Introduction
An RLC Series Circuit is a fundamental AC circuit that includes a Resistor (R), Inductor (L), and Capacitor (C) connected in series. It plays a vital role in understanding resonance, impedance, and power factor in electrical engineering. When powered by an AC source, the circuit exhibits unique behavior depending on the frequency, such as resonant frequency where inductive and capacitive reactance cancel each other out.
This circuit is commonly used in filters, oscillators, and signal tuning applications. Learning the voltage-current relationships, phase angles, and impedance in an RLC series circuit helps students grasp key concepts in both electronics and power systems.
How to find impedance of RLC circuit ?
- In RLC Series Circuit, there is voltage drop across R is $V_{R}=IR$, in voltage and current there is no phase difference.
- Voltage drop across L is $V_{L}=IX_{L}$, in voltage and current there is $90^{\circ}$ phase difference, Voltage lead by $90^{\circ}$
- Voltage drop across C is $V_{C}=IX_{C}$, in
voltage and current there is $90^{\circ}$ phase difference, Voltage
lag by $90^{\circ}
Current is taken as the reference phasor. The voltage across the inductor $(V_{L})$ leads the current by 90°, while the voltage across the capacitor $(V_{C})$ lags the current by 90°. Therefore, $(V_{L})$ and $(V_{C})$ are in opposition to each other.
Voltage and Current Relations in RLC Series Circuit
In a series RLC circuit, the same current $I$ flows through all components. The voltage across each component is given by:
Voltage across Resistor:
$V_R = I R$
Voltage across Inductor:
$V_L = I X_L = I \omega L$
Voltage across Capacitor:
$V_C = I X_C = \dfrac{I}{\omega C}$
The total voltage $V$ applied to the circuit is the phasor sum of the voltages across each component:
$V = V_R + j(V_L - V_C)$
Or, in magnitude form:
$|V| = \sqrt{V_R^2 + (V_L - V_C)^2}$
Substituting the expressions:
$V = \sqrt{(IR)^2 + \left( I \omega L - \dfrac{I}{\omega C} \right)^2}$
Factoring out $I$:
$V = I \sqrt{R^2 + \left( \omega L - \dfrac{1}{\omega C} \right)^2}$
Therefore, the **current** in the circuit is:
$I = \dfrac{V}{\sqrt{R^2 + \left( \omega L - \dfrac{1}{\omega C} \right)^2}} = \dfrac{V}{Z}$
Impedance in RLC Series Circuit
$Z = \sqrt{R^2 + \left( \omega L - \dfrac{1}{\omega C} \right)^2}$
Power Factor and Energy Analysis in RLC Series Circuit
In an RLC series circuit powered by an AC source, the total current $I$ and total voltage $V$ are generally not in phase due to the presence of inductive and capacitive reactance. The phase angle $\phi$ determines their phase difference.
Power Factor
The power factor (PF) is defined as the cosine of the phase angle between the voltage and current:
$\text{Power Factor} = \cos\phi$
The phase angle $\phi$ is given by:
$\phi = \tan^{-1} \left( \dfrac{X_L - X_C}{R} \right)$
Therefore, the power factor becomes:
$\cos\phi = \dfrac{R}{Z} = \dfrac{R}{\sqrt{R^2 + (X_L - X_C)^2}}$
Types of Power Factor
- Lagging Power Factor: If $X_L > X_C$, the circuit is inductive and the current lags the voltage.
- Leading Power Factor If $X_C > X_L$, the circuit is capacitive and the current leads the voltage.
- Unity Power Factor: If $X_L = X_C$, the circuit is purely resistive at resonance and $\phi = 0^\circ$.
Energy and Power Analysis
Apparent Power (S)
$S = V \cdot I \quad \text{(in volt-amperes, VA)}$
Real or Active Power (P)
$P = V I \cos\phi \quad \text{(in watts, W)}$
Reactive Power (Q)
$Q = V I \sin\phi \quad \text{(in vars)}$
Energy Stored
Energy stored in Inductor: $E_L = \dfrac{1}{2} L I^2$
Energy stored in Capacitor: $E_C = \dfrac{1}{2} C V_C^2$