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1. Fundamentals of Electrical Engineering
RESONANCE IN AC CIRCUITS

2.  Resonance in Series RLC Circuit
 Quality Factor (Q)
 Bandwidth and Half-Power Frequencies
 Resonance in Parallel RLC Circuit
 Parallel LC Circuit (Tank Circuit)
 Dynamic Impedance (or) Dynamic Resistance
2
Outline

3. PHENOMENON OF RESONANCE IN VARIOUS DOMAINS
4
RFAmplifiers

4. PHENOMENON OF RESONANCE IN VARIOUS DOMAINS
Radio
4

5. PHENOMENON OF RESONANCE IN VARIOUS DOMAINS
Pendulum
5

6. PHENOMENON OF RESONANCE IN VARIOUS DOMAINS
Swing
6

7. WHY STUDY RESONANCE?
• Resonance is the frequency response of a circuit or network when it is
operating at its natural frequency called “ Resonance Frequency”.
• For many applications, the supply (defined by its voltage and frequency)
is constant. e.g. The supply to residential homes is 230 V, 50 Hz.
• However, many communication systems involve circuits in which the
supply voltage operates with a varying frequency.
• To understand communication systems, one requires a knowledge of how
circuits are affected by a variation of the frequency. Examples of such
communication systems are,
Radio, television, telephones, and machine control systems.
7

8. WHEN RESONANCE OCCURS? AND WHAT IT RESULTS?
• Resonance occurs in any circuit that has energy storage elements, at least
one inductor and one capacitor.
• Under resonance, the total supply voltage and supply current are in phase.
So, the power factor (PF) becomes unity.
• At resonance, L and C elements exchange energy freely as a function of
time, which results in sinusoidal oscillations either across L or C.
TYPES OF RESONANCE
• Series resonance.
• Parallel resonance. L
8
C

9. RESONANCE IN SERIES RLC CIRCUIT
Resonance is a condition in an RLC circuit in which the capacitive and
inductive reactances are equal in magnitude, thereby resulting in a purely
resistive impedance.
The input impedance is as follows,
At resonance, the net reactance becomes zero. Therefore,
Series resonant RLC
circuit
1 1 
jC
Z  R  j
L   R  j
L 


C 
 
1
r r r
r
1
LC
C
L    rad/s; f 
1
Hz
2 LC
where r and fr represent resonant frequency in rad/s and in Hz, respectively
9

10. VARIATION OF REACTANCE AND IMPEDANCE WITH
FREQUENCY
• At resonant frequency fr, |Z| = R, the
power factor is unity (purely resistive).
• Below fr, |XL| < |XC |, so the circuit is more
capacitive and the power factor is leading.
• Above fr, |XL| > |XC |, so
the circuit is more
inductive and the power
factor is lagging. Variation of resistance, reactance and
impedance with frequency
XL + XC
10

11. VARIATION OF MAGNITUDE AND PHASE OF
CURRENT WITH FREQUENCY
• The current is maximum at resonant
frequency (fr).
Variation of magnitude |I|
and phaseof current with
frequency in a series RLC
circuit
11

12. QUALITY FACTOR (Q)
• The “sharpness” of the resonance in a resonant circuit is measured
quantitatively by the quality factor Q.
• The quality factor relates the maximum or peak energy stored to the
energy dissipated in the circuit per cycle of oscillation:
• It is also regarded as a measure of the energy storage property of a circuit
in relation to its energy dissipation property.
 Peak energy stored in the circuit 
Q  2Energy dissipated by the circuit in one period at resonance 
 
12

13. QUALITY FACTOR (Q)
• In the series RLC circuit, the quality factor (Q) is,
2
1
r
1
LI2
R
I2
R(
fr
2f L





Q  2 1 
 ) 
 2 
Q 

r L

1

1 L
R C
r R R C
13

14. QUALITY FACTOR (Q)
• The Q factor is also defined as the ratio of the reactive power, of either
the capacitor or the inductor to the average power of the resistor at
resonance:
• For capacitive reactance XL at resonance:
Q 
 Reactive power 
 Average power 
 
• For inductive reactance XL at resonance:
 Reactive power  I2
X L
Q     L
 r
 Average power  I2
R R
 Reactive power  I2
X 1
Q     C

 Average power  I2
R CR
r 14

15. VOLTAGES IN A SERIES RLC CIRCUIT
(a) f < f
15
r
Capacitive,
I leads V
(c) f > fr
Inductive,
I lags V
(b) f = fr
Resistive,
V and I in
phase

16. VOLTAGES ACROSS RLC ELEMENTS AT RESONANCE
26
The voltage across resistor at fr is,
The voltage across inductor at fr is,
R R m R
R
V  I  R  I  R 
V
 R V V
L L r m r
R R
VL  X  I L I L
V


r L
V  QV
 VL  QV
The voltage across capacitor at fr is,
1
m C
r r r
C C R CR
1 1 V
VC  XC  IC   I    V  QV  V  QV

17. VOLTAGES ACROSS RLC ELEMENTS AT RESONANCE
27
• Q is
magnification, because VC or VL
termed as Q factor or voltage
equals Q
multiplied by the source voltage V.
• In a series RLC circuit, values of VL and VC
can actually be very large at resonance and
can lead to component damage if not
recognized and subject to careful design.
r
R CR R
 L 
Q 

r L

1

1
 C 
 
Voltage magnification
Q in series resonant circuit

18. VOLTAGES ACROSS RLC ELEMENTS
28
Effect of frequency variation on voltages across R, L and C

19. BANDWIDTH AND HALF POWER FREQUENCIES
• The bandwidth of a circuit is also defined as
the frequency range between the half-power
points when I = Imax/√2.
• In a series RLC circuit, at resonance, maximum power is drawn. i.e.,
P  I 2
 R; where I 
V
at resonance
r max max
R
• Bandwidth represents the range of frequencies for which the power level
in the signal is at least half of the maximum power.
2 2
P I 2
 R  I 
2
r
 max
 
max
  R
 2 
19

20. BANDWIDTH AND HALF POWER FREQUENCIES
• Thus, the condition for half-power is given when
• The vertical lines either side of |I | indicate
that only the magnitude of the current is
under consideration – but the phase angle
will not be neglected.
• The impedance corresponding to half
power-points including phase angle is
I 
Imax V
2 R 2

Z(
1,2 )  R 2  45
The resonance peak, bandwidth
and half-power frequencies
20

21. BANDWIDTH AND HALF POWER FREQUENCIES
I 
V
R1 j1
• The impedance in the complex form
Z(,12 )  R1 j1
• Thus for half power,
and Z  R1 j1
• At the half-power points, the phase angle of the current is 45°. Below the
resonant frequency, at ω1, the circuit is capacitive and Z(ω1) = R(1 − j1).
• Above the resonant frequency, at ω2, the circuit is inductive and
Z(ω2) = R(1 + j1).
21

22. BANDWIDTH AND HALF POWER FREQUENCIES
• Now, the circuit impedance is given by,
• At half power points,
• By comparison of above two equations, resulting in
• As we know,
Z  R1 j1
Z  R  j
L 
1   R

1 j
L


C 
  R
 
CR 
  

1 

L  1  1
R CR
1
R 
C
r R
Q 

r L

22

23. BANDWIDTH AND HALF POWER FREQUENCIES
• Now, by multiplying and dividing with ωr :
• For ω2 :
• For ω1 :

L
r  1 r  1 Q r Q  1 Q
  
r 
 1
R r 
CR 
r r 

 r


r
Q

2


r 
1
  
 2 
Q
1

r
 1
 
 r 1 
23

24. BANDWIDTH AND HALF POWER FREQUENCIES
• The half-power frequencies ω2 and ω1 are obtained as,
• The bandwidth is obtained as:
i.e.
• Resonant frequency in terms of ω2 and ω1, is expressed as:
2 r
2Q
1
4Q2
 
r
 1 1 r
2Q
1
4Q2


r
 1
2 1
Q
BW  

r
Q factor
Bandwidth 
Resonant frequency
r  
1

2
24

25. BANDWIDTH AND HALF POWER FREQUENCIES
The bandwidth is also expressed as:
For Q >> 1,
2 1 2 1
(or)
Hz
Q L
R
2L
  
r
  
R
rad/s
f2  f1 
rad/s
r 1 1 r 1 r
2 2 2L
  
BW
  
BW
  
R
rad/s
2 r 2 r 2 r
2 2 2L
  
BW
  
BW
  
R
25

26. CONCLUSIONS
Resonance in series RLC circuit:
• The voltages which appear across the reactive
components can be many times greater than that of the
supply. The factor of magnification, the voltage
magnification in the series circuit, is called the Q factor.
• An RLC series circuit accepts maximum current from
the source at resonance and for that reason is called an
acceptor circuit.
26

27. RESONANCE IN PARALLEL RLC CIRCUIT
•
is the net impedance of the three
parallel branches.
• In parallel circuits, it is simpler to
consider the total admittance Y of the
three branches. Thus,
where
The supply voltage: V  IZ where Z
V  IZ 
I
Y
1
jL
j
L
Y  G   j
C  G   jC G  jC
1 


L 
 
27

28. RESONANCE IN PARALLEL RLC CIRCUIT
• At resonance (ω = ωr), the net susceptance is zero.
i.e.
• Therefore, the resonant frequency (ωr) :
• At the resonant frequency, Y = G = 1/R, the
conductance of the parallel resistance, and I = VG.
C
1   0


L 
 
r
 
1
rad/s
LC
28

29. CURRENT THROUGH RESISTANCE
• The supply voltage magnitude:
• At resonance, ω = ωr,
The three-branch
parallel resonant circuit
V 
I
1  1 
2
R2  
C 

L 
 
I
R2
V  |V || I |R
1
 02
r R R
• Current through the resistance at ω : I 
VR

V

I  R
 I
R R R
 I
29

30. CURRENT MAGNIFICATION
• Magnitude of current through inductor at ωr :
• Magnitude of current through capacitor at ωr :
where Q is the current magnification i.e.,
L
L r
X L
| I |
V 
I  R

 R 
 I  Q I
L 
 r 
1
C r
C
X
rC
| I |
V

I  R
 CR I  Q I
r
Q 
 R 
 CR
L 
 r  The three-branch parallel resonant circuit
30

31. CURRENT MAGNIFICATION
Current magnification Q is also expressed in terms of inductive or capacitive
susceptance (B), inductive or capacitive reactance (X ) and conductance (G) :
By substituting ωr = 1/√(LC) in Q :
C
L
Q 
1 C
 R
G L
1 r B R
C
      
Q 

  
 G   G   X 
LG 
     
 r 
The three-branch parallel resonant circuit
31

32. BANDWIDTH AND HALF POWER FREQUENCIES
49
The parallel RLC circuit is the dual of the series RLC circuit. Therefore, by
replacing R, L, and C in the expressions for the series circuit with 1∕R, C, and
L respectively, we obtain for the parallel circuit, the Ymin/21/2 frequencies:
1 1


1 
2RC
  2RC  
LC
 
1
RC
• Bandwidth: BW 21
1 1 1 1


2

2
2 
2RC
  2RC  
LC
 
BW
r
r
R
L
• Relation between BW and Q: Q 
r
RC 

33. BANDWIDTH AND HALF POWER FREQUENCIES
The half-power frequencies in terms of quality factor:
For Q >> 1,
 1 
2

1 
r 1    r
 2Q  2Q
  1 
2


2 
r 1    r
 2Q  2Q
2
r 1
 
BW
2
2 r
  
BW
The three-branch parallel resonant circuit
33

34. PRACTICAL PARALLEL RESONANCE CIRCUIT
Computation of resonant frequency of a “tank circuit”:
• The Figure shown is the two branch parallel resonant circuit.
Also called tank circuit.
• The total admittance (YT) of the circuit shown is:
The two-branch
parallel resonant circuit
(or) tank circuit
1 1
Y  Y Y 
 

 
,
T 1 2  R  
 jX   jX
 S L   C 
2
L
  j 
,
 jX
Y 
 RS

T    X 
R  X 2
 S L   C 
2 2 2
S C S
RS
R2

,
Y 
 
 j
 1

T    X 
R  X
XL
 X
 L   L  34

35. PRACTICAL PARALLEL RESONANCE CIRCUIT
Computation of resonant frequency of a “tank circuit”:
At resonance (ω = ωr), the net susceptance is zero.
i.e.
•
The two-branch
parallel resonant circuit
(or) tank circuit
2 2 2
S C S
RS
R2

,
Y 
 
 j
 1

T    X 
R  X
XL
 X
 L   L 
1
S L r
C S L r
X R2
C
  0  R2
 X 2
L ,
XL 1
 X 2
S L
L
R2
C
 X 2
 2 2 2
rad/s
S r
R2
L
C
1
LC
S
L2
R L  r  
35

36. PRACTICAL PARALLEL RESONANCE CIRCUIT
Computation of resonant frequency of a “tank circuit”:
• The resonant frequency in Hz is:
• The admittance at resonance is:
The two-branch parallel
resonant circuit (or) tank circuit
R2
S
L2
1 1
2 LC
fr   Hz
2 2
L
Y ( f  f ) 
 RS 

RSC
,
T r  
R  X
 S L 
36

37. IDEAL TANK CIRCUIT
Computation of resonant frequency of a “ ideal tank circuit”:
• In ideal tank circuit, the series resistance RS is made zero.
• The total admittance (YT) of the circuit shown (Rs = 0) is:
• At resonance (ω = ωr), the net susceptance is zero.
i.e. Ideal tank circuit
with RS = 0
1 1 
,
YT  YL YC

 



 jX    jX
 L   C 
T
C X
Y   j
 1

1 
.
 X 
 L 
Hz
r r
C L
X X LC 2 LC
1

1 1 1
 0   rad/s; f 
37

38. CONCLUSIONS
Resonance in parallel RLC circuit:
• The lowest current from the source occurs at the resonant frequency of
a parallel circuit hence it is called a rejector circuit.
• At resonance, the current in the branches of the parallel circuit can be
many times greater than the supply current.
• The factor of magnification, the current magnification in the parallel
circuit, is again called the Q factor.
• At the resonant frequency of a resonant parallel network, the impedance
is wholly resistive. The value of this impedance is known as the
dynamic resistance or dynamic impedance.
38

39. APPLICATIONS OF RESONANCE
• Resonant circuits (series or parallel) are used in many applications
such as selecting the desired stations in radio and TV receivers.
• Most common applications of resonance are based on the frequency
dependent response. (“tuning” into a particular frequency/channel)
• Aseries resonant circuit is used as voltage amplifier.
• Aparallel resonant circuit is used as current amplifier.
• Aresonant circuit is also used as a filter.
39

40. DYNAMIC IMPEDANCE (OR) DYNAMIC RESISTANCE
• The dynamic impedance (dynamic resistance) is the resistance offered by
the circuit to the input signal under resonance condition.
Q1. What is the dynamic impedance in a standard series RLC circuit?
Ans. In a standard series RLC circuit, at resonance the
net reactance becomes zero. Therefore, the input supply
see only resistance. Hence,
dynamic impedance Zdynamic = R.
40

41. DYNAMIC IMPEDANCE (OR) DYNAMIC RESISTANCE
Q2. What is the dynamic impedance in a general parallel RLC circuit?
Ans. In a general parallel RLC circuit, at
resonance, the net susceptance becomes zero.
Therefore, the input supply see only resistance.
Hence, dynamic impedance Zdynamic = R.
41

42. DYNAMIC IMPEDANCE (OR) DYNAMIC RESISTANCE
Ideal tank circuit with RS = 0
Q3. What is the dynamic impedance in a ideal tank circuit?
Ans. In ideal tank circuit, at resonance, the
circuit acts like a open circuit. Because, in ideal
tank circuit, the RS = 0. Therefore, the
dynamic impedance Zdynamic  
42

43. DYNAMIC IMPEDANCE (OR) DYNAMIC RESISTANCE
Q4. Determine the dynamic impedance of a practical tank circuit?
Ans.
Practical tank circuit
dynamic
S
L
Z
R C

43