17.12 Navigating AC Through RLC-Series Circuit

17.12 Navigating AC Through RLC-Series Circuit

Consider a RLC-series circuit having an alternating voltage source with variable frequency as shown in Fig. 17.11(a). The impedance diagram of RLC-series circuit is shown in Fig. 17.11(b) in which inductive reactance $X_{L} = \omega L$ and capacitive reactance $X_{C} = 1/\omega C$ are directed opposite to each other. The impedance of the RLC-series circuit is:

or

(Fig. 17.11(a): RLC-series circuit with its AC source phasor diagram)

(Fig. 17.11 (b): The impedance vector diagram of RLC-series circuit.)

When frequency is high, then $X_{L} = \omega L$ is much greater than $X_{C} = 1/\omega C$, and hence the inductance $X_{L}$ dominates at high frequency, so RLC-series circuit behaves as RL-series circuit. When frequency is low, then $X_{C} = 1 / \omega C$ is much greater than $X_{L} = \omega L$, and hence the capacitance $X_{C}$ dominates at low frequency, hence RLC series circuit behaves as RC series circuit as shown in Fig.17.11(b). In between these frequencies, there will be a frequency $\omega_{r}$ at which $X_{L} = X_{C}$. At this condition, RLC-series circuit behaves as pure resistive $R$-circuit as shown in Fig.17.11(c). This condition is called resonance. Thus, at resonance, the inductive reactance $X_{L}$ being equal and opposite to capacitive reactance $X_{C}$, cancel each other. The value of resonance frequency $f_{r}$ can be obtained by applying the condition: $X_{L} = X_{C}$ $\omega_{r} L = \dfrac{1}{\omega_{r} C}$ $\omega_{r}^{2} = \dfrac{1}{LC}$ $\omega_{r} = \dfrac{1}{\sqrt{LC}}$ As $\omega_{r} = 2\pi f_{r}$ , so $2\pi f_{r} = \dfrac{1}{\sqrt{LC}}$

(Fig. 17.11(c): Vector diagram of RLC-series circuit showing that it behaves as pure resistive $R$-circuit.at $X_{L} = X_{C}$)

(Fig. 17.11 (d): Phasor diagram of RLC-series circuit resonance frequency at $X_{L} = X_{C}$)

Brain teaser

lightbulb

Guess the name of the AC circuit used in the ceiling fan?