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:

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

Separating this out, we get:

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

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]

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

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]

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

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.

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

It is important to remember that the $\mathbb {Z}$ 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 ($\mathbb {Z} _{t}$) must be the complex conjugate of the load impedance ($\mathbb {Z} _{l}$):

[Maximum Power Transfer, with Phasors]

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