Cut-insertion theorem

The Cut-insertion theorem, also known as Pellegrini's theorem,[1] 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

Generic linear network N.
Equivalent linear network N'.
Implementation of the three-terminal circuit by means of an independent source Wr and an immittance Xp.

Let e, h, u, w, q=q', and t=t' be six arbitrary nodes of the network N and S be an independent voltage or current source connected between e and h, while U is the output quantity, either a voltage or current, relative to the branch with immittance X_{u}, connected between u and w. Let us now cut the qq' connection and insert a three-terminal circuit ("TTC") between the two nodes q and q' and the node t=t' , as in figure b (W_{r} and W_{p} are homogeneous quantities, voltages or currents, relative to the ports qt and q'q't' of the TTC).

In order for the two networks N and N' to be equivalent for any S, the two constraints W_{r}=W_{p} and \bar{W_{r}}=\bar{W_{p}}, 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 W_{p} between q' and t' , and an immittance X_{p} between q and t.

Network functions

With reference to the network N', the following network functions can be defined:

 A\equiv\frac{U}{W_{p}}|_{S=0} \!\, ;  \beta\equiv\frac{W_{r}}{U} |_{S=0} \!\, ;  X_{i}\equiv\frac{W_{p}}{\bar{W_{p}}} |_{S=0} \!\,

 \gamma\equiv\frac{U}{S} |_{W_{p}=0} \!\, ;  \alpha\equiv\frac{W_{r}}{S} |_{W_{p}=0} \!\, ;  \rho\equiv\frac{\bar{W_{p}}}{S} |_{W_{p}=0} \!\,

from which, exploiting the Superposition theorem, we obtain:

W_{r}=\alpha S+\beta AW_{p}

\bar{W_{p}}=\rho S+\frac{W_{p}}{X_{i}}.

Therefore the first constraint for the equivalence of the networks is satisfied if W_{p}=\frac{\alpha}{1-\beta A}S.

Furthermore,

\bar{W_{r}}=\frac{W_{r}}{X_{p}}

\bar{W_{p}}=\left(\frac{1}{X_{i}}+\frac{\rho}{\alpha}(1-\beta A)\right)W_{r}

therefore the second constraint for the equivalence of the networks holds if \frac{1}{X_{p}}=\frac{1}{X_{i}}+\frac{\rho}{\alpha}(1-\beta A)[2]

Transfer function

If we consider the expressions for the network functions \gamma and A, the first constraint for the equivalence of the networks, and we also consider that, as a result of the superposition principle, U=\gamma S+AW_{p}, the transfer function A_{f}\equiv \frac{U}{S} is given by

A_{f}=\frac{\alpha A}{1-\beta A}+\gamma.

For the particular case of a feedback amplifier, the network functions \alpha, \gamma and \rho take into account the nonidealities of such amplifier. In particular:

If the amplifier can be considered ideal, i.e. if \alpha=1, \rho=0 and \gamma=0, the transfer function reduces to the known expression deriving from classical feedback theory:

A_{f}=\frac{A}{1-\beta A}.

Evaluation of the impedance and of the admittance between two nodes

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

Impedance

Cut for the evaluation of the impedance between the nodes k=h and j=e=q.

Let us insert a generic source S between the nodes j=e=q and k=h between which we want to evaluate the impedance Z. By performing a cut as shown in the figure, we notice that the immittance X_{p} is in series with S and the current through it is thus the same as that provided by S. If we choose an input voltage source V_{s}=S and, as a consequence, a current I_{s}=\bar{S}, and an impedance Z_{p}=X_{p}, we can write the following relationships:

Z=\frac{V_{s}}{I_{s}}=\frac{V_{s}}{I_{r}}=Z_{p}\frac{V_{s}}{V_{r}}=Z_{p}\frac{V_{s}}{V_{p}}=Z_{p}\frac{1-\beta A}{\alpha}.

Considering that \alpha=\frac{V_{r}}{V_{s}} |_{V_{p}=0}=\frac{Z_{p}}{Z_{p}+Z_{b}}, where Z_{b} is the impedance seen between the nodes k=h and t if remove Z_{p} and short-circuit the voltage sources, we obtain the impedance Z between the nodes j and k in the form:

Z=\left(Z_{p}+Z_{b}\right)\left(1-\beta A \right)

Admittance

Cut for the evaluation of the impedance between the nodes k=h=t and j=e=q.

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 S is now in parallel to X_{p}. If we consider an input current source I_{s}=S (as a result we have a voltage V_{s}=\bar{S}) and an admittance Y_{p}=X_{p}, the admittance Y between the nodes j and k can be computed as follows:

Y=\frac{I_{s}}{V_{s}}=\frac{I_{s}}{V_{r}}=Y_{p}\frac{I_{s}}{I_{r}}=Y_{p}\frac{I_{s}}{I_{p}}=Y_{p}\frac{1-\beta A}{\alpha}.

Considering that \alpha=\frac{I_{r}}{I_{s}} |_{I_{p}=0}=\frac{Y_{p}}{Y_{p}+Y_{b}}, where Y_{b} is the admittance seen between the nodes k=h and t if we remove Y_{p} and open the current sources, we obtain the admittance Y in the form:

Y=\left(Y_{p}+Y_{b}\right)\left(1-\beta A \right)

Comments

Implementation of the three-terminal circuit by means of an independent source W_{p} and a dependent source \bar{W_{p}}.

The implementation of the TTC with an independent source W_{p} and an immittance X_{p} 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 X_{p} from the equivalence equation. Such difficulty can be avoided using a dependent source \bar{W_{p}} in place of X_{p} and using the Blackman formula[3] for the evaluation of X. 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.

Notes

  1. Bruno Pellegrini has been the first Electronic Engineering graduate at the University of Pisa where is currently Professor Emeritus. He is also author of the Electrokinematics theorem, that connects the velocity and the charge of carriers moving inside an arbitrary volume to the currents, voltages and power on its surface through an arbitrary irrotational vector.
  2. Notice that for the evaluation of Xp we need network functions that depend, in their turn, on Xp. To proceed with calculations, it is therefore appropriate to perform a cut for which ρ=0, so that Xp=Xi.
  3. R. B. Blackman, Effect of Feedback on Impedance, Bell System Tech. J. 22, 269 (1943).

References

See also

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