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**Notes 18 ECE 5317-6351 Microwave Engineering Multistage Transformers**

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

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**Single-stage Transformer**

The transformer length is arbitrary in this analysis. Step Z1 line Load From previous notes: Step Impedance change

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**Single-stage Transformer (cont.)**

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

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**Single-stage Transformer (cont.)**

We then have Putting both terms over a common denominator, we have or

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**Single-stage Transformer (cont.)**

Note: It is also true that But

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**Multistage Transformer**

Assuming small reflections:

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**Multistage Transformer (cont.)**

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

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**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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**Multistage Transformer (cont.)**

Hence, for symmetric reflections we then have Note that this is a finite Fourier cosine series.

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**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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**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.

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**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.)

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**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.

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**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.

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**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 .

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**Binomial Multistage Transformer (cont.)**

Example showing a microstrip line A three-stage transformer is shown.

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**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.

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**Binomial Multistage Transformer (cont.)**

Use a series approximation for the ln function: Recall Hence

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**Binomial Multistage Transformer (cont.)**

Bandwidth Maximum acceptable reflection The bandwidth is then: Hence

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**Binomial Multistage Transformer (cont.)**

Summary of Design Formulas Reflection coefficient response A coefficient Design of line impedances Bandwidth

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Example Example: three-stage binomial transformer Given:

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Example (cont.)

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**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

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Example (cont.) Response from Ansoft Designer

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**Chebyshev Multistage Matching Transformer**

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

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**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).

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**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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**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.

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**Chebyshev Multistage Transformer (cont.)**

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

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**Chebyshev Multistage Transformer (cont.)**

Hence

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**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 .

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**Chebyshev Multistage Transformer (cont.)**

Bandwidth Hence

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**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

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**Example Example: three-stage Chebyshev transformer Given Equate**

(finite Fourier cosine series form) Equate

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**Example: 3-Section Chebyshev Transformer**

Equating coefficients from the previous equation on the last slide, we have

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**Example: 3-Section Chebyshev Transformer**

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**Example: 3-Section Chebyshev Transformer**

Alternative method:

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**Example: 3-Section Chebyshev Transformer**

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**Example: 3-Section Chebyshev Transformer**

Response from Ansoft Designer

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**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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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.

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