Resonance

A circuit is said to be “at resonance” when its power factor becomes 1.

Resonance occurs when:

$$\omega L = \frac{1}{\omega C}$$

For series circuit, occurs when current through a circuit for a given voltage source reaches its maximum.

For parallel circuit, occurs when voltage across a circuit for a given current source reaches its maximum. Or admittance becomes minimum.

Resonance frequency

Frequency at resonance. Denoted by $f_0$. $$

$$f_0 = \frac{1}{2\pi \sqrt{LC}}$$

Stored energy

In an electric circuit, energy is stored in both:

• electrostatic force in the capacitance
• electromagnetic force in the inductance

Oscillations occur when the energy gets transferred between the 2. Resistance causes energy losses and decreases the oscillation.

• Maximum energy stored in the electromagnetic field occurs in the inductor when current is a maximum
• Maximum energy stored in the electrostatic field occurs in the capacitor when voltage is a maximum.

For calculation purposes, energy stored at either the peak current or the peak voltage is considered. For a series circuit, it is easier to consider the current through the inductance rather than the voltage across the capacitor.

Quality factor

A relationship between maximum energy stored and energy dissipation.

$$Q_s = 2\pi\frac{\text{maximum stored energy}}{\text{energy dissipation per cycle}}$$

For series circuit:

$$Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 CR}$$

For parallel cricuit:

$$Q = \frac{R}{\omega_0 L} = \omega_0 CR$$