Lcr Circuits Analytical Solution Resonance
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.