NETWORK SYNTHESIS
EE-1736
B.TECH 7 SEMESTER
EED
July-December 2020
Prof. Richa Negi
(Course Co-ordinator)
LECTURE PLAN
B.TECH 7 SEMESTER EE
NETWORK SYNTHESIS (EE-1736)
S.No. Topic No. of Lectures
Unit I Elements of Network Synthesis
1. Synthesis of L- C Driving –point Immitances 2
2. Synthesis of R-C Impedances 1
3. R-L Admittances 1
4. Synthesis of certain R-L -C Functions 1
Unit II Elements of Transfer function Synthesis
5. Properties of Transfer function 2
6. Synthesis of Y21 Z 21 with 1-Ω termination 3
7. Synthesis of Constant Resistance Networks 2
UNIT III Filter Design
8. Filter design problem 1
9. Low Pass Filter Approximations 1
10. Synthesis of Low Pass Filter 2
11. Magnitude and Frequency Normalization 1
12. Frequency Transformations 1
UNIT IV Biquad Circuits
13. Biquad Circuits 1
14. Four Op-Amp Biquad Circuit 1
15. Frequency and Phase Response of Biquad Circuit 1
16. Butterworth Low Pass filter 2
17. Chebychev, Bessel Thomson Filter 2
UNIT V Leapfrog Simulation of Ladders
18. Ladder Simulation 2
19. Bandpass Leapfrog Filters 1
20. Active Resonators 1
21. Bandpass Leapfrog Design 1
22. Girling-Good Form of Leapfrog 1
UNIT VI Switched Capacitor Filters
23. Switched Capacitor,Analog Operations 1
24. Range of Circuit elements sizes 1
25. Bandpass Switched –Capacitor Filters 1
26. OP Amp Oscillators: Loop gain, Conditons for Third – 3
Order Circuit Oscillations Amplitude Stabilization
Text/ Reference Books:
Franklin F. Kuo, “Network Analysis and Synthesis” ,John Wile
M E Valkenberg, “Analog Filter Design” ,Oxford University Press.
A S Sedra and P O Brackett, “Filter Theory and design: Active and Passive”,
Matrix Publishers.
F.W.Stephenson, “RC Active Filter Design Handbook”, John Wiley & Sons
Wai-Kai Chen, “Passive and Active Filters Theory and Implementations”, John
Wiley & Sons.
UNIT I
LC Driving Point Immitance Function
ZLC (s) or YLC (s) is the ratio of odd to even or even to odd polynomials.
Consider the impedance Z(s) of passive one-port network.
M ( s ) N1 ( s )
Z ( s) 1 (M is even, N is odd)
M 2 (s) N 2 (s)
As we know, when the input current is I, the average power dissipated by one-port
network is :
1
Average Power= Re Z ( j ) I
2
2
For pure reactive network, power dissipated is zero. Therefore,
Re Z ( j ) Ev Z ( j ) 0
where
M 1 ( s ) M 2 ( s ) N1 ( s ) N 2 ( s )
Ev Z ( s )
M 22 ( s ) N 22 ( s )
In order for Ev Z ( j ) 0 , that is M1 ( j )M 2 ( j ) N1 ( j ) N2 ( j ) 0
Either of the following cases must hold:
(a)M1 0 N2 or (b) M 2 0 N1
N1
In case (a) Z ( s)
M2
M1
In case (b) Z ( s )
N2
Properties-
1. Z(s) or Y(s) is the ratio of even to odd or odd to even.
2. The poles and zeros are simple and lie j on the axis.
N1 M
Z ( s) , Z (s) 1
M2 N2
Since both M and N are Hurwitz, they have only imaginary roots, and it follows
that the poles and zeros of Z(s) or Y(s) are on the imaginary axis.
a4 s 4 a2 s 2 a0
Consider the example Z ( s)
b5 s 5 b3 s 3 b1s
In order for the impedance to be positive real, the coefficients must be real and
positive.
Impedance function cannot have multiple poles or zeros on the axis.
The highest orders of the numerator and the denominator polynomials can differ
by, at most, unity.
Example-If highest order of the numerator : 2n, highest order of the denominator
can either be 2n-1 (simple pole at s ) or the order can be 2n+1 (simple zero at
s ).
The lowest orders of the numerator and the denominator polynomials can differ
by, at most, unity else there would be multiple poles or zeros of Z(s) at s=0.
3. The poles and zeros interlace on the jω axis.
If highest power is 2n, next highest power must be 2n-2. There cannot be missing
term.
We can write a general L-C impedance or admittance as-
Since these poles are all on the jω axis, the residues must be real and positive in
order for Z(s) to be positive real . S=jω implies Z(jω)=jX(ω) (no real part).
Since all the residues Ki are positive, it is easy to see that for an L-C function
dX ( )
0
d
Ks( s 2 32 ) K ( 2 32 )
Example: Z ( s) 2 jX ( ) j
( s 22 )( s 2 42 ) ( 2 22 )( 2 42 )
4. The highest powers of the numerator and denominator must differ by unity; the
lowest powers also differ by unity.
5. There must be either a zero or a pole at the origin and infinity.
Summary of properties
1. ZLC (s) or YLC (s) is the ratio of odd to even or even to odd
polynomials.
2. The poles and zeros are simple and lie on the jω axis.
3. The poles and zeros interlace on the jω axis.
4. The highest powers of the numerator and denominator must differ
by unity; the lowest powers also differ by unity.
5. There must be either a zero or a pole at the origin and infinity.
Examples
The following functions are not LC Immitance Functions
The following function is an LC Immitance Function
Synthesis of L-C Driving point Immittance
L-C immittance is a positive real function with poles and zeros on the jω axis only.
The synthesis is accomplished directly from the partial fraction.
If F(s) is an impedance Z(s), then term
K0/s = Capacitor of 1/ K0 farads
K∞s = Inductance of K∞ Henrys
2Kis/(s2+ωi2)=parallel tank circuit
It consists of a capacitor of 1/2Ki farads in parallel with an inductance of
2Ki/ ωi .
2
Example Consider the LC function
For Z(s) partial fraction
Synthesized Network
For Y(s) partial fraction
Synthesized Network
Impedance Form is called Foster Series Network
Admittance Form is called Foster Parallel Network.
Ladder Network
Using property 4 “The highest powers of numerator and denominator must differ by
unity; the lowest powers also differ by unity.”
Therefore, there is always a zero or a pole at s=∞.
Suppose Z(s) numerator:2n ,denominator:2n-1
This network has pole at ∞
we can remove this pole by removing an impedance L1 s
Z2 (s) = Z(s) - L1 s
Degree of denominator : 2n-1 , numerator:2n-2
Z2 (s) has zero at s=infinite.
Y2 (s) =1/ Z2 (s) , Y3 (s) = Y2 (s) - C2s
This infinite term removing process continues until the remainder is zero.
Each time we remove the pole, we remove an inductor or capacitor depending upon
whether the function is impedance or admittance.
Finally synthesized is a ladder whose series arms are inductors and shunt arms are
capacitors.
This circuit (Ladder) called as Cauer because Cauer discovered the continuous fraction
method.
Both the Foster and Cauer form the minimum number of elements for a specified L-C
network.
Example of Cauer Method
Reference
Franklin F. Kuo, “Network Analysis and Synthesis”, John Wile
