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How can the S21 to transmission impedance (Zt) conversion equation on the 4395A Operation Manual be derived?

The equivalent circuit model of transmission measurement is as follows.

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.

Therefore,

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

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