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 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 = αS + βAW_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 $\frac{1}{X_{p}}=\frac{1}{X_{i}}+\frac{\rho}{\alpha}(1-\beta A)$

Transfer function

If we consider the expressions for the network functions γ and A, the first constraint for the equivalence of the networks, and we also consider that, as a result of the superposition principle, U = γS + 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 α, γ and ρ take into account the nonidealities of such amplifier. In particular:

• α takes into account the nonideality of the comparison network at the input
• γ takes into account the non unidirectionality of the feedback chain
• ρ takes into account the non unidirectionality of the amplification chain.

If the amplifier can be considered ideal, i.e. if α = 1, ρ = 0 and γ = 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

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 = S̄, 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 = (Z_p+Z_b)(1−βA)

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 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 = S̄) 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 = (Y_p+Y_b)(1−βA)

Comments

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 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.

References

• B. Pellegrini, Considerations on the Feedback Theory, Alta Frequenza 41, 825 (1972).
• B. Pellegrini, Improved Feedback Theory, IEEE Transactions on Circuits and Systems 56, 1949 (2009).

See also

• Feedback
• Control theory