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How can I derive the following equation which converts the S21 into the transmission impedance (Zt), shown on page A-14 of the 4395A Operation Manual?

formula 1

 

The equivalent circuit model of transmission measurement is as follows.

circuit model of transmission measurement

Where;
Vs: Source voltage [V]
Z0: System impedance [Ω]
Zt: Transmission impedance of the DUT [Ω]

 

Since the relationship between the Vs and the V1 is Vs = 2 x V1, the S21 is expressed as equation (1).

(1) S21 = V2/V1 = V2/(0.5 x Vs) = 2 x V2/Vs

 

Also, following equations are derived from the above model.

(2) V1 = I x (Zt + Zo)

(3) V2 =I x Zo

(4) Vs-V1 = I x Zo

 

By rearranging equation (4), we get the equation (5) as follows.

(5) Vs = I x Zo + V1

Putting (2), (3), and (5) into (1), we can transform the equation for S21 as follows.

formula 2

Therefore,

formula 3

This is equivalent to the formula on page A-14 of the 4395A Operation Manual.