THE IMPEDANCE MATCHING AND CALCULATION OPTIMIZATION OF MICROSTRIP LINE ARE REALIZED BY MATLAB

At high frequencies, above 1 GHz where we are building amplifiers, there is no such thing as a lumped capacitor or a coil inductor like you are used to in low frequency circuits (like we did in circuits).   So for your impedance matching (IM) network, you have to operate above 1 GHz, and it must be all based on transmission lines.  You can have hardware like that depicted in Figure 8-26.  There the hardware is based on microstrip, a good choice for frequencies between 500 MHz and 6 GHz. 

The idea is to use computer optimization to determine all the lengths and characteristic impedances of all section of the hardware, such that the reflection coefficients are within a certain range over a wide bandwidth (as wide as possible). 

Make sure your report shows all the mathematics of the optimization, the code where you implemented it, and all the results.  Also describe how you did the optimization practically.  How long did it take, how did you choose the initial settings and all these details.  I need to see how your cost function (that you minimize) looks, and you must explain why and how you chose it. 

1) I had a lot of Q's about caps and inductors.  I'll say it again, at the frequencies where we are working, you cannot make caps and inductors.  They dont work at such high frequencies.   All energy storage devices at high frequencies take the form of transmission line "stubs", an example system for high frequencies is shown in Figure 8-26 in the book.  The stubs are either shorted (load 0 Ohm) or they are open (infinity Ohm).   So they cant burn energy, they can only store energy, so their reactance is imaginary.  

2) The other Q I am getting is how to compute the input reflection coefficient.  I do it with a signal flow graph.  It gives a nice closed form expression for the reflection coefficient.  I input unity (1), then I just compute the reflected term, that gives gamma.  Then I take the absolute value squared of gamma, which is a function of the frequency.  Then I sample this term at a few frequencies over the range I am interested in.  I sum all these terms, as in Alpha = |gamma(f1)|^2 + |gamma(f2)|^2 + |gamma(f3)|^2 ......... 

This sum, or Alpha, is my "cost function", this I give to the optimization function in matlab.  It finds the lengths and Z_0's of all the lines in the system to get the reflection coefficient as low as possible.  So for this assignment we are trying to get the reflection coefficient over a range of frequencies (bandwidth) as low as possible. 

One of the most potent techniques to model electronics systems at high frequencies (above 1 GHz) is the signal flow graph.  In fact I use it exclusively, and have done that for decades and it gives me answers fast, and has never failed me.  I have never regretted using a signal flow graph to compute something, even when I could have used a basic network analysis combined with the transmission line equation. 

I am told you do cover a little bit of signal flow graph theory here at UTAS (in control?), but in case you are not comfortable with this very powerful method, review the material in the book I wrote on linear systems theory.  If you don't have a copy of the book, download one from the internet (its free, just google it for the download link).  

In the book chapter 11, find an entire chapter with theory and examples so you can learn to model circuits and systems using signal flow graphs.  Mason's rule or Mason's theorem (see https://en.wikipedia.org/wiki/Mason%27s_gain_formula for more info and examples) gives you the ability to write down the answer to any query on a graph almost by inspection.  It is an incredibly powerful theorem invented during the 1950's by Sam Mason - make use of it, dont make yourself suffer by doing things through slow  complex algebra.  Make use of methods that have been invented by very smart people.  

I had a lot of queries as to how to model RF systems with transmission lines.   Here is a tip. 

A) Look at the first figure (one_line).  See how I modelled that case.   Get the input reflection coefficient with Mason's rule. 

B) Now look at the second figure (three_line).  Can you model that case with a graph?  Z_o1 not equal to Zo2 not equal Zo3.  So there are reflections where the different transmission lines connect.  For both incoming waves and backward waves.  Get the input reflection coefficient with Mason's rule.

C) Now add a stub somewhere, that is further transmission line that takes a wave away from the "a" where it connects, and returns a wave to the corresponding "b". Now model that with a graph.  Get the input reflection coefficient with Mason's rule.  You will see  that now the length of the stub, and its Zo modifies the input reflection coefficient.  So it gives you 2 additional levers to pull.  You can optimize the length and Zo of the stub to make the input reflection coefficient smaller (at least at 1 frequency).  

D) Now add another stub.  Get the input reflection coefficient with Mason's rule.  Now you have 4 more levers to pull with the 2 stubs.  Of course you can also change l1, l2 l3, and Z01, Z02 and Z03, that give you 6 more levers to pull. 

You could do this system without graphs.  You just compute the input impedance at the generator side, including the stubs.  So you start at the load, then move back section for section, using the transmission line equation.  Each time combining admittances, then converting to impedance, then moving along a transmission line, etc etc.  This gives the same answer.  

I did an example for you, where there are two transmission lines, different Z0's in series.  Solved it with graph, see screenshot.  You can see how fast the graph is.  I choose Z_load = 3 + j 17.  

If I added a stub at the intersection, to modify the graph is easy, Mason's rule will pop out the updated gamma fast.  

You go and solve with transmission line analysis, then compare.