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RESONANCE
ELE101/102 Dept of E&E,MIT Manipal 2
SERIES RESONANCE - RLC Series Circuit
Is said to be in resonance, when
Current is in phase with voltage
Inductive reactance =Capacitive reactance; XL = XC
Circuit Power factor is unity
Effective Circuit impedance is resistive; Z = R
v(t), variable frequency
C
1
XC
ω
=
)XX(jRZ CL −+=
LXL ω=
ELE101/102 Dept of E&E,MIT Manipal 3
SERIES RESONANCE
Resonant frequency (f0):
Imaginary part of Zeq = 0
sec/radians
LC
1
o =ω
hertz
2
1
LC
fo
π
=
CL XX =
C
1
L
0
0
ω
=ω
LC
12
0 =ω
v(t), variable frequency
ELE101/102 Dept of E&E,MIT Manipal 4
Variation of reactance with frequency
ωo
XL
=ωL
Z
XC=-1/ωC
R
ωo
ω0 is the resonance frequency.
Series Resonance
ELE101/102 Dept of E&E,MIT Manipal 5
SERIES RESONANCE
ωo ω
I
Imax
valuerms;
R
V
II maxo ==
Acceptor circuit
Variation of current
ELE101/102 Dept of E&E,MIT Manipal 6
Series Resonance - Half Power frequencies and Bandwidth
Band Width = ω2- ω1
ω1 = lower half power frequency
ω2 = upper half power frequency
ω1 ωo ω2
i
IO
2
Io
ω
2
I
I,ωandωatTherefore, o
21 =
R
I
RIPower o
2
2
0
22
1






==
Half Power Frequency =
Frequency at which the power is
half of the power at resonant
frequency
ELE101/102 Dept of E&E,MIT Manipal 7
Half power frequencies and Band Width
R
2
I
V
ZandatpedanceIm
o
o
21 2==ωω
LC
1
L2
R
L2
R
2
1 +





+
−
=ω
)(Z1 LC XXjR −−=
R2)XX(R 2
LC
2
=−+
;RXX LC =−
RL
C
1
1
1
=ω−
ω
At ω1:
ELE101/102 Dept of E&E,MIT Manipal 8
L
R
=− 12Then, ωω 2
012
1
and ωωω ==
LC
)(Z2 CL XXjR −+=
R)XX(R CL 222
=−+
;RXX CL =−
R
C
L =−
2
2
1
ω
ω
LCL
R
L
R 1
22
2
2 +





+=ω
At ω2:
Half power frequencies and Band Width
ELE101/102 Dept of E&E,MIT Manipal 9
Series Resonance: - Q-factor
Quality factor- a measure of selectivity
( )
WidthBand
frequencyResonant
ωω
ω
L
R
ω
R
Lω
Q
12
ooo
=
−
=






==
;
C
L
R
1
CRω
1
R
Lω
Q
o
o
===
Cω
1
Lωresonanceat-:Note
o
o =
ELE101/102 Dept of E&E,MIT Manipal 9
Series Resonance: - Q-factor
Quality factor- a measure of selectivity
( )
WidthBand
frequencyResonant
ωω
ω
L
R
ω
R
Lω
Q
12
ooo
=
−
=






==
;
C
L
R
1
CRω
1
R
Lω
Q
o
o
===
Cω
1
Lωresonanceat-:Note
o
o =

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L24 resonance

  • 2. ELE101/102 Dept of E&E,MIT Manipal 2 SERIES RESONANCE - RLC Series Circuit Is said to be in resonance, when Current is in phase with voltage Inductive reactance =Capacitive reactance; XL = XC Circuit Power factor is unity Effective Circuit impedance is resistive; Z = R v(t), variable frequency C 1 XC ω = )XX(jRZ CL −+= LXL ω=
  • 3. ELE101/102 Dept of E&E,MIT Manipal 3 SERIES RESONANCE Resonant frequency (f0): Imaginary part of Zeq = 0 sec/radians LC 1 o =ω hertz 2 1 LC fo π = CL XX = C 1 L 0 0 ω =ω LC 12 0 =ω v(t), variable frequency
  • 4. ELE101/102 Dept of E&E,MIT Manipal 4 Variation of reactance with frequency ωo XL =ωL Z XC=-1/ωC R ωo ω0 is the resonance frequency. Series Resonance
  • 5. ELE101/102 Dept of E&E,MIT Manipal 5 SERIES RESONANCE ωo ω I Imax valuerms; R V II maxo == Acceptor circuit Variation of current
  • 6. ELE101/102 Dept of E&E,MIT Manipal 6 Series Resonance - Half Power frequencies and Bandwidth Band Width = ω2- ω1 ω1 = lower half power frequency ω2 = upper half power frequency ω1 ωo ω2 i IO 2 Io ω 2 I I,ωandωatTherefore, o 21 = R I RIPower o 2 2 0 22 1       == Half Power Frequency = Frequency at which the power is half of the power at resonant frequency
  • 7. ELE101/102 Dept of E&E,MIT Manipal 7 Half power frequencies and Band Width R 2 I V ZandatpedanceIm o o 21 2==ωω LC 1 L2 R L2 R 2 1 +      + − =ω )(Z1 LC XXjR −−= R2)XX(R 2 LC 2 =−+ ;RXX LC =− RL C 1 1 1 =ω− ω At ω1:
  • 8. ELE101/102 Dept of E&E,MIT Manipal 8 L R =− 12Then, ωω 2 012 1 and ωωω == LC )(Z2 CL XXjR −+= R)XX(R CL 222 =−+ ;RXX CL =− R C L =− 2 2 1 ω ω LCL R L R 1 22 2 2 +      +=ω At ω2: Half power frequencies and Band Width
  • 9. ELE101/102 Dept of E&E,MIT Manipal 9 Series Resonance: - Q-factor Quality factor- a measure of selectivity ( ) WidthBand frequencyResonant ωω ω L R ω R Lω Q 12 ooo = − =       == ; C L R 1 CRω 1 R Lω Q o o === Cω 1 Lωresonanceat-:Note o o =
  • 10. ELE101/102 Dept of E&E,MIT Manipal 9 Series Resonance: - Q-factor Quality factor- a measure of selectivity ( ) WidthBand frequencyResonant ωω ω L R ω R Lω Q 12 ooo = − =       == ; C L R 1 CRω 1 R Lω Q o o === Cω 1 Lωresonanceat-:Note o o =