Document Type


Publication Date

Fall 2010


A practical parallel resonant circuit has a resistor in series

with an inductor, and that combination is in parallel with a

capacitor. For such a circuit, it is well known that there are

two possible definitions for the resonant frequency: (i) the

resonant frequency , p f which is the frequency at which the

phase of the total impedance is zero, and (ii) the resonant

frequency m f , which is the frequency that achieves maximum

magnitude of the total impedance. To find the latter

traditionally requires calculus. However, in this paper, the

authors show how m f

can be found exactly without using

calculus. By modifying a formula that is given as an approximation

to m f in a popular technology textbook, an improvement

in the accuracy of the approximation was

achieved. Furthermore, a novel expression for the exact

maximum impedance, as a function of Q = L /C / R.was

derived. This has been approximated by previous authors

as 2 RQ forQ ³ 10. However, in this report, the authors show

that this approximation has a percentage error less than _2%

forQ ³ 5, and less than −10% forQ ³ 2.Furthermore, it can

be shown that the maximum impedance is also accurately

approximated by ( ) 2 2 R Q 1+Q , which has an excellent

percentage error performance, even forQ = 1, with a percentage

error of only −4% for this value, and less than −0.6%

forQ ³ 1.5. Finally, the authors used PSpice simulations to

verify their results.

Journal Name

Technology Interface International