# Viewing blog entries in category: microwave engineering

• #### How to solve Microwave Engineering Pozar chapter 5 exercise 13 with MATLAB

Question:

Literature source [POZAR] available here:
https://www.amazon.co.uk/Microwave-Engineering-David-M-Pozar/dp/0470631554

There's also a solutions manual available here:
https://www.scribd.com/doc/176505749/Microwave-engineering-pozar-4th-Ed-solutions-manual

MATLAB script source pozar_05_exercise_13.m and all necessary support functions included in .zip here attached file pozar_05_exercise_13.zip

MATLAB, an ADS circuit, the ADS LinCalc Utility and the online free microstrip calculator, these tools are used to dimension microstrip tracks as required in the question. The higher ZL/Z0 the sharper, more selective, becomes the frequency response.

John BG
jgb2012@sky.com
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• #### How to solve Microwave Engineering Pozar chapter 05 exercise 15 with ADS

Question:

Literature source [POZAR] available here:
https://www.amazon.co.uk/Microwave-Engineering-David-M-Pozar/dp/0470631554

There's also a solutions manual available here:
https://www.scribd.com/doc/176505749/Microwave-engineering-pozar-4th-Ed-solutions-manual

Basic MATLAB script source pozar_05_exercise_15.m

I call this one 'basic' because for this exercise the MATLAB script doesn't go much further than getting the cut-off details.

As the reader can appreciate for this exercise I have used ADS instead to supply the result of the matching waveguide section because
it's in chapter 3 of [POZAR] where it makes more sense to review 3D electromagnetic models but I chose to start in chapter 5, so this exercise
can be reviewed once for instance a working Yee model in MATLAB is available, that allows enough flexibility and accuracy
needed to change waveguide dimensions and fill-up dielectric characteristics in order to reach a valid affordable solution like it's the case here to be a dielectric fin instead of a complete fill-up.

John BG
jgb2012@sky.com
• #### How to solve Microwave Engineering Pozar chapter 05 exercise 16 with MATLAB

Question:

Literature source [POZAR] available here:
https://www.amazon.co.uk/Microwave-Engineering-David-M-Pozar/dp/0470631554

There's also a solutions manual available here:
https://www.scribd.com/doc/176505749/Microwave-engineering-pozar-4th-Ed-solutions-manual

MATLAB script source pozar_05_exercise_16.m and all necessary support functions included in .zip here attached file pozar_05_exercise_16.zip

The more sections added the flatter is the pass-band but the size of the circuit increases. Despite Binomial has as flat frequency response around f0 it doesn't have the best insertion loss, not .6 at 2*f0.

John BG
jgb2012@sky.com
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1.1 MB
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• #### How to solve Microwave Engineering Pozar chapter 05 exercise 17 with MATLAB

Question:

Literature source [POZAR] available here:
https://www.amazon.co.uk/Microwave-Engineering-David-M-Pozar/dp/0470631554

There's also a solutions manual available here:
https://www.scribd.com/doc/176505749/Microwave-engineering-pozar-4th-Ed-solutions-manual

MATLAB script source pozar_05_exercise_17.m and all necessary support functions included in .zip here attached file pozar_05_exercise_17.zip

This question is again the kind that one thinks the Symbolic Toolbox is going to be the perfect tool but again
the Symbolic Toolbox returns 9 DIN4 pages full of a Symbolic expression that cannot be called a solution. let alone
handled without a crane.

Luckily it can be solved numerically, as shown in the attached script and notes.

Firstly it's convenient to find out Z1 and Z2 to validate the sought expressions
• Z1(ZL,Z0)
• Z2(ZL,Z0)
z1=Z1/Z0;z2=Z2/Z0

z1 z2 turn out to be 1.1067 and 1.3554

the frequency response of |Γ| is

As shown in the attached script, the ABCD expressions can be manipulated to obtain the following

that in turn reveals (Z2/Z1) Z2_ov_Z1 = 1.224563802132642

From here all that's left is to calculate the characteristic impedance of the λ/4 section next to load Z2, from [POZAR] 253pg.

gamma_n(2)==(ZL_ov_Z0-z2)/(ZL_ov_Z0+z2)

John BG
jgb2012@sky.com
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• #### How to solve Microwave Engineering Pozar chapter 05 exercise 18 with MATLAB

Question

Literature source [POZAR] available here:
https://www.amazon.co.uk/Microwave-Engineering-David-M-Pozar/dp/0470631554

There's also a solutions manual available here:
https://www.scribd.com/doc/176505749/Microwave-engineering-pozar-4th-Ed-solutions-manual

This one is easy

ZL_ov_Z0=[1.5:.01:6];
for n=N
A=2^(-n)*(ZL_ov_Z0-1)./(ZL_ov_Z0+1);
df_over_f0=2-4./pi*acos(.5*(gamma_m./abs(A)).^(1/n));
plot(ZL_ov_Z0,df_over_f0);hold all
end
title('\Deltaf/f0');xlabel('ZL/Z0');grid on

John BG
jgb2012@sky.com
• #### How to solve Microwave Engineering Pozar chapter 05 exercise 12 with MATLAB

Question:

Literature source [POZAR] available here:
https://www.amazon.co.uk/Microwave-Engineering-David-M-Pozar/dp/0470631554

There's also a solutions manual available here:
https://www.scribd.com/doc/176505749/Microwave-engineering-pozar-4th-Ed-solutions-manual

The question only asks for the short circuit cases, but all open circuit results have been added.

John BG
jgb2012@sky.com
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• #### How to solve Microwave Engineering Pozar chapter 05 exercise 11 with MATLAB

Question:

Literature source [POZAR] available here:
https://www.amazon.co.uk/Microwave-Engineering-David-M-Pozar/dp/0470631554

There's also a solutions manual available here:
https://www.scribd.com/doc/176505749/Microwave-engineering-pozar-4th-Ed-solutions-manual

the solutions manual erroneously solves for single stub, ignoring X2, the reactance of the 2nd stub.

The sought solutions have to be of the following shape

· D_OCstub1(d,RL,XL)

· D_OCstub2(d,RL,XL)

· D_SCstub1(d,RL,XL)

· D_SCstub2(d,RL,XL)

the RL constraint would imply that there would be need for a really long transmission line for really small RL values, or that there would be need for really short transmission line when attempting to match large RL values.

Numerical and symbolic equations are supplied. For the numerical case 2D slice

is not as meaningful as visualizing the volume with both a static and a shifting plane

John BG
jgb2012@sky.com
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• #### How to solve Microwave Engineering Pozar chapter 05 exercise 10 with MATLAB

Question

Literature source [POZAR] available here:
https://www.amazon.co.uk/Microwave-Engineering-David-M-Pozar/dp/0470631554

There's also a solutions manual available here:
https://www.scribd.com/doc/176505749/Microwave-engineering-pozar-4th-Ed-solutions-manual

MATLAB script source pozar_05_exercise_09.m and all necessary support functions included in .zip here attached file pozar_05_exercise_09.zip

Reducing |s11| the resolution can be improved from 0.001 to 0.0002 without significant increase in processing delay.

For this exercise, the peaks do not show as clearly as in 5.9 so the funciton kmeans comes handy to decide where the |s11| nulls are exactly located.

Verifying the frequency responside directly connecting impedance equations for each component

Like in 5.9 some additional ADS simluations and optimiser results are included, but ADS has more accurate models of all circuit components, even for such basic ones, so the results obtained with an accurate but basic MATLAB script, and using ADS, are not going to always agree.

John Bofarull Guix
jgb2012@sky.com
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• #### How to solve Microwave Engineering Pozar chapter 05 exercise 09 with MATLAB

Question

Literature source [POZAR] available here:
https://www.amazon.co.uk/Microwave-Engineering-David-M-Pozar/dp/0470631554

There's also a solutions manual available here:
https://www.scribd.com/doc/176505749/Microwave-engineering-pozar-4th-Ed-solutions-manual

MATLAB script source pozar_05_exercise_09.m and all necessary support functions included in .zip here attached file pozar_05_exercise_09.zip
In the attached zip there is a detailed explanation in Adobe format detailing all lines of the solution.

An analytic solution with |s11| surface is show and the frequency response to valide the calculated lengths is obtained with different cross-sections of such surface.

The sought Open Ciruit shunt stub lengths are the shortest D1 D2 that null |s11|

The obtention of the |s11| surface nulls, the exact location where D1 D2 cancel reflections is found reversing the surface and catching peaks.
This is because when attemting too small D1 D2 steps, with conventional computer platforms MATLAB may reach default limits or borrow excessive
time.

The Laplacian obtained with del2 is here conveniently used to reduce the amount of peaks returned by command findpeaks.

1st solution, stubs with lengths below 0.5:

D11=0.086

D12=0.375

2nd solution
D21=0.199

D22=0.375

Some of the support functions used in previous questions have been ugraded to solve this exercise with the Smith chart.

1st stub, next to load, 1st length. The red is ZL and the green marker is YL=1/ZL.
the 1st stub has to bring YL to the magenta circle, to the intersection blue marker to the left hand side of YL.

1st stub, 2nd length, reaching the 2nd intersection now with opposite sign reactance.

2nd stub, 1st length, 3rd and 4th intersections again are the blue markers.

2nd stub, 2nd length

The resulting stub lengths match those obtained analytically.

Verifying results with frequency response:

It turns out that for a given pair D1 D2 the frequency resonse is already a particular cross section of surface |s11|.
Choosing 2 cross sections including those pairs D1 D2 reaching |s11| nulls,

And the extracted contours are

and

John Bofarull Guix
jgb2012@sky.com
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• #### How to solve Microwave Engineering Pozar chapter 05 exercise 07 with MATLAB.

Question:

Literature source [POZAR] available here:
https://www.amazon.co.uk/Microwave-Engineering-David-M-Pozar/dp/0470631554

There's also a solutions manual available here:
https://www.scribd.com/doc/176505749/Microwave-engineering-pozar-4th-Ed-solutions-manual

Certain loads can be matched at a single frequency with just a transmission line with the correct characteristic impedance and length.

Attached MATLAB script to to calculate Z1 and L of such transmission load to match to Z0, not all loads can be matched with such simple circuit.

John BG
jgb2012@sky.com
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• #### How to solve Microwave Engineering Pozar chapter 05 exercise 08 with MATLAB

Question

Literature source [POZAR] available here:
https://www.amazon.co.uk/Microwave-Engineering-David-M-Pozar/dp/0470631554

There's also a solutions manual available here:
https://www.scribd.com/doc/176505749/Microwave-engineering-pozar-4th-Ed-solutions-manual

For Open Circuit and Short circuit lossy transmission line stubs there are peaks of valleys of reactance close to multiples of λ/4 and 3λ/4 (equivalent electric lengths π/2 and 3π/2).

Since there's always some real part of the input impedance, it's about choosing a length that supplies high enough |reactance| while keeping the resistive part low enough for the OC/SC stub to be considered an equivalent 'good' capacitor or inductor.

Attached MATLAB script that shows how to find such reactance peaks and valleys for a section of lossy transmission line.
File size:
865.5 KB
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• #### How to solve Microwave Engineering Pozar chapter 05 exercise 05 with ADS and MATLAB

Question

the literature source [POZAR] is available here:
https://www.amazon.co.uk/Microwave-Engineering-David-M-Pozar/dp/0470631554

There's also a solutions manual available here:
https://www.scribd.com/doc/176505749/Microwave-engineering-pozar-4th-Ed-solutions-manual

this is the OC series stub impedance match circuit on single frequency.

The arcs of the transmission line lengths on the Smith chart are:

And the OC series stub arcs:

There's also an alternative graph to find the required transmission line and stub lengths; |s11| surface.

clc;clear all;close all
ZL=90+1j*60;Z0=75;
Dstep=.001;drange=[0: Dstep:1];
D1=drange;D2=D1;
[D1,D2]=meshgrid(drange);

Z1=Z0*(ZL+1j*Z0*tan(2*pi*D1))./(Z0+1j*ZL*tan(2*pi*D1));
Zin_stub=Z0./(1j*tan(2*pi*D2));
% series oc stub
Zin=Z1+Zin_stub;
s11=(Zin-Z0)./(Zin+Z0);

surf(abs(s11),'LineStyle','none')
ax=gca
xlabel('D1');ylabel('D2');
ax.XTickLabelMode='manual'; ax.YTickLabelMode='manual';
ax.XTickLabel=[0:0.1:1];ax.YTickLabel=[0:0.1:1];
ax.XTick=[0:100:1000];ax.YTick=[0:100:1000];
ax.PlotBoxAspectRatio=[1 1 1]

hold all;x0=find(drange==.5) % plotting corner to box D1<.5 D2<.5
plot3([x0 x0 0],[0 x0 x0],[5 5 5],'Color',[1 0 0],'LineWidth',3)

% moving camera birdeye view
ax.CameraPosition=[500 500 10]

camzoom(ax,1.5) % zoom in a bit, camzoom is cumulative
% zoom_factor within [0 1) zooms out zoom_factor>1 zooms in

ax.CameraUpVector = [0 1 0]; % camera attitude
ax.CameraTarget = [500 500 0]; % centring
ax.CameraViewAngle =8*pi; % focus

automating peaks capture: findpeaks with MinPeakProminence=2.5 returns the right amount of peaks. MinPeakProminence larger or smaller than 2.5 then either too few or too many peaks.

With MinPeakHeight=2 command findpeaks doesn't catch the right amount of peaks for any MinPeakDistance, going from 6 peaks only to too many peaks.

With Threshold=2 doesn't catch a single peak but Threshold=1 gets the right amount of peaks.

To get zeros exact locations, it's useful to invert |s11| surface just plotted, with the Laplacian of the surface, command del2

V=1e3*del2(abs(s11));
figure(2);ax=gca;surf(V,'Lines','none');
xlabel('D1');ylabel('D2');
ax.XTickLabelMode='manual';
ax.YTickLabelMode='manual'; ax.XTickLabel=[0:0.1:1];ax.YTickLabel=[0:0.1:1];
ax.XTick=[0:100:1000];ax.YTick=[0:100:1000];
[pks,locs]=findpeaks(V( : ),'Threshold',1);
[nd1,nd2]=ind2sub(size(V),locs);

hold all;figure(2);plot3(nd2,nd1,V(nd2,nd1)+2,'ro');
% plot peaks
ax.PlotBoxAspectRatio=[1 1 1]

x0=find(drange==.5) % plot corner to box D1<.5 D2<.5
figure(2);
plot3([x0 x0 0],[0 x0 x0],[.5 .5 .5],'Color',[1 0 0],'LineWidth',3)
ax2=gca

ax2.CameraPosition=[500 500 10]
% moving camera bird-eye view
ax2.CameraUpVector = [0 1 0]; % camera attitude

camzoom(ax,1.5) % zoom in a bit, camzoom is cumulative

% zoom_factor within [0 1) zooms out zoom_factor>1 zooms in
ax2.CameraTarget = [500 500 0]; % centring

abs(s11(sub2ind(size(V),nd1,nd2)))

= 0.001390798787950
0.001390798787950
0.002883920448033
0.002883920448033
0.001390798787950
0.001390798787950
0.002883920448033
0.002883920448033

unique(sort(drange(nd1)))
= 0.1470 0.3530 0.6470 0.8530

numel(nd1)
= 8

among these stub lengths

D1=unique(drange(nd1))
= 0.1470 0.3530 0.6470 0.8530

D2=unique(drange(nd2))

= 0.1740 0.4820 0.6740 0.9820

the stub lengths inside D<.5 are the smallest ones

Dstub1= D1([1 2])
= 0.1470 0.3530

Dstub2= D2([1 2])

= 0.1740 0.4820

The resulting frequency response is the same as the one obtained with the MATLAB solution

So the problem is simple enough to find transmission line and stub length on a surface.

Quick check how short circuit and open circuit stubs behave over frequency:

Zin_sc_stub=1j*Z0*tan(beta*L)

Zin_oc_stub=-1j*Z0*cot(beta*L)

da=pi/2000;a=[-2*pi:da:2*pi];
y_OC=-cot(a); y_SC=tan(a);
figure;plot(a,y_OC,a,y_SC);
grid on;axis([-2*pi 2*pi -10 10])
legend('SC','OC')
ylabel('Stub Z_i_n');xlabel('\betaL')
xticks([-2*pi -2*pi*3/4 -pi -pi/2 0 pi/2 pi 3*pi/2 2*pi])
xticklabels({'-2\pi','-3\pi/4','-\pi','-\pi/2','0','\pi/2','\pi','3\pi/2','2\pi'})

John BG
jgb2012@sky.com
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1.4 MB
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• #### How to solve Microwave Engineering Pozar chapter 05 example 03 with ADS and MATLAB

Question

the literature source [POZAR] is available here:
https://www.amazon.co.uk/Microwave-Engineering-David-M-Pozar/dp/0470631554

There's also a solutions manual available here:
https://www.scribd.com/doc/176505749/Microwave-engineering-pozar-4th-Ed-solutions-manual

The attached pack .pdf shows the solution with Keysight ADS only, but all included .m scripts also show how to solve this example with MATLAB.

With MATLAB the following Transmission Line arcs on Smith chart are obtained

and the OC series stub lengths on Smith chart are

the frequency responses obtained with the MATLAB solution are

John BG
jgb2012@sky.com
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940.4 KB
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4
• #### How to solve Microwave Engineering Pozar chapter05 example 02 with MATLAB

Question

the literature source [POZAR] is available here:
https://www.amazon.co.uk/Microwave-Engineering-David-M-Pozar/dp/0470631554

There's also a solutions manual available here:
https://www.scribd.com/doc/176505749/Microwave-engineering-pozar-4th-Ed-solutions-manual

In this example, the 1st attached pack pozar_05_Example_02.zip uses intersections.m while the 2nd attached pack pozar_05_example_02_2.zip uses the simplified circles_intersect.m instead.

A 3rd attachment to just test this alternative support function circles_intersect.m
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• #### How to solve Microwave Engineering Pozar chapter 5 exercise 4 with MATLAB

Question:

Literature source [POZAR] is available here:
https://www.amazon.co.uk/Microwave-Engineering-David-M-Pozar/dp/0470631554

There's a solutions manual available here:
https://www.scribd.com/doc/176505749/Microwave-engineering-pozar-4th-Ed-solutions-manual