#### Document Type

Article

#### Publication Date

Fall 2010

#### 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*Q *³ 10. However, in this report, the authors show

that this approximation has a percentage error less than _2%

for*Q *³ 5, and less than −10% for*Q *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 for*Q *= 1, with a percentage

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

for*Q *³ 1.5. Finally, the authors used PSpice simulations to

verify their results.

#### Journal Name

Technology Interface International

#### Recommended Citation

Cartwright, K., E. Joseph, and E. Kaminsky, “Finding the Exact Maximum Impedance Resonant Frequency of a Practical Parallel Resonant Circuit without Calculus,” Technology Interface Internat. J., vol.11, no. 1, Fall/Winter 2010, pp. 26-36.