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

• 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.