General Four-Terminal Network Analysis

Author: Leonard Krugman

The general four-terminal active network is fully described by the relationship between the input and output currents and voltages. Referring to Fig. 3-1, the general voltage (loop) equations are:

E₁ = Z₁₁I₁ + Z₁₂I₂

and

E₂ = Z₂₁I₁ + Z₂₂I₂

where

Z₁₁ is the input impedance with the output open.
Z₁₁ = E₁/I₁, when I₂ = 0.
Z₁₂ is the feedback or reverse transfer impedance with the input open.
Z₁₂ = E₁/I₂, when I₁ = 0.
Z₂₁ is the forward transfer impedance with the output open.
Z₂₁ = E₂/I₁, when I₂ = 0.
Z₂₂ is the output impedance with the input open.
Z₂₂ = E₂/I₂, when I₁ = 0.

The equivalent current (nodal) equations are

I₁ = Y₁₁E₁ + Y₁₂E₂

and

I₂ = Y₂₁E₁ + Y₂₂E₂

where

Y₁₁ is the input admittance with the output shorted.
Y₁₁ = I₁/E₁, where E₂ = 0.
Y₁₂ is the feedback or reverse transfer admittance with the input shorted.
Y₁₂ = I₁/E₂, when E₁ = O.
Y₂₁ is the forward transfer admittance with the output shorted.
Y₂₁ = I₂/E_l, when E₂ = O.
Y₂₂ is the output admittance with the input shorted.
Y₂₂ = I₂/E₂, when E₁ = 0.

Amplification factors are the best general index of an active network. Since the general case may have amplification in both directions, definitions are included for forward and reverse directions.

The forward current amplification factor, α₂₁, is equal to the negative ratio of the current at the shorted output terminals to the current at the input terminals.

a₂₁ = -I₂/I₁ when E₂ = 0

Then 0 = E₂ = Z₂₁I₁ + Z₂₂I₂.

Solving these equations a₂₁ = -I₂/I₁ = Z₂₁/Z₂₂ and in terms of admittance

a₂₁ = -Y₂₁/Y₁₁

The reverse current amplification factor, α₁₂, is equal to the negative ratio of the current at the shorted input terminals to the current at the output terrminals:

a₁₂ = -I₁/I₂ when E₁ = 0

Then 0 = E₁ = Z₁₁I₁ + Z₁₂I₂.

Solving as before, a₁₂ = -I₁/I₂ = Z₁₂/Z₁₁, and in terms of admittances

a₁₂ = -Y₁₂/Y₂₂

The forward voltage amplification factory, μ₂₁, is equal to the ratio of the open circuit output voltage to the input voltage. , when I₂ = 0. On this basis, E₁ = Z₁₁I₁ and E₂ = Z₂₁I₁.

Thus and on an admittance basis

The reverse voltage amplification factor μ₁₂ is equal to the ratio of the open circuit input voltage to the output voltage. when I₁ = 0. Then E₁ = Z₁₂I₂ and E₂ = Z₂₂I₂. Thus In terms of admittance .