1. Network Analysis and Synthesis
Chapter 6
Synthesis of two port networks

2. 5.1 Introduction
In this chapter we will discuss 2 of the most widely used two port synthesis methods
Coefficient matching and
Darlington method (insertion loss method)

3. 5.2 Coefficient matching
It is one of the most simple and effective method for two port synthesis.
In this method,
We compare the transfer function we want to synthesize with the transfer function of a circuit we already know.
If they have the same form, we try to match the coefficients of the two functions. (Remember that the coefficients of the second transfer function is a function of the circuit elements.)
Once we match the coefficients, we know the values of the elements in the circuit.

4. Example 1 The voltage transfer function of the following circuit is
R1-source impedance
R2-load impedance

5. If we are required to synthesize the following voltage transfer function.
Comparing the two equations and assuming equal termination and normalizing the resistors to 1Ω, R1=R2=1Ω

6. Example 2 Consider the following network
The voltage transfer function is

7. If we were to synthesize the following voltage transfer function assuming equal termination with 1Ω resistor.

8. Pros and Cons of Coefficient matching
Simple
Effective
Cons
When the order of the transfer function increases, the number of simultaneous equations we have to solve for increases.
Doesn’t demonstrate sophisticated network design methods.

9. 5.3 Insertion Loss (Darlington Method)
A low pass characteristics can be obtained by using RC, RL or LC networks.
Low cost, low sensitivity to component variations and simplicity of design make LC two port networks the most widely used filter networks.
Here the network is assumed to be doubly terminated. (A valid assumption in almost all cases.)

10. A powerful method for designing doubly terminated LC two port networks is the Darlington method.
It is one of the most effective method of realizing a two port network: Insertion loss method (Darlington method).

11. In Darlington method of filter design,
The specifications of the insertion loss of the filter is converted to the reflection coefficients (related to the maximum power that can be delivered by the source vs. the actual delivered power to the load) of the filter.
From the reflection coefficients the driving point impedance of the terminated networks is obtained.
Then this driving-point impedance is developed into resistively terminated LC ladder network.

12. Procedures of Darlington synthesis
The derivation of Darlington method is complicated, hence, we will just discuss the procedure for using the Darlington method.
Procedure
From F(s) obtain the reflection coefficient p(s)

13. From ρ(s) determine the normalized Z(s).
The zeros of q1(s) are the left plane zeroes of q(s).
The zeros of p(s) are equally distributed between the zeros of ρ (s) and ρ (-s), with restrictions that conjugate zeros must be together.
From ρ(s) determine the normalized Z(s).
Expand Z(s) into continued fraction expansion about infinity and obtain the ladder. The two impedances defined above lead to 2 (dual) ladders, one terminated with R2 and the other 1/R2.

14. Example 3
Synthesize the following voltage transfer function using Insertion loss method
Solution:
To find the reflection coefficient
The zeros of q(s) are z1=1, z2=-1, z3=-0.5+j0.866, z4=-0.5-j0.866, z5=0.5+j0.866, z6=0.5-j0.866
Hence, poles of ρ(s) are z2=-1, z3=-0.5+j0.866, z4=-0.5-j0.866

15. The zeros of p(s) are 6 multiple poles at s=0.
Hence, the zeros of ρ(s) are 3 multiple poles at s=0.
Hence
The driving point impedance is

16. Using continued fraction expansion

17. Example 4
Synthesize the following voltage transfer function using Insertion loss method with R1=R2=1Ω
Solution:
To find the reflection coefficient
The zeros of p(s) are 8 multiple poles at s=0, hence the zeros of ρ(s) are 4 multiple poles at s=0.

18. The angle between the two zeros is
(Refer at the end for a more detailed explanation on how to get roots of polynomials of the form sn+a)
The zeros of q(s) are evenly distributed on the unit circle on the s plane.
The angle between the two zeros is
Since no zero on real axis or jw plane
and because the zeros of q(s)
have to be conjugate complex,
the angle of one of the roots
from the real axis should be equal
for two conjugate roots.

19. The zeros of q(s) are then
The roots of ρ(s) are the left hand zeros of q(s)

20. ρ(s) becomes
The driving point impedance becomes:

21. Taking the continuous fraction expansion
Hence, the network becomes

22. Finding roots of polynomials of the form sn+a
The roots will be located on the circle with radius of a1/n on the s plane.
Separation between them is
Check if the polynomial has root at s=a1/n or s=-a1/n or s=ja1/n or s=-ja1/n.
Case 1: root at one of these
Start at that root and plot
the roots with separation of θ.

23. Case 2: no root at those locations
Because the roots have to be conjugate complex, the angle one of the roots makes with the real axis will be equal to the negative angle of the conjugate root.
Hence, one of them will be
Start from this root and plot the rest
with separation angle of θ.

24. Example What are the roots of s8+1 Separation between the roots is
Root on one of the axis? No
Hence one of the root is
Plot the rest starting from this one.

25. Example s6-1 The separation angle is
Root at one of the axis? Yes at r1=1.
Hence, plot the rest with separation of 600 starting from r1=1.
The roots are then