RLC Series Circuit: At Large and Small Frequencies; Phasor Diagram

In previous Atoms we learned how an RLC series circuit, as shown in , responds to an AC voltage source. By combining Ohm's law (I_rms=V_rms/Z; I_rms and V_rms are rms current and voltage) and the expression for impedance Z, from:

Series RLC Circuit

A series RLC circuit: a resistor, inductor and capacitor (from left).

$Z = \sqrt{R^2 + (X_L - X_C)^2}$$(X_L = 2\pi \nu L, X_C = \frac{1} {2\pi \nu C})$,

we arrived at: $I_{rms} = \frac{V_{rms}}{\sqrt{R^2 + (X_L - X_C)^2}}$.

From the equation, we studied resonance conditions for the circuit. We also learned the phase relationships among the voltages across resistor, capacitor and inductor: when a sinusoidal voltage is applied, the current lags the voltage by a 90º phase in a circuit with an inductor, while the current leads the voltage by 90^∘ in a circuit with a capacitor. Now, we will examine the system's response at limits of large and small frequencies.

At Large Frequencies

At large enough frequencies $(\nu \gg \frac{1}{\sqrt{2\pi LC}})$, X_L is much greater than X_C. If the frequency is high enough that X_L is much larger than R as well, the impedance Z is dominated by the inductive term. When $Z \approx X_L$, the circuit is almost equivalent to an AC circuit with just an inductor. Therefore, the rms current will be V_rms/X_L, and the current lags the voltage by almost 90^∘. This response makes sense because, at high frequencies, Lenz's law suggests that the impedance due to the inductor will be large.

At Small Frequencies

The impedance Z at small frequencies $(\nu \ll \frac{1}{\sqrt{2\pi LC}})$ is dominated by the capacitive term, assuming that the frequency is high enough so that X_C is much larger than R. When $Z \approx X_C$, the circuit is almost equivalent to an AC circuit with just a capacitor. Therefore, the rms current will be given as V_rms/X_C, and the current leads the voltage by almost 90^∘.