Design of Quarter-Wave Transformer for Impedance Matching Applications

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 ${\mathit{Z}}_1=\sqrt{\left({\mathit{Z}}_0{\mathit{Z}}_{\mathit{L}}\right)}$[1]. This example is to design a single section quarter-wave transformer to match the$100\Omega$ load to a $50\Omega$transmission line at an operating frequency of $2\mathrm{GHz}$. The calculated characteristic impedance of the quarter-wave transformer ${\mathit{Z}}_1$is $70.71\Omega$.

Design of Single Section Quarter-Wave Transformer

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

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 $100\Omega$ 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)

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)

It is observed from the ${\mathit{S}}_{11}$ values that the impedance is perfectly matched at the desired frequency of $2\mathrm{GHz}$ where the value of magnitude is less than $-22\mathrm{dB}$ with the bandwith of $2800\mathrm{MHz}$. Matching alsol occurs at frequencies where the ${\mathit{Z}}_1$has a length of $\left(2\mathit{n}+1\right)\raisebox{1ex}{${\lambda }_0$}\!\left/ \!\raisebox{-1ex}{$4$}\right.,\mathit{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 $100\Omega$ load to a $50\Omega$ with ripple level = $0.05$. Each section's characteristic impedances are computed using the formulas given in [1] and the values are ${\mathit{Z}}_1=57.5\Omega ,{\mathit{Z}}_2=70.7\Omega$ and ${\mathit{Z}}_3=87\Omega$. 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)

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')

From the comparison the s-parameters plot for single section and three section quarter-wave transformers designed at $2\mathrm{GHz}$, it is observed that the bandwidth achieved for three section quarter-wave transformer is $7750\mathrm{MHz}$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, $4^{\mathrm{th}}$ Edition, John Wiley & Sons, 2012.