Franclim F. Ferreira, Pedro Guedes de Oliveira, Vítor G. Tavares

Equivalent two-port network technique

The equivalent two-port network technique consists in replacing a two-port network inserted in a given circuit by another one in which the internal transmission between the two ports is represented by controlled sources.

What we do is to implement the linear equations that relate the variables (voltage and current) associated to the two ports. These equations may assume four different forms, depending upon the choice of the independent variables, each of them associated, obviously, to different ports:

V₁, V₂ = f (I₁, I₂)

V₁, I₂ = f (I₁, V₂)

I₁, V₂ = f (V₁, I₂)

I₁, I₂ = f (V₁, V₂)

We should stress that each equation, being the sum of voltages or currents, may respectively be represented by a Thévenin or Norton configuration.

Let us take, as an example, the system of equations V₁, I₂ = f (I₁, V₂), that corresponds to

V₁ = a₁₁ I₁ + a₁₂ V₂

I₂ = a₂₁ I₁ + a₂₂ V₂

where a₁₁ is a resistor, a₁₂ and a₂₁ are dimensionless transmission factors and a₂₂ is a conductance, and that can be represented by the circuit on the right.

This is what we look for. Let us see how to proceed with an example.

Suppose we want to analyse the amplifier circuit on the right, in which the existence of R₂ makes the analysis a little bit more difficult and we decided, thereafter, to apply the equivalent two-port network technique to this resistance.

The choice of the independent variables should clearly be the voltages on both sides, bearing in mind that our final objective is to obtain an equivalent circuit that will allow an easy evaluation of gain and input and output resistances.

Looking into each one of the ports we can draw the circuit depicted on the right, where the independent sources are represented by controlled voltage sources.

It should be noticed that this circuit does not implement any of the four systems of equations indicated above, that are coherent in the sense that the independent variables are the same in both equations. However, if we just replace the Thévenin circuits by their Norton equivalent, we get a coherent circuit in the sense we are using the term.

The replacement of the two-port equivalent network of R₂ in the global circuit results in the following:

This circuit can now be easily simplified leading to an equivalent circuit for the amplifier. When we use numerical examples, some of the simplifying procedures are still more obvious.