RLC Series Circuit: Impedance, Phasor Diagram, Power Factor and Energy Analysis
An RLC series circuit is a fundamental topic in AC circuit analysis and is frequently asked in engineering examinations. It consists of a Resistor (R), Inductor (L), and Capacitor (C) connected in series to an AC supply. This circuit is important for understanding impedance, phasor diagrams, resonance, power factor, and energy analysis.
What is an RLC Series Circuit?
In an RLC series circuit, the same current flows through all the components because they are connected in series. The circuit behavior depends on the frequency of the AC supply. At a particular frequency known as the resonant frequency, the inductive reactance and capacitive reactance cancel each other.
Impedance of RLC Series Circuit
Impedance (Z) is the total opposition offered by the RLC circuit to alternating current.
- Voltage across resistor: $$V_R = IR$$ (Current and voltage are in phase)
- Voltage across inductor: $$V_L = IX_L = I\omega L$$ (Voltage leads current by $90^\circ$)
- Voltage across capacitor: $$V_C = IX_C = \dfrac{I}{\omega C}$$ (Voltage lags current by $90^\circ$)
Since $V_L$ and $V_C$ are opposite in phase, the net reactive voltage is $(V_L - V_C)$.
$$Z = \sqrt{R^2 + (X_L - X_C)^2}$$
$$Z = \sqrt{R^2 + \left( \omega L - \dfrac{1}{\omega C} \right)^2}$$
Phasor Diagram of RLC Series Circuit
In the phasor diagram of an RLC series circuit, current is taken as the reference phasor. The voltage across the resistor is in phase with current, whereas the voltages across the inductor and capacitor are perpendicular to it.
Power Factor in RLC Series Circuit
The power factor is defined as the cosine of the phase angle between voltage and current.
$$\cos\phi = \dfrac{R}{Z}$$
Phase angle: $$\phi = \tan^{-1}\left( \dfrac{X_L - X_C}{R} \right)$$
Types of Power Factor
- Lagging Power Factor: $X_L > X_C$ (Inductive circuit)
- Leading Power Factor: $X_C > X_L$ (Capacitive circuit)
- Unity Power Factor: $X_L = X_C$ (Resonance condition)
Energy and Power Analysis in RLC Series Circuit
Apparent Power: $$S = VI \quad \text{(VA)}$$
Active Power: $$P = VI\cos\phi \quad \text{(W)}$$
Reactive Power: $$Q = VI\sin\phi \quad \text{(VAR)}$$
Energy stored in Inductor: $$E_L = \dfrac{1}{2}LI^2$$
Energy stored in Capacitor: $$E_C = \dfrac{1}{2}CV_C^2$$
Numerical Example on RLC Series Circuit
An RLC series circuit has the following values:
- Resistance, $R = 20 \, \Omega$
- Inductance, $L = 0.2 \, H$
- Capacitance, $C = 100 \, \mu F$
- Supply voltage = 230 V, 50 Hz
Calculate inductive reactance, capacitive reactance, impedance, and current.
Frequently Asked Questions (FAQ)
Why is impedance minimum at resonance?
At resonance, inductive reactance equals capacitive reactance, so they cancel each other. Only resistance remains.
What is the power factor at resonance?
At resonance, voltage and current are in phase, therefore the power factor is unity.
Why is RLC series circuit important?
RLC series circuits are widely used in filters, tuning circuits, oscillators, and communication systems.