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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.
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 3dB 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 =
Qfactor :
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 SecondOrder 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:
Some Conclusions
at resonant frequency Y_{min} = G
or
Case2:
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
Case3:
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
⇒
⇒
⇒ . ..(6.32)
put
...(6.33)
⇒ Q = RωC ...(6.34)
Qfactor of Tank Circuit
Tank circuit Fig.(6.9)
⇒ ...(6.35)
Antiresonance Curve
Fig. (6.10)
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