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**Resonance**

It is the condition when the voltage across a circuit becomes in phase with the current supplied to the circuit.

At resonance, the circuit behaves like a resistive circuit.

Power factor of the circuit at resonance becomes = "1"

The resonance may be classified into two groups

1. Series resonnat circuit

2. Parallel resonant circuit.

**Series Resonance (RLC series circuit)**

The total impedance of series n/w is given by

where

X_{L} = ωL

At resonance

lm{z} = 0

X_{L }– X_{C} = 0

where f_{0} is the frequency of resonance in Hertz

At resonance, the current is I_{0} = V/R

Variation in Z with respect to 'ω'

Note : For ω < ω_{0} series RLC behaves like RC capacitive circuit.

ω > ω_{0} , behaves like RL inductive circuit

ω = ω_{0} , behaves like resistive circuit.

- Impedance of series RLC circuit is minimum at res onance frequ ency ω = ω
_{0}at resonant frequency Z = R - Current in series RLC circuit is maximum at resonance frequency

**Selectivity and Bandwidth :** At frequency of resonance, the impedance of a series RLC circuit is minimum. Hence the current is maximum. As the frequency of the applied voltage deviates on either side of the series resonant frequency, the impedance increases and the current falls. Figure shows the variation of current I with frequency for small values of R. Thus, a series RLC circuit possesses frequency selectivity. The frequencies f_{1} and f_{2} at which the current I falls to (1/√2) times its maximum values I_{0}( = V/R) are called halfpower frequencies of 3 - dB frequencies. The bandwidth (f_{2}-f_{1}) is called the halfpower bandwidth or 3-dB bandwidth or simply bandwidth (BW) of the circuit.

Selectivity of a resonant circuit is defined as the ratio of resonant frequency to the BW. Thus,

Selectivity =

Q-factor :

- Q of an inductor L with internal resistor R = ωL/R
- Q of a capacitor C with effective internal resistor R = 1/ωCR
- Q of a leaky capacitor which is represented by a capacitor C with a high resistance R
_{P}in shunt = ωCR_{P}.

Selectivity increases with decreasing bandwidth For series Resonant circuit at ω = ω_{0}

For series RLC circuit

BW = ( w_{2} – w_{1}) = R/L

Parallel RLC resonnace circuit:

Admittance of circuit

At resonance lm (Y) = 0

at resonance

Z = R

Properties of Second-Order parallel RLC

**Resonant Circuit : **A circuit consisting of a parallel connection of a resistor R, an inductor L, and a capacitor C is called a second - order parallel resonant circuit.

The important proporties of such a circuit are as follows:

- The resonant frequency is
- Below resonance, the circuit acts like an RL circuit.
- Above resonance, the circuit acts like an RC circuit.
- If the conductance is zero (the resistnace is infinite) ate resonance the circuit acts like an open circuit.
- The bandwidth is 1/RC.
- The quality factor Q
_{OP}is

Some Conclusions

- Y = G + j(B
_{L }– B_{C})

at resonant frequency Y_{min} = G

or

**Current will be minimum**

- Power factor will be unity i.e. cos ø = 1.
- In parallel resonance net reactive current is equal to zero
- Under resonance condition current flowing through inductor or capacitor is greater than total current.
- This phenomenon is called as current magnification.
- Parallel resonant circuit is also called as antiresonant circuit.

**Case-2: **

Consider the circuit shown in figure

Equivalent phasor diagram Fig. (b)

at resonance I_{1} sinθ_{1} = I_{2} sinθ_{2}

In figure (b),

OA = I_{1} cosθ_{1}

OC = AB = I_{1} sinθ_{1}

OK = I_{2} cosθ_{2}

OM = FK = I_{2} sinθ_{2}

Corresponding to figure (a)

Y = (G_{1} + G_{2}) + j(B_{C} – B_{L})

At resonance, B_{C} = B_{L}

Y = G_{1} + G_{2}

Current I = VY

Resonant frequency is given by

**Case-3: **

This is a very important case. Observe the tank circuit shown in figure (6.7).

Equivalent phasor diagram Fig. (6.7)

In figure 6.7(b),

OA = I_{1} cosθ_{1 }

OC = AB = I_{1} sinθ_{1}

and I_{1} sinθ_{1} = I_{2}

Now at resonance, I = I_{1} cosθ_{1}

[From equation (6.28)]

....(6.29)

Here L/RC is defined as dynamic impedance of tank circuit i.e.

And resonance frequency for tank circuit is given by

Q.factor for Parallel Resonance Circuit

- Q-factor for parallel resonant circuit is defined as the ratio of current flowing through inductor (or capacitor) to total current

⇒

⇒

⇒ . ..(6.32)

put

...(6.33)

- In another way Q-factor can be obtained as

⇒ Q = RωC ...(6.34)

**Q-factor of Tank Circuit**

Tank circuit Fig.(6.9)

**Q-factor is given by**

⇒ ...(6.35)

**Anti-resonance Curve **

- Anti-resonance curve is shown in figure (6.10)

Fig. (6.10)

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