Circuit Theory/Phasor Theorems

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Circuit Theorems[edit]

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[edit]

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[edit]

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

Kirchoff's Current Law[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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