Circuit Analysis Theorem For the Number of Independent Equations

• rtareen
In summary, this book claims that there are six equations given by the current law and three equations given by the voltage law, but only one independent equation.
rtareen
I attached a screenshot of the book (sorry no pdf available for this book). Right above the somewhat central line they give the theorem that if there are m currents and n nodes, then there will be n - 1 independent equations from the current law and m - n - 1 from the voltage law.

I count 4 (branching) nodes and 6 currents. So the number of equations given by the current law is 4 - 1 = 3. Thats fine. But then we should get one more (6- 4 - 1) from the voltage law. But they end up with three more from the voltage law leading to a total of six equations, which they will need because there are six unknown currents.

So what's going on here?

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rtareen said:
I count (...) 5 currents (...) because there are six unknown currents.

So what's going on here?

Good question.

Borek said:
Good question.
Well I fixed that. It still leads to only one more equation according to this theorem. What else is wrong?

What is reference to the book?

I think the issue here is "independent equations". You can make lots of equations from KVL & KCL, but you don't need all of them. For example, in the bridge circuit shown, if I told you the values for I1, I2, and I3, couldn't you solve the whole circuit?

I've just made a post that ties circuits to polyhedrons. Essentially every node is a vertex, every branch is an edge, and every indivisible loop is a face. There is a current continuity EQ for every node/vertex, and a voltage loop EQ for every loop/face. Because the continuity EQs are all equal to zero, the resultant vector is the zero vector, and thus there is 1 nullspace DOF in that set. And since a polyhedron can topographically be stretched so that 1 face could have the image of all the rest (i.e., a Schlegal diagram), it is a combination of loops, and thus there is 1 nullspace DOF in the set of loop/face EQs - and since there must be an EMF element in some branch to have a non-null system, the resultant vector is not the zero vector, and thus there is no other nullspace DOF. Therefore, the net rank of the combined set is ( # of node/vertices + # of loop/faces - 2 ), which of course as per Euler's Law of Polyhedra is equal to the # of branch/edge currents.

What is the Circuit Analysis Theorem for the Number of Independent Equations?

The Circuit Analysis Theorem for the Number of Independent Equations states that in a circuit with N nodes and M independent voltage sources, the number of independent equations needed to solve for all the node voltages is N-M+1.

How is the number of independent equations determined in a circuit?

The number of independent equations is determined by subtracting the number of independent voltage sources from the number of nodes in the circuit, and adding 1. This is based on the Circuit Analysis Theorem for the Number of Independent Equations.

Why is the Circuit Analysis Theorem for the Number of Independent Equations important?

This theorem is important because it helps determine the minimum number of equations needed to solve for all the node voltages in a circuit. This can save time and effort in circuit analysis and design.

Can the Circuit Analysis Theorem be applied to circuits with dependent sources?

No, the Circuit Analysis Theorem for the Number of Independent Equations only applies to circuits with independent sources. In circuits with dependent sources, additional equations may be needed to solve for all the node voltages.

What is the relationship between the number of independent equations and the number of unknown node voltages in a circuit?

The number of independent equations needed to solve for all the node voltages is equal to the number of unknown node voltages in the circuit. This is based on the Kirchhoff's Current Law and Kirchhoff's Voltage Law, which state that the number of unknowns and the number of equations must be equal for a unique solution to be obtained.

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