This article presents a derivation of the expression for the unknown impedance in an s21 series through measurement.

Recall that S parameters are with respect to some reference impedance Zref, commonly 50+jΩ, but not necessarily. This article and the references assume Zref=50+jΩ.

The diagram above is from (Agilent 2009) and illustrates the configuration of a series-through impedance measurement.

Above is a simplified schematic of the voltage division from source to load in such a test configuration.

Calculate Vl1 where Zu=0:

\({V_l}_1=\frac{Z_L}{Z_s+Z_l}\\\)Calculate Vl2 where Zu≠0:

\({V_l}_2=\frac{Z_L}{Z_s+Zu+Z_l}\\\)Calculate s21 with Zu:

\(s_{21}=\frac{{V_l}_2}{{V_l}_1}=\frac{Zs+Zl}{Zl}\frac{Zl}{Zs+Zu+Zl}=\frac{Zs+Zl}{Zs+Zu+Zl}\\\)Make Zu the subject:

\(Zu=\frac{Zs+Zl}{s_{21}}-(Zs+Zl)\\Zu=(Zs+Zl)(\frac{1}{s_{21}}-1)\\\)Software often assumes Zref=50+jΩ and Zs+Zl=2*Zref. This may be a reasonable assumption for a 50Ω VNA corrected for source and load mismatch.

## References

- Agilent. Feb 2009. Impedance Measurement 5989-9887EN.
- Agilent. Jul 2001. Advanced impedance measurement capability of the RF I-V method compared to the network analysis method 5988-0728EN.