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MHT-CET : Physics Entrance Exam

MHT - CET : Physics - Electomagnetic Induction Page 7

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Resonant Circuits:

 

  • The current i in this circuit is given by

i = i0 sin wt. . (1)

  • The e.m.f. e at any instant is the sum of voltage drops across R, across L and across C.

\ e = L  

di

 + Ri + 

q

 

...(2)

dt

C

  •  
  • Substituting (1) in (2) we get,

    \ e = I0 [ R sin wt + (XL - Xc ) cos wt ] (3)
  • Multiplying and dividing R.H.S. of (3) by
    e = I0 x

Let tan f 

XL - Xc

 , so that sin f =

XL - Xc

R

  •  

and 

cos f =

R

 

substituting we get e - E0 sin [ wt + f ]
where E0 = I0

 

Impedance Z:

z =  

e0

 =

I0

 

=

 

 

 

Phase Difference : The instantaneous current and e.m.f. are
given by i = I0 sin wt and e = e0 sin (
wt + f ).

These equations show that e.m.f. e leads the current by a phase angle
f given by

f = tan -1 

XL - XC

R

 

Resonant Frequency:

 

  • When the impedance offered by the circuit is minimum, the current through the circuit is maximum. This condition is known as series resonance.
  • Z = is minimum when XL = XC and
    the minimum value of Z = R.

 

 

Series resonance curve

 

XL = XC

 

wL =  

1

wC

 

 

w2

1

LC

 

 

4p2f2r

1

LC

 

 

fr2

1

4pLC

 

 fr =  

1

2p

...(Resonance frequency)

 

L-C-R Parallel Resonance:

 


L - Inductance,
C - Capacitance,
e - Applied alternating e.m.f.

  • Current in inductor lags behind e.m.f. by 90

\ iL =  

e0

 sin (wt- p / 2)

XL

  •  
  • Current in capacitor leads e.m.f. by 90

\ iC =  

e0

 sin (wt+ p / 2)

XC

  •  
  • Current drawn from source

i = iL + iCe0  

(

1

-

1

)

 cos wt.

XC

XL

  •  
  • If XL = XC, i = 0 No current is drawn from the source, alternating current goes on circulating in the LC loop. This condition is called the parallel resonance.

  • Parallel resonance curve

At resonance,

XL= XC

 

wL =  

1

wC

 

\ w

1

 

Resonant fresquency fr =   

w

=

1

2p

2p

 

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