Design of Quarter-Wave Transformer for Impedance Matching Applications

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This example shows how to design the Quarter-wave transformer for impedance matching applications by using the pcbComponent, microstripLine, and traceRectangular object in the RF PCB Toolbox.

Quarter-wave transformer is a simple and useful circuit for matching the real impedance of a terminating load ( $Z_{L}$ ) to the characteristic impedance of the feeding transmission line ( $Z_{0}$ ) as depicted in the given figure. The characteristic impedance of the quarter-wave transformer is calculated as $Z_{1} = \sqrt{(Z_{0} Z_{L})}$ [1]. This example is to design a single section quarter-wave transformer to match the $100 Ω$ load to a $50 Ω$ transmission line at an operating frequency of $2 GHz$ . The calculated characteristic impedance of the quarter-wave transformer $Z_{1}$ is $70.71 Ω$ .

Design of Single Section Quarter-Wave Transformer

Use the design function on the microstripLine object to create the $50 Ω$ input transmission line and $70.71 Ω$ quarter-wave transformer's Length and Width dimension for the operating frequency of $2 GHz$ . The default substrate for microstripLine is Teflon with thickness of $1.6 mm$ .

freq = 2e9;
m50 = design(microstripLine,freq,Z0=50,LineLength=0.05); % input transmission line
m70 = design(microstripLine,freq,Z0=70.71,LineLength=0.25); % section 1

Use the traceRectangular object to create the groundplane, input transmission line, and quarter-wave transmission line shapes.

% ground plane dimension
gndL = 2*m50.Length+m70.Length;
gndW = 5*m50.Width;

ground = traceRectangular("Length",gndL,"Width",gndW);
% input transmission line
Z0 = traceRectangular("Length",m50.Length,"Width",m50.Width,...
    "Center",[-gndL/2+m50.Length/2 0]);
% First section Quarter-wave transformer
Z1 = traceRectangular("Length",m70.Length,"Width",m70.Width,"Center",...
    [-gndL/2+m50.Length+m70.Length/2 0]);

qline = Z0 + Z1;

Use the pcbComponent to create the quarter-wave transformer and use the lumpedElement for the terminating load of $100 Ω$ and place it at the end of the quarter-wave transformer and use a via to connect the load to ground.

pcb =pcbComponent;
pcb.BoardShape = ground;
pcb.BoardThickness = m50.Height;
pcb.Layers = {qline,m50.Substrate,ground};
pcb.FeedDiameter = m50.Width/2;
pcb.FeedLocations = [-gndL/2 0 1 3];
pcb.ViaLocations = [-gndL/2+m50.Length+m70.Length,0,1,3];
pcb.ViaDiameter = m70.Width/2;
% Load
ZL = lumpedElement;
ZL.Impedance = 100;
ZL.Location = [-gndL/2+m50.Length+m70.Length,0,pcb.BoardThickness];
pcb.Load = ZL;
% show the single section quarter-wave transformer
figure;show(pcb)

Figure contains an axes object. The axes object with title pcbComponent element, xlabel x (mm), ylabel y (mm) contains 9 objects of type patch, surface. These objects represent PEC, feed, Teflon, load.

Use the sparameters function to calculate the S parameters and plot it using the rfplot function.

sparams = sparameters(pcb,linspace(100e6,8e9,51));
figure; rfplot(sparams)

$Figure contains an axes object. The axes object with xlabel Frequency (GHz), ylabel Magnitude (dB) contains an object of type line. This object represents dB(S_{11}).$

It is observed from the $S_{11}$ values that the impedance is perfectly matched at the desired frequency of $2 GHz$ where the value of magnitude is less than $- 22 dB$ with the bandwith of $2800 MHz$ . Matching alsol occurs at frequencies where the $Z_{1}$ has a length of $(2 n + 1) \frac{λ_{0}}{4}, n = 0, 1, 2, 3 . . .$

Design of Multisection Quarter-Wave Transformer

In general, the single section transformer may be sufficient for narrow band impedance match. This transformer can be extended to multisection in a methodical manner to yield optimum matching characteristics over a wider bandwidth [1]. The following example is for the design of three section Chebyshev matching transformer to match a $100 Ω$ load to a $50 Ω$ with ripple level = $0.05$ . Each section's characteristic impedances are computed using the formulas given in [1] and the values are $Z_{1} = 57.5 Ω, Z_{2} = 70.7 Ω$ and $Z_{3} = 87 Ω$ . The three section Chebyshev matching transformer is designed by using same steps described in the design of single section quarter-wave transformer.

m50 = design(microstripLine,freq,"Z0",50,"LineLength",0.05); % input transmission line
m57 = design(microstripLine,freq,"Z0",57.5,"LineLength",0.25); % section 1
m70 = design(microstripLine,freq,"Z0",70.7,"LineLength",0.25); % section 2
m87 = design(microstripLine,freq,"Z0",87,"LineLength",0.25); % section 3
% ground plane dimension
gndL = 2*m50.Length+m57.Length+m70.Length+m87.Length;
gndW = 5*m50.Width;

ground = traceRectangular("Length",gndL,"Width",gndW);
% inpt transmission line
Z0 = traceRectangular("Length",m50.Length,"Width",m50.Width,...
    "Center",[-gndL/2+m50.Length/2 0]);
% First section Quarter-wave transformer
Z1 = traceRectangular("Length",m57.Length,"Width",m57.Width,"Center",...
    [-gndL/2+m50.Length+m57.Length/2 0]);
% Second section Quarter-wave transformer
Z2 = traceRectangular("Length",m70.Length,"Width",m70.Width,"Center",...
    [-gndL/2+m50.Length+m57.Length+m70.Length/2 0]);
% Third section Quarter-wave transformer
Z3 = traceRectangular("Length",m87.Length,"Width",m87.Width,"Center",...
    [-gndL/2+m50.Length+m57.Length+m70.Length+m87.Length/2 0]);

qline = Z0+Z1+Z2+Z3;
% create pcbComponent
pcb =pcbComponent;
pcb.BoardShape = ground;
pcb.BoardThickness = m50.Height;
pcb.Layers ={qline,m50.Substrate,ground};
pcb.FeedDiameter = m50.Width/2;
pcb.FeedLocations = [-gndL/2 0 1 3];
pcb.ViaLocations = [-gndL/2+m50.Length+m57.Length+m70.Length+m87.Length,0,1,3];
pcb.ViaDiameter = m87.Width/2;
% Load
ZL = lumpedElement;
ZL.Impedance = 100;
ZL.Location = [-gndL/2+m50.Length+m57.Length+m70.Length+m87.Length,0,pcb.BoardThickness];
pcb.Load = ZL;
% show the three section quarter-wave transformer
figure;show(pcb)

% Analysis
sparams3 = sparameters(pcb,linspace(100e6,8e9,51));
figure; rfplot(sparams3)

$Figure contains an axes object. The axes object with xlabel Frequency (GHz), ylabel Magnitude (dB) contains an object of type line. This object represents dB(S_{11}).$

Comparison of S parameters

figure; rfplot(sparams,'-.')
hold on
rfplot(sparams3)
legend('dB(S_{11}) for N=1','dB(S_{11}) for N=3','Location','northeast')

$Figure contains an axes object. The axes object with xlabel Frequency (GHz), ylabel Magnitude (dB) contains 2 objects of type line. These objects represent dB(S_{11}) for N=1, dB(S_{11}) for N=3.$

From the comparison the s-parameters plot for single section and three section quarter-wave transformers designed at $2 GHz$ , it is observed that the bandwidth achieved for three section quarter-wave transformer is $7750 MHz$ with improved impedance matching characteristics.Therefore, it is evident that wider bandwidth is achieved for quarter-wave transformers with multiple sections.

References

[1] David M. Pozar, Microwave Engineering, pp. 246-261, $4^{th}$ Edition, John Wiley & Sons, 2012.