1. Design and Analysis of RF and Microwave Systems IMPEDANCE TRANSFORMERS AND TAPERS Lecturers: Lluís Pradell (pradell@tsc.upc.edu)pradell@tsc.upc.edu Francesc Torres (xtorres@tsc.upc.edu) March 2010

2. Design and Analysis of RF and Microwave Systems The quarter-Wave Transformer* (i) Z in Z1Z1 ZLZL Z0Z0 A quarter-wave transformer can be used to match a real impedance Z L to Z 0 If The matching condition at f o is At a different frequency and the input reflection coefficient is The mismatch can be computed from: *Pozar 5.5

3. Design and Analysis of RF and Microwave Systems The quarter-Wave Transformer (ii) If Return Loss is constrained to yield a maximum value, the frequency that reaches the bound can be computed from: Where for a TEM transmission line And the bound frequency is related to the design frequency as:

4. Design and Analysis of RF and Microwave Systems The quarter-Wave Transformer (iii) Finally, the fractional bandwdith is given by

5. Design and Analysis of RF and Microwave Systems Multisection transformer* (i) That is, in the case of small reflections the permanent reflection is dominated by the two first transient terms: transmission line discontinuity and load The theory of small reflections In the case of small reflections, the reflection coefficient can be approximated taking into account the partial (transient) reflection coefficients: *Pozar 5.6

6. Design and Analysis of RF and Microwave Systems Multisection transformer (ii) The theory of small reflections can be extended to a multisection transformer It is assumed that the impedances Z N increase or decrease monotically The reflection coefficients can be grouped in pairs (Z N may not be symmetric)

7. Design and Analysis of RF and Microwave Systems for N even for N odd Finite Fourier Series: periodic function (period:  ) Multisection transformer (iii) The reflection coefficient can be represented as a Fourier series Any desired reflection coefficient behaviour over frequency can be synthesized by properly choosing the coefficients and using enough sections: Binomial (maximally flat) response Chebychev (equal ripple) response

8. Design and Analysis of RF and Microwave Systems Binomial multisection matching transformer (i) Binomial function The constant A is computed from the transformer response at f=0: The transformer coefficients are computed from the response expansion: The transformer impedances Z n are then computed, starting from n=0, as:

9. Design and Analysis of RF and Microwave Systems Binomial multisection matching transformer (ii)

10. Design and Analysis of RF and Microwave Systems Binomial multisection matching transformer (iii) Bandwidth of the binomial transformer The maximum reflection at the band edge is given by: 1 The fractional bandwitdh is then:

11. Design and Analysis of RF and Microwave Systems Chebyshev multisection matching transformer Chebyshev polynomial

12. Design and Analysis of RF and Microwave Systems Chebyshev transformer design

13. Design and Analysis of RF and Microwave Systems Chebyshev transformer design Application: Microstrip to rectangular wave-guide transition: both source and load impedances are real. Rectangular guide Ridge guide: five λ/4 sections: Chebychev design Steped ridge guideMicrostrip line Ridge guide section

14. Design and Analysis of RF and Microwave Systems TRANSFORMER EXAMPLE (1): ADS SIMULATION Chebyshev transformer, N = 3, |  M |=0.05 (l total = 3 /4) 87,14  70,71  100  57,37  50 

15. Design and Analysis of RF and Microwave Systems TRANSFORMER EXAMPLE (2): ADS SIMULATION BW = 102 % microstrip loss

16. Design and Analysis of RF and Microwave Systems Tapered lines (i) In the limit, when  z 0: Taper: transmission line with smooth (progressive) varying impedance Z(z) The transient ΔΓ for a piece Δz of transmission line is given by: This expression can be developed taking into account the following property:

17. Design and Analysis of RF and Microwave Systems Tapered lines (ii) Taking into account the theory of small reflections, the input reflection coefficient is the sum of all differential contributions, each one with its associated delay: Fourier Transform Taper electrical length Exponential taper Triangular taper Klopfenstein taper

18. Design and Analysis of RF and Microwave Systems Exponential Taper for 0 <z < L (sinc function) LL Fourier Transform

19. Design and Analysis of RF and Microwave Systems Triangular taper (squared sinc function) - lower side lobes - wider main lobe LL

20. Design and Analysis of RF and Microwave Systems Klopfenstein Taper LL Shortest length for a specified |  M | Lowest |  M | for a specified taper length l taper = Based on Chebychev coefficients when n→∞. Equal ripple in passband

21. Design and Analysis of RF and Microwave Systems Microstrip to rectangular wave-guide transition Example of linear taper: ridged wave-guide Microstrip line Ridged guide Rectangular guide SECTION A-A’ SECTION B-B’ SECTION C-C’

22. Design and Analysis of RF and Microwave Systems Rectangular wave-guide to finline to transition Example of taper: finline wave guide Finline mixer configuration

23. Design and Analysis of RF and Microwave Systems TAPER EXAMPLE (1): ADS SIMULATION ADS taper model

24. Design and Analysis of RF and Microwave Systems TAPER EXAMPLE (2): ADS SIMULATION Aproximation to exponential taper using ADS : 10 sections of  50  53,59  57,44  61,56  65,97  70,71  75,79  81,22  87,05  93,30  100 

25. Design and Analysis of RF and Microwave Systems TAPER EXAMPLE (3): ADS SIMULATION Aproximation to exponential taper using ADS : 10 sections of  50  53,59  57,44  61,56  65,97  70,71  75,79  81,22  87,05  93,30  100 

26. Design and Analysis of RF and Microwave Systems TAPER EXAMPLE (4): ADS SIMULATION − 10 section approx. − ADS model

27. Design and Analysis of RF and Microwave Systems TAPER EXAMPLE (5): ADS SIMULATION l taper = @ 10 GHz − 10 section approximation − ADS model

28. Design and Analysis of RF and Microwave Systems TAPER EXAMPLE (6): ADS SIMULATION (l i = /2) (l i = /10) − ADS model − 10 section approximation is periodic.

29. Design and Analysis of RF and Microwave Systems MATCHING NETWORKS LEVY DESIGN Lecturers: Lluís Pradell (pradell@tsc.upc.edu)pradell@tsc.upc.edu Francesc Torres (xtorres@tsc.upc.edu)

30. Design and Analysis of RF and Microwave Systems Matching Network ( passive lossless ) Z0Z0  f  VsVs    f  Minimize |  1 (f)| Maximize G t (  2 ) P d1 P dL MATCHING NETWORKS

31. Design and Analysis of RF and Microwave Systems CONVENTIONAL CHEBYSHEV FILTER (1) Conversion from Low-Pass to Band- Pass filter LC low-pass filter Center frequency Relative bandwidth

32. Design and Analysis of RF and Microwave Systems CONVENTIONAL CHEBYSHEV FILTER (2) Pass-band ripple Chebychev polynomials

33. Design and Analysis of RF and Microwave Systems CONVENTIONAL CHEBYSHEV FILTER (3) Fix pass-band ripple and filter order “n” g 0, g 1,.., g n+1 are the low-pass LC filter coefficients:

34. Design and Analysis of RF and Microwave Systems APPLICATION TO A MATCHING NETWORK Solution (?): increase  n (n constant)a, x decrease or increase n (  n constant) a, x decrease Transistor modeled with a dominant RLC behaviour in the pass-band to be matched The final design may be out of specifications: n too high (too many sections) or r too large

35. Design and Analysis of RF and Microwave Systems LEVY NETWORK (1) SOLUTION: An additional parameter is introduced: K n <1

36. Design and Analysis of RF and Microwave Systems LEVY NETWORK (2) Example: n = 2 SOLUTION: Additional design equations

37. Design and Analysis of RF and Microwave Systems LEVY NETWORK (3) Design procedure a) Choose C s1 or L s1 taking into account the load to be matched c) Compute x-y from the parameter g 1 b) Choose network order (n) and compute g 1

38. Design and Analysis of RF and Microwave Systems LEVY NETWORK (4) OPTIMAL DESIGN: minimize For n=2: Select L s1 (or C s1 ) and n. Compute g 1. and x-y. Then determine x, y and K n,  n : x  y  b   a  a  d) Choose x, compute y, Example: usual case n=2: Optimum x The matched bandwith can be increased from ~5% to ~20% with n=2, with moderate Return Loss requirements (~20 dB)

39. Design and Analysis of RF and Microwave Systems LEVY NETWORK EXAMPLE (1)

40. Design and Analysis of RF and Microwave Systems LEVY NETWORK EXAMPLE (2)

41. Design and Analysis of RF and Microwave Systems LEVY NETWORK EXAMPLE (3): ADS SIMULATION A transformer is necessary since g 3 ≠1 (R 3 ≠50 Ω). This transformed must be eliminated from the design

42. Design and Analysis of RF and Microwave Systems Norton Transformer equivalences STEPS:1) the capacitor C 2 is pushed towards the load through the transformer 2) The transformer is eliminated using Norton equivalences

43. Design and Analysis of RF and Microwave Systems LEVY NETWORK EXAMPLE (4): ADS SIMULATION

44. Design and Analysis of RF and Microwave Systems SMALL SERIES INDUCTANCES AND PARALLEL CAPACITANCES IMPLEMENTED USING SHORT TRANSMISSION LINES L, C elements are then synthesized by means of short transmission lines:

45. Design and Analysis of RF and Microwave Systems SMALL SERIES INDUCTANCES AND PARALLEL CAPACITANCES IMPLEMENTED USING SHORT TRANSMISSION LINES: EXAMPLE

46. Design and Analysis of RF and Microwave Systems LEVY NETWORK EXAMPLE ADS SIMULATION (5):

47. Design and Analysis of RF and Microwave Systems LEVY NETWORK EXAMPLE: ADS SIMULATION (6):

48. Design and Analysis of RF and Microwave Systems LEVY NETWORK EXAMPLE (7): ADS SIMULATION: optimization

49. Design and Analysis of RF and Microwave Systems LEVY NETWORK EXAMPLE (8): ADS SIMULATION: optimization

50. Design and Analysis of RF and Microwave Systems LEVY NETWORK EXAMPLE (9): ADS SIMULATION: optimization

51. Design and Analysis of RF and Microwave Systems LEVY NETWORK EXAMPLE (10): ADS SIMULATION: optimization