1. For a single quarter wavelength transformer, how can the equation forits impedance be achieved, if the characteristic and the load impedancesare known and real?
2. Find the characteristic impedances of the quarter wavelength trans-formers which match a load ZL = 100 Ω to a transmission line Z0 =50 Ω at f = 1.5 GHz for:
(a) a single transformer.
(b) 3 binomial transformers.
(c) 3 Chebyshev transformers with Γm = 0.2.
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2 Background Theory
2.1 Introduction
In this experiment we will exercise a broadband matching technique based onseries of quarter wavelength transformers. This type of matching is suitableespecially for a nearly constant and real impedance load in a relevant fre-quency range. As the number of matching elements is increased, the matchinglevel can be improved and/or frequency range can be increased. For a givenmatching level and frequency range, one can find the number of transformsneeded in order to get the relevant performance. The main algorithms arethe binomial and Chebyshev matching. The matched load circuit is shownin Figure 1. More details are presented in [1]
Z0 Z02 Z03Z01 RL
4λ
4λ
4λ
Figure 1 - Matching the load RL with 3 quarter wavelength transformers.
Γ0 =ZL − Z0ZL + Z0
The fractional bandwidth of the transformers is given by:
∆f
f0= 2− 4θm
π= 2− 4
πcos−1
"Γmp1− Γ2m
· 2√Z0ZL
|ZL − Z0|
#Where Γm is the maximum value of the reflection coefficient magnitude,
which can be tolerated at the input, and θm is the θ = βl where Γ = Γm (thelower edge of the passband).
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The fractional bandwidth is usually expressed in percentage, 100 · ∆ff0[%].
Note that the bandwidth increases as ZL becomes closer to Z0 (a less mis-matched load).
2.2 Binomial Matching
In this algorithm, the reflection coefficient is a smooth function of frequencyand has a maximal flat response near the design frequency.The values of the characteristic impedances of the quarter wavelength
transformers are given by:
Z0k = Z0
µRL
Z0
¶Mk2N
; k = 1, 2, ...N
Where N is the number of transformers and
Mk = C1 + C2 + ...+ Ck
Where
C1 = 1, Ck =N !
(N − k + 1)!(k − 1)!The maximum value of the reflection coefficient magnitude, Γm, which
can be tolerated at the input is given by:
Γm = 2N |A| cosN(θm)
Where:
A = 2−N · Γ0Thus, θm can be computed using Γm by:
θm = cos−1
"1
2·µΓm|A|
¶ 1N
#When possible, one should use a table. Table 1 shows the values of the
characteristic impedances for the binomial transformers.
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Table 1: Values of the characteristic impedances of binomial transformers.
2.2.1 Example
Match the load ZL = 75 Ω by two quarter wavelength microstrip transformersto a transmission line with Z0 = 50 Ω. Use binomial matching for a matchat f = 1 GHz. The microstrip has a permittivity of εr = 4.9, height ofsubstrate is h = 1.6 mm, height of conductive layer is T = 0.02 mm and aloss tangent of tan δ = 0.018.
2.2.2 Solution
Calculating the length of each transform:f = 1 GHz → λ = c
f ·√εr = 0.138 m → L = λ4= 3.4 cm.
Using Table 1 for ZLZ0= 1.5 and N = 2 will yield Z1 = 55.3 Ω and
Z2 = 67.77 Ω.Using the tool LineCalc (or calculating by the formula for w
dfor mi-
crostrip) will yield w1 = 2.44 mm and w2 = 1.65 mm.An example for the layout is shown in Figure 2.
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0 50Z = Ω 75LZ = Ω
3.4L cm= 3.4L cm=
1 2.44w mm= 1 1.65w mm=
Figure 2 - An example of a load matched by 2 qarter wavelength microstriptransformers.
2.3 Chebyshev Matching
In this case, a ripple in the matched frequency range is defined, but thenumber of transformers can be slightly reduced relative to the binomial casefor a given bandwidth.
sec θm = cosh
∙1
Ncosh−1
µΓ
Γm
¶¸Table 2 shows the values of the characteristic impedances for the Cheby-
shev transformers.
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Table 2: Values of the characteristic impedances of Chebyshev transformers.
In case that the value of the load is smaller than the value of the char-acteristic impedance of the transmission line - replace the values, find thecharacteristic impedance of the transformers by using the table, and thenchange the order of the transformers: the left one will be now the right one,etc.
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3 Experiment Procedure
3.1 ADS Simulation
Match a load ZL = 100 Ω to a transmission line Z0 = 50 Ω at f = 1.5 GHz.Use ’Prelab Exercise’ question 1 and match for the cases:a single transformer.3 binomial transformers.3 Chebyshev transformers with Γm = 0.2.
1. Match the load to the transmission line by an ideal transmission lines,as shown for a single ideal transformer in Figure 1.
TLINTL1
F=1.5 GHzE=90Z=
S_ParamSP1
Step=10 MHzStop=3 GHzStart=0.1 GHz
S-PARAMETERS
TermTerm1
Z=50 OhmNum=1
TermTerm2
Z=75 OhmNum=2
Figure 1 - An ADS example of a single ideal transformer.
Draw the graph of S11 in the frequency range of 100MHz−3 GHz foreach case and measure the bandwidth. Why can we treat the graph ofS11 as the graph of the reflection coefficient?
2. Now, each transformer will be a microstrip with a substrate FR4 witha permittivity of εr = 4.9, height of substrate is h = 1.6 mm, height ofconductive layer is T = 0.02 mm and a loss tangent of tan δ = 0.018.
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An ADS example of two microstrip transformers is shown in Figure 2.
S_ParamSP1
Step=10 MHzStop=3 GHzStart=0.1 GHz
S-PARAMETERS
MLINTL2
L=W=Subst="MSub1"
MLINTL1
L=W=Subst="MSub1"
TermTerm2
Z=75 OhmNum=2
TermTerm1
Z=50 OhmNum=1
MSUBMSub1
Rough=0 mmTanD=0.0018T=0.02 mmHu=1.0e+033 mmCond=1.0E+50Mur=1Er=4.9H=1.6 mm
MSub
Figure 2 - An ADS example of a single microstrip transformer.
1. Draw the graph of S11 in the frequency range of 100 MHz − 3 GHzfor each case and measure the bandwidth.
Compare the bandwidths for all cases, and compare them to theoreticalbandwidth with Γm = 0.2. What is the value of SWR for this case?
3. Draw the graph of S11 in the frequency range of 100MHz−20 GHz forthe case of 3 binomial transformers (microstrip and ideal) and comparethe graphs.
3.2 CST Simulation
CST - Microwave Studio is a software package for solving 3D electromagneticequations. We will learn how to use this software by an example of a simu-lation of a T-type microstrip divider, step by step. Figure 3 shows a splitterwith and without a transformer.
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(a)(b)
4λ
Figure 3 - Microstrip splitter (a) without matching, and (b) with matching.
3.2.1 Starting the simulation
Double click CST icon, as shown in Figure 4.
Figure 4 - Double click on the CST icon.
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Choose the CST Microwave Studio option, as shown in Figure 7.
Figure 7 - Choose CST Microwave Studio icon.
Choose antenna (planar) template. Each template has a special choice ofboundary conditions and geometry, as described in Figure 5.
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Figure 5 - Choose antenna (planar) template.
3.2.2 Defining the Parameters
First, define the parameters, as shown in Table 3.Name V alue Descriptionxg 50 Ground x dimensionyg 100 Ground y dimensiont 0.02 Metal thicknesshs 1.6 Substrate thicknessw 2.84 Microstrip line widthLL 70 Length of main lineTable 3 - Parameters Definition
3.2.3 Building the Component
Building the Ground We will start with the ground component. Lowerleft window is for the parameters and their values. For the ground we choosea brick and, press the escape button and then entering the component name,
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dimensions and material, as shown in Figure 7.
Figure 7 - Building the ground.
The material of the ground brick is PEC (Perfect Electric Conductor).The dimensions of the brick are −xg ≤ x ≤ xg, 0 ≤ y ≤ yg, −t < z < 0.
Building the substrate Next we build the FR4 substrate, again with abrick. We now have to choose the material of the substrate. You can chooseit from the library, as shown in Figure 8.
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Figure 8 - Choosing the material for the substrate.
Find the Width of the Microstrip Line Choose Macros -> Calculate
-> Calculate analytic Line Impedance, then you get a the needed window.You can find that the needed width of the microstrip line is 2.84 mm for50 Ω, as shown in Figure 9.
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Figure 9 - Impedance calculation.
Building the Divider We use the Extrude option, which enables us todefine a polygon in a plane, and then inverting it to a body, as shown inFigure 10.
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Figure 10 - Building the divider.
3.2.4 Define the Ports
In this case, we will choose simple waveguide ports, as shown in Figure 11.Another option is a coaxial port, but then we have to first build coaxialconnectors.
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Figure 11 - Define a port.
The model should look like Figure 12 (make sure that the numbers of theports are facing out).
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Figure 12 - The divider with all 3 ports.
3.2.5 Running the Simulation
Choosing a frequency range by entering the ˜ icon, and choose the frequencyrange of 0− 2 GHz, as shown in Figure 13.
Figure 13 - Choosing the frequency range of the simulation.
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Choose the transient solver and then start. Here we choose to calculateonly the input port (no. 1), as shown in Figure 14.
Figure 14 - Running the simulation using the transient solver.
3.2.6 Results
You can see the scattering parameters by pressing 1D results from the navi-gating tree, and then |S| [dB]. The scattering parameters magnitude (in dB)should look like Figure 15.
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Figure 15 - S1,1 / S2,1 / S3,1 magnitude (in dB).
It is shown that the matching level is about S11 ≈ −11 dB and thetransmission to the ports are approximately S21 = S31 ≈ −4 dB at f =1 GHz. The scattering parameters phase (in degrees) should look like Figure16.
Calculate the impedance of a single quarter-wave transformer to match animpedance of 50 Ω to 25 Ω and find the width of the strip for the microstriptransformer. Add two new parameters: ’wtransformer’ with the value ofthe width you found, and ’Ltransformer’ with the length of the transformerwhich correspond to f = 1 GHz. Construct the transformer, as shown inFigure 17, and simulate again.
Figure 17 - Adding a transformer to the spliter.
Observe the graphs of the scattering parameters (magnitude and phase)and compare them to the graphs without the matching.
3.3 Measurements
1. Measure the S-parameters of the 1:2 microstrip divider (magnitude andphase).
2. Measure the S-parameters of the 1:2 microstrip divider matched by theparasitic transformer (magnitude and phase).
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3. Compare your measurements results.
References
[1] David M. Pozar, "Microwave Engineering", John Wiley & Sons, 3rd ed.,2005, ch. 5.
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