Stepped Impedance A

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TECHNICAL NOTES

Stepped and Tapered

Impedance Matching Transformers

ABSTRACT

This technical note outlines the theory used in Rfutils_M toolbox for MATLAB. This toolboxis basically an adjunct to the existing Rfutils toolbox and facilitates the design andevaluation of multi-step impedance matching transformers. The designs include n-sectionquarter-wave matches and splitters based on an approximation to the Chebyshevdistribution, as well as tapered matches using the Klopfenstein taper. In addition to theimpedance profiles, routines are included to calculate Microstrip realisations of thetransformers and output these in Autocad DXF format.





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CONTENTS

ABSTRACT ................................................................................................................................................................................ 1

CONTENTS ................................................................................................................................................................................ 2

1. INTRODUCTION.............................................................................................................................................................. 3

2. MULTI-SECTION MATCHES........................................................................................................................................ 4

2.1 Binomial .................................................................................................................................................................... 4

2.2 Chebyshev ................................................................................................................................................................... 5

2.3 Response Evaluation .................................................................................................................................................. 7

3. TAPERED MATCHES ..................................................................................................................................................... 8

3.1 Exponential ................................................................................................................................................................ 8

3.2 Triangular .................................................................................................................................................................. 8

3.3 Klopfenstein................................................................................................................................................................ 9

3.4 Comparison .............................................................................................................................................................. 10

REFERENCES ......................................................................................................................................................................... 10





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1. INTRODUCTION

One of the commonest problems facing the Rf/Microwave engineer is matching, whetherfor maximum power transfer, minimum noise figure or optimum bandwidth, matchingusually crops up somewhere in the design. Lumped element and stub type matching is themost common and very well documented, less well documented is stepped and taperedline matching.

Lumped element and stub type matching can provide very effective and compact matchingsolutions. The price paid for this compactness is that the impedance must be movedrapidly to the required value, requiring relatively

1 high currents and voltages to exist in the

matching components. High current densities result in ohmic losses in the conductors,while high field intensities result in dielectric losses. Bandwidth can be improved by usingmore complex networks with increased the component count, but accounting for ‘strays’can make a practical design increasingly difficult.

Stepped and tapered line matching solutions are unlikely to be the most compact but arewell suited to applications requiring wide bandwidth, low loss or high power operation.Typical applications include receiver front ends, transmitter outputs and general test andmeasurement.

A variety of stepped and tapered line impedance transformer designs have been explored,the following sections outline the associated theory used to design and evaluate them.

Note 1 Relative to those on the matched line carrying the same power.





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2. MULTI-SECTION MATCHES

The simplest transmission line match is usually quoted as being of a single quarter-wavesection of line, characteristic impedance Zmatch as given by equation 2-1

loadmatch ZZZ ⋅= 0 Eq 2-1

Where : Z0 = Characteristic to be matched to

Zload = Load impedance to be matched

If we take the example of matching a load impedance of 100ohms to a characteristicimpedance of 50ohms, equation 2-1 gives us a matching impedance of 70.7ohms and afractional bandwidth of 29%, see [1] for derivation and calculations.

To increase the bandwidth of the transformer we need to add more quarter-wave sections,the trick is deciding what impedance to make them.

2.1 Binomial

The Binomial or maximally flat design, as its name suggests, gives the flattest response inthe band of operation for a given quality of match. The calculation of the appropriatetransformer impedances is not that arduous and well documented in [1]. The essentialcalculations are summarised in equations 2.1-1 and 2.1-2 below.

The binomial coefficients are given by :

!)!(

!

nnN

NC N

n⋅−

= Eq 2.1-1

Where : N = Number of matching sections

n = Section number

The coefficients are obtained by evaluating Eq 2-2 for values of n=0 to (N-1)

The natural log of the impedances is then given by :

⋅⋅+= −

+

0

)()1( ln2)ln()ln(Z

ZCZZ LN

n

N

nn Eq 2.1-2

The values for ln(Z(n+1)) and hence Z(n+1), are obtained by evaluating Eq 2.1-2 for values ofn= 0 to (N-1)





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2.2 Chebyshev

The Chebyshev design gives the best bandwidth for a given number of transformersections, at the expense of ripple in the band of operation. The calculation of theappropriate transformer sections is not so easy in this case. The only methods I havefound seem to involve a fair amount of algebra followed by comparing terms in theresulting expansions, a method that is not easy to automate. I must confess my resolve todig through the maths was not that strong.

Ultimately any design is probably going to need some ‘tuning’, either with a scalpel andcopper tape or more likely in an em-simulator. So, do you spend all your time searching formathematical perfection only to let an optimiser loose on it. Or, settle for a good firstattempt, so at least your optimiser will not end up stuck on a false summit.

Out of interest I decided to plot some of the tabulated Chebyshev design values for the100ohm to 50ohm match, operating reflection coeff 0.05, for N=2,3..7 sections. Instead ofplotting the section impedance values, the equivalent reflection coefficients for eachsection are plotted (y-axis). The x-axis is simply the impedance section number, figure 2.2-1 shows the resulting curves (solid, coloured).

0 1 2 3 4 5 6 7 8 90

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Linear X-axis

Refle

ction C

oeffi

cie

nt

N-section Transformer Reflection Coefficient (Zo=50 ZL=100)

Figure 2.2-1 Chebyshev curves plotted as reflection coefficient

(reference chapprox.m in rfutils_m toolbox)





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Somewhat unsurprisingly, all the curves converge (if extrapolated) to a point atapproximately (0.333,0.333), the reflection coefficient of the 100ohm load impedancemeasured in a 50ohm system. The other end of the curves finish at very approximately(N,0.333/(N+2)), where N is the number of sections. Effectively we have a series of curveswhose start and finish points are determined purely by the reflection coefficient of the loadimpedance (referenced to Zo) and the number of sections in the transformer.

With a little a ‘fit by eye’ curves for N=2,3 and 4 can be approximated by a straight line.Curves for N=5,6 and 7 sections are approximated using an x-shifted cosine curve. For 8or more sections a shifted and scaled (in x-axis) cosine curve is used. See figure 2.2-1(dashed curves).

cxmy +⋅= for 2 ≤ N ≤ 4 Eq 2.2-1

295.0

212)52(cos ldld TT

Nxy +⋅⋅

⋅−+=

πfor 5 ≤ N ≤ 7 Eq 2.2-2

295.0

2)1()1(cos ldld TT

Nxy +⋅⋅

−⋅−=

πfor 8 ≤ N Eq 2.2-3

Where the load reflection coefficient 0

0

ZZ

ZZT

load

load

ld+

−=

Depending on the number of sections required (N), the appropriate curve is selected andthe reflection coefficients for each section Tn is calculated. The impedance for each sectionis then calculated using equation 2.2-4, below.

01

1Z

T

TZ

n

n

n ⋅−

+= Eq 2.2-4

Whilst lacking the precision of the formal solution, the resulting transformer values will giveoperating band reflection coefficients of around 0.1 (-20dB) or better for any impedanceratios up to 4:1, for any number of sections. Bearing in mind the practical limitations ofrealising very high or low impedances, the degradation beyond the 4:1 ratio doesn’t seemtoo much of a problem.





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2.3 Response Evaluation

Once the impedance for each section has been established, the frequency response iseasily calculated by successively applying equation 2.3-1 at each frequency of interest,working from the load impedance back towards the input.

)tanh(

)tanh(

lZZ

lZZZZ

Ltr

trL

trin⋅⋅−

⋅⋅+⋅=

γ

γEq 2.3-1

Where : γ = α+βj

α = LossdB/8.686 and β = 2π/λ

LossdB = Loss in dB per unit length along line

l = Transformer line length (m)

λ = Wavelength at the frequency of interest (m)

ZL = Load impedance (ohms)

Ztr = Transformer impedance (ohms)

Once the last transformer has been taken into account, Zin represents the matched inputimpedance as a function of frequency.





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3. TAPERED MATCHES

Although the stepped impedance transformer designs described in section 2 are normallysequences of quarter-wave sections (at frequency = f0), an increased number of shortersections can also be used to approximate a taper. If we then look at the overall length ofthe taper instead of the individual sections, the frequency at which this becomes a half-wavelength is roughly the lowest frequency of operation.

Since the response of these tapered lines can be calculated in the same way as outlined in

section 2.3. Design can simply be a matter of setting the transformer lengths to be λ0/100

and the number of sections to N to around 50, thus giving an overall length of roughly λ0/2.

Note that f0 is not the definitive lower cut-off frequency, it just defines a convenientwavelength to divide up into small sections and define the taper. Although the optimum

length for tapered matches is around λ0/2 (N=50), it does vary depending on taper type.

The length of the taper can be altered in λ0/100 increments by simply changing the numberof sections used.

3.1 Exponential

The Exponential taper gives the lowest cut-off frequency for a given length but at theexpense of ripple in the operating band. The impedance as a function of distance (z) alongthe taper (length L) is given by equation 3.1-1 below.

azeZzZ ⋅= 0)( for 0 ≤ z ≤ L Eq 3.1-1

Where :

=

0

ln1

Z

Z

La L

3.2 Triangular

The Triangular taper has the highest cut-off frequency for a given length but has fewerripples that die-off rapidily at higher frequencies. The impedance as a function of distance(z) along the taper (length L) is given by equation 3.1-1 below.

)/ln()/(2

00

2

)(ZZLz LeZzZ ⋅= for 0 ≤ z ≤ L/2 Eq 3.1-1

)/ln()1/2/4(

00

22

)(ZZLzLz LeZzZ

⋅−−⋅= for L/2 ≤ z ≤ L





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3.3 Klopfenstein

Tapered matches could be based on either of the profiles already described, or indeed anyprofile you could imagine. However, the Klopfenstein taper is generally considered to offerthe ‘best’ performance in that for a fixed length, the ripple is minimum over the pass band.Or, for a given ripple value, it is the taper that can be realised in the shortest length. Thenatural log of the impedance as a function of distance (z) along the taper (length L) isgiven by equation 3.3-1 below.

),1/2(cosh

ln2

1)(ln

2

0 ALzAA

TZZzZ load

L −⋅+= φ for 0 ≤ z ≤ L Eq 3.3-1

Where :

=

ripple

load

T

TaA cosh

≅

+

−=

00

0 ln2

1

Z

Z

ZZ

ZZT L

L

L

load and 20/

10RdB

rippleT =

Although cited as approximately equivalent, I found that better results are obtained byusing the log-quotient approximation for Tload. This is probably due to the mathematics ofthe derivation of equation 3.1-1 itself.

∫−

−=

x

dyyA

yAIAx

0 2

2

1

1

)1(),(φ

Where : x=2z/L-1 and I1(x) is the modified Bessel function

The function Phi needs to be evaluated numerically, but note that the limit x is dependenton the position along the taper z.

The optimum length for this taper design is 0.565 lambda.





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3.4 Comparison

The graphs in figure 3.4-1 shows the output from running the example file excomp1.m. Theplots are for the Exponential, Triangular and Klopfenstein tapers implemented as half-wave

structures at f0=1000MHz using 50 x λ0/100 sections, used to match 100ohm to 50ohmlines.

0 2000 4000 6000 8000 10000-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0 Return Loss (Zo=50)

Frequency MHz

Re

turn

Lo

ss

dB

Exponential Triangular Klopfenstein

0 5 10 15 20 25 30 35 40 45 5050

60

70

80

90

100

110

Zo Matching Section Number Zload

Impedance (

Ohm

s)

Transformer Impedances

Exponential Triangular Klopfenstein

Figure 3.4-1 Taper response comparisons

REFERENCES

[1] “Microwave Engineering“ 2nd

Edition by David M. Pozar Published by WileyISBN 0-471-17096-8

[2] “RF Circuit Design” by Chris Bowick Published Newnes ISBN 0-7506-9946-9

[3] “FastCAD32D” A drafting package used to text the DXF export.

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