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

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Single-stage Transformer Step Impedance change 2 From previous notes: Step Z 1 line Load The transformer length is arbitrary in this analysis.

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

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

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

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Assuming small reflections: 6 Multistage Transformer

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

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

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

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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.

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

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

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

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

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15 Note: The table only shows data for Z L > Z 0 since the design can be reversed (Ioad and source switched) for Z L < Z 0. Binomial Multistage Transformer (cont.)

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

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17 Binomial Multistage Transformer (cont.) Figure 5.15 (p. 250) Reflection coefficient magnitude versus frequency for multisection binomial matching transformers of Example 5.6. Z L = 50Ω and Z 0 = 100Ω. Note: Increasing the number of lines increases the bandwidth.

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

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

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

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Example: three-stage binomial transformer 21Example Given:

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22 Example (cont.)

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

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Response from Ansoft Designer 24 Example (cont.)

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

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

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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.

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

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

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30 Chebyshev Multistage Transformer (cont.) Hence

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31 Chebyshev Multistage Transformer (cont.) Note: The table only shows data for Z L > Z 0 since the design can be reversed (Ioad and source switched) for Z L < Z 0.

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32 Chebyshev Multistage Transformer (cont.) Bandwidth Hence

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Summary of Design Formulas 33 Reflection coefficient response A coefficient Design of line impedances Bandwidth Chebyshev Multistage Transformer (cont.) 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 ). m term

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

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

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36 Example: 3-Section Chebyshev Transformer

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Alternative method: 37 Example: 3-Section Chebyshev Transformer

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38 Example: 3-Section Chebyshev Transformer

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

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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.

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41 Tapered Transformer The Pozar book also talks about using continuously tapered lines to match between an input line Z 0 and an output load Z L. (pp ). Please read this.

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