Resonance in RLC Circuits

In this article we will discuss on Resonance in RLC Circuits, We have already seen from Article Resistance Inductance and Capacitance in Series that net reactance in RLC circuits of Figure (A) is

X = X_L−X_C and

Let such a circuit be connected across an A.C. source of constant voltage V but of frequency varying from zero to infinity. There would be a certain frequency of the applied voltage which would make X_L equal to X_C in magnitude. In that case, X = 0 an Z = R as shown in Figure (C). Under this condition, the circuit is said to be in electrical resonance.

As shown in Figure (C), V_L = I.X_L and V_C = I.X_C and the two are equal in magnitude but opposite in phase. Hence, they cancel each other out. The two reactance taken together act as a short-circuit since no voltage develops across them. Whole of the applied voltage drops across R so that V = V_R. The circuit impedance Z = R. The phasor diagram for series resonance is shown in Figure (D).

Calculation of Resonant Frequency

The frequency at which the net reactance of the series circuit is zero is called the resonant frequency f₀.

Its value can be found as under : X_L − X_C = 0 or X_L = X_C or ω₀L = 1/ω₀C
or

If L is in henry and C in farad, then f₀ is given in Hz.
When a series R-L-C circuit is in resonance, it possesses minimum impedance Z = R.

Hence, circuit current is maximum, it being limited by value of R alone. The current I₀ =V/R and is in phase with V.

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Since circuit current is maximum, it produces large voltage drops across L and C. But these drops being equal and opposite, cancel each other out. Taken together, L an C from part of a circuit across which no voltage develops, however, large the current flowing. If it were not for the presence of R, such a resonant circuit would act like a short-circuit to currents of the frequency to which it resonates. Hence, a series resonant circuit is sometimes called acceptor circuit and the series resonance is often referred to as voltage resonance.

In fact, at resonance the series RLC circuit is reduced to a purely resistive circuit, as shown in Figure (A), (B), (C) & (D).

Incidentally, it may be noted that if X_L and X_C are shown at any frequency f, that the value of the resonant frequency of such a circuit can be found by the relation

Summary

When an R-L-C circuit is in resonance
1. Net reactance of the circuit is zero i.e. (X_L − X_C) = 0. or X = 0.
2. Circuit impedance is minimum i.e. Z = R. Consequently, circuit admittance is maximum.
3. Circuit current is maximum and is given by I₀= V/Z₀ = V/R.
4. Power dissipated is maximum i.e. P₀ = I₀²R = V²/R
5. Circuit power factor angle θ = 0. Hence, power factor cos θ = 1.
6. Although V_L = V_C yet V_coil is greater than V_C because of its resistance.
7. At resonance, ω²L_C = 1
8. Q = tan θ = tan 0° = 0

Graphical Representation of Resonance

Suppose an alternating voltage of constant magnitude, but of varying frequency is applied to an R-L-C circuit. The variations of resistance, inductive reactance X_L and capacitive reactance X_C with frequency are shown in Figure (E).

(i) Resistance : It is independent of f, hence, it is represented by a straight line.

(ii) Inductive Reactance : It is given by X_L = ωL = 2π fL. As seen, X_L is directly proportional to f i.e. X_L increases linearly with f. Hence, its graph is a straight line passing through the origin.

(iii) Capacitive Reactance : It is given by X_C = 1/ωC = 1/2π fC. Obviously, it is inversely proportional to f. Its graph is a rectangular hyperbola which is drawn in the fourth quadrant because X_C is regarded negative. It is asymptotic to the horizontal axis at high frequencies and to the vertical axis at low frequencies.

(iv) Net Reactance : It is given by X = X_L ~ X_C. Its graph is a hyperbola (not rectangular) and crosses the X-axis at point A which represents resonant frequency f₀.

(v) Circuit Impedance : It is given by

At low frequencies Z is large because X_C is large. Since X_C > X_L, the net circuit reactance X is capacitive and the p.f. is leading [Figure (F)]. At high frequencies, Z is again large (because X_L is large) but is inductive because X_L > X_C. Circuit impedance has minimum values at f₀ given by Z = R because X = 0.

(vi) Current I₀ : It is the reciprocal of the circuit impedance. When Z is low, I₀ is high and vice versa. As seen, I₀ has low value on both sides of f₀ (because Z is large there) but has maximum value of I₀ = V/R at resonance. Hence, maximum power is dissipated by the series circuit under resonant conditions. At frequencies below and above resonance, current decreases as shown in Figure (F).

Now, I₀ = V/R and

Hence where X is the net circuit reactance at any frequency f.

(vii) Power Factor : As pointed out earlier, X is capacitive below f_o. Hence, current leads the applied voltage. However, at frequencies above f_o, X is inductive. Hence, the current lags the applied voltage as shown in Figure (F). The power factor has maximum value of unity at f_o.

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