In this article we will discuss about series and parallel resonance in R-L-C circuits.

Resonance is identified with engineering situations which involve energy-storing elements subjected to a forcing function of varying frequency. Specifically, resonance is the term employed for describing the steady-state operation of a circuit or system at that frequency for which the resultant response is in time phase with the source function despite the presence of energy-storing elements.

Resonance cannot take place when only one type of energy-storing element is present, e.g., capacitance or spring. There must exist two types of independent energy-storing elements capable of interchanging energy between them—for example, inductance and capacitance or mass and spring. Thus resonance is a phenomenon found in any system involving two independent energy-storing elements, be they electrical, mechanical, pneumatic, hydraulic or whatever.

If we have an ac circuit having a resistance R, an inductance L, and a capacitance C, connected in series (Fig. 6.1) and apply a small voltage V from a source that can keep the magnitude of V constant but can vary its frequency, we find that the magnitude of current drawn from the source of supply varies with the variation in frequency of supply source. There will be a value of frequency at which the current is maximum. Electrical resonance is said to exist when this condition is reached.

In this article we will discuss this phenomenon (more precisely called the series or voltage resonance) and also the situation when a constant voltage of variable frequency is applied to a parallel circuit. The simplest parallel circuit encountered in practice is a coil having resistance R and inductance L connected in parallel with a capacitor C. The resonant condition in this case is called parallel resonance, and sometimes anti-resonance. The latter name is prompted due to the fact that at resonance the input current to the parallel circuit is minimum.

Under the conditions of resonance, such a network becomes purely resistive in its effects, and the voltage and the current in the network are in phase. For this to occur, the inductive reactance and the capacitive reactance should be made equal.

**Series or Voltage Resonance in R-L-C Circuits****: **

Consider an ac circuit containing a resistance R, an inductance L and a capacitance C connected in series, as shown in Fig. 6.1.

**If for some frequency of applied voltage, X _{L} = X_{C} in magnitude then:**

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(i) Net reactance is zero i.e. X = 0

(ii) Impedance of the circuit, Z = R

(iii) The current flowing through the circuit is maximum and in phase with the applied voltage. The magnitude of the current will be equal to V/R

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(iv) The voltage drop across the inductance is equal to the voltage drop across capacitance and is maximum

(v) The power factor is unity, and

(vi) The power expended = VI watts.

When this condition exists, the circuit is said to be in resonance and the frequency at which it occurs is known as resonant frequency.

From the above expression it is obvious that the value of resonance frequency depends on the parameters of the two energy-storing elements.

Phasor diagrams for the series R-L-C circuit shown in Fig. 6.1 at three different frequencies i.e. (a) f < f_{r} (b) f = f_{r} and (c) f > f_{r} with L and C kept constant are shown in Figs. 6.2 (a), (b) and (c) respectively.

For any frequency lower than resonant frequency f_{r}, inductive reactance X_{L} is lesser than the capacitive reactance X_{C} and so the circuit behaves as a capacitive circuit. Similarly for any frequency higher than resonant frequency inductive reactance is larger than capacitive reactance and so the circuit behaves as an inductive circuit.

ADVERTISEMENTS:

When the frequency of applied voltage is equal to the resonant frequency, the inductive reactance is equal to the capacitive reactance, the voltage drop across inductor is equal to voltage drop across capacitor in magnitude, but opposite in phase and, therefore, the circuit current I is in phase with the applied voltage i.e., the circuit behaves as a resistive circuit.

When the circuit is in resonance, the current is too large and will produce large voltage drop across inductance and capacitance, which will be equal in magnitude but opposite in phase and each may be several times greater than the applied voltage. If resistance R would have not been present in the circuit, such a circuit, would act like a short-circuit to currents of frequency to which it resonates.

Since in this resonance the voltage is maximum, it is called the voltage resonance. The series resonance is also called an acceptor circuit because such a circuit accepts currents at one particular frequency but rejects currents of other frequencies. Such circuits are used in radio-receivers.

**Graphical Representation of Resonance in a R-L-C Series Circuit:**

**The circuit can be made resonant in two different ways, namely: **

(i) By varying L and C parameters (either one or both) at a constant supply frequency or

(ii) By varying the supply frequency/with parameters L and C constant. In our study of series resonance phenomenon we shall keep the applied voltage to the circuit and parameters L and C constant and allow the frequency to vary.

The circuit resistance R is independent of supply frequency and, therefore, remains constant. It has been represented by a straight line, parallel to the X-axis (or frequency axis) in Figure (Fig. 6.3). Inductive reactance X_{L}, being equal to ωL, increases in direct proportion to the supply frequency and is represented by a straight line passing through the origin (as X_{L} is regarded positive, so it lies in the first quadrant). Capacitive reactance, being equal to 1/ωC decreases inversely with the increase in frequency and is represented by a rectangular hyperbola lying in the fourth quadrant, below the frequency axis (capacitive reactance is considered negative).

The net reactance is the difference of inductive reactance X _{L} and capacitive reactance X _{C} and the curve drawn between the net reactance (X_{L} ~ X_{C}) and frequency will be a hyperbola (not rectangular) as shown in Fig. 6.3. The frequency at which the reactance curve crosses the frequency axis is called the resonant frequency, f_{r} (or f_{0}).

The impedance of the circuit, Z, being equal to √R^{2} + (x_{L} – X_{c})^{2} is minimum at resonant frequency f_{r}.

At frequencies lower than resonant frequency f_{r} the impedance Z is large and capacitive as X _{C} > X _{L} and the power factor is leading and at frequencies higher than resonant frequency f_{r}, the impedance Z is again large but inductive as X_{L} > X_{C} and the power factor is lagging. The power factor has the maximum value of unity at resonant frequency.

**Resonance Curve:**

Current varies inversely with the variation in impedance and, therefore, it is maximum at resonant frequency when impedance is minimum and decreases with the variation in frequency on both sides of the resonant frequency (as impedance Z is large), as illustrated in Fig. 6.4.

The curve drawn between circuit current and the frequency of the applied voltage is called the resonance curve and its shape depends upon the value of circuit resistance R, as shown in the figure For smaller values of R, the resonance curve is sharply peaked, but for larger values of R, the curve is flat (Fig. 6.4).

**Selectivity and Bandwidth:**

We have seen that for low resistance circuit the resonance curve is sharply peaked and such a circuit is said to be sharply-resonant or highly- selective. On the other hand, high resistance circuit have flat resonance curve and is said to have poor selectivity. Selectivity of different resonant circuits are compared in terms of their bandwidths, which are given by the bands of frequencies which lie between two points on either side of the resonant frequency where current is 1/√2 times of maximum current I_{max}.

Bandwidth, Δ f = f_{2 }– f_{1} … (6.2)

**The actual power input at frequencies f _{1} and f_{2}:**

That is why the frequencies f_{1} and f_{2} at the limits of the bandwidth are called the half- power points on the frequency scale, and the corresponding value of the bandwidth is termed as half-power bandwidth (B_{hp}) or -3 dB bandwidth.

**The worth-noting points regarding half-power points are:**

**Quality Factor of a Series Resonant Circuit:**

**The Q-factor of an R-L-C series circuit can be defined in any of the following ways:**

It may be given as the voltage magnification that the circuit produces at resonance. We have seen that at resonance, the circuit current is maximum and is equal to V/R or supply voltage, V = I_{max} R.

In case of series resonance, higher value of Q-factor means not only higher voltage magnification, but also a higher selectivity of the tuning coil and that is why it is necessary that the coil be of high inductance and low resistance.

**In fact, Q-factor of a series resonant circuit may be defined as the ratio of resonant frequency to bandwidth:**

**Current or Parallel Resonance in R-L-C Circuits: **

When an inductive reactance and a capacitive reactance are connected in parallel, as shown in Fig. 6.8, conditions may reach under which current resonance (also known as parallel or anti-resonance) will take place. Consider a practical case of a coil in parallel with a condenser, as shown in Fig. 6.8. Let the coil be of resistance R ohms and inductance L henrys and the condenser of resistance R ohms and capacitance C farads.

Such a circuit is said to be in electrical resonance when the reactive (or wattless) component of line current becomes zero. The frequency at which this happens is known as resonant frequency.

Circuit will be in electrical resonance if reactive component of R-L branch current, I_{R} _{–} _{L} sin ɸ_{R –} _{L} = Reactive component of R-C branch current, I_{R – C} sin ɸ_{R} _{–} _{c}

**Current at Resonance:**

**Assuming R _{1} = 0, as usually, at resonant frequency:**

The denominator L/CR is known as effective or equivalent or dynamic impedance of the parallel circuit CR at resonance.

**Important Points about Current or Parallel Resonance****: **

1. Net susceptance is zero i.e. 1/X_{C} = X_{L}/X_{Z} or ω_{r} C = ω_{r}L/Z^{2} or Z = √L/C

2. The admittance is equal to conductance.

3. Reactive or wattles component of line current is zero hence circuit power factor is unity.

4. Impedance is purely resistive, maximum in magnitude and is equal to L/CR.

5. Line current is minimum and is equal to V/L/CR magnitude and is in phase with the applied voltage.

6. Frequency is equal to 1/2π √1/LC – R^{2}/L^{2} Hz.

**Note: **

Parallel resonant circuit is sometimes called the rejecter circuit because at resonant frequency the line current is minimum or it almost rejects it.

Since in parallel resonant circuits circulating current between the branches is many times the line current, such type of resonance is sometimes known as current resonance.

An inductive coil of inductance L connected in parallel with a capacitance C is called the tank circuit.

**Graphical Representation of Current or Parallel Resonance:**

Now we will discuss the effect of variation in frequency on the susceptances of the two parallel branches. The variations are illustrated in Fig. 6.10.

Inductive susceptance being equal to 1/ω L or 1/ 2π f L, decreases inversely with the increase in frequency and is represented by a rectangular hyperbola lying in the fourth quadrant, below the frequency axis (inductive susceptance being considered negative).

Capacitive susceptance, being equal to ωC or 2π f C, increases in direct proportion to the supply frequency and is represented by a straight line passing through origin. As capacitive susceptance is considered positive, so it lies in first quadrant.

The net susceptance B is the difference of capacitive susceptance and inductive susceptance and the curve drawn between net susceptance and frequency of the applied voltage is a hyperbola (not rectangular) as illustrated in Fig. 6.10.

The frequency at which the net susceptance curve crosses the frequency axis is called the resonant frequency. At this point impedance is maximum or admittance is minimum and is equal to G, consequently line current is minimum.

Obviously at frequency lower than resonant frequency the inductive susceptance is more than capacitive susceptance, therefore, the circuit is inductive and the line current lags behind the applied voltage. But for frequencies exceeding resonant frequency capacitive susceptance predominates, therefore, the circuit is capacitive and the line current leads the applied voltage.

If the resistance is relatively low the current will fall considerably at resonant frequency and if the resistance is high, decrease in current will be less pronounced, as illustrated in Fig. 6.11.

**Bandwidth in Case of a Parallel Resonant Circuit: **

The bandwidth in case of a parallel circuit is defined in the same way as in case of a series circuit. In this case also there are upper and lower half-power points where power absorbed is half of that at resonant frequency.

At bandwidth frequencies, the net susceptance B is equal to the conductance G. So at frequency f_{1} the net susceptance B_{L1} – B_{C1} = G and at frequency f_{2}, B_{C2} – B_{L2} = G. Thus admittance Y = √G^{2} + B^{2} = √2 G and phase angel ɸ = tan^{-1} 1 = 45˚ or π/4 radian.

**Q-Factor or Current Magnification Factor: **

Q-factor of a parallel circuit is defined as the ratio of the circulating current to the line current or as the current magnification.