RLC Parallel Circuit

Finding the impedance of a parallel RLC circuit is considerably more difficult than finding the series RLC impedance. This is because each branch has a phase angle and they cannot be combined in a simple way. The impedance of the parallel branches combine in the same way that parallel resistors combine:

But although the branch impedance magnitudes can be calculated from

 and
RL Circuit
RC Circuit
they cannot be directly combined as suggested by the expression above because they are different in phase - like vectors in different directions cannot be added directly. This dilemma is most easily solved by the complex impedance method.

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RLC Parallel: Complex Impedance Method

When the complex impedances of the branches of the parallel RLC circuit are combined, the equivalent impedance is of the form

When this expression is rationalized and put in the standard form

Calculation

then the impedance in ohms and the phase can be determined. By setting the = 0, the resonant frequency can be calculated. The expressions for these calculations are quite lengthy.

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RLC Parallel Expressions

The complex impedances of the parallel RLC circuit takes the form

when rationalized, and the components have the form

Calculation
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