Lecture 9

1. Lecture 9 • Insertion Loss Method • Filter Transformation • Filter Implementation • Stepped-Impedance Low-Pass Filter • Coupled Line Filters • Filters Using Coupled Resonators Microwave Technique

2. Proof of Kuroda’s Identity • consider the second identity in which a series stub is converted into a shunt stub Microwave Technique

3. Proof of Kuroda’s Identity • for the short-circuited series stub has an series impedance of Microwave Technique

4. Proof of Kuroda’s Identity • the ABCD matrix of the series stub is • the ABCD matrix of the transmission line is Microwave Technique

5. Proof of Kuroda’s Identity • the cascaded ABCD matrix is given by Microwave Technique

6. Proof of Kuroda’s Identity • the ABCD matrix for the transmission line with a characteristic impedance Z1 + Zo is given by Microwave Technique

7. Proof of Kuroda’s Identity • the ABCD matrix of the open-circuit shunt stub is given by Microwave Technique

8. Proof of Kuroda’s Identity • the composite ABCD matrix for the figure on the right is given by Microwave Technique

9. Proof of Kuroda’s Identity • this implies that these two setups are identical • Design a low-pass third-order maximally flat filter using only shunt stubs. The cutoff frequency is 8GHz and the impedance is 50 W. • from table 9.3, g1 = 0.7654, g2 = 1.8478, g3=1.8478, g4 = 0.7654, g5 = 1 Microwave Technique

10. Proof of Kuroda’s Identity • the lowpass filter prototype • applying Richard’s transform Microwave Technique

11. Proof of Kuroda’s Identity • Add unit elements Microwave Technique

12. Proof of Kuroda’s Identity • use second Kuroda identity on left, (1+0.765)=1.765, • use first Kuroda identity on right, (1/1+1/1.307)=1.765. 1/1.765 = 0.566, ½.307=0.433 Microwave Technique

13. Proof of Kuroda’s Identity Microwave Technique

14. Proof of Kuroda’s Identity • use the second Kuroda identity twice, (1.848+0.567)=2.415, • (0.567+0.5672/1.848)= 0.741, (1+0.433)=1.433, (1+12/0.433)=3.309 Microwave Technique

15. Proof of Kuroda’s Identity • scale to 50 W • all lines are l/8 long at 8 GHz Microwave Technique

16. Impedance and Admittance Inverters • to transform series connected elements to shunt-connected elements and vice versa Microwave Technique

17. Impedance and Admittance Inverters Microwave Technique

18. Impedance and Admittance Inverters Microwave Technique

19. Impedance and Admittance Inverters Microwave Technique

20. Stepped-Impedance Low-Pass Filters • stepped impedance, or hi-Z, low-Z filters are easier to design and take up less space than a similar filter using stubs • limited to application when a sharp cutoff is not required Microwave Technique

21. Stepped-Impedance Low-Pass Filters • consider the lowpass filter depicted below • we will replace the inductance and capacitance with a short length of transmission line • consider the ABCD parameters of a transmission line with length Microwave Technique

22. Stepped-Impedance Low-Pass Filters Microwave Technique

23. Stepped-Impedance Low-Pass Filters • AD-BC=1, A=D, • for an equivalent T-circuit, its series elements are ) Microwave Technique

24. Stepped-Impedance Low-Pass Filters • The shunt element is • For small , • Resembles an inductor when • Resembles a capicator Microwave Technique

25. Stepped-Impedance Low-Pass Filters • therefore a short transmission line with a large impedance yields Microwave Technique

26. Stepped-Impedance Low-Pass Filters • a short transmission line with a small impedance yields Microwave Technique

27. Coupled Line Filters • a parallel coupled line section is shown below Microwave Technique

28. Coupled Line Filters • the even- and odd-mode currents are • and are the even-mode currents while and are the odd-mode currents Microwave Technique

29. Coupled Line Filters • we will look at the open-circuit impedance matrix which has a bandpass response Microwave Technique

30. Coupled Line Filters • by superposition, we have • if Ports 1 and 2 are driven by an even-mode current when Ports 3 and 4 are open, the impedance seen at Ports 1 and 2 is Microwave Technique

31. Coupled Line Filters • the voltage on either conductor due to source current i1 can be written as • , = 1 for OC Microwave Technique

32. Coupled Line Filters • the voltage at Port 1 or 2 is • Therefore, • , similarly, the voltage due to i3 can be written as Microwave Technique

33. Coupled Line Filters • using the same treatment, for the odd-mode excitation, we have Microwave Technique

34. Coupled Line Filters • the total voltage is therefore given by the sum of all four contributions Microwave Technique

35. Coupled Line Filters • from the relation between i and I, we have • the above equation represent the first row of the impedance matrix for the open-circuit; from symmetry, all other matrix element can be found Microwave Technique

36. Coupled Line Filters Microwave Technique

37. Coupled Line Filters • a two-port network can be formed from the coupled line section by terminating two of the four ports in either open or short circuits • their performances are summarized in Table 9.8 • we will pay more attention to the case that I2 = I4 = 0 Microwave Technique

38. Coupled Line Filters • the impedance matrix equation now becomes Microwave Technique

39. Coupled Line Filters • the filter characteristic can be determined from the image impedance and the propagation constant • if the line section is l/4 long Microwave Technique

40. Design of Coupled Line Bandpass Filters • narrowband bandpass filter can be designed by cascading open-circuit coupled line sections • we first show that a single couple line section can be approximated by the equivalent circuit Microwave Technique

41. Design of Coupled Line Bandpass Filters • note the the admittance inverter is a transmission line of characteristic impedance 1/J and electrical length of -90o Microwave Technique

42. Design of Coupled Line Bandpass Filters • we calculate the image impedance and propagation constant of the equivalent circuit and show that they are approximately equal to those of the coupled line section for q = p/2, which will correspond to the center frequency of the bandpass filter Microwave Technique

43. Design of Coupled Line Bandpass Filters • the ABCD matrix of the equivalent circuit is given cascading those of three sections of transmission line Microwave Technique

44. Design of Coupled Line Bandpass Filters • recall that the image impedance is given by • as A=D Microwave Technique

45. Design of Coupled Line Bandpass Filters • for q = p/2 • the propagation constant is Microwave Technique

46. Design of Coupled Line Bandpass Filters • therefore, and for • equating these equations yields • and Microwave Technique

47. Design of Coupled Line Bandpass Filters • we have related the coupled line parameters with its equivalent circuit • Design a four-section coupled line bandpass filter with a maximally flat response. The passband is 3.00 to 3.50 GHz, and the impedance is 50 W. What is the attenuation at 2.9 GHz? Microwave Technique

48. Design of Coupled Line Bandpass Filters • N=3, • transform 2.9 GHz to normalized lowpass filter form: • , from Figure 9.26, = 10.5 dB Microwave Technique

49. Design of Coupled Line Bandpass Filters • the prototype values are given in Table 9.3 and Microwave Technique

50. Design of Coupled Line Bandpass Filters • n gn ZoJn Zoe Zoo • 1 1.00 0.492 86.7 37.5 • 2 2.00 0.171 60.0 42.9 • 3 1.00 0.171 60.0 42.9 • 4 1.00 0.492 86.7 37.5 Microwave Technique