Three Simple Circuits — Question 5

Question 5

Equal reactances also equalize peak stored energies under equal voltage

Separate ideal L and C elements are each driven by the same sinusoidal peak voltage Vp at adjustable frequency. Assume the inductor current has zero DC component.

1. Find the frequency at which their peak currents are equal.

2. Compare their maximum stored energies at this frequency and at twice it.

Question 5 - Solution

Equal reactances also equalize peak stored energies under equal voltage

(a) Method and calculation

$$I_{L,p}=V_p/(\omega L),\quad I_{C,p}=\omega CV_p.$$$$\boxed{\omega_*=1/\sqrt{LC}}.$$

(b) Interpretation and further calculation

$$U_{L,\max}=V_p^2/(2\omega^2L),\quad U_{C,\max}=CV_p^2/2.$$$$\boxed{U_{L,\max}/U_{C,\max}=\omega_*^2/\omega^2=1\ \text{at}\ \omega_*;\quad1/4\ \text{at}\ 2\omega_*}.$$

Physical check

Capacitor peak energy is set by voltage amplitude, while inductor peak energy decreases with drive frequency.