Download No Slide Title

VARIABLE-FREQUENCY NETWORK
PERFORMANCE
Resonant Circuits
The resonance phenomenon and its characterization
Filter Networks
Networks with frequency selective characteristics:
low-pass, high-pass, band-pass
RESONANT CIRCUITS
These are circuits with very special frequency characteristics…
And resonance is a very important physical phenomenon
Parallel RLC circuit
Series RLC circuit
Z ( j )  R  jL 
1
jC
Y ( j )  G  jC 
1
jL
The reactance of each circuit is zero when
L 
1
 0 
C
1
LC
The frequency at which the circuit becomes purely resistive is called
the resonance frequency
Properties of resonant circuits
At resonance the impedance/admittance is minimal
Z ( j )  R  jL 
| Z |2  R 2  (L 
1
jC
Y ( j )  G 
1 2
)
C
1
jL
| Y |2  G 2  (C 
 jC
1 2
)
L
Current through the serial circuit/
voltage across the parallel circuit can
become very large (if resistance is small)
Quality Factor : Q 
0 L
R

1
 0CR
Given the similarities between series and parallel resonant circuits,
we will focus on serial circuits
EXAMPLE
Determine the resonant frequency, the voltage across each
element at resonance and the value of the quality factor
I
1
  0 L  50
 0C
VC 
1
j 0 C
I   j 50  5  250  90
Q
1
1

 2000rad / sec
3
6
LC
(25  10 H )(10  10 F )
At resonance Z  2
V
100
I S 
 5A
Z
2
0 
0 L  (2 103 )(25 103 )  50
VL  j0 LI  j50  5  25090 (V )
0 L
R

50
 25
2
At resonance
VS
 Q | VS |
R
| VC | Q | VS |
| VL |  0 L
M ( ) 
Resonance for the series circuit
Z ( j )  R  jL 
| Z |2  R 2  (L 
1
jC
1 2
)
C
1
1/ 2

0 2 
2 
1

Q
(

) 





0
BW 
0
Q
Claim : The voltage gain is
V
1
Gv  R 
V1 1  jQ (    0 )
0
Gv 
At resonance :
 0 L  QR,  0C 
R
1
R  jL 
jC
1
QR
Z ( j )  R  j
Gv 
R
Z


R
Z ( j )
Half power frequencie s
 
 ( )   tan 1 Q (  0 )
0 


QR  j 0 QR
0


  
 R 1  jQ(  0 )
0  

M ( ) | Gv |,  ( ) | Gv
 LO
2
 1

 1 
  0 
 
  1
 2Q

 2Q 
The Q factor
0 L
1
R 0CR
For series circuit : High Q  Low R
For parallel circuit : High Q  High R (low G)
Q
High Q  Small BW
M

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
EXAMPLE
The Tacoma Narrows Bridge
Opened: July 1, 1940
Collapsed: Nov 7, 1940
Likely cause: wind
varying at frequency
similar to bridge
natural frequency
0  2  0.2
FILTER NETWORKS
Networks designed to have frequency selective behavior
COMMON FILTERS
Low-pass filter
High-pass filter
Band-reject filter
Band-pass filter
Simple low-pass filter
1
V
1
jC
Gv  0 

V1 R  1
1  jRC
jC
1
Gv 
;   RC
1  j
M ( ) | Gv |
1
1   2
Gv   ( )   tan 1 
1 1

M max  1, M    

2

1
   half power frequency

BW 
1

Simple high-pass filter
Gv 
V0
R
jCR


V1 R  1
1  jCR
jC
Gv 
j
;   RC
1  j

M ( ) | Gv |
Gv   ( ) 
1   2

2
 tan 1 
1 1

M max  1, M    

2

1
   half power frequency

 LO 
1

Simple band-pass filter
Band-pass
V
Gv  0 
V1
M ( ) 
 LO 
R
1 

R  j  L 

C 

RC
 HI 
RC 2   2 LC  1
2
1 

M  
  1 M (  0)  M (  )  0
LC 

0 
M ( LO ) 
 ( R / L) 
1
LC
1
 M ( HI )
2
 R / L2  4 20
2
( R / L) 
 R / L2  4 20
2
BW   HI   LO 
R
L
Simple band-reject filter

1
1 
  0
 j0 L 
LC

C

0 
at   0 the capacitor acts as open circuit  V0  V1
0 
at    the inductor acts as open circuit  V0  V1
 LO ,  HI are determined as in the
band - pass filter
Sketch the magnitude characteristic of the Bode plot for Gv ( j )
1
1
jC
Gv ( j ) 

1
1  jRC
R
jC
  RC  (10 103 )(20 106 F )  0.2rad / s
Break/corner frequency : 5rad/s
low frequency asymptote of 0dB/dec
High frequency asymptote of - 20dB/dec
Sketch the magnitude characteristic of the Bode plot for Gv ( j )
20dB/dec. Crosses 0dB at  
Gv ( j ) 
  RC  (25 103 )(20 106 F )  0.5rad / s
R
R
1
jC

1

 2rad / s
jRC
1  jRC
Break/corner frequency : 2rad/s
low frequency asymptote of 0dB/dec
High frequency asymptote of - 20dB/dec