Finding the Exact Maximum Impedance Resonant Frequency of a Practical Parallel Resonant Circuit without Calculus

Abstract

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 for 10. Q ≥ However, in this report, the authors show that this approximation has a percentage error less than 2% for 5, Q ≥ and less than −10% for 2. Q ≥ Furthermore, it can be shown that the maximum impedance is also accurately approximated by ( ) 2 2 1 R Q Q + , which has an excellent percentage error performance, even for 1, Q = with a percentage error of only −4% for this value, and less than −0.6% for 1.5. Q ≥ Finally, the authors used PSpice simulations to verify their results.

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Cite this paper

@inproceedings{Cartwright2016FindingTE, title={Finding the Exact Maximum Impedance Resonant Frequency of a Practical Parallel Resonant Circuit without Calculus}, author={Kenneth V. Cartwright and Elton Joseph and Edit J. Kaminsky}, year={2016} }