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1. Circuit Analysis – II (EE-201)
Variable-frequency Networks
Part 3

2. Resonant Circuits
The resonance phenomenon and its characterization
Filter Networks
Networks with frequency selective characteristics:
low-pass, high-pass, band-pass
VARIABLE-FREQUENCY NETWORK
PERFORMANCE
LEARNING GOALS

3. RESONANT CIRCUITS - SERIES RESONANCE
Im{ } 0Z 

RESONANT FREQUENCY
PHASOR DIAGRAM
QUALITY FACTOR

4. RESONANT CIRCUITS
These are circuits with very special frequency characteristics…
And resonance is a very important physical phenomenon
Cj
LjRjZ


1
)( 
circuitRLCSeries
Lj
CjGjY


1
)( 
circuitRLCParallel
LCC
L
11
0  


whenzeroiscircuiteachofreactanceThe
The frequency at which the circuit becomes purely resistive is called
the resonance frequency

5. Properties of resonant circuits
At resonance the impedance/admittance is minimal
Current through the serial circuit/
voltage across the parallel circuit can
become very large (if resistance is small)
CRR
L
Q
0
0 1


:FactorQuality
222
)
1
(||
1
)(
C
LRZ
Cj
LjRjZ






222
)
1
(||
1
)(
L
CGY
Cj
Lj
GjY







Given the similarities between series and parallel resonant circuits,
we will focus on serial circuits

6. Properties of resonant circuits
At resonance the power factor is unity
CIRCUIT BELOW RESONANCE ABOVE RESONANCE
SERIES CAPACITIVE INDUCTIVE
PARALLEL INDUCTIVE CAPACITIVE
Phasor diagram for series circuit Phasor diagram for parallel circuit


RV



C
I
jVC



Lj
1GV
1CVj
L
V
j

1


7. LEARNING EXAMPLE Determine the resonant frequency, the voltage across each
element at resonance and the value of the quality factor
LC
1
0  sec/2000
)1010)(1025(
1
63
rad
FH




I
A
Z
V
I S
5
2
010



 2ZresonanceAt
 
50)1025)(102( 33
0L
)(902505500 VjLIjVL  


90250550
1
50
1
0
0
0
jI
Cj
V
L
C
C



R
L
Q 0
 25
2
50

||||
|||| 0
SC
S
S
L
VQV
VQ
R
V
LV


resonanceAt

8. LEARNING EXTENSION Find the value of C that will place the circuit in resonance
at 1800rad/sec
LC
1
0  2
18001.0
1
)(1.0
1
1800



 C
CH
FC 86.3

9. Resonance for the series circuit
222
)
1
(||
1
)(
C
LRZ
Cj
LjRjZ






QR
CQRL
1
, 00  
:resonanceAt








)(1
)(
0
0
0
0









jQR
QRjQRjRjZ
)(1
1
0
0
1






jQV
V
G R
v
isgainvoltageThe:Claim
)(1 

 jZ
R
Cj
LjR
R
Gv 


vv GGM  |)(|,|)( 
2/1
20
0
2
)(1
1
)(













Q
M
)(tan)( 0
0
1




  
Q
Q
BW 0

sfrequenciepowerHalf
Z
R
Gv 

10. The Q factor
CRR
L
Q
0
0 1



RLowQHigh:circuitseriesFor 
G)(lowRHighQHigh:circuitparallelFor 
M
BWSmallQHigh 
dissipates
Stores as E
field
Stores as M
field
Capacitor and inductor exchange stored
energy. When one is at maximum the
other is at zero
Q can also be interpreted from an
energy point of view

11. FILTER NETWORKS
Networks designed to have frequency selective behavior
COMMON FILTERS
Low-pass filter
High-pass filter
Band-pass filter
Band-reject filter
We focus first on
PASSIVE filters

12. Simple low-pass filter
RCj
Cj
R
Cj
V
V
Gv







1
1
1
1
1
0
RC
j
Gv 

 

;
1
1
 



1
2
tan)(
1
1
||)(




v
v
G
GM
2
11
,1max 







MM
frequencypowerhalf


1

1
BW

13. Simple high-pass filter
CRj
CRj
Cj
R
R
V
V
Gv







111
0
RC
j
j
Gv 

 


;
1
 






1
2
tan
2
)(
1
||)(




v
v
G
GM
2
11
,1max 







MM
frequencypowerhalf


1


1
LO

14. Simple band-pass filter
Band-pass








C
LjR
R
V
V
Gv


11
0
   222
1
)(


LCRC
RC
M



1
1







LC
M  0)()0(   MM
 
2
4/)/( 2
0
2




LRLR
LO
LC
1
0 
 
2
4/)/( 2
0
2




LRLR
HI
L
R
BW LOHI  
)(
2
1
)( HILO MM  

15. Simple band-reject filter
0
11
0
00 






C
Lj
LC 

10 VV  circuitopenasactscapacitorthe0at 
10 VV  circuitopenasactsinductortheat 
filterpass-band
theinasdeterminedareHILO  ,

16. LEARNING EXAMPLE Depending on where the output is taken, this circuit
can produce low-pass, high-pass or band-pass or band-
reject filters
Band-pass
Band-reject filter








C
LjR
Lj
V
V
S
L



1   1)(,00  
S
L
S
L
V
V
V
V
High-pass








C
LjR
Cj
V
V
S
C



1
1
  0)(,10  
S
C
S
C
V
V
V
V
Low-pass
FCHLR  159,159,10 forplotBode