Question

A.C. Fundamentals - Resonance in A.C. Circuits

Answer 1

Resonance in AC circuits is a phenomenon that occurs when the inductive reactance (XL) and capacitive reactance (XC) in a circuit become equal, resulting in a condition where the impedance (Z) of the circuit becomes purely resistive. This leads to a maximum flow of current through the circuit for a given applied AC voltage. Resonance is an important concept in AC circuit analysis and has various applications in electronics and engineering.

Here are some key points about resonance in AC circuits:

Frequency of Resonance: The resonant frequency (f₀) of an AC circuit is the frequency at which the inductive reactance (XL) is equal to the capacitive reactance (XC). It can be calculated using the formula:


0
=
1
2



f
0
    ​

=
2π
LC
    ​

1
    ​


where:


0
f
0
    ​

 = Resonant frequency (in Hertz)

π = Pi (approximately 3.14159)

L = Inductance of the circuit (in Henries)

C = Capacitance of the circuit (in Farads)

Impedance at Resonance: At resonance, the impedance of the circuit (

Z) becomes purely resistive (

R), and it is given by:


=

Z=R

This means that the circuit behaves as if it only contains a resistor, and the current flow is maximum at this frequency.

Current at Resonance: When the impedance of the circuit is purely resistive, the current flowing through the circuit reaches its maximum value. The current (

I) can be calculated using Ohm's law:


=


I=
Z
V
    ​


where:


I = Current (in Amperes)

V = Voltage (in Volts)

Z = Impedance (in Ohms)

Voltage across Components: At resonance, the voltage across the inductor (


VL) and the voltage across the capacitor (


VC) are equal in magnitude but 180 degrees out of phase with each other. This cancellation of voltage drops across these components contributes to the purely resistive behavior of the circuit.

Applications: Resonance has various applications in electronics and engineering. It is used in designing filters, tuning circuits, and various communication systems. For example, radio receivers use resonance to select and amplify specific frequencies, while transformers and resonant circuits are used in power distribution and impedance matching.

Bandwidth: The bandwidth of a resonant circuit is the range of frequencies around the resonant frequency for which the circuit's impedance remains close to its minimum value (purely resistive). It is related to the quality factor (Q) of the circuit, which indicates how selective the circuit is in responding to specific frequencies.

Understanding resonance in AC circuits is important for designing and analyzing circuits involving inductors and capacitors. It allows engineers to optimize circuit performance for specific frequency ranges and applications.