Exploring Quadripole Interconnected with External Circuit

In summary, the conversation discusses the use of a two-port 'external' representation for a quadripole (four-terminal electrical network) when connected to an external circuit. The speaker suggests using a tree spanning the internal structure of the quadripole and extending it to the external network to write equations for equilibrium of currents and voltage. They also mention the need for constrain equations for port currents and auxiliary unknowns for port voltages. They explain that a solution for the first set of equations is also a solution for the network with the two-port representation, but the reverse may not be true. The speaker suggests explicitly checking for KVL equations to ensure the solution is unique when using the two-port representation. The conversation ends with a question about whether this
  • #1
cianfa72
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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.
 
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  • #2
Any idea? I'm aware of this is a theoretical question...
 
  • #3
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 :wink:) 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.

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Related to Exploring Quadripole Interconnected with External Circuit

1. What is a quadripole and how is it connected to an external circuit?

A quadripole is a circuit element that has four terminals - two input terminals and two output terminals. It is typically represented by a box with four terminals, and is commonly used to model complex circuits. The external circuit refers to the circuit elements connected to the quadripole's input and output terminals.

2. Why is it important to explore quadripole interconnected with external circuit?

Exploring quadripole interconnected with external circuit allows us to understand the behavior and characteristics of complex circuits. By analyzing the input and output signals, we can determine the transfer function and other parameters of the quadripole, which can help us design and troubleshoot circuits.

3. How do you determine the transfer function of a quadripole interconnected with external circuit?

The transfer function of a quadripole can be determined by applying an input signal to the input terminals and measuring the output signal at the output terminals. By comparing the input and output signals, we can calculate the transfer function, which represents the relationship between the input and output signals.

4. What is the difference between a series and parallel quadripole interconnected with external circuit?

In a series quadripole, the external circuit elements are connected in series with the quadripole's input and output terminals. This means that the input and output signals share the same current path. In a parallel quadripole, the external circuit elements are connected in parallel with the quadripole's input and output terminals, meaning that the input and output signals share the same voltage source.

5. How does the external circuit affect the behavior of a quadripole?

The external circuit can affect the behavior of a quadripole in various ways. It can change the input and output impedances, alter the frequency response, and introduce nonlinearity. The external circuit can also interact with the quadripole, causing feedback and affecting stability. Therefore, it is important to consider the external circuit when analyzing and designing quadripole circuits.

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