An RLC series circuit has a 40.0 Ω resistor, a 3.00 mH inductor, and a 5.00 μF capacitor. (a) Find the circuit’s impedance at 60.0 Hz and 10.0 kHz, noting that these frequencies and the values for L and C are the same as in Example 23.10 and Example 23.11. (b) If the voltage source has Vrms = 120

Question

An RLC series circuit has a 40.0 Ω resistor, a 3.00 mH inductor, and a 5.00 μF capacitor. (a) Find the circuit’s impedance at 60.0 Hz and 10.0 kHz, noting that these frequencies and the values for L and C are the same as in Example 23.10 and Example 23.11. (b) If the voltage source has [latex]V_{ rms }=120 V[/latex], what is [latex]I_{ rms }[/latex] at each frequency?

Strategy
For each frequency, we use [latex]Z=\sqrt{R^{2}+\left(X_{L}-X_{C}\right)^{2}}[/latex] to find the impedance and then Ohm’s law to find current. We can take advantage of the results of the previous two examples rather than calculate the reactances again.

Accepted Answer

Solution for (a)
At 60.0 Hz, the values of the reactances were found in Example 23.10 to be [latex]X_{L}=1.13 \Omega[/latex] and in Example 23.11 to be [latex]X_{C}=531 \Omega[/latex].

Entering these and the given 40.0 Ω for resistance into [latex]Z=\sqrt{R^{2}+\left(X_{L}-X_{C} ight)^{2}}[/latex] yields

[latex]Z=\sqrt{R^{2}+\left(X_{L}-X_{C} ight)^{2}}[/latex] (23.67)

[latex]=\sqrt{(40.0 \Omega)^{2}+(1.13 \Omega-531 \Omega)^{2}}[/latex]

= 531 Ω at 60.0 Hz.

Similarly, at 10.0 kHz, [latex]X_{L}=188 \Omega ext { and } X_{C}=3.18 \Omega[/latex], so that

[latex]Z=\sqrt{(40.0 \Omega)^{2}+(188 \Omega-3.18 \Omega)^{2}}[/latex] (23.68)

= 190 Ω at 10.0 kHz.

Discussion for (a)
In both cases, the result is nearly the same as the largest value, and the impedance is definitely not the sum of the individual values. It is clear that [latex]X_{L}[/latex] dominates at high frequency and [latex]X_{C}[/latex] dominates at low frequency.

Solution for (b)
The current [latex]I_{ rms }[/latex] can be found using the AC version of Ohm’s law in Equation [latex]I_{ rms }=V_{ rms } / Z[/latex] :

[latex]I_{ rms }=\frac{V_{ rms }}{Z}=\frac{120 V }{531 \Omega}=0.226 A ext { at } 60.0 Hz[/latex]

Finally, at 10.0 kHz, we find

[latex]I_{ rms }=\frac{V_{ rms }}{Z}=\frac{120 V }{190 \Omega}=0.633 A ext { at } 10.0 kHz[/latex]

Discussion for (b)
The current at 60.0 Hz is the same (to three digits) as found for the capacitor alone in Example 23.11. The capacitor dominates at low frequency. The current at 10.0 kHz is only slightly different from that found for the inductor alone in Example 23.10. The inductor dominates at high frequency.

Question 23.12: An RLC series circuit has a 40.0 Ω resistor, a 3.00 mH induc...

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