1. Variable-Frequency Response Analysis
Network performance as function of frequency.
Transfer function
Sinusoidal Frequency Analysis
Bode plots to display frequency response data
VARIABLE-FREQUENCY NETWORK
PERFORMANCE
2. 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
0
R
R
ZR
Resistor
5. Frequency dependent behavior of series RLC network
C
j
RC
j
LC
j
C
j
L
j
R
Zeq
1
)
(
1 2
C
LC
j
RC
j
j
)
1
( 2
C
LC
RC
Zeq
2
2
2
)
1
(
)
(
|
|
RC
LC
Zeq
1
tan
2
1
sC
sRC
LC
s
s
Z
s
j
eq
1
)
(
2
notation"
in
tion
Simplifica
"
6. For all cases seen, and all cases to be studied, the impedance is of the form
0
1
1
1
0
1
1
1
...
...
)
(
b
s
b
s
b
s
b
a
s
a
s
a
s
a
s
Z n
n
n
n
m
m
m
m
sC
Z
sL
s
Z
R
s
Z C
L
R
1
,
)
(
,
)
(
Simplified notation for basic components
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
sL
sC
1
R
S
o V
sC
sL
R
R
s
V
/
1
)
(
S
V
sRC
LC
s
sRC
1
2
S
o V
RC
j
LC
j
RC
j
V
j
s
1
)
( 2
0
10
1
)
10
53
.
2
15
(
)
10
53
.
2
1
.
0
(
)
(
)
10
53
.
2
15
(
3
3
2
3
j
j
j
Vo
7. NETWORK FUNCTIONS
INPUT OUTPUT TRANSFER FUNCTION SYMBOL
Voltage Voltage Voltage Gain Gv(s)
Current Voltage Transimpedance Z(s)
Current Current Current Gain Gi(s)
Voltage Current Transadmittance Y(s)
When voltages and currents are defined at different terminal pairs we
define the ratios as Transfer Functions
If voltage and current are defined at the same terminals we define
Driving Point Impedance/Admittance
Some nomenclature
admittance
Transfer
tance
Transadmit
)
(
)
(
)
(
1
2
s
V
s
I
s
YT
gain
Voltage
)
(
)
(
)
(
1
2
s
V
s
V
s
Gv
To compute the transfer functions one must solve
the circuit. Any valid technique is acceptable
EXAMPLE
8. EXAMPLE
admittance
Transfer
tance
Transadmit
)
(
)
(
)
(
1
2
s
V
s
I
s
YT
gain
Voltage
)
(
)
(
)
(
1
2
s
V
s
V
s
Gv
We will use Thevenin’s theorem
sL
R
sC
s
ZTH ||
1
)
( 1
1
1
1
R
sL
sLR
sC
)
(
)
(
1
1
1
2
R
sL
sC
R
sL
LCR
s
s
ZTH
)
(
)
( 1
1
s
V
R
sL
sL
s
VOC
)
(s
VOC
)
(s
ZTH
)
(
2 s
V
2
R
)
(
2 s
I
)
(
)
(
)
(
2
2
s
Z
R
s
V
s
I
TH
OC
)
(
)
(
1
1
1
2
2
1
1
R
sL
sC
R
sL
LCR
s
R
s
V
R
sL
sL
1
2
1
2
1
2
2
)
(
)
(
)
(
R
C
R
R
L
s
LC
R
R
s
LC
s
s
YT
)
(
)
(
)
(
)
(
)
(
)
( 2
1
2
2
1
2
s
Y
R
s
V
s
I
R
s
V
s
V
s
G T
v
)
(
)
(
1
1
R
sL
sC
R
sL
sC
9. POLES AND ZEROS (More nomenclature)
0
1
1
1
0
1
1
1
...
...
)
(
b
s
b
s
b
s
b
a
s
a
s
a
s
a
s
H n
n
n
n
m
m
m
m
Arbitrary network function
Using the roots, every (monic) polynomial can be expressed as a
product of first order terms
)
)...(
)(
(
)
)...(
)(
(
)
(
2
1
2
1
0
n
m
p
s
p
s
p
s
z
s
z
s
z
s
K
s
H
function
network
the
of
poles
function
network
the
of
zeros
n
m
p
p
p
z
z
z
,...,
,
,...,
,
2
1
2
1
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
1
)
0
(
2
2
,
2
2
,
1
2
1
1
H
j
p
j
p
z
:
poles
:
zeros
)
2
2
)(
2
2
(
)
1
(
)
( 0
j
s
j
s
s
K
s
H
8
4
1
2
0
s
s
s
K
1
8
1
)
0
( 0
K
H
8
4
1
8
)
( 2
s
s
s
s
H
10. SINUSOIDAL FREQUENCY ANALYSIS
)
(s
H
Circuit represented by
network function
)
cos(
0
)
(
0
t
B
e
A t
j
)
(
cos
|
)
(
|
)
(
0
)
(
0
j
H
t
j
H
B
e
j
H
A t
j
)
(
)
(
)
(
)
(
)
(
|
)
(
|
)
(
j
e
M
j
H
j
H
j
H
M
Notation
stics.
characteri
phase
and
magnitude
called
generally
are
of
function
as
of
Plots
),
(
),
(
M
)
(
log
)
(
))
(
log
20
10
10
PLOTS
BODE vs
(M
.
of
function
a
as
function
network
the
analyze
we
frequency
the
of
function
a
as
network
a
of
behavior
the
study
To
)
( j
H
11. HISTORY OF THE DECIBEL
Originated as a measure of relative (radio) power
1
2
)
2 log
10
(
|
P
P
P dB
1
P
over
2
1
2
2
2
1
2
2
)
2
2
2
log
10
log
10
(
|
I
I
V
V
P
R
V
R
I
P dB
1
P
over
By extension
|
|
log
20
|
|
|
log
20
|
|
|
log
20
|
10
10
10
G
G
I
I
V
V
dB
dB
dB
Using log scales the frequency characteristics of network functions
have simple asymptotic behavior.
The asymptotes can be used as reasonable and efficient approximations
12. ]...
)
(
)
(
2
1
)[
1
(
]...
)
(
)
(
2
1
)[
1
(
)
(
)
( 2
2
3
3
3
1
0
b
b
b
a
N
j
j
j
j
j
j
j
K
j
H
General form of a network function showing basic terms
Frequency independent
Poles/zeros at the origin
First order terms Quadratic terms for
complex conjugate poles/zeros
..
|
)
(
)
(
2
1
|
log
20
|
1
|
log
20
...
|
)
(
)
(
2
1
|
log
20
|
1
|
log
20
|
|
log
20
log
20
2
10
10
2
3
3
3
10
1
10
10
0
10
b
b
b
a j
j
j
j
j
j
j
N
K
D
N
D
N
B
A
AB
log
log
)
log(
log
log
)
log(
|
)
(
|
log
20
|
)
(
| 10
j
H
j
H dB
2
1
2
1
2
1
2
1
z
z
z
z
z
z
z
z
...
)
(
1
2
tan
tan
...
)
(
1
2
tan
tan
90
0
)
(
2
1
1
2
3
3
3
1
1
1
b
b
b
a
N
j
H
Display each basic term
separately and add the
results to obtain final
answer
Let’s examine each basic term
13. Constant Term
Poles/Zeros at the origin
90
)
(
)
(
log
20
|
)
(
|
)
( 10
N
j
N
j
j N
dB
N
N
line
straight
a
is
this
log
is
axis
-
x
the 10
14. Simple pole or zero
j
1
1
2
10
tan
)
1
(
)
(
1
log
20
|
1
|
j
j dB
asymptote
frequency
low
0
|
1
|
dB
j
(20dB/dec)
asymptote
frequency
high
10
log
20
|
1
|
dB
j
frequency)
ak
corner/bre
1
when
meet
asymptotes
two
The (
Behavior in the neighborhood of the corner
FrequencyAsymptoteCurve
distance to
asymptote Argument
corner 0dB 3dB 3 45
octave above 6dB 7db 1 63.4
octave below 0dB 1dB 1 26.6
1
2
5
.
0
0
)
1
(
j
90
)
1
(
j
1
1
Asymptote for phase
High freq. asymptote
Low freq. Asym.
18. 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
A. different from 0dB.
There is a constant Ko
B
B. Simple pole at 0.1
1
)
1
1
.
0
/
(
j
C
C. Simple zero at 0.5
)
1
5
.
0
/
(
j
D
D. Simple pole at 3
1
)
1
3
/
(
j
E
E. Simple pole at 20
1
)
1
20
/
(
j
)
1
20
/
)(
1
3
/
)(
1
1
.
0
/
(
)
1
5
.
0
/
(
10
)
(
j
j
j
j
j
G
20
|
0
0
0
10
20
|
dB
K
dB K
K
If the slope is -40dB we assume double real pole. Unless we are given more data
