{01}\sigma}{1-e_{11}\sigma}

We can also write it as:

\sigma=\frac{\rho-e_{00}}{e_{01}e_{10}+e_{11}(\rho-e_{00})}

For 1-port calibration, a popular method is SOL, which stands for short-open-load calibration. Before measuring our DUT, we attach short, open, load to the port. The theoretic short has an S11 of –1, open of 1, and load of 0. Therefore, we have 3 equations directly from the last equation, where ρs denotes measured S11 of short, ρo of open, and ρl of load:

-1=\frac{\rho_s-e_{00}}{e_{01}e_{10}+e_{11}(\rho_s-e_{00})}
1=\frac{\rho_o-e_{00}}{e_{01}e_{10}+e_{11}(\rho_o-e_{00})}
0=\frac{\rho_l-e_{00}}{e_{01}e_{10}+e_{11}(\rho_l-e_{00})}

From the above, we have e00 = ρl. With further derivation, we can finally get the formula we want to calculate the calibrated S11 value of our DUT:

\sigma_{DUT}=\frac{\rho_{DUT}-\rho_l}{e_{01}e_{10}+e_{11}(\rho_{DUT}-\rho_l)}
where
e_{01}e_{10}=\frac{2(\rho_o-\rho_l)(\rho_l-\rho_s}{\rho_o-\rho_s}
and
e_{11}=\frac{\rho_s+\rho_o-2\rho_l}{\rho_o-\rho_s}

If your cal kit is not SOL for some reason, just substitute the theoretical values into the equations, and you can still get the calibration formula!

Of course, your VNA will do the calculation for you. Normally.

References
1. D. M. Pozar, Microwave Engineering, 4th edition, John Wiley & Sons, 2012.
2. B. Bianco et al., “Evaluation of Errors in Calibration Procedures for Measurements of Reflection Coefficient”, IEEE Trans. Instrum. Meas., vol. IM-27, no. 4, Dec. 1978.
3. K. W. Whites, EE481/581 Class Notes, South Dakota School of Mines and Technology, 2016.

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