Concepts in LCR Circuits- Analytical Solution

Resonance

Natural/resonant frequency (ω0)

• Formula: $$ \omega_0 = \frac{1}{\sqrt{LC}} $$

• Description: The frequency at which an undamped LCR circuit naturally oscillates or resonates.

Quality factor (Q )

• Formula: $$ Q = \frac{\omega_0 L}{R}= \frac{1}{R\sqrt{LC}} $$

• Description: Represents the amount of energy stored in the circuit relative to the energy dissipated per cycle. Higher Q means lower energy loss.

Bandwidth (BW )

• Formula: $$ \text{BW} = \frac{\omega_0}{Q} = R \sqrt{\frac{C}{L}} $$

• Description: The range of frequencies around the resonant frequency within which the circuit’s response is significantly affected.

Sharpness of resonance

• Description: The sharpness or selectivity of the circuit’s response around the resonant frequency. A higher Q value indicates sharper resonance.

Q- factor and bandwidth relation

• Inverse relationship: Higher Q results in a narrower bandwidth and vice versa.

Power factor (pf)

• Formula: $$ \text{pf} = \cos \phi = \frac{R}{\sqrt{R^2 + ( \omega L - \frac{1}{\omega C})^2}} $$

• Description: Represents the effectiveness of the circuit in utilizing power. It’s maximum at resonance.

Current at resonance (I0)

• Formula: $$I_0 = \frac{V_s}{R}$$

• Description: The maximum current that flows through the circuit at resonance.

Other quantities at resonance

• Voltage across inductor (VL) : $$ V_L = I_0 \omega_0 L = V_s Q$$
• Voltage across capacitor (VC) : $$V_C = I_0 \frac{1}{\omega_0 C} = V_s Q$$

Locus diagrams

• Graphical representations that show the behavior of circuit variables (voltage and current) in the complex plane as frequency varies. Useful for analyzing resonance and circuit behavior.