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VARIABLE-FREQUENCY NETWORK
PERFORMANCE
Variable-Frequency Response Analysis
Network performance as function of frequency.
Transfer function
Sinusoidal Frequency Analysis
Bode plots to display frequency response data
VARIABLE FREQUENCY-RESPONSE ANALYSIS
In AC steady state analysis the frequency is assumed constant (e.g., 60Hz).
Here we consider the frequency as a variable and examine how the performance
varies with the frequency.
Variation in impedance of basic components
Resistor
Z R  R  R0
Inductor
Z L  jL  L90
Capacitor
Zc 
1
1

  90
jC C
Frequency dependent behavior of series RLC network
2
1
( j ) 2 LC  jRC  1  j RC  j ( LC  1)


Z eq  R  jL 

j
C
jC
jC
" Simplifica tion in notation" j  s
s 2 LC  sRC  1
Z eq ( s) 
sC
(RC )  ( LC  1)
| Z eq |
C
2
2
2
1  
LC  1 

Z eq  tan 
 RC 
2
Simplified notation for basic components
Z R ( s)  R, Z L ( s)  sL, ZC 
1
sC
For all cases seen, and all cases to be studied, the impedance is of the form
am s m  am 1s m 1  ...  a1s  a0
Z ( s) 
bn s n  bn1s n1  ...  b1s  b0
Moreover, if the circuit elements (L,R,C, dependent sources) are real then the
expression for any voltage or current will also be a rational function in s
1
sC
sL
R
sRC
VS  2
VS
R  sL  1/ sC
s LC  sRC  1
s  j
jRC
Vo 
VS
2
( j ) LC  jRC  1
Vo ( s) 
R
j (15  2.53  103 )
Vo 
100
2
3
3
( j ) (0.1 2.53  10 )  j (15  2.53  10 )  1
NETWORK FUNCTIONS
Some nomenclature
When voltages and currents are defined at different terminal pairs we
define the ratios as Transfer Functions
INPUT
Voltage
Current
Current
Voltage
OUTPUT TRANSFER FUNCTION SYMBOL
Voltage
Voltage Gain
Gv(s)
Voltage
Transimpedance
Z(s)
Current
Current Gain
Gi(s)
Current
Transadmittance
Y(s)
If voltage and current are defined at the same terminals we define
Driving Point Impedance/Admittance
EXAMPLE
To compute the transfer functions one must solve
the circuit. Any valid technique is acceptable
I 2 ( s)  Transadmittance

V1 ( s)  Transfer admittance
V ( s)
Gv ( s )  2
Voltage gain
V1 ( s)
YT ( s) 
EXAMPLE
VOC ( s) 
sL
V1 ( s)
sL  R1
We will use Thevenin’s theorem
1
sLR1
1

 R1 || sL 
sC sL  R1
sC
s 2 LCR1  sL  R1
ZTH ( s ) 
sC ( sL  R1 )
ZTH ( s) 
I 2 ( s)  Transadmittance

V1 ( s)  Transfer admittance
V ( s)
Gv ( s )  2
Voltage gain
V1 ( s)
YT ( s) 
ZTH (s)

VOC (s)


sL
V1 ( s )
sL  R1
VOC ( s)
sC ( sL  R1 )
I 2 ( s) 


s 2 LCR1  sL  R1 sC ( sL  R1 )
R2  ZTH ( s)
R2 
sC ( sL  R1 )
I 2 ( s)
R2 V2 ( s )

s 2 LC
YT ( s)  2
s ( R1  R2 ) LC  s( L  R1R2C )  R1
Gv ( s) 
V2 ( s ) R2 I 2 ( s)

 R2YT ( s )
V1 ( s )
V1 ( s)
POLES AND ZEROS
(More nomenclature)
am s m  am 1s m 1  ...  a1s  a0
H ( s) 
bn s n  bn1s n1  ...  b1s  b0
Arbitrary network function
Using the roots, every (monic) polynomial can be expressed as a
product of first order terms
H ( s)  K 0
( s  z1 )( s  z2 )...( s  zm )
( s  p1 )( s  p2 )...( s  pn )
z1, z2 ,..., zm  zeros of the network function
p1, p2 ,..., pn  poles of the network function
The network function is uniquely determined by its poles and zeros
and its value at some other value of s (to compute the gain)
EXAMPLE
zeros : z1  1,
poles : p1  2  j 2, p2  2  j 2
H (0)  1
H ( s)  K 0
( s  1)
s 1
 K0 2
( s  2  j 2)( s  2  j 2)
s  4s  8
1
H (0)  K 0  1 
8
H ( s)  8
s 1
s 2  4s  8
SINUSOIDAL FREQUENCY ANALYSIS
A0e j ( t  ) 

B0 cos( t   )
H (s )

A0 H ( j )e j ( t  )

 B0 | H ( j ) | cos t    H ( j ) 
Circuit represented by
network function
To study the behavior of a network as a function of the frequency we analyze
the network function H ( j ) as a function of .
Notation
M ( ) | H ( j ) |
 ( )  H ( j )
H ( j )  M ( )e j ( )
Plots of M ( ), ( ), as function of  are generally called
magnitude and phase characteristics.
20 log10 (M ( ))
BODE PLOTS
vs log10 ( )

(

)

HISTORY OF THE DECIBEL
Originated as a measure of relative (radio) power
P2 |dB (over P1 )  10 log
P2
P1
V2
V22
I 22
PI R
 P2 |dB (over P1 )  10 log 2  10 log 2
R
V1
I1
2
V |dB  20 log10 | V |
By extension
I |dB  20 log10 | I |
G |dB  20 log10 | G |
Using log scales the frequency characteristics of network functions
have simple asymptotic behavior.
The asymptotes can be used as reasonable and efficient approximations
General form of a network function showing basic terms
Poles/zeros at the origin
Frequency independent
K 0 ( j ) N (1  j1 )[1  2 3 ( j3 )  ( j3 )2 ]...
H ( j ) 
(1  ja )[1  2 b ( jb )  ( jb )2 ]...
log( AB )  log A  log B First order terms
N
log( )  log N  log D
D
Quadratic terms for
complex conjugate poles/zeros
| H ( j ) |dB  20 log10 | H ( j ) |  20 log10 K 0  N 20 log10 | j |
 20 log10 | 1  j1 | 20 log10 | 1  2 3 ( j3 )  ( j3 ) 2 | ...
 20 log10 | 1  ja | 20 log10 | 1  2 b ( jb )  ( jb ) 2 | ..
z1z2  z1  z2 H ( j )  0  N 90
Display each basic term
z1
2


  z1  z2
1
1
separately and add the
3
3

tan


tan

...
1
z2
results to obtain final
1  (3 ) 2
2 bb
 tan 1 a  tan 1
 ...
1  (b ) 2
answer
Let’s examine each basic term
Constant Term
the x - axis is log10
this is a straight line
Poles/Zeros at the origin
( j )
N
| ( j )  N |dB   N  20 log10 ( )

( j )  N   N 90

| 1  j |dB  20 log10 1  ( ) 2
1

j

Simple pole or zero


(1  j )  tan 1 
 1  | 1  j |dB  0 low frequency asymptote
(1  j )  0
 1  | 1  j |dB  20 log10  high frequency asymptote (20dB/dec)
The two asymptotes meet when   1(corner/break frequency)
(1  j )  90
Behavior in the neighborhood of the corner
corner
octave above
octave below
distance to
FrequencyAsymptoteCurve asymptote Argument
3dB
3
45
  1 0dB

 2
6dB
7db
1
63.4
  0 .5
0dB
1dB
1
26.6
Asymptote for phase
Low freq. Asym.
High freq. asymptote
Simple zero
Simple pole
Quadratic pole or zero t 2  [1  2 ( j )  ( j ) ]  [1  2 ( j )  ( )2 ]
2
| t 2 |dB  20 log10
1  ( ) 
2 2
t 2  tan 1
 2 
2
  1 | t2 |dB  0 low frequency asymptote
2
1  ( ) 2
t2  0
  1 | t2 |dB  20 log10 ( )2 high freq. asymptote 40dB/dec t2  180
  1 | t2 |dB  20 log10 (2 ) Corner/break frequency
t2  90
  1  2 2 | t2 |dB  20 log10 2 1   2 Resonance frequency
These graphs are inverted for a zero
Magnitude for quadratic pole

t 2  tan
1
1  2 2

Phase for quadratic pole

1
2Q
2
2
LEARNING EXAMPLE
Draw asymptotes
for each term
Generate magnitude and phase plots
Gv ( j ) 
10(0.1 j  1)
( j  1)(0.02 j  1)
Breaks/corners : 1,10,50
Draw composites
dB
40
20
10 |dB
20dB / dec
0
 20dB / dec
 20
90
45 / dec
 45 / dec
0.1
1
10
100
90
1000
asymptotes
DETERMINING THE TRANSFER FUNCTION FROM THE BODE PLOT
This is the inverse problem of determining frequency characteristics.
We will use only the composite asymptotes plot of the magnitude to postulate
a transfer function. The slopes will provide information on the order
A. different from 0dB.
There is a constant Ko
A
B
C
K 0 |dB  20  K 0
D
E
K 0 |dB
 10 20
B. Simple pole at 0.1
( j / 0.1  1)1
C. Simple zero at 0.5
( j / 0.5  1)
D. Simple pole at 3
( j / 3  1)1
E. Simple pole at 20
G ( j ) 
10( j / 0.5  1)
( j / 0.1  1)( j / 3  1)( j / 20  1)
( j / 20  1)1
If the slope is -40dB we assume double real pole. Unless we are given more data
LEARNING EXAMPLE
A function with complex conjugate poles
t 2  [1  2 ( j )  ( j )2 ]
Put in standard form
G ( j ) 
G ( j ) 
Draw composite asymptote
25 j
( j  0.5) ( j ) 2  4 j  100


0.5 j
( j / 0.5  1) ( j / 10) 2  j / 25  1


2  1 / 25
    0.2
  0.1 
dB
40
20
  1 | t2 |dB  20 log10 (2 )
0
8dB
 20
90
90
Behavior close to corner of conjugate pole/zero
is too dependent on damping ratio.
Computer evaluation is better
0.01
0.1
1
10
 270
100
Evaluation of frequency response using MATLAB
G ( j ) 
Using default options
25 j
( j  0.5) ( j ) 2  4 j  100


» num=[25,0]; %define numerator polynomial
» den=conv([1,0.5],[1,4,100]) %use CONV for polynomial multiplication
den =
1.0000
4.5000 102.0000
50.0000
» freqs(num,den)
> pzmap(num,den)
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