JEE Physics/ Lakshmikanta Satapathy/ Alternating Current Theory Part 4/ Phase relationship between current and voltage in a series LCR circuit discussed in details

1. Physics Helpline
L K Satapathy
Alternating Current Theory 4
LCR Series Circuit
L C R
E

2. Physics Helpline
L K Satapathy
Cells in Parallel
Alternating Current Theory 4
L C R
E
sin . . . (1)o
dI q
L RI E E t
dt C
    
sinoE E t
sin( )oI I t  
LCR Series Circuit
Applied voltage
Current
Phase difference =  & Current amplitude = Io
If dI/dt be the rate of change of current , then voltage across L
dI
L
dt

Voltage across the resistor = RI
If q be the charge on the capacitor , voltage across C
q
C

Inductive reactance LX L
Capacitive reactance 1CX C

3. Physics Helpline
L K Satapathy
Cells in Parallel
Alternating Current Theory 4
VC
Io
t
Capacitance
t
Inductance
VL
Io
t
Resistance
VRIo
R oV I R
L o L oV I X I L 
o
C o C
I
V I X
C
 
Phasor Method
Voltage across R is
Current is in phase with voltage
Voltage across L is
Current is (/2) behind voltage
Voltage across C is
Current is (/2) ahead of voltage
For single circuit elements (discussed in AC Theory 3) :

4. Physics Helpline
L K Satapathy
Cells in Parallel
Alternating Current Theory 4
VR
VL
VC
t+
Io
VL & VC are out of phase
The corresponding Phasors act along the same line
but in opposite directions
Eo
-VLVC
VR

t
Io
We assume that voltage across C > voltage across L
The Phasors of (VC – VL) & VR are shown in the figure
The resultant voltage = Eo
Current leads voltage by phase angle  (as shown)
For LCR series circuit :
Let the phase of current = (t + )

5. Physics Helpline
L K Satapathy
Cells in Parallel
Alternating Current Theory 4
2 2 2
( )o R C LE V V V   2 2 2
( )o o C o LI R I X I X  
2 2 2 2 2
[ ( ) ]o C L oI R X X I Z    2 2
( )C Lwhere Z R X X  
o
o o o
E
E I Z I
Z
   
1
tan tanC L C LX X X X
Also
R R
    
    
 
(i) XC  XL :  is +ve  current leads voltage (Capacitive behavior)
(ii) XC  XL :  is ve  current lags behind voltage (Inductive behavior )
Adding the voltages vectorially, we get

6. Physics Helpline
L K Satapathy
Cells in Parallel
Alternating Current Theory 4
Analytical Method :
(1) sino
dI q
L RI E t
dt C
   
2
2
sin . . . (2)o
d q dq q
L R E t
dt dt C
   
sin( )oq q t  
cos( ) . . . (3)o
dq
q t
dt
    
2
2
2
& sin( )o
d q
q t
dt
    
2
2
&
dq dI d q
I
dt dt dt
 
  
 
Let the solution be
 o oq I 

7. Physics Helpline
L K Satapathy
Cells in Parallel
Alternating Current Theory 4
2
(2) sin( ) cos( ) sin( ) sino
o o o
q
q L t q R t t E t
C
               
1
[ sin( ) cos( ) sin( )] sino oq L t R t t E t
C
        

       
[( )sin( ) cos( )] sino C L oq X X t R t E t          
cos( ) sin( ) sin . . . (4)C L
o o
X XR
q Z t t E t
Z Z
     
 
     
 
cos & sinC LX XR
Put
Z Z
 

 
tan C LX X
R


 

8. Physics Helpline
L K Satapathy
Cells in Parallel
Alternating Current Theory 4
(4) [cos( )cos sin( )sin ] sino oq Z t t E t           
[cos( )] sino oq Z t E t       
 o oI Z E
2 2
( )C LZ R X X   
22
2 2
2 2
( )
cos sin 1C LX XR
Also
Z Z
 

   
[cos( )] sino oI Z t E t        o oq I 
[cos( )] sino oE t E t      
cos( ) sint t      
sin( 2) sint t         sin cos
2

 
  
   
  

9. Physics Helpline
L K Satapathy
Cells in Parallel
Alternating Current Theory 4
2 2t t              
(3) cos( ) cos( )o o
dq
I q t I t
dt
         
cos( 2 )oI I t     
( ) sin( )oii I I t  
1
( ) tan C LX X
iii
R
   
  
 
( ) sinoi E E t
cos( 2)oI I t     
sin( )oI I t   
 cos( ) cos  
Results obtained :

10. Physics Helpline
L K Satapathy
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