1) A series RLC circuit exhibits resonance when the inductive reactance equals the capacitive reactance. At resonance, the circuit's impedance is purely resistive and the power factor is unity. 2) The resonant frequency is defined as the frequency at which the imaginary part of the equivalent impedance is zero. 3) The quality factor Q is a measure of selectivity near the resonant frequency, defined as the ratio of the resonant frequency to the bandwidth.

1. 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 =