Resonance in R-L-C circuit: SERIES & PARALLEL

1. RESONANCE IN R-L-C
CIRCUIT
1. SERIES CIRCUIT
2. PARALLEL CIRCUIT

2. RESONANCE IN R-L-C
SERIES CIRCUIT

3. Let us consider as R-L-C series circuit
We know that the impedance in R-L-C series circuit is
Where XL =2πfL & XC =
1
2πfL
Such a circuit shown in figure is connected to an a.c. source of
constant supply voltage V but having variable frequency. The
frequency can be varied from zero, increasing and approaching
infinity.
 
22
| | L CZ R X X  
𝓋

4. • Since XL and Xc are functions of frequency, at a particular
frequency of the applied voltage, XL and Xc will became equal
in magnitude.
Since XL = Xc
XL - Xc = 0
∴ 𝑍 = 𝑅2 + 0 = 𝑅
The circuit, when XL = Xc and hence 𝑍 = 𝑅 , is said to be in
resonance. In a series circuit current I remains the same
throughout we can write,

5. 𝐼𝑋 𝐿 = 𝐼𝑋 𝐶
i.e. 𝑉𝐿 = 𝑉𝐶
So, at resonance 𝑉𝐿 𝑎𝑛𝑑 𝑉𝐶 will cancel out each other.
∴ the supply voltage
𝑉 = 𝑉𝐿
2
+ (𝑉𝐿 − 𝑉𝐶)2
𝑉 = 𝑉𝑅
2
∴ 𝑉 = 𝑉𝑅
𝑖. 𝑒. The entire supply voltage will drop across the resistor R

6. Resonant frequency
• At resonance XL = Xc
∴ 2π𝑓𝑟 𝐿 =
1
2π 𝑓𝑟 𝐶
(𝑓𝑟 is the resonant frequency)
∴ 𝑓𝑟
2
=
1
(2π)2
L𝐶
∴ 𝑓𝑟 =
1
2π L𝐶
Where L is the inductance in henry, C is the capacitance in farad
and 𝑓𝑟 the resonant frequency in Hz

7. Under resonance condition the net reactance is zero . Hence the
impedance of the circuit.
This is the minimum possible value of impedance. Hence, circuit
current is maximum for the given value of R and its value is given
by
The circuit behaves like a pure resistive circuit because net
reactance is zero . So, the current is in phase with applied voltage
.obviously, the power factor of the circuit is unity under resonance
condition.
as current is maximum it produces large voltage drop across L
and C.
 2 2
0 0L CZ R X R X orX X     
m
V V
I
Z R
   Z R

8. Voltage across the inductance at resonance is given by
 
2
2
1
2
2
L m L m r
m r
m m m
m
V I X I L
I f L
L L
I L I I
LCLC LC
L
I
C




 

   

At resonance, the current the current flowing in the
circuit is equal to V
R
2L
V L L
V V
R C CR
 

9. Similarly voltage across capacitance at resonance is given by
Thus voltage drop across L and C are equal and many times the
applied voltage. Hence voltage magnification occurs at the
resonance condition.so series resonance condition is often refers to
as voltage resonance.
2
1 1
2
1
1
2
2
C m C
m m
r r
m
m m
m
V I X
I I
C f C
I LC LC
I I
C CC
LC
L V L
I
C R C
 



 
  
 
 

10. Q-FACTOR IN R-L-C SERIES CIRCUIT
Q-FACTOR: In case of R-L-C series circuit Q-Factor is
defined as the voltage magnification of the circuit at resonance.
Current at resonance is given by
And voltage across inductance or capacitor is given by =
OR
Voltage magnification = voltage across L or C /applied voltage
OR
OR
m m
V
I V I R
R
  
m LI X m CI X
LV
V
 CV
V
  L CV V
m L
m
I X
I R
 m C
m
I X
I R
CL XX
OR
R R
 

11. Thus Q-factor
2
2
1
2 1
2
2 1
12 2
2
1 1
1
CL
r
r
r
r
XX
OR
R R
L
OR
R CR
f L
OR
R f CR
L
OR
LCR CR
LC
L LC
OR
R LC R C
L
R C





 

 
 
 
 

 

21 1
2
r
r
f LL
Q factor
R C R f CR


   

12. Effects of series resonance
1. When a series in R-L-C circuit attains resonance 𝑋 𝐿 = 𝑋 𝐶
i.e., the next reactance of the circuit is zero.
2. 𝑍 = 𝑅 𝑖. 𝑒. , the impedance of the circuit is minimum.
3. Since Z is minimum, 𝐼 =
𝑉
𝑍
will be minimum.
4. Since I is maximum, the power dissipated would be
maximum 𝑃 = 𝐼2 R .
5. Since 𝑉𝐿 = 𝑉𝐶 , 𝑉 = 𝑉𝑅. 𝑖. 𝑒., the supply voltage is in phase
with the supply current

13. RESONANCE IN R-L-C
PARALLEL CIRCUIT

14. In R-L-C series circuit electrical resonance takes place when the
voltage across the inductance is equal to the voltage across the
capacitance. Alternatively, resonance takes place when the power
factor of the circuit becomes unity. this is the basic condition of
resonance. It remains the same for parallel circuits also .thus
resonance will occur in parallel circuit when the power factor of the
entire circuit becomes unity . let us consider R-L-C parallel circuit

15. For parallel circuit, the applied voltage is taken as reference phasor.
The current drawn
𝑑𝑦
𝑑𝑥
an inductive coil lags the applied voltage by an
phase angle 𝜃 .
The current drawn by capacitor leads the applied voltage by 90° .
Now power factor of the entire circuit is in phase with the applied
voltage. This will happen when the current drawn by the capacitive
branch, equals to the reactive component of current of inductive
branch.

16. Hence the resonance takes place the necessary condition is
……………(1)
Current in a capacitive branch, and ………...(2)
Current in inductive branch,
Where = impedance of the inductive branch
angle of lag
and
Now …………….(3)
sinC LI I 
C
C
V
I
X

L
L
V
I
Z

LZ
2 2
L LR jX R X   
1
2
tan
sin
sin
L
L
L
L L
L
L L L
X
R
X
Z
X VXV
I
Z Z Z






  

17.  
 
2
2
22
2 2
2
2 2
1
1
L
C L
L L C
r r
r
r
r
VXV
X Z
Z X X
L
R L L
C C
L
L R
C
L R
L
C
 




 
   
  
 
2
2
2
2
2
2
2
2
2
1
1
1
2
1 1
2
r
r
r
r
R
LC L
R
LC L
R
f
LC L
R
f
LC L




 
 
  
  
Now substituting the equations 2 and 3 in equation 1,we get

18. Where 𝑓𝑟 is a resonant frequency in Hertz. The expression is different
from that of series circuit. However if the resistance (R) of the coil is
negligible the expression of resonant frequency reduces to
1 1 1
2 2
rf
LC LC 
 

19. From the phasor diagram it is clear that, the current in the inductive
and capacitive branches may be many times greater then the
resultant current under the condition of resonance. As the reactive
component is 0 under resonance condition in order to satisfy the
condition of unity power factor, the resultant current is minimum
under this condition. From the above, it is observed that the current
taken from the supply can be greatly magnified by means of a
parallel resonant circuit. This current magnification is termed as
Q-factor of the circuit.
sin
tan
cos
C L
L
I I
Q factor
I I



    

20. At resonance the resultant current drawn by parallel circuit is in
phase with the applied voltage.
resultant current
But under the condition of resonance
Resultant current
2L rX f L
R R

 
2
cos .L
L L L
V R VR
I I
Z Z Z
  
2 1
L L C r
r
L
Z X X L
C C


   
/ /
VR VR
L C L CR
 

21. Thus the impedance offered by a parallel resonant circuit .
This impedance is purely resistive and generally known as
equivalent or dynamic impedance of the circuit.as the current at
resonance is minimum, the dynamic impedance represents the
maximum impedance offered by the circuit at resonance .
So the parallel circuit is consider as a condition of maximum
impedance or minimum admittance.
The current at resonance is minimum , hence the circuit is
sometimes called as rejecter circuit because it rejects that frequency
to which it resonant.
The phenomenon of resonance is parallel circuits is of
great importance because it forms the basis of tuned circuits in
electronics.
L
CR


22. COMPARISIONOFSERIESANDPARALLEL
RESONANCE

23. Description
Impedance at resonance
Current at resonance
Resonant frequency
When 𝒇 < 𝒇 𝒓
When 𝒇 > 𝒇 𝒓
Power factor at resonance
Q- factor
It magnifies at resonance
Series circuit
Minimum given by 𝒁 = 𝑹
Maximum =
𝑽
𝑹
𝒇 𝒓 =
𝟏
𝟐𝝅 𝑳𝑪
Circuit is capacitive (as the net
reactance is negative)
Circuit is inductive (as the net
reactance is positive)
Unity
𝑿 𝑳
𝑹
voltage
Parallel circuit
Maximum given by 𝒁 =
𝑳
𝑪𝑹
Minimum =
𝑽
𝑳 𝑪𝑹
𝒇 𝒓 =
𝟏
𝟐𝝅
𝟏
𝑳𝑪
−
𝑹 𝟐
𝑳 𝟐
Circuit is inductive (as the net
susceptance is negative)
Circuit is capacitive (as the net
susceptance is positive)
Unity
𝑿 𝑳
𝑹
current

24. • MADE BY
• SIDDHI SHRIVAS
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