Notes 18

1. ECE 5317-6351 Microwave Engineering Fall 2011 Prof. David R. Jackson Dept. of ECE Notes 18 Multistage Transformers

2. Single-stage Transformer The transformer length is arbitrary in this analysis. Step Z1 line Load From previous notes: Step Impedance change

3. Single-stage Transformer (cont.) From the self-loop formula, we have (as derived in previous notes) For the numerator: Next, consider this calculation: Hence

4. Single-stage Transformer (cont.) We then have Putting both terms over a common denominator, we have or

5. Single-stage Transformer (cont.) Note: It is also true that But

6. Multistage Transformer Assuming small reflections:

7. Multistage Transformer (cont.) Hence Note that this is a polynomial in powers of z = exp(-j2).

8. Multistage Transformer (cont.) If we assume symmetric reflections of the sections (not a symmetric layout of line impedances), we have Last term

9. Multistage Transformer (cont.) Hence, for symmetric reflections we then have Note that this is a finite Fourier cosine series.

10. Multistage Transformer (cont.) Design philosophy: If we choose a response for () that is in the form of a polynomial (in powers of z =exp(-j2)) or a Fourier cosine series, we can obtain the needed values of n and hence complete the design.

11. Binomial (Butterworth*) Multistage Transformer Consider: Choose all lines to be a quarter wavelength at the center frequency so that (We have a perfect match at the center frequency.) *The name comes from the British physicist/engineer Stephen Butterworth, who described the design of filters using the binomial principle in 1930.

12. Binomial Multistage Transformer (cont.) Use the binomial expansion so we can express the Butterworth response in terms of a polynomial series: A binomial type of response is obtained if we thus choose We want to use a multistage transformer to realize this type of response. Set equal (Both are now in the form of polynomials.)

13. Binomial Multistage Transformer (cont.) Hence Note: A could be positive or negative. Equating responses for each term in the polynomial series gives us: Hence This gives us a solution for the line impedances.

14. Binomial Multistage Transformer (cont.) Note on reflection coefficients Note that Hence Although we did not assume that the reflection coefficients were symmetric in the design process, they actually come out that way.

15. Binomial Multistage Transformer (cont.) Note: The table only shows data for ZL > Z0since the design can be reversed (Ioad and source switched) for ZL < Z0 .

16. Binomial Multistage Transformer (cont.) Example showing a microstrip line A three-stage transformer is shown.

17. Binomial Multistage Transformer (cont.) Figure 5.15 (p. 250)Reflection coefficient magnitude versus frequency for multisection binomial matching transformers of Example 5.6. ZL = 50Ω and Z0 = 100Ω. Note: Increasing the number of lines increases the bandwidth.

18. Binomial Multistage Transformer (cont.) Use a series approximation for the ln function: Recall Hence

19. Binomial Multistage Transformer (cont.) Bandwidth Maximum acceptable reflection The bandwidth is then: Hence

20. Binomial Multistage Transformer (cont.) Summary of Design Formulas Reflection coefficient response A coefficient Design of line impedances Bandwidth

21. Example Example: three-stage binomial transformer Given:

22. Example (cont.)

23. Example (cont.) Using the table in Pozar we have: (The above normalized load impedance is the reciprocal of what we actually have.) Hence, switching the load and the source ends, we have Therefore

24. Example (cont.)  Response from Ansoft Designer

25. Chebyshev Multistage Matching Transformer Chebyshev polynomials of the first kind: We choose the response to be in the form of a Chebyshev polynomial.

26. Chebyshev Multistage Transformer (cont.) Figure 5.16 (p. 251)The first four Chebyshev polynomials Tn(x).

27. Chebyshev Multistage Transformer (cont.) A Chebyshev response will have equal ripple within the bandwidth. This can be put into a form involving the terms cos (n) (i.e., a finite Fourier cosine series). Note: As frequency decreases, x increases.

28. Chebyshev Multistage Transformer (cont.) We have that, after some algebra, Hence, the term TN (sec, cos) can be cast into a finite cosine Fourier series expansion.

29. Chebyshev Multistage Transformer (cont.) Transformer design From the above formula we can extract the coefficients n (no general formula is given here).

30. Chebyshev Multistage Transformer (cont.) Hence

31. Chebyshev Multistage Transformer (cont.) Note: The table only shows data for ZL > Z0since the design can be reversed (Ioad and source switched) for ZL < Z0 .

32. Chebyshev Multistage Transformer (cont.) Bandwidth Hence

33. Chebyshev Multistage Transformer (cont.) Summary of Design Formulas Reflection coefficient response mterm A coefficient No formula given for the line impedances. Use the Table from Pozar or generate (“by hand”) the solution by expanding () into a polynomial with terms cos(n). Design of line impedances Bandwidth

34. Example Example: three-stage Chebyshev transformer Given Equate (finite Fourier cosine series form)

35. Example: 3-Section Chebyshev Transformer Equating coefficients from the previous equation on the last slide, we have

36. Example: 3-Section Chebyshev Transformer

37. Example: 3-Section Chebyshev Transformer Alternative method:

38. Example: 3-Section Chebyshev Transformer

39. Example: 3-Section Chebyshev Transformer  Response from Ansoft Designer

40. Example: 3-Section Chebyshev Transformer Comparison of Binomial (Butterworth) and Chebyshev • The Chebyshev design has a higher bandwidth (100% vs. 69%). • The increased bandwidth comes with a price: ripple in the passband. Note: It can be shown that the Chebyshev design gives the highest possible bandwidth for a given N and m.

41. Tapered Transformer The Pozar book also talks about using continuously tapered lines to match between an input line Z0 and an output load ZL. (pp. 255-261). Please read this.