Resonance in LC Circuit

Resonance in an LC circuit occurs when the magnitude of inductive reactance and capacitive reactance is equal and they have a phase difference of 180 degrees i.e. they are equal and opposite to each other. It means that the resonance is a condition when the inductance and capacitance cancel out each other. Now we shall discuss resonance in series and parallel LC circuit in detail.

Resonance in Series LC Circuit

Consider a series LC circuit with V_L and V_C as the potential difference across inductor and capacitor respectively and I is the current through the circuit and V is the net potential difference across the circuit.

We know that in a series circuit the current across the circuit is the same. Thus current across inductor I_L and current across capacitor I_C are the same.

I = I_L = I_C

The total potential difference across the circuit is the sum of the potential differences across inductor and capacitor.

V = V_L + V_C

The condition for resonance is that magnitude of inductive reactance X_L and magnitude of capacitive X_C are equal. Thus, resonance occurs when:

X_L = X_C

We know that,

X_L = ωL = 2πfL

X_C = 1/(ωC) = 1/(2πfC)

where,

• ω is the angular frequency of the circuit
• f is the frequency of the circuit

Resonance Frequency In LC Series Circuit

Resonance occurs when:

X_L = X_C

ωL = 1/( ωC)

ω² = 1/LC

ω = 1/√LC

As ω = 2πf

f = ω/2π

f = 1/2π√LC

Thus, at resonance the frequency of circuit is 1/2π√LC and is denoted by f₀.

Impedance in Series LC Circuit

Impedance of an LC circuit is the net resistance of the LC circuit. It is the effective resistance offered by the inductor as well as capacitor in the LC circuit. Impedance is represented by symbol Z.

We know that when two resistors are connected in series then effective resistance is the sum of resistances of the individual resistors. Consider a series LC circuit with R_L and R_C as the resistance of the inductor and capacitor respectively. Thus impedance or net resistance of series LC circuit is given by:

Z = R_L + R_C

Z = jωL + 1/jωC

where j² = -1

Z = jωL – j²/jωC

Z = jωL – j/ωC

At resonance ω = ω_o= 1/√LC. Thus impedance of series LC circuit becomes zero as follows:

Resonance in Parallel LC Circuit

Consider a parallel LC circuit with I_L and I_C as the currents across inductor and capacitor respectively and I is the current through the circuit and V is the net potential difference across the circuit.

We know that in a parallel circuit the potential difference across the circuit is the same. Thus potential difference across inductor V_L and potential difference across capacitor V_C are the same.

V = V_L = V_C

The total current across the circuit is the sum of the currents across inductor and capacitor.

I = I_L + I_C

The condition for resonance is that magnitude of inductive X_L and magnitude of capacitive X_C are equal. Thus, resonance occurs when:

X_L = X_C

We know that,

X_L = ωL = 2πfL

X_C = 1/( ωC) = 1/(2πfC)

where ω is the angular frequency of the circuit and f is the frequency of the circuit.

Resonance Frequency In Parallel LC Circuit

Resonance occurs when:

X_L = X_C

ωL = 1/( ωC)

ω² = 1/LC

ω = 1/√LC

As ω = 2πf

f = ω/2π

f = 1/2π√LC

Thus, at resonance the frequency of circuit is 1/2π√LC and is denoted by f0. Some important points about resonant frequency in LC circuit are:

• If f < f₀ then X_C >> X_L. Thus the circuit is capacitive in nature.
• If f < f₀ then X_C << X_L. Thus the circuit is inductive in nature.

Impedance in Parallel LC Circuit

We know that when two resistors are connected in parallel then effective resistance is the sum of reciprocal of the resistances of the individual resistors. Consider a parallel LC circuit with R_L and R_C as the resistance of the inductor and capacitor respectively. Thus impedance or net resistance of parallel LC circuit is given by:

1/Z = 1/R_L + 1/R_C

1/Z = (R_L+R_C)/R_LC_L

Z = R_LR_C/(R_L+R_C)

At resonance ω = ω_o = 1/√LC. Thus impedance of parallel LC circuit becomes infinity as follows:

Voltage and Current in LC Circuit

Let I(t) be the current in LC circuit at any instant. Thus potential difference across inductor becomes:

V_L = LdI(t)/dt

The potential difference across capacitor becomes:

V_C = q/C

According to Kirchoff’s law, the sum of potential differences across different components in a closed circuit is zero and as LC circuit is closed circuit:

V_L + V_C = 0

Dividing LHS and RHS by L and differentiating both sides w.r.t ‘t’, we get:

As an LC circuit follows SHM and we know that current in SHM is given by:

Differentiating both sides w.r.t ‘t’, we get:

The potential difference across an LC circuit is given by the equation:

LC Circuit is a special type of electric circuit that is made up of an Inductor and a Capacitor. The inductor is represented by using the symbol L whereas the capacitor is represented using the symbol C. Hence, the name LC Circuit. LC Circuit acts as a major electric component in various devices such as oscillators, tuners, and filters. Thus LC circuit finds a number of applications in daily life. In this article, we shall discuss the LC circuit in detail.