Question 21.6: A Circuit in Resonance Goal Understand resonance frequency a...
A Circuit in Resonance Goal Understand resonance frequency and its relation to inductance, capacitance, and the rms current. Problem Consider a series RLC circuit for which R = 1.50 × 10² Ω, L = 20.0 mH, \Delta V_{\text {rms }}=20.0 V , \text { and } f=796 s ^{-1} . (a) Determine the value of the capacitance for which the rms current is a maximum. (b) Find the maximum rms current in the circuit.
Strategy The current is a maximum at the resonance frequency f_{0} , which should be set equal to the driving frequency, 796 s ^{-1} . The resulting equation can be solved for C. For part (b), substitute into Equation 21.18 to get the maximum rms current.
I_{ rms }=\frac{\Delta V_{ rms }}{Z}=\frac{\Delta V_{ rms }}{\sqrt{R^{2}+\left(X_{L}-X_{C}\right)^{2}}}
(a) Find the capacitance giving the maximum current in the circuit (the resonance condition).
Solve the resonance frequency for the capacitance:
\begin{aligned}f_{0} &=\frac{1}{2 \pi \sqrt{L C}} \rightarrow \quad \sqrt{L C}=\frac{1}{2 \pi f_{0}} \quad \rightarrow \quad L C=\frac{1}{4 \pi^{2} f_{0}^{2}} \\C &=\frac{1}{4 \pi^{2} f_{0}^{2} L}\end{aligned}
Insert the given values, substituting the source frequency for the resonance frequency, f_{0}:
C=\frac{1}{4 \pi^{2}(796 Hz )^{2}\left(20.0 \times 10^{-3} H \right)}=2.00 \times 10^{-6} F
(b) Find the maximum rms current in the circuit.
The capacitive and inductive reactances are equal, so Z=R=1.50 \times 10^{2} \Omega . Substitute into Equation 21.18 to find the rms current:
I_{ rms }-\frac{\Delta V_{ rms }}{Z}-\frac{20.0 V }{1.50 \times 10^{2} \Omega}-0.133 A
Remark Because the impedance Z is in the denominator of Equation 21.18, the maximum current will always occur when X_{L}=X_{C} , since that yields the minimum value of Z.