chapter-5-synthesis-of-one-port-network-2016.pptx

ppt

1. Network Analysis and Synthesis Chapter 5 Synthesis of deriving point functions (one port networks) 1

2. Synthesis of one port networks with two kinds of elements In this section we will focus on the synthesis of networks with only L-C, R-C or R-L elements. The deriving point impedance/admittance of these kinds of networks have special properties that makes them easy to synthesize. 2

3. 1 . L-C immittance functions These networks have only inductors and capacitors . 3

4. Properties of L-C function The driving point impedance/admittance of an L-C network is even/odd or odd/even. Both are Hurwitz, hence only simple imaginary zeros and poles on the jw axis. Poles and zeros interlace on the jw axis. Highest power of the numerator and denominator may only differ by 1. Either a zero or a pole at origin or infinity. Re[F( jw )]=0 4

5. Synthesis of LC Network There are two kinds of network realization types for two element only networks. Foster I and foster II form Cauer I and cauer II form 5

6. Foster I synthesis The driving point function of a reactive one-port Z(s) is given by or Uses decomposition of the given z(s) into simpler two element impedances This is because Z(s) has poles on the jw axis only. Using the above decomposition, we can realize Z(s) as 6

7. Foster II synthesis The second foster form, known parallel network, is a parallel combination of series LC circuits. Because all branches in the network of foster I are connected in parallel, the network can be simplified by taking the driving point admittance Y(s). Therefore, we have Uses decomposition of the given Y(s) into simpler two element impedances Using the above decomposition, we can realize Y(s) as 7

8. Example : The driving point impedance of a one-port reactive network is given by Obtain the first and second Foster networks. Foster I realization becomes Foster II realization To find the second Foster form, first we have to take the function into admittance form. 8

9. Synthesis of One-Ports LC by the Cauer Method In the Cauer method, there are two types of ladder networks to realize the one-port network. In one type of network, the series arms are inductors and the shunt arms are capacitors as shown in Fig. (a). From the driving point function Z(s) or Y(s), there is always a zero or a pole at s = ∞. We can remove this pole or zero by remaining an impedance Z1(s) or admittance Y1(s). Then from each remainder left, an inductor or a capacitor is removed, depending upon the driving point function. It may be an impedance or an admittance function. This process continues until the remainder is zero. 9

10. From the above, the impedance Z(s) may be written as a continued fraction as under. The first Cauer form. Consider a driving point function having a pole at infinity. This implies that the degree of the numerator is greater than that of the denominator. We always remove pole at infinity by inverting the remainder, and dividing. That means an LC driving point function can be synthesized by the continued fraction expansion. If Z(s) is the function to be synthesized, then the continued fraction expansion is as follows. Therefore, in the first Cauer network shown in Fig. (a), the inductors are connected in series and the capacitors are connected in shunt. 10

11. second Cauer network If the driving point function, Z(s) has zero at infinite, that is, if the degree of its numerator is less than that of its denominator, the driving point function is inverted. To realize the second Cauer network, we have to take ascending powers of impedance function In this case, the continued fraction will give a capacitive admittance as first element, and a series inductance. Now let us realize the second Cauer network. In this case, the removal of the pole at zero gives the network shown in Fig. (b), where the capacitors are connected in series and the inductors are connected in shunt. If Z(s) is the function to be synthesized, then the continued fraction expansion is In case of second Cauer network, the first element is a series capacitor when the driving point function consists of a pole at zero and it is shunt inductance when the function consists of a zero at zero. 11

12. Example Using Cauer I and II realization synthesize Solution: This is an impedance function . 12

13. Example: Find the two Cauer realizations of driving point function given by By taking the continued fraction expansion, we get the first Cauer form To realize the second Cauer network, we have to take ascending powers of impedance function. Then we obtain 13

14. The driving point impedance of an LC network is given by Determine the second Cauer form of the network Solution: To obtain the second Cauer form, we have to arrange the numerator and the denominator of given Z(s) in ascending powers of s before starting the continued fraction expansion. 14

15. Synthesis of R-C Impedance /R-L Admittance Networks R-C impedance and R-L admittance driving point functions have the same properties. By replacing the inductor in LC by a resistor an R-C driving point impedance or R-L driving point admittance, it can be written as 15

16. Properties of R-C Impedance or R-L Admittance Networks The poles and zeros are simple. There are no multiple poles and zeros. The poles and zeros are located on negative real axis. The poles and zeros interlace (alternate) each other on the negative real axis. The lowest critical frequency nearest to the origin is always a pole . This may be located at the origin. The highest critical frequency at a greatest distance away from the origin is always a zero. The partial fraction expansion of Z RC (s) gives the residues which are always real and positive. 16

17. Properties of R-C Admittance or R-L Impedance Networks The poles and zeros are simple. There are no multiple poles and zeros. The poles and zeros are located on negative real axis. The poles and zeros interlace (alternate) each other on the negative real axis. The lowest critical frequency nearest to the origin is always a Zero. This may be located at the origin. The highest critical frequency at a greatest distance away from the origin is always a pole. The partial fraction expansion of Y RC (s) gives the residues which are always real and negative. But the residues of Y RC (s)/s are real and positive 17

18. Synthesis of R-C impedance Foster I The driving point impedance RC network, Z(s) is given by For R-C impedance 18

19. Synthesis of R-C admittance Foster II As mentioned in the properties of RC admittance function ,the residues at the poles of YRC(s) are real and negative while the residues at the poles of [YRC(s)]/s are real and positive. Therefore =   19

20. Example : Consider a function The first Foster form can be realized by taking the partial fraction of Z(s) 20

21. The second Foster form can be realized by taking the reciprocal of the impedance function and partial fraction expansion as 21

22. Example : Find the first and second Foster forms of the function. First Foster form of RC network shown second Foster form of RC network shown 22

23. First and second cauer form The first form of continued fraction expansion is called the first Cauer form, and is given by The second form of continued fraction expansion is The Cauer network for realizing the above function is shown in Fig. In the network shown, if Z(s) has a zero at s =∞, the first element is C1. degree of numerator is greater than denominator. If Z(s) is a constant at s =∞ the first element is R1. If Z(s) has a pole at s = 0, the last element is Cn . If Z(s) is constant at s = 0, the last element is Rn . In the network shown in Fig. if Z(s) has a pole at s = 0, the first element is C1. If Z(s) is a constant at s = 0, the first element is R1. If Z(s) has a zero at s =∞, the last element is Cn . If Z(s) is constant at s =∞, the last element is Rn . 23

24. Example Consider a function Z(s) = (s + 2) (s + 4)/s(s + 3). find the first and second Cauer form First cauer :The continued fraction expansion is Similarly, the second Cauer network can be obtained by arranging the numerator and denominator polynomials of Z(s) in ascending powers of s. The continued fraction expansion is 24

25. Example :Find the first and second Cauer forms of the function The first cauer network can be realized by taking continued fraction expansion The second Cauer network can be realized by arranging the numerator and denominator polynomials of Z(s) in ascending power of s and taking continued fraction expansion, we get 25

26. Example : A function F(s) has poles and zeros as under: poles at: 0,-4,-6 and zeros at: -2,-5 Taking scale factor as unity synthesize F(s) As an impedance in foster form As an admittance in cauer I form 26

27. Synthesis of R-L impedance Network Properties of R-L Impedance or R-C Admittance Networks The poles and zeros are simple. There are no multiple poles and zeros. The poles and zeros are located on negative real axis. The poles and zeros interlace (alternate) each other on the negative real axis. The lowest critical frequency nearest to the origin is always a Zero. This may be located at the origin. The highest critical frequency at a greatest distance away from the origin is always a pole. The residues at the poles of Z(s) are real and negative. The residues of Z(s)/s are real and positive. 27

28. Properties of R-L Admittance or R-C Impedance Networks The poles and zeros are simple. There are no multiple poles and zeros. The poles and zeros are located on negative real axis. The poles and zeros interlace (alternate) each other on the negative real axis. The lowest critical frequency nearest to the origin is always a pole . This may be located at the origin. The highest critical frequency at a greatest distance away from the origin is always a zero The partial fraction expansion of Y(s) gives the residues which are always real and positive. 28

29. Foster I method The driving point impedance function of an RL network Z(s) is given by The first form of the Foster network is shown in Fig 29

30. Example: Consider a function obtain foster I Z(s) represents RL impedance, because it satisfies all the properties, but the signs of Z(s) at its poles are negative. Z(s)/s 30

31. Foster II method The second form of the Foster network is shown in fig 31

32. Example : Consider an admittance function. obtain fosterII The poles and zeros are positive, real and simple. The poles are at - 2, - 4, and the zeros are at - 3, and -5. For the second Foster form of realization by partial fraction expansion, 32

33. Example: Find the first Foster form of the driving point function of If we take the partial fraction of Z(s), the signs of the function and its poles are negative. 33

34. Synthesis of R-L Network by Cauer I and cauer II Method The first form of continued fraction expansion is called the first Cauer form, which is The first form of the Cauer network can be obtained by continued fraction expansion and arranging the numerator and denominator polynomials of Z(s) in descending powers of s. In the network shown above, if Z(s) has a pole at s =∞, the first element is L1. If Z(s) is a constant at s =∞, the first element is R1. If Z(s) has a zero at s = 0, the last element is Ln. If Z(s) is a constant at s = 0, the last element is Rn . 34

35. The second form of continued fraction expansion is called second cauer form The second form of the Cauer network can be obtained by continued fraction expansion and arranging the numerator and denominator polynomials of Z(s) in ascending powers of s . Here also the presence of the first and the last element depends on the characteristics of impedance function, Z(s). If Z(s) has a zero at s = 0, the first element is L1. If Z(s) is a constant at s = 0, the first element is R1. If Z(s) has a pole at s =∞, the last element is Ln. If Z(s) is a constant at s =∞, the last element is Rn . 35

36. Example: Find the second Foster form and the first Cauer form of the network whose driving point admittance is By taking partial fraction expansion, we get Therefore, the second Foster network is To get the first cauer realization, we take continued fraction expansion from the expression. 36

37. Example Synthesize as R-C impedance and R-L admittance in foster realization. Solution: Note that the singularity near origin is a pole and a zero near infinity. The poles and zeros alternate We can expand F(s) as ZR-C Y R-L 37

38. Example Synthesize using Cauer realization as R-C impedance and R-L admittance. Solution: Note that the singularity near origin is a pole. The singularity near infinity is a zero. The zeros and the poles alternate. Note that the power of the numerator and denominator is equal, hence, we remove the resistor first. F(s) is R-L impedance or R-C admittance 38

39. For R-C impedance For R-L admittance 39

40. Example Synthesize as R-L impedance and R-C admittance using Foster realization. Solution: Note that the singularity near origin is a zero. The singularity near infinity is a pole. The zeros and the poles alternate. F(s) is R-L impedance or R-C admittance 40

41. R-L impedance R-C admittance We divide F(s) by s. 41

42. Example Synthesize as R-L impedance and R-C admittance using Cauer realization. Solution: We write P(s) and M(s) as 42

43. R-L impedance R-C admittance 43

44. Quiz 2 A function Z(s) has the following poles and zeros. Poles at -2 , -6 and zeros at -1 ,-3. Taking scale factor two obtain foster I and cauer I. 44