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Series resonance 
For resonance 
? ?
?
m
I Z 0 
? ?
? ? ? ?
?
1
Z R j L
c
 
? ? ? ? ?
?
1
1
L rad s
c
LC
 
The frequency at which impedance of circuits is purely resistive is called resonant frequency. 
??
0
1
rad s
LC
 
At resonance 
S
V
I
R
?
 
??
RS
V IR V 
??
??
??
??
? ? ? ?
S
L o o
V
V j LI j L
R
 
??
??
??
??
??
??
??
S
c
0o
V jj
VI
c C R
 
 
At ? ? ? ?
0 L C
; X X , net reactance is capacitive so circuits operates at leading pf. 
At ? ? ? ?
0 L C
; X X , net reactance is zero, so circuits operates at unity pf. 
At ? ? ? ?
0 L C
; X X , net reactance is inductive, so circuits operates at lagging pf. 
Bandwidth: range of frequency for which power delivered to R is half of power at resonance. 
R
Bw
L
?
 
Quality factor 
? ? ?
22
22
LC
11
I X I X
Reactive power
22
Q
11 Average power
I R I R
22
 
? ? ?
LC
XX
1L
Q
R R R C
 
Page 2


 
 
 
 
 
 
Series resonance 
For resonance 
? ?
?
m
I Z 0 
? ?
? ? ? ?
?
1
Z R j L
c
 
? ? ? ? ?
?
1
1
L rad s
c
LC
 
The frequency at which impedance of circuits is purely resistive is called resonant frequency. 
??
0
1
rad s
LC
 
At resonance 
S
V
I
R
?
 
??
RS
V IR V 
??
??
??
??
? ? ? ?
S
L o o
V
V j LI j L
R
 
??
??
??
??
??
??
??
S
c
0o
V jj
VI
c C R
 
 
At ? ? ? ?
0 L C
; X X , net reactance is capacitive so circuits operates at leading pf. 
At ? ? ? ?
0 L C
; X X , net reactance is zero, so circuits operates at unity pf. 
At ? ? ? ?
0 L C
; X X , net reactance is inductive, so circuits operates at lagging pf. 
Bandwidth: range of frequency for which power delivered to R is half of power at resonance. 
R
Bw
L
?
 
Quality factor 
? ? ?
22
22
LC
11
I X I X
Reactive power
22
Q
11 Average power
I R I R
22
 
? ? ?
LC
XX
1L
Q
R R R C
 
 
 
 
 
 
 
Parallel Resonance 
? ? ? ?
?
1
1
Y j C
R
jL
 
At resonance 
? ?
?
m
I Y 0 
??
o
1
rad s
LC
 
?
S
At resona e V c I n R 
?
RS
II 
??
??
S
L
00
IR V
I
j L j L
 
? ? ? ?
? ? ? ?
C 0 S 0
I V j C I R j C 
? Bandwidth 
1
RC
 
 
?
Reactive P
Quality fa
ower
Q
Average
ctor, 
Power
 
2 2
C L
22
LC
V V
2X 2X
R R C
QR
X X L
VV
2R 2R
? ? ? ? ?
 
 
 
 
 
 
 
 
 
 
 
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