3-terminal device external modeling

[cianfa72]

TL;DR Summary

About the number of independent equations needed to externally model a 3-terminal device

Very basic question. Consider a 3-terminal device with terminals say A,B,C. Kirchhoff Current Law (KCL) and Kirchhoff Voltage Law (KVL) establish two relationships between the 3 currents entering the terminals and the 3 terminal's voltage pairs respectively.

So we have 2 equations in 6 unknowns. To proceed further we need two more (independent) equations in order to solve the circuit the 3-terminal device is connected to (basically one treats such a device as an unbalanced two-port element).

My question: from a theoretical point to view, is it possible, aside from KCL and KLV, that the number of equations describing the 3-terminal device could be less or more than 2 ?

 

[Baluncore]

I think you need to identify the meaning of the terms terminal and port.

A three terminal device could be a two port black box, with the third terminal representing the reference ground. As an example, a BJT can be modelled as a three terminal device, but in three different ways, as common-emitter, common-collector, or common-base. That reduces the number of equations needed to fully describe the device.

Three terminal, two port networks, can be modelled as a scattering matrix with four parameters, S11, S12, S21, and S22.
https://en.wikipedia.org/wiki/Scattering_parameters#Two-port_S-parameters

 

[cianfa72]

A three terminal device could be a two port black box, with the third terminal representing the reference ground. As an example, a BJT can be modeled as a three terminal device, but in three different ways, as common-emitter, common-collector, or common-base. That reduces the number of equations needed to fully describe the device.

Three terminal, two port networks, can be modeled as a scattering matrix with four parameters, S11, S12, S21, and S22.
https://en.wikipedia.org/wiki/Scattering_parameters#Two-port_S-parameters

Yes, the three terminal device can be modeled as an unbalanced two port black box with the terminal picked as reference ground representing the terminal shared between the two ports.

But the question is: is such a three terminal device as black box always described externally by two equations in 4 unknowns (two currents + two port's voltages) ?

 

[Baluncore]

But the question is: is such a three terminal device as black box always described externally by two equations in 4 unknowns (two currents + two port's voltages) ?

Nothing is forever, always.

If the impedance of the ports are defined, then only one voltage or current is needed. The other ports can be shorted, terminated, or left open.

 

[cianfa72]

If the impedance of the ports are defined, then only one voltage or current is needed. The other ports can be shorted, terminated, or left open.

Right, but it isn't my point.

Consider a three terminal electronic device, whatever. What is the number of equations that the 3 current through terminals plus the 3 voltages across terminals must obey ?

We have 1 KCL and 1 KVL plus a number of independent equations where 6 - 2 = 4 unknown enter.

The question is: from a theoretical standpoint is always two the number of such equations ?

 

[DaveE]

I think the only thing you can say for the most general case is that the terminal currents all sum to zero (KCL) and that there are two independent voltage potentials between the three nodes (KVL). Other constructs, like two port networks, have extra requirements.

PS: $i_1+i_2+i_3=0$ and $v_{12}+v_{23}+v_{31}=0$ are the two equations.

 

[DaveE]

My question: from a theoretical point to view, is it possible, aside from KCL and KLV, that the number of equations describing the 3-terminal device could be less or more than 2 ?

You can't really describe a network without seeing what's inside of it.

For a passive LTI network you should be able, with source transformations, to reduce it to the Thevenin (or Norton) equivalent. This could look like a voltage source with a series impedance between each node, for example.

 

[cianfa72]

You can't really describe a network without seeing what's inside of it.

Exactly.

Suppose now to have a new type of electronic device with three terminals. Are we sure we can always describe it externally by using two equations in 4 unknowns (currents and voltages of two unbalanced ports sharing a common terminal) ?

 

[DaveE]

Suppose now to have a new type of electronic device with three terminals. Are we sure we can always describe it externally by using two equations in 4 unknowns (currents and voltages of two unbalanced ports sharing a common terminal) ?

I fail to see the value of hypothetical questions like this. Your "new type of electronic device" is completely undefined. Suppose it contains Radium and detector? Write that equation...

 

[cianfa72]

I fail to see the value of hypothetical questions like this. Your "new type of electronic device" is completely undefined. Suppose it contains Radium and detector? Write that equation...

Even without knowing how it works internally, we can always trace its characteristic curves.

To do that define a common terminal (say C) and attach two voltage sources between A-C and B-C. For each pair of applied voltages I think we'll get two definite currents flowing into A and C terminals.

From above it seems that in any circumstance we'll always be able to describe it externally by two equations in 4 unknowns.

 

[Baluncore]

From above it seems that in any circumstance we'll always be able to describe it externally by two equations in 4 unknowns.

Only if it is linear, and you have the four scattering parameters at a number of different frequencies.

 

[cianfa72]

Only if it is linear, and you have the four scattering parameters at a number of different frequencies.

Why ? Take for instance the case of BJT. It isn't linear, yet we can draw its characteristic curves.

Note that by the term "equation" above I mean basically mathematical relationships between electric variables (including non linear ones).

 

[Baluncore]

Why ? Take for instance the case of BJT. It isn't linear, yet we can draw its characteristic curves.

The characteristic curve does not show resonances, nor charge or information storage.

The S21 parameter is the forward transfer function, a spectrum.

You seem determined to deny complexity.

 

[cianfa72]

The characteristic curve does not show resonances, nor charge or information storage.

Ah yes, you're right.

Forget for a moment device's internal energy storage (or assume it operates in DC). Then characteristic curves contain all the the information needed to analyze/solve the circuit the device is attached to.

 

[Baluncore]

Forget for a moment device's internal energy storage (or assume it operates in DC). Then characteristic curves contain all the the information needed to analyze/solve the circuit the device is attached to.

For a circuit to be without stored energy, it must be a matrix of resistors.
For such a three terminal device, you would be unable to identify if it contained more than three resistors, or if the minimum three resistors were wired as a star or a delta.