Cut-insertion theorem


The Cut-insertion theorem, also known as Pellegrini's theorem, is a linear network theorem that allows transformation of a generic network N into another
network N' that makes analysis simpler and for which the main properties are more apparent.

Statement

Let e, h, u, w, q=q, and t=t' be six arbitrary nodes of the network N and ' be an independent voltage or current source connected between e and h, while is the output quantity, either a voltage or current, relative to the branch with immittance, connected between u and w. Let us now cut the qq' connection and insert a three-terminal circuit between the two nodes q and q' and the node t=t' , as in figure b.
In order for the two networks N and N' to be equivalent for any, the two constraints and, where the overline indicates the dual quantity, are to be satisfied.
The above-mentioned three-terminal circuit can be implemented, for example, connecting an ideal independent voltage or current source between q' and t' , and an immittance between q and t''.

Network functions

With reference to the network N', the following network functions can be defined:
; ;
; ;
from which, exploiting the Superposition theorem, we obtain:
Therefore, the first constraint for the equivalence of the networks is satisfied if.
Furthermore,
therefore the second constraint for the equivalence of the networks holds if

Transfer function

If we consider the expressions for the network functions and, the first constraint for the equivalence of the networks, and we also consider that, as a result of the superposition principle,, the transfer function is given by
For the particular case of a feedback amplifier, the network functions ', ' and ' take into account the nonidealities of such amplifier. In particular:
If the amplifier can be considered ideal, i.e. if, and, the transfer function reduces to the known expression deriving from classical feedback theory:

Evaluation of the impedance and of the admittance between two nodes

The evaluation of the impedance between two nodes is made somewhat simpler by the cut-insertion theorem.

Impedance

Let us insert a generic source between the nodes j=e=q and k=h between which we want to evaluate the impedance. By performing a cut as shown in the figure, we notice that the immittance is in series with and the current through it is thus the same as that provided by. If we choose an input voltage source and, as a consequence, a current, and an impedance, we can write the following relationships:
Considering that, where is the impedance seen between the nodes k=h and t if remove and short-circuit the voltage sources, we obtain the impedance between the nodes j and k in the form:

Admittance

We proceed in a way analogous to the impedance case, but this time the cut will be as shown in the figure to the right, noticing that is now in parallel to. If we consider an input current source and an admittance, the admittance between the nodes j and k can be computed as follows:
Considering that, where is the admittance seen between the nodes k=h and t if we remove and open the current sources, we obtain the admittance in the form:

Comments

The implementation of the TTC with an independent source and an immittance is useful and intuitive for the calculation of the impedance between two nodes, but involves, as in the case of the other network functions, the difficulty of the calculation of from the equivalence equation. Such difficulty can be avoided using a dependent source in place of and using the Blackman formula for the evaluation of. Such an implementation of the TTC allows finding a feedback topology even in a network consisting of a voltage source and two impedances in series.