The Series RLC Circuit — Question 5

Question 5

Bandwidth measurements determine inductance and capacitance

A series RLC circuit with R=9.00 ohms has half-power angular frequencies 400 and 625 radians per second.

1. Infer L, C, and the resonance angular frequency.

2. Find its quality factor and distinguish arithmetic from geometric center.

Question 5 - Solution

Bandwidth measurements determine inductance and capacitance

(a) Method and calculation

$$\Delta\omega=625-400=225\,s^{-1}\Rightarrow\boxed{L=R/\Delta\omega=.0400\,H}.$$$$\boxed{\omega_0=\sqrt{400\cdot625}=500\,s^{-1},\quad C=1/(L\omega_0^2)=100\,\mu F}.$$

(b) Interpretation and further calculation

$$\mathcal Q=\omega_0/\Delta\omega=500/225=2.22222.$$$$\boxed{\omega_{\mathrm{arithmetic}}=512.5\,s^{-1}\ne\omega_0}.$$

Physical check

A broad resonance makes the midpoint error visible even with exact measurements.