Exploring Quadripole Interconnected with External Circuit

[cianfa72]

TL;DR Summary

Conditions to be fulfilled to employ (one of ) the 2-port network representation of a quadripole (four-terminal electrical network) in the analysis of a complete 'external' electrical network.

Hello,

I'm struggling with the conditions under which makes sense employ a two-port 'external' representation of a quadripole (four-terminal electrical network) when interconnected to an external circuit (to take it simple assume a linear + permanent electrical network).

Starting from circuit theory I elaborated the following:

Take a quadripole (four-terminal network) interconnected to an 'external' circuit. Do not place any constrains about the current entering in each of the four terminal (no 'port' constrains for the currents). From a network analysis point of view we can proceed as follows:

1. choose a tree spanning just the quadripole internal structure (directed graph) up to its four terminals
2. extend this tree to the overall 'external' network starting from those 4 terminals
3. write the equations for the equilibrium of currents at each node belonging to the complete network (actually N-1 nodes suffices)
4. write the KVLs for the voltage equilibrium at the fundamental loops (f-loop) w.r.t the chosen tree
5. write the BCEs (Branch Constitutive Equations) for each element branch

We can now proceed as follows:

• add 2 constrain equations for the 'port' current condition at each of the 'coupled' terminal pair (port)
• add 2 auxiliary unknowns for the port voltages + the 2 related equations defining them w.r.t the branches of the chosen tree spanning the quadripole internal structure

The set of equations involving only the unknowns for branches inside the quadripole, is formally the same as the set of equations for the same quadripole closed on 2 external indeterminate bipoles (one for each port). A solution of the first system of equations is actually also a solution for the network you get replacing the quadripole with (one of) its 2-port network 'representation' (note that the set of equations for the last one is actually obtained as linear combinations of the equations belonging to the first one).

The other way around, a solution of the second one (the complete network you get replacing the quadripole with its 2-port representation) might not be a solution of the first one (the fundamental loops involving the not 'coupled' quadripole's terminals are actually not included in the equations set)

Thus, we have to explicitly check for those KVL when taking in account any solution of the last network to be sure it is actually a solution of the network we started with.

What do you think about, does it make sense ?

ps. same question shows up in other (italian) forum.

 

[cianfa72]

Any idea? I'm aware of this is a theoretical question...

 

[cianfa72]

Just to take an example consider the following circuit having unique solution. If we choose to employ the quadripole (four-terminal network) two-port V(I) representation we get a system of equations having this time infinte-1 solutions (we can pick for instance I₂ as parameter).

As shown below (sorry it is in Italian ) adding the disregarded KVL equation the solution became unique (it is the solution of the first system for I₂ parameter equals to $\frac {3} {4}$).

This actually seems to be an example of what I said above: we need to explicitly check the solutions we get solving the network when two-port network model is employed.