1. Notes 18 ECE 5317-6351 Microwave Engineering Multistage Transformers
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 > Z0 since 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.)
Note: Increasing the number of lines increases the bandwidth.
Figure (p. 250) Reflection coefficient magnitude versus frequency for multisection binomial matching transformers of Example 5.6. ZL = 50Ω and Z0 = 100Ω.

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. Figure 5.16 (p. 251) The first four Chebyshev polynomials Tn(x).
Chebyshev Multistage Transformer (cont.)
Figure (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 > Z0 since 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
m term
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)
Equate

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 ). Please read this.