The Series RLC Circuit — Question 7

Question 7

Source resistance reduces both resonant current and loaded quality factor

A source with rms open-circuit voltage Vs and internal resistance Rs drives an external series RLC circuit with resistor R. All losses are resistive and frequency-independent.

1. Find the resonant current and the quality factor of the complete loop.

2. Find the fraction of total average dissipation delivered to the external resistor and the terminal rms voltage at resonance.

Question 7 - Solution

Source resistance reduces both resonant current and loaded quality factor

(a) Method and calculation

$$R_{\mathrm{total}}=R+R_s,\quad\omega_0=1/\sqrt{LC}.$$$$\boxed{I_0=V_s/(R+R_s),\quad\mathcal Q_{\mathrm{loaded}}=\omega_0L/(R+R_s)}.$$

(b) Interpretation and further calculation

$$P_R=I_0^2R,\quad P_{\mathrm{total}}=I_0^2(R+R_s).$$$$\boxed{P_R/P_{\mathrm{total}}=R/(R+R_s),\quad V_{\mathrm{terminal}}=V_sR/(R+R_s)}.$$

Physical check

Using only the external resistor would overestimate current and underestimate bandwidth.