Resonance in an LCR Circuit

As the inductor, capacitor and the resistor are all connected in series, the same amount of current will flow through them. Resistance can reduce alternating current (AC) as well as direct current (DC). Inductance can reduce AC but not DC. Moreover, capacitance blocks DC but allows AC current to pass through. Furthermore, through resistance, inductance, and capacitance, AC is in phase with the alternating voltage,  lags it by a phase angle of 90° and leads it by a phase angle of 90 degree respectively. 

Resonance occurs when the value of inductive and capacitive reactances have equal magnitude but a phase difference of 180°. In this condition, they cancel each other. This is known as the resonance frequency of a series LCR circuit. Thus, the circuit is a resonance LCR circuit.

LCR Circuit

According to the definition of resonance in LCR circuit, LCR circuit consists of an Inductor L, a Capacitor C,  and a  Resistor R.  As the resistor, capacitor and inductor are connected in series, this implies the same amount of current will flow through each of the components.

Considering source voltage V = V_msinωt

Applying Kirchhoff’s rule:- 

Net EMF = 

V (source voltage)+e (self induced emf) = IR (voltage drop across the resistor)+ q/C (Voltage drop across the capacitor).

V_msinωt – L(dl/dt) = IR + q/C

V_msinωt = L(dl/dt) + IR + q/C

By putting I = dl/dt : – V_msinωt = R(dq/dt) + (q/C) + L(d²q/dt²)

By rearranging, R(dq/dt) + L(d²q/dt²) + (q/C)= V_msinωt

 LCR circuit connected to ac source

Electrical Resonance

The process of resonance is quite common among different systems that have a tendency to oscillate at a specific frequency. This frequency is termed as the natural frequency of oscillation of the system. If a specific system is using an energy source, in which the frequency is equal to the natural frequency of the system, then resonance indicates the phenomenon of increased amplitude when the applied force frequency is equal or nearly equal to the natural frequency of the system.

Resonance is exhibited by a circuit when both L and C are present in the circuit. Only then the voltages across L and C may cancel each other, being opposite in phase with a phase difference of 180°. The total source voltage appears across R, giving maximum current I₀ = E₀/R.

We cannot obtain resonance in RL or RC circuits.

Series Resonance Circuit

A circuit that includes an inductor of inductance L, a capacitor of capacitance C, and a resistor of resistance R are all connected in series. The circuit that allows the highest amount of current at a specific frequency of AC is known as a series resonance circuit or series resonance LCR circuit.

Impedance

Consider that the voltage amplitude across resistor R, inductor L, and capacitor C is given as V_R, V_L, and V_C, respectively. The values are:

V_R = iR where i is current, R is resistance

V_L = iX_L = iωL where X_L is inductive reactance, is angular frequency, L is inductance

V_C = iX_C = i(1/ωC) where X_C is capacitive reactance, C is capacitance 

We have:

V² = V_R² + (V_L-V_C)²

On substitution we get:

V² = (iR)² + (iX_L – iX_C)²

V² = i²[R² + (X_L-X_C)²]

Then current:

i = v/√[R²+ (X_L-X_C)²] = v/Z where Z is impedance

Z = √[R²+ (X_L-X_C)²] 

On substituting the value of inductive and capacitive reactance:

The impedance (Z) of an LCR circuit is given by Z = √[R²+ {ωL – 1/ωC}²]

At very low frequencies, inductive reactance X_L= ωL is negligible, but capacitive reactance (X_C = 1/ωC) is very high.

The frequency of alternating e.m.f. Applied to the circuit is increased, X_L, goes on increasing X_C goes on decreasing. For a specific value of ω = ω_r , say 

X_L= X_C

I.e. ω_rL = 1/ω_rC or ω_r = 1/√(LC)

2πv_r = 1/√(LC) or v_r = 1/2π√(LC)

At this frequency v_r ,  as X_L= X_C; therefore,

Z = √(R²+0) = R = minimum

Here, impedance of the LCR circuit is minimum. Hence, the current I₀ = E₀/Z = E₀/R becomes maximum. This frequency is called resonance frequency.

From the above relation, for frequencies greater than or less than ω_r, the values of current are less than the maximum value (I₀).

Furthermore, at ω = ω_r, current is maximum (I₀ ). Here the value of current I₀ is inversely proportional to the resistance R. For lower R values, I₀ is large and vice-versa.

The series resonance circuit is known as the acceptor circuit. A series resonance circuit or resonance LCR circuit provides the maximum response to currents at its resonance frequency. It is called an acceptor circuit.

Conclusion

According to the definition of resonance in an LCR circuit, an RLC circuit, which is an oscillating circuit, has a resistor, capacitor, and inductor connected in series. The voltages in the capacitor are responsible for the current to flow or to stop and then flow in the opposite direction. This gives the rise to the resonance.