Resonance in Long LC-Ladder Circuits
We investigated very long but finite "ladder" circuits composed of alternating identical inductors and capacitors connected in series and parallel and derived an exact expression for the equivalent impedance of such circuits of arbitrary size. The remarkable simplicity of the impedance formula allowed us to directly obtain all resonance and anti-resonance frequencies. We tested our analytical results by constructing the corresponding circuits using the standard circuit simulation software (Multisim ©). Specifically, we focused on circuits ranging from as few as one element to as many as a hundred elements. The equivalent impedance of these modeled circuits and the relevant voltage readings were in an excellent agreement with our theoretical calculations. In addition, we resolved the well known paradoxical phenomenon arising from a naïve calculation of the equivalent impedance of an infinite LC-ladder: for driving frequencies below some critical value, the impedance of a purely reactive circuit seemed to acquire a non-zero active part. Using our formula, we demonstrated that there was no paradox and investigated the behavior of the equivalent impedance as the circuit size increased. We did so for various representative values of driving frequency and again found our theoretical predictions to be in agreement with the modeled circuits.
|Work Title||Resonance in Long LC-Ladder Circuits|
|License||Attribution-NonCommercial-NoDerivs 3.0 United States|
|Publication Date||April 20, 2016|
|Deposited||April 20, 2016|
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