Series resonance

SERIES RESONANCE

I = E / Z ... (i)

At resonance, the power factor being unity, Z = R.

Therefore, reactive part of the complex impedance must be zero i.e., (X_L – X_C) = 0.

Equation (ii) indicates that in a series RLC circuit resonance may be produced by (a) varying frequency for a given L and C, (b) varying either L or C or both for given frequency. So if frequency is varied, keeping

L and C constant, the condition for resonance will be given by

ω = 1/√LC

ωo is called the angular resonance frequency in radian / sec.

Therefore, the resonance frequency in

ωo is not dependent on R. At resonance frequency, the inductive reactance = capacitive reactance i.e., the net reactance of the circuit at resonance is equal to zero.

The figure shows a series RLC circuit. E is the RMS value of the impressed voltage.

I = the RMS value of the current.

ω = angular frequency of supply in radians / sec

Series Resonance

In a R-L-C series circuit, when the current is in phase with the applied voltage, the circuit is said to be in resonance. Then, phase angle is zero and hence power factor is unity. The circuit acts as purely resistive

f, or fo is called resonance frequency. At resonance, in a series circuit the following main effects are to be observed.

(i) Since X_L = X _C, the impedance is minimum and equal to resistance i.e., Z _r = R where Z _r is impedance at resonance.

(ii) Since the impedance is minimum, the current is maximum.

The current at resonance = I_t

= V / Z_r = V / R

(iii) The power taken by the circuit is maximum because I, is maximum.

Power taken by the circuit = P_t

= I_r² R

V²/R

(iv) The voltage across L = V_L = IX_L

The voltage across C = Vc = IXc

At resonance, I = Ir, which is very large.

So, V_L and Vc are also very large.

(v) In R-L-C circuit, at resonance both V and I are in phase. Hence, in the polar form we can express them as V = |V| ∠0 and I= |I| ∠0.

In otherwords, for R-L-C circuit if the applied voltage and source current are in phase, the circuit is at resonance.

In power systems, at resonance the excessive voltage built up across L and C components (such as circuit breakers, reactors etc.) may cause damage. Hence, in power systems, series resonance must be avoided.

But the principle of series resonance is used to increase the signal voltage and current at desired frequency, in some of the electronic circuits. Examples of such electronic devices are antenna circuit of radio and T.V. receiver, tuning circuits etc.

A series resonant circuit has the capability to draw heavy currents and power from the mains. So, it is often regarded as an acceptor circuit.

 

1. Phasor Diagram

Figure 5.2 shows phasor diagram for the series RLC circuit at three different frequencies i.e., (i) f < fo, (ii) ffo, (iii) f > fo with L and C kept constant.

For any frequency lower than fo, X_L < X_C so, the circuit behaves as a capacitive circuit. For any frequency, higher than fo, X_L > X_C and the circuit behaves as an inductive one.

At f=fo, X_L = X_C and hence current and source voltage are in phase i.e., the circuit behaves as a resistive circuit.

 [Note:

1. For a series combination E, and Ec are always in anti phase.

2. At resonance EL = Ec. Voltmeter connected across L and that across C read same magnitude. 3. If the voltmeter is connected across the series combination of L and C, it reads zero at resonance.

4. At resonance, in a series circuit, the impedance is minimum and hence current maximum. In es to other words, if the current is maximum in a RLC series circuit, we can definitely say that the circuit is in resonance. awon ni sonst gemsbo

5. At ffo, the power factor is unity (u.p.f). At f<fo, I leads E. So power factor is leading in nature. For f > fo. I lags behind E. So the power factor is lagging in nature.]

 

2. Reactance Curves

In a series RLC circuit, three reactance curves can be plotted. Reactance curves are the graphs between reactance and frequency. They are:

(a) Inductive reactance (X_L) versus frequency.

(b) Capacitive reactance (X_C) versus frequency.

(c) Total reactance [X = (X_L – X_C)] versus frequency.

X_L = ωL=2лƒL. It shows linear variation.

Xc = 1/ωC = 1/2πfC

It shows a rectangular hyperbolic shape.

For lesser frequency than fo, X is capacitive. At f = fo, X = 0. For f>fo, X is inductive. The various reactance curves are shown in the following figures.

 

3. Variation of Impedance with Frequency for RLC Series Circuit

From equation (vi), it is evident that at f = fo, | Z |= R. At all other frequencies | Z❘ > R.

For lesser frequency than fo, the impedance is capacitive and hence the power factor is leading in nature. At frequencies above fo, the impedance is inductive and hence the power factor is lagging in nature.

 

4. Variation of Current and Voltage Distribution in a Series RLC Circuit with Frequency

At resonance frequency, Z is minimum and hence current is maximum. The variations of E_L and E_C are also shown in fig.5.5. Neither E_C nor E, has the maximum value at resonance frequency. They are equal in magnitude and opposite in phase at f₀.

The variation of current with frequency can be shown as in fig. 5.6. As the resistance is less, current is more. As the resistance increases, current becomes less. So the curves in the fig. 5.6 show as how the maximum current varies with resistance. The current varies more sharply around fo for low values of resistance. When the resistance is increased, this sharpness is not visible. Hence, a series resonant circuit is said to be sharply tuned for lower resistance and broadly tuned for higher resistance.

 

5. Selectivity-Q-Half Power Frequencies - Bandwidth

The current versus frequency curve of a RLC series circuit is symmetrical about the resonance frequency. At frequency fo, the current is maximum and is given by E/R . There will be two frequencies f1, f2 on either side of the resonant frequency at which the power is half the power at resonance. So, they are called half power frequencies. f, is called lower half power frequency and f2 is called upper half power frequency.

Power at points 1 and 2 = I₁² R = I₂² R. Power at resonance = I² R

By definition of half-power frequencies,

i.e., at half-power points 1 and 2, the current is 70.7% of the current at resonance.

Let Z be the impedance at half-power points.

i.e., at half-power points, the power factor is 0.707 or = 45°. At f1, power factor is leading in nature and at f2, it is lagging in nature.

The bandwidth is defined as the band of frequencies between f2 and f1.

The bandwidth = f2 - f1 = R / 2πL … (xi)

The sharpness of the tuning depends on ratio R/L A small value of R/L indicates a high degree of  sensitivity.

Ratio fo to (f2-f1) is called the quality factor of a coil represented by Q.

From the above equation, we can say that f1, fo, f2 are in Geometric progression.

[Note: Q can be expressed in terms of R, L and C as below.]