Class 12 > Unit # 20:AC Circuits > RLC Series and Parallel Circuits & Resonance

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AC Through RLC Series and Parallel Circuits & Resonance

Video Lecture

Watch the complete video lecture below to understand alternating current flowing through RLC series and parallel circuits, impedance and current triangles, and resonance principles.

Describe the Flow of AC Through RLC Series Circuit and Explain Resonant Frequency

"An RLC series circuit consists of a resistor ($R$), inductor ($L$), and capacitor ($C$) connected in series to an AC voltage source where the same current flows through all components."

In an RLC series circuit, electrical components influence current behavior [00:01:09]. Since all elements are connected in series, the same current flows through them [00:02:08]. Phasor and impedance diagrams help analyze circuit properties [00:02:15], with inductive reactance ($X_L$) and capacitive reactance ($X_C$) defined as:

$$X_L = 2\pi f L \quad \text{and} \quad X_C = \frac{1}{2\pi f C}$$

Depending on the operating frequency:

1. Smaller Frequency ($X_C > X_L$): Capacitive reactance exceeds inductive reactance, causing the circuit to behave akin to an RC circuit [00:16:03].
2. Larger Frequency ($X_L > X_C$): Inductive reactance dominates over capacitive reactance, making the circuit behave more inductively like an RL circuit [00:16:11].

Resonant Frequency

"For a certain frequency, the capacitive and inductive reactances become equal ($X_C = X_L$). This frequency is called the resonant frequency ($f_r$), and the circuit is said to be in resonance state."

To calculate the resonant frequency:

$$X_L = X_C \implies 2\pi f_r L = \frac{1}{2\pi f_r C}$$

$$f_r^2 = \frac{1}{4\pi^2 L C} \implies f_r = \frac{1}{2\pi \sqrt{LC}}$$

At resonant frequency, inductive and capacitive reactances cancel each other out [00:16:53]. Opposition to current flow is solely offered by resistance ($R$), allowing the maximum current to flow through the circuit [00:14:07].

Fig: RLC series circuit configuration and frequency response curve.

Describe Flow of AC Through RLC Series Circuit and Explain Resonance of Parallel RLC Circuit

"A Parallel RLC AC Circuit is one where the resistor, inductor, and capacitor are connected in parallel to each other and the AC source, sharing the same voltage but having different branch currents."

In a parallel RLC circuit, supply voltage $V_{rms}$ is common across all three parallel branches, while supply current $I_s$ consists of three components: $I_R$, $I_L$, and $I_C$. The total current drawn from the supply is the vector sum of these individual branch currents rather than their arithmetic sum.

Phasor Diagram of a Parallel RLC Circuit

Current vectors form a right triangle known as a Current Triangle, where the hypotenuse represents $I_s$, the horizontal axis represents $I_R$, and the vertical axis represents the net reactive current ($|I_L - I_C|$). Applying Pythagoras' theorem gives:

$$I_s = \sqrt{I_R^2 + (I_L - I_C)^2}$$

Resonance of Parallel RLC AC Circuit

"A parallel circuit containing resistance ($R$), inductance ($L$), and capacitance ($C$) produces a parallel resonance (anti-resonance) circuit when the resultant current through the parallel combination is in phase with the supply voltage."

At resonance:

A large circulating current flows between the inductor and capacitor due to stored oscillating energy being constantly transferred between the inductor's magnetic field and the capacitor's electric field.
This energy exchange results in zero reactive current being drawn from the external supply.
Since $I_L = I_C$ (or $X_L = X_C$), the circuit current at this frequency reaches its minimum value of $\frac{V}{R}$.

The parallel resonant frequency is likewise given by:

$$f_r = \frac{1}{2\pi \sqrt{LC}}$$

Fig: Parallel RLC circuit current triangle and frequency response showing minimum current at resonance.

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