# Circuit Theory/Phasor Theorems

## Circuit Theorems

Phasors would be absolutely useless if they didn't make the analysis of a circuit easier. Luckily for us, all our old circuit analysis tools work with values in the phasor domain. Here is a quick list of tools that we have already discussed, that continue to work with phasors:

• Ohm's Law
• Kirchoff's Laws
• Superposition
• Thevenin and Norton Sources
• Maximum Power Transfer

This page will describe how to use some of the tools we discussed for DC circuits in an AC circuit using phasors.

## Ohm's Law

Ohm's law, as we have already seen, becomes the following equation when in the phasor domain:

${\displaystyle \mathbb {V} =\mathbb {Z} \mathbb {I} }$

Separating this out, we get:

${\displaystyle M_{V}\angle \phi _{V}=(M_{Z}\times M_{I})\angle (\phi _{Z}+\phi _{I})}$

Where we can clearly see the magnitude and phase relationships between the current, the impedance, and the voltage phasors.

## Kirchoff's Laws

Kirchoff's laws still hold true in phasors, with no alterations.

### Kirchoff's Current Law

Kirchoff's current law states that the amount of current entering a particular node must equal the amount of current leaving that node. Notice that KCL never specifies what form the current must be in: any type of current works, and KCL always holds true.

[KCL With Phasors]

${\displaystyle \sum _{n}\mathbb {I} _{n}=0}$

### Kirchoff's Voltage Law

KVL states: The sum of the voltages around a closed loop must always equal zero. Again, the form of the voltage forcing function is never considered: KVL holds true for any input function.

[KVL With Phasors]

${\displaystyle \sum _{n}\mathbb {V} _{n}=0}$

## Superposition

Superposition may be applied to a circuit if all the sources have the same frequency. However, superposition must be used as the only possible method to solve a circuit with sources that have different frequencies. The important part to remember is that impedance values in a circuit are based on the frequency. Different reactive elements react to different frequencies differently. Therefore, the circuit must be solved once for every source frequency. This can be a long process, but it is the only good method to solve these circuits.

## Thevenin and Norton Circuits

Thevenin Circuits and Norton Circuits can be manipulated in a similar manner to their DC counterparts: Using the phasor-domain implementation of Ohm's Law.

${\displaystyle \mathbb {V} =\mathbb {Z} \mathbb {I} }$

It is important to remember that the ${\displaystyle \mathbb {Z} }$ does not change in the calculations, although the phase and the magnitude of both the current and the voltage sources might change as a result of the calculation.

## Maximum Power Transfer

The maximum power transfer theorem in phasors is slightly different then the theorem for DC circuits. To obtain maximum power transfer from a thevenin source to a load, the internal thevenin impedance (${\displaystyle \mathbb {Z} _{t}}$) must be the complex conjugate of the load impedance (${\displaystyle \mathbb {Z} _{l}}$):

[Maximum Power Transfer, with Phasors]

${\displaystyle \mathbb {Z} _{l}=R_{t}-jX_{t}}$