Download - Network Analysis and Synthesis
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Network Analysis and Synthesis
Chapter 2Network transform representation
and analysis
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2.1 The transformed circuit
• When analyzing a network in time domain we will be dealing with– Derivation and – Integration
• However, when transformed to complex frequency domain these become– Derivation -> multiplication by ‘s’– Integration -> division by ‘s’
• Hence, it is easier to do network analysis in complex frequency domain.
![Page 3: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/3.jpg)
• The voltage current relationships of network elements in time domain and complex frequency domain are given as:
• Resistor
)()( tRitv )()( sRIsV
![Page 4: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/4.jpg)
• Inductor– The time domain relation ships are
– In frequency domain they become
)0()(1
)(
)()(
0
idvL
ti
dt
tdiLtv
t
s
i
sL
sVsI
LissLIsV
)0()()(
)0()()(
![Page 5: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/5.jpg)
![Page 6: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/6.jpg)
• An inductor is represented in frequency domain as– An impedance sL in series with a voltage source
Used in mesh analysis.or
– An admittance 1/sL in parallel with a current source
Used in nodal analysis.
![Page 7: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/7.jpg)
• Capacitor– The time domain relation ships are
– In frequency domain they becomedt
tdvCti
vdiC
tvt
)()(
)0()(1
)(0
)0()()(
)0()()(
CvssCVsI
s
v
sC
sIsV
![Page 8: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/8.jpg)
![Page 9: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/9.jpg)
• A capacitor is represented in frequency domain as– An impedance 1/sC in series with a voltage
sourceUsed in mesh analysis.or
– An admittance sC in parallel with a current sourceUsed in nodal analysis.
![Page 10: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/10.jpg)
Example 1
• In the figure below, the switch is switched from postion 1 to 2 at t=0. Draw its transformed circuit and write the transformed equations using mesh analysis.
![Page 11: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/11.jpg)
• The transformed circuit is
![Page 12: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/12.jpg)
• The transformed equations become
![Page 13: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/13.jpg)
Example 2
• The switch is thrown to position 2 at t=0. Find i(t). Vv
ampi
C
L
2)0(
2)0(
![Page 14: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/14.jpg)
• The transformed circuit is
![Page 15: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/15.jpg)
• Writing the transformed equation
• Solving for I(s)• Inverse transforming
)(2
32
25
sIs
sss
1
1
2
1)(
)1)(2(
32)(
sssI
ss
ssI
tt eeti 2)(
![Page 16: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/16.jpg)
Example 3
• At t=0, the switch is opened. Find the node voltages v1 and v2
v1 mho 1
f1 2
1
VG
ChL
![Page 17: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/17.jpg)
• The transformed circuit becomes
![Page 18: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/18.jpg)
• The transformed equations become
• Solving these 2 equations
![Page 19: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/19.jpg)
2.2 System function
• The excitation , e(t), and response, r(t), of a linear system are related by a linear differential equation.
• When transformed to complex frequency domain the relationship between excitation and response is algebraic one.
• When the system is initially inert, the excitation and response are related by the system function H(s) given by
)()()( sEsHsR
![Page 20: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/20.jpg)
• The system function may have many different forms and may have special names. Such as:– Driving point admittance– Transfer impedance– Voltage or current ratio transfer function
• This is because the excitation and response may be taken from the same port or different ports and the excitation and response can be either voltage or current.
![Page 21: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/21.jpg)
Impedance
• Transfer impedance is when the excitation is a current source and the response is a voltage.
• When both the excitation and response is at the same port it is called driving point impedance.
)(
)()( 0
sI
sVsH
g
sLsC
sLsCRsH
1
1
)(
![Page 22: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/22.jpg)
Admittance
• Transfer admittance is when the excitation is a voltage source and the response is a current.
)(
)()( 0
sV
sIsH
g
RsC
sLsH
11
)(
![Page 23: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/23.jpg)
Voltage ratio transfer function
• When the excitation is a voltage source and the response is a voltage.
)(
)()( 0
sV
sVsH
g
)()(
)()(
21
2
sZsZ
sZsH
![Page 24: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/24.jpg)
Current ratio transfer function
• When the excitation is a current source and the response is a current.
)(
)()( 0
sI
sIsH
g
sCsLR
sCsH
RsLsC
RsLsH
1
1
)(
1
1
)(
![Page 25: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/25.jpg)
• Note that, the system function is a function of the system elements only.
• It is obtained from the network by using the standard circuit laws. Such as:– Kirchhoffs law– Nodal analysis– Mesh analysis
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Example 4
• Obtain the driving point impedance of the network. Then using the following excitations determine the response.1. 2. The square pulse on figure b3. The waveform on figure c
)()( tutSinwti og
a b c
![Page 27: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/27.jpg)
• First lets find the driving point impedance• Note that it is the equivalent impedance of
the 3 elements
CLs
CG
sC
s
GsL
sCsH
111
)(2
![Page 28: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/28.jpg)
1. Its transform is
Hence, the response is
)()( tutSinwti og
220)(
ows
wSI
LCCGssC
s
ws
wsHsIsV
o
ogo 1
.)()()(2
22
![Page 29: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/29.jpg)
2. The excitation is given as
Hence, the response is
asess
sI
atututi
11
)(
)()()(
LCCGssC
s
s
esHsIsV
as
go 1.
1)()()(
2
![Page 30: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/30.jpg)
3. The excitation is given as
)()()()( atua
attu
a
ttutig
22
11)(
as
e
asssI
as
![Page 31: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/31.jpg)
• Consider the partial fraction expansion of R(s)where si are the poles of H(s) and sj are the poles of E(s).
• Taking the inverse Laplace transform of R(s)
• The terms are associated with the system H(s) and are called the free response terms.
tsi
ieA
![Page 32: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/32.jpg)
• The terms are due to the excitation E(s) and are called the forced response terms.
• The frequencies si are the natural frequency of the system, while the frequencies sj are the frequencies of the excitation.
tsj
jeB
![Page 33: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/33.jpg)
Problem
• Find the free response and the forced response for the circuit below. The system is inert before applying the source.
)()(cos2
1)( tuttvg
![Page 34: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/34.jpg)
2.3 Poles and zeros of system
• We will discuss the relationship between the poles and zeros of a system function and its steady state sinusoidal response.
• In other words, we will investigate the effect of positions of poles and zeros upon H(s) on the jw axis.
![Page 35: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/35.jpg)
• To find the steady-state sinusoidal response of a system function we replace ‘s’ by ‘jw’.
• Hence, the system function becomes
Where M(w) is the amplitude or magnitude response φ(w) is the phase response
)()()(
|)()(wj
jws
ewMjwH
sHjwH
![Page 36: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/36.jpg)
• The amplitude and phase response of a system provide valuable information in the analysis and design of transmission circuits.
• Consider the low pass filter• Observe that– It passes only frequency
below wc– The phase response is
almost linear till wc
![Page 37: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/37.jpg)
• Hence, if all the significant harmonic terms are less than wc , then the system will produce minimum phase distortion.
• In the rest of this section, we will concentrate on methods to obtain amplitude and phase response curves.
![Page 38: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/38.jpg)
R-C network
•
• To obtain H(jw) we substitute s by jw.RCs
RC
sCR
sCsV
sVsH
1
1
1
1
)(
)()(
1
2
RCjw
RCjwH1
1
)(
![Page 39: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/39.jpg)
• In polar form H(jw) becomes
)(tan
2
1
222
)(1
1
)(1 wjwRCj ewMe
CRw
RCjwH
wRCw
CRw
RCwM
1
2
1
222
tan)(
1
1
)(
![Page 40: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/40.jpg)
• The amplitude is unity and the phase is zero degrees at w=0.
• The amplitude and phase decrease monotonically as we increase w.
• When w=1/RC, the amplitude is 0.707 and phase is -450.
• As w increases to infinity M(w) goes to zero and the phase approaches -900.
Half power point
![Page 41: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/41.jpg)
Amplitude and phase from pole-zero diagram
• For the system function
• H(jw) can be written as
• Each one of the or represent a vector from zi or pj to the jw axis at w.
))()((
))(()(
210
100
pspsps
zszsAsH
))()((
))(()(
210
100
pjwpjwpjw
zjwzjwAjwH
)( izjw )( jpjw
![Page 42: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/42.jpg)
• If we express
• Then H(jw) can be given as
jij
jjj
ii eMpjweNzjw ,
210210
210
210)( jeMMM
NNAjwH
![Page 43: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/43.jpg)
![Page 44: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/44.jpg)
• In general,
![Page 45: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/45.jpg)
Example
• For find the magnitude and phase for w=2.
• Solution– First let us find the zeros and poles
– Zero at jw=0– Poles at
22
4)(
2
ss
ssF
)1)(1(
4)(
jjwjjw
jwjwF
)1( )1( jjwandjjw
![Page 46: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/46.jpg)
![Page 47: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/47.jpg)
• Magnitude
• Phase
5
4
10*2
2*4)2( jM
0000 8.26458.7190)2( j
![Page 48: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/48.jpg)
Exercise
• Examine the property of F(s) around the poles and zeroes.
![Page 49: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/49.jpg)
Bode plots
• In this section we turn our attention to semi logarithmic plots of system function, called Bode plots.
• In these plots we take the logarithm of the amplitude and plot it on linear frequency scale.
• For amplitude M(jw), if we express in terms of decibel it becomes )(log20 jwM
![Page 50: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/50.jpg)
• For system function
• If we express the amplitude in terms of decibels we have
|)(|
|)(||)(|)(
)(
)()(
jwD
jwNjwHjwM
sD
sNsH
|)(|log20|)(|log20)(log20 jwDjwNjwM
![Page 51: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/51.jpg)
• In factored from both N(s) and D(s) are made up of 4 kinds of terms1. Constant K2. A root at origin, s3. A simple real root, s-a4. A complex set of roots,
• To understand the nature of log-amplitude plots, we only need to discuss the amplitude response of these 4 terms.
• If the term is on the numerator it carries positive sign, if on denominator negative sign.
222 2 ss
![Page 52: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/52.jpg)
1. Constant K
• The dB gain or loss is
• K2 is either positive |K|>1 or negative |K|<1.• The phase is either 00 for K>0, or 1800 for K<0.
2log20 KK
![Page 53: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/53.jpg)
Single root at origin, s
• The loss or gain of a single root at origin is
• Thus the plot of magnitude in dB vs frequency is a straight line with slope of 20 or -20.
• 20 when s is in the numerator.• -20 when s is in the denominator.
• The phase is either 900 or -900.• 900 when s is in the numerator.• -900 when s is in the denominator.
wjw log20||log20
![Page 54: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/54.jpg)
![Page 55: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/55.jpg)
The factor s+α
• For convenience lets set α=1. Then the magnitude is
• The phase is
• A straight line approximation can be obtained by examining the asymptotic behavior of the factor jw+1.
2
12 1log20|1|log20 wjw
wjw 1tan)1arg(
![Page 56: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/56.jpg)
• For w<<1, the low frequency asymptote is
• For w>>1, the high frequency asymptote is
Which has a slope of • These 2 asymptotic approximations meet at w=1.
dBw 01log201log20 2
12
ww log201log20 2
12
cadedecibel/de log20 w
![Page 57: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/57.jpg)
![Page 58: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/58.jpg)
• Note that the maximum error is for w=1 or for the non normalized one w=α.
• For the general case α different from 1, we normalize the term by dividing by α.
• The low frequency asymptote is
• The high frequency asymptote isdB
w01log201log20
2
1
2
2
log20log201log202
1
2
2
w
w
![Page 59: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/59.jpg)
For complex conjugates
• For complex conjugates it is convenient to adopt a standard symbol.
• We describe the pole (zero) in terms of magnitude ω0 and angle θ measured from the negative real axis.
• These parameters that describe the pole (zero) are ω0, the undamped frequency of oscillation, and ζ, the damping factor.
![Page 60: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/60.jpg)
• If the pole (zero) pair is given as
• α and β are related to ω0 and ζ with
• Substituting these terms in the conjugate equation
jp 2,1
200
00
1sin
cos
))(( 21 psps
2
002
200
200
2
11))((
jww
jjwjjwjjwjjw
![Page 61: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/61.jpg)
• For ω0=1 (for convenience), the magnitude of conjugate pairs can be expressed as
• The phase is
2122222 41log2021log20 wwwjw
21
1
22tan)(
ww
![Page 62: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/62.jpg)
• The asymptotic behavior is – For low frequency, w<<1
– For high frequency, w>>1
which is a straight line with slope of 40dB/decade.• These 2 asymptotes meet at w=1.
dBww 01log2041log20 2
12222
www log4041log20 2
12222
![Page 63: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/63.jpg)
![Page 64: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/64.jpg)
Example
• Using Bode plot asymptotes, draw the magnitude vs. frequency for the following system function
1
1010*161
50
1.0)(
34
2 sss
ssG
![Page 65: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/65.jpg)
![Page 66: Network Analysis and Synthesis](https://reader030.vdocuments.us/reader030/viewer/2022012404/56815303550346895dc12584/html5/thumbnails/66.jpg)
Actual plot