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:

 X_L = ωL

and when current flows through capacitor, the capacitive reactance becomes:

X_C = 1/ (ωC)

1. At High Frequency:
   Inductive Reactance  X_L  is greater than capacitive reactance X_C 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 ∝ X_L and f ∝ 1/X_C. So, when f increases  X_L  > X_C  ).
   
   
2. At Low Frequency:
   Capacitive Reactance X_C is greater than inductive reactance  X_L 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 ∝ X_L and f ∝ 1/X_C . So, when f decreases  X_L< X_C ).
   
   
3. At Intermediate Frequency
   In between these frequencies, there will be a certain value of frequency at which X_L becomes equal to X_C and this frequency is called resonance Frequency.  

Characteristics:

• At resonance,  X_L = X_C, 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  X_L= X_C,This circuit becomes the resonant circuit.

1/(ω_rc)=ω_rL

ω_r ²=1/LC

ω_r=1/√(LC)

2πf_r=1/√(LC)

f_r=1/√(2π(LC))

‘f_r’ 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

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

  

  At resonance frequency f_r, if the amplitude of source voltage V_o remains constant, the current becomes maximum and is given as:
  

                                                                  I_r= V_o/R.

• 
  
  
  V = V_C + V_L + V_R 

There are 3 cases :

1. V_C > V_L (At high frequency)
2. V_C= V_L (At resonance frequency)
3. V_C < V_L (At low frequency)

1. At Vc>VL:

                        

By applying Pythagoras Theorem,

V=[V_R ² + (V_C - V_L ) ²]^1/2

(At resonance frequency, V_C = V_L )

V=[(V_R) ² + 0]^1/2

 V= V_R

ii. At Vc<VL:

                       

By applying Pythagoras Theorem,

V = [V_R ² + (V_L  - V_C) ²] ^1/2

(At resonance frequency, V_C = V_L )

V=[(V_R) ² + 0]^1/2

V= V_R

Therefore, at resonance, the voltage drop across inductance V_L and the voltage drop across capacitance V_C may be much larger than the source voltage V_R .

Entry # 11

By Rabia Khalid