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# Series Resonance Circuit

RESONANCE

DEFINITION:
“A condition exists when an oscillating circuit responds with maximum amplitude of an external source of angular frequency ‘ω’ called resonance.”

RESONATING CIRCUITS:

DEFINITION:
“The circuits that contain both inductors and capacitors along with a resistor so arranged that the circuit is capable of resonance are called resonating circuits.”

SERIES RESONATING CIRCUIT:
Consider a resister, inductor and capacitor which are connected in series with A.C. source whose frequency can be varied. When current flows through inductor, the inductive reactance becomes:
XL = ωL
and when current flows through capacitor, the capacitive reactance becomes:
XC = 1/ (ωC)
1. At High Frequency:
Inductive Reactance  XL  is greater than capacitive reactance XC due to which circuit behaves as R-L Series Circuit in which voltage V leads to the current I by 90° or π/2 (As, f ∝ XL and f ∝ 1/XC. So, when f increases  XL  > XC  ).

2. At Low Frequency:
Capacitive Reactance XC is greater than inductive reactance  XL due to which circuit behaves as R-C Series Circuit in which voltage V lags to the current I by 90° or π/2 (As, f ∝ XL and f ∝ 1/XC . So, when f decreases  XL< XC ).

3. At Intermediate Frequency
In between these frequencies, there will be a certain value of frequency at which XL becomes equal to XC and this frequency is called resonance Frequency.

Characteristics:
• At resonance,  XL = XC, so they are equal and opposite to each other and cancel each other’s effect and circuit behaves as resistive circuit.
• By changing frequency if  XL= XC,This circuit becomes the resonant circuit.
1/(ωrc)=ωrL
ωr 2=1/LC
ωr=1/√(LC)
2πfr=1/√(LC)
fr=1/√(2π(LC))
fr’ is known as ‘resonant frequency’.

• Impedance of circuit becomes minimum and is equal to R (resistive circuit).

• Power loss at resonance is maximum.
P=IV cosθ
θ=0
Hence, power factor cos0° =1

There are 3 cases :

1. VC > VL (At high frequency)
2. VC= VL (At resonance frequency)
3. VC < VL (At low frequency)

1. At Vc>VL:

V=[VR 2 + (VC - VL ) 2]^1/2
(At resonance frequency, VC = VL )
V=[(VR) + 0]^1/2
V= VR

ii. At Vc<VL:

(At resonance frequency, VC = VL )
V=[(VR) + 0]^1/2
V= VR

Therefore, at resonance, the voltage drop across inductance VL and the voltage drop across capacitance VC may be much larger than the source voltage VR .

Entry # 11
By Rabia Khalid