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

A practical parallel resonant circuit has a resistor in serieswith an inductor, and that combination is in parallel with acapacitor. For such a circuit, it is well known that there aretwo possible definitions for the resonant frequency: (i) theresonant frequency , p f which is the frequency at which thephase of the total impedance is zero, and (ii) the resonantfrequency m f , which is the frequency that achieves maximummagnitude of the total impedance. To find the lattertraditionally requires calculus. However, in this paper, theauthors show how m fcan be found exactly without usingcalculus. By modifying a formula that is given as an approximationto m f in a popular technology textbook, an improvementin the accuracy of the approximation wasachieved. Furthermore, a novel expression for the exactmaximum impedance, as a function of Q = L /C / R.wasderived. This has been approximated by previous authorsas 2 RQ forQ ³ 10. However, in this report, the authors showthat this approximation has a percentage error less than _2%forQ ³ 5, and less than −10% forQ ³ 2.Furthermore, it canbe shown that the maximum impedance is also accuratelyapproximated by ( ) 2 2 R Q 1+Q , which has an excellentpercentage error performance, even forQ = 1, with a percentageerror of only −4% for this value, and less than −0.6%forQ ³ 1.5. Finally, the authors used PSpice simulations toverify their results.