Question

A.C. Fundamentals - Resonance between Parallel R-L and R-C Circuit

Answer 1

Resonance is an important concept in AC (alternating current) circuits, particularly when dealing with parallel R-L (resistor-inductor) and R-C (resistor-capacitor) circuits. Resonance occurs when the impedance of the circuit becomes purely real, i.e., the reactance cancels out, leading to a peak in the current and/or voltage across the components.

Let's discuss the resonance phenomenon in both parallel R-L and R-C circuits:

Parallel R-L Circuit Resonance:

In a parallel R-L circuit, you have a resistor (R) and an inductor (L) connected in parallel. The total admittance (Y) of the circuit is given by the sum of the admittances of the individual components:


total
=


+


=
1

+



Y
total
    ​

=Y
R
    ​

+Y
L
    ​

=
R
1
    ​

+jωL

Where:



Y
R
    ​

 is the admittance of the resistor (


=
1

Y
R
    ​

=
R
1
    ​

)


Y
L
    ​

 is the admittance of the inductor (


=



Y
L
    ​

=jωL)

ω is the angular frequency of the AC signal (

=
2


ω=2πf)

j is the imaginary unit

To find the resonant frequency, we need to equate the imaginary part of the total admittance to zero, which gives us:



=
1

ωL=
R
1
    ​


This is the condition for resonance in a parallel R-L circuit. At resonance, the reactance of the inductor cancels out the reactance of the resistor, resulting in a purely real impedance. The circuit is most conductive at resonance, leading to a peak in current.

Parallel R-C Circuit Resonance:

In a parallel R-C circuit, you have a resistor (R) and a capacitor (C) connected in parallel. The total admittance (Y) of the circuit is given by the sum of the admittances of the individual components:


total
=


+


=
1

+



Y
total
    ​

=Y
R
    ​

+Y
C
    ​

=
R
1
    ​

+jωC

Where:



Y
R
    ​

 is the admittance of the resistor (


=
1

Y
R
    ​

=
R
1
    ​

)


Y
C
    ​

 is the admittance of the capacitor (


=



Y
C
    ​

=jωC)

To find the resonant frequency, we need to equate the imaginary part of the total admittance to zero, which gives us:



=
1

ωC=
R
1
    ​


This is the condition for resonance in a parallel R-C circuit. At resonance, the reactance of the capacitor cancels out the reactance of the resistor, resulting in a purely real impedance. The circuit is most conductive at resonance, leading to a peak in current.

In summary, resonance in parallel R-L and R-C circuits occurs when the reactance of the reactive component cancels out the reactance of the resistor, resulting in a purely real impedance. This leads to a peak in current and/or voltage at the resonant frequency.