Derivation of the expression for the unknown impedance in an s21 series through measurement

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

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 for 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:

\(Zu=\frac{Zs+Zl}{s21}-(Zs+Zl)\)

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.