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 W_r and an immittance X_p.

Let e, h, u, w, q=q', and t=t' be six arbitrary nodes of the network N and ${\displaystyle S}$ be an independent voltage or current source connected between e and h, while ${\displaystyle U}$ is the output quantity, either a voltage or current, relative to the branch with immittance ${\displaystyle X_{u}}$, 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 (${\displaystyle W_{r}}$ and ${\displaystyle W_{p}}$ 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 ${\displaystyle S}$, the two constraints ${\displaystyle W_{r}=W_{p}}$ and ${\displaystyle {\bar {W_{r}}}={\bar {W_{p}}}}$, 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 ${\displaystyle W_{p}}$ between q' and t' , and an immittance ${\displaystyle X_{p}}$ between q and t.

Network functions

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

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

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

from which, exploiting the Superposition theorem, we obtain:

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

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

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

Furthermore,

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

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

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

Transfer function

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

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

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

• ${\displaystyle \alpha }$ takes into account the nonideality of the comparison network at the input
• ${\displaystyle \gamma }$ takes into account the non unidirectionality of the feedback chain
• ${\displaystyle \rho }$ takes into account the non unidirectionality of the amplification chain.

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

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

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

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

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

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

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

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

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

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

Comments

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

The implementation of the TTC with an independent source ${\displaystyle W_{p}}$ and an immittance ${\displaystyle X_{p}}$ 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 ${\displaystyle X_{p}}$ from the equivalence equation. Such difficulty can be avoided using a dependent source ${\displaystyle {\bar {W_{p}}}}$ in place of ${\displaystyle X_{p}}$ and using the Blackman formula ^[3] for the evaluation of ${\displaystyle X}$. 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 X_p we need network functions that depend, in their turn, on X_p. To proceed with calculations, it is therefore appropriate to perform a cut for which ρ=0, so that X_p=X_i.
3. R. B. Blackman, Effect of Feedback on Impedance, Bell System Tech. J. 22, 269 (1943).

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