Series & Parallel Resonance

Definition

Resonance in electrical circuits is a condition where the inductive reactance ($X_L$) and the capacitive reactance ($X_C$) of a circuit become equal in magnitude, causing the circuit to behave as a purely resistive network. This occurs at a specific frequency known as the resonant frequency ($f_r$).

Main Content

1. Series Resonance

Occurs when an inductor (L), a capacitor (C), and a resistor (R) are connected in series.
At resonance, the circuit impedance is at its minimum value (equal to R), resulting in maximum current flow.

      L      C
  +--[ ]--+--||--+
  |       |      |
--+-------+------+--
     R    V(source)

2. Parallel Resonance

Occurs when an inductor (L) and a capacitor (C) are connected in parallel across an AC source.
At resonance, the circuit impedance is at its maximum value, resulting in minimum current flow from the source.

      +---[ ]---+  (Inductor)
      |    L    |
  ----+---------+----
      |         |
      |   ||    |  (Capacitor)
      +---||----+
           C

3. Resonant Frequency ($f_r$)

The frequency at which the energy stored in the inductor and capacitor oscillates back and forth.
It is calculated using the formula: $f_r = \frac{1}{2\pi\sqrt{LC}}$.

Working / Process

1. Reactance Balancing

In a series circuit, the inductive reactance $X_L = 2\pi fL$ and capacitive reactance $X_C = \frac{1}{2\pi fC}$ are calculated.
Resonance is achieved when $X_L = X_C$, cancelling each other out because they are $180^\circ$ out of phase.

2. Impedance Adjustment

For series circuits, total impedance $Z = \sqrt{R^2 + (X_L - X_C)^2}$. When $X_L = X_C$, $Z$ becomes just $R$.
For parallel circuits, the admittance is calculated; at resonance, the reactive components cancel, leaving only the conductance of the branches.

3. Energy Transfer

At the resonant frequency, energy oscillates between the magnetic field of the inductor and the electric field of the capacitor.
The external source only needs to supply enough power to overcome the resistance (losses) in the circuit.

Advantages / Applications

Radio Tuning: Series resonance is used to select a specific frequency from a broad range of radio waves.
Power Factor Correction: Parallel resonance is used in industrial power systems to improve energy efficiency.
Band-pass Filters: These circuits allow a narrow band of frequencies to pass while blocking others, essential for signal processing.

Summary

Resonance is the state in an AC circuit where inductive and capacitive reactances cancel out, leaving the circuit purely resistive. In series, this creates a peak in current; in parallel, it creates a peak in impedance.

Resonant Frequency: The frequency where $X_L = X_C$.
Series Resonance: Minimum impedance, maximum current.
Parallel Resonance: Maximum impedance, minimum current.
Important terms: Impedance, Reactance, Frequency, Phase Angle.