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Friday, January 17

Series Resonance Circuit


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


                        “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.”

                                                        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.  

  • 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.
ωr 2=1/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
Hence, power factor cos0° =1

  • The variation of current with frequency is giver by the graph given below:

    At resonance frequency fr, if the amplitude of source voltage Vo remains constant, the current becomes maximum and is given as:
                                                                    Ir= Vo/R.

  • V = VVVR 
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:
By applying Pythagoras Theorem,
V=[VR 2 + (VC - VL ) 2]^1/2
(At resonance frequency, VC = VL )
V=[(VR) + 0]^1/2
 V= VR

ii. At Vc<VL:

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

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


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