1Notes 18 ECE 5317-6351 Microwave Engineering Multistage Transformers Fall 2011Prof. David R. JacksonDept. of ECENotes 18Multistage Transformers
2Single-stage Transformer The transformer length is arbitrary in this analysis.StepZ1 lineLoadFrom previous notes:Step Impedance change
3Single-stage Transformer (cont.) From the self-loop formula, we have (as derived in previous notes)For the numerator:Next, consider this calculation:Hence
4Single-stage Transformer (cont.) We then havePutting both terms over a common denominator, we haveor
5Single-stage Transformer (cont.) Note: It is also true thatBut
6Multistage Transformer Assuming small reflections:
7Multistage Transformer (cont.) HenceNote that this is a polynomial in powers of z = exp(-j2).
8Multistage Transformer (cont.) If we assume symmetric reflections of the sections (not a symmetric layout of line impedances), we haveLast term
9Multistage Transformer (cont.) Hence, for symmetric reflections we then haveNote that this is a finite Fourier cosine series.
10Multistage 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.
11Binomial (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.
12Binomial 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 chooseWe want to use a multistage transformer to realize this type of response.Set equal(Both are now in the form of polynomials.)
13Binomial Multistage Transformer (cont.) HenceNote: A could be positive or negative.Equating responses for each term in the polynomial series gives us:HenceThis gives us a solution for the line impedances.
14Binomial Multistage Transformer (cont.) Note on reflection coefficientsNote thatHenceAlthough we did not assume that the reflection coefficients were symmetric in the design process, they actually come out that way.
15Binomial 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 .
16Binomial Multistage Transformer (cont.) Example showing a microstrip lineA three-stage transformer is shown.
17Binomial 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.
18Binomial Multistage Transformer (cont.) Use a series approximation for the ln function:RecallHence
19Binomial Multistage Transformer (cont.) BandwidthMaximum acceptable reflectionThe bandwidth is then:Hence
20Binomial Multistage Transformer (cont.) Summary of Design FormulasReflection coefficient responseA coefficientDesign of line impedancesBandwidth
25Chebyshev Multistage Matching Transformer Chebyshev polynomials of the first kind:We choose the response to be in the form of a Chebyshev polynomial.
26Figure 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).
27Chebyshev 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.
28Chebyshev 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.
29Chebyshev Multistage Transformer (cont.) Transformer designFrom the above formula we can extract the coefficients n (no general formula is given here).
33Chebyshev Multistage Transformer (cont.) Summary of Design FormulasReflection coefficient responsem termA coefficientNo 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 impedancesBandwidth
34Example Example: three-stage Chebyshev transformer Given Equate (finite Fourier cosine series form)Equate
35Example: 3-Section Chebyshev Transformer Equating coefficients from the previous equation on the last slide, we have
39Example: 3-Section Chebyshev Transformer Response from Ansoft Designer
40Example: 3-Section Chebyshev Transformer Comparison of Binomial (Butterworth) and ChebyshevThe 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.
41Tapered TransformerThe 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.