Impedance Matching by Using Smith Chart Associated with A Proprietary Spreadsheet, Part I-The Start Sheet



This is one of the articles of our Smith chart and impedance matching sequence. If you feel need to know more basics of Smith chart or impedance matching, you should start with “Smith Charts-Basics, Parameters, Equations, and Plots.” and follow the instructions to visit all articles.

Then you can come back here and continue to read further.

In this article you’ll learn step-by-step guide how to use both Smith chart and spreadsheet together to complete your impedance matching tasks, and you don’t even need to to know all those complicate formulas to use this guide.

At the end of this article, you’ll be directed to download a proprietary spreadsheet that you can use to do impedance matching for all types of impedance and get the answers you look for within a fraction of second.

This is a unique spreadsheet you never find anywhere else. It’s created by the author by applying together those sophisticate Smith chart formulas and accumulating a large amount of graph plot data.

Since all heavy jobs have already been done, you can easily use the spreadsheet by only input the needs-to-be-matched impedance and you’ll get all answers you are looking for as soon as you finish entering the data. You’ll be provided the Smith chart plots, matching circuits,as well as component values.

If you are only looking for a program that you can use to get you quick answers without knowing much about how or why behind it, then this program will also satisfy you perfectly.

Without further ado, let’s introduce this wonderful program.

The Smith Chart

Fig. 1    The Basic parameters of Smith chart

The Smith chart consists of 3 basic parameters, Reflection Coefficient $$Γ=Γ_r+jΓ_i$$, normalized impedance $$z=r+jx$$, and normalized admittance $$y=g+jb$$.

$$Γ$$ is a typical rectangular $$(x,y)$$ function, $$r$$ and $$g$$ are 2 sets of circles representing the real parts of impedance $$z$$ and admittance $$y$$, while $$x$$ and $$b$$ are 2 sets of arcs representing the imaginary parts of impedance and admittance.

The whole chart graph is a plot of a sequence of data in the spreadsheet.

You don’t really need a very fine Smith chart to work with to get accurate answers. Smith chart is only a tool to help you understand the process of impedance matching.

We’ll discuss only the meaningful shaded area in Fig. 1.

Impedance matching using Smith chart and spreadsheet

Suppose you find a power amplifier and would like to get the most power output by impedance matching both input port and output port. You most likely obtain input return loss $$S_{11}$$ and output return loss $$S_{22}$$ from the datasheets of this particular component and you wonder what is the next step you should take.

Fig. 2   Input and output impedance matching

For the sake of simplicity, we only work on input port at single frequency $$F$$. We’ll discuss wideband impedance matching in different articles.

The “Start” sheet

The only part you need to take care of is the “Start” sheet in which you need to enter data to get things started.

All other sheets provide only answers and you need not to modify anything at all.

However, even though this is a very user friendly program, you should take a few precautions to avoid any confusion or mistakes.

We’ll discuss each parameter first before learning how to enter required data to get the job done.

Fig. 3    The “Start” sheet data entry

1.   $$Z_o$$, in ohms, the characteristic impedance of the network, is 50 ohms in most RF applications. Do not leave this space blank.

2.   Operation frequency $$F$$, in MHz, do not leave this space blank.

3.   Return loss amplitude $$|S_{11}| \le 1$$

4.   Return loss angle $$\angle S_{11}$$, it’s between 0° and 360°, or between -180° and 180°.

5. & 6.  Reflection coefficient $$Γ=Γ_r+jΓ_i$$, $$Γ_r$$ is equivalent to $$S_{11x}$$ and $$Γ_i$$ is equivalent to $$S_{11y}$$. Both $$Γ_r$$ & $$Γ_i$$ are derived from $$|S_{11}|$$ and $$\angle S_{11}$$.

$$Γ_r=S_{11x}=|S_{11}|\cos \angle S_{11}$$, it is the real part of $$Γ$$ on the Smith chart.

$$Γ_i=S_{11y}=|S_{11}|\sin \angle S_{11}$$, it is the imaginary part of $$Γ$$ on the Smith chart.

7. & 8.  Normalized impedance $$z=r+jx$$. $$r$$ and $$x$$ are the only data needed for all other worksheets to get final answers.

Both $$r$$ and $$x$$ are derived from $$Γ$$ by formulas.

9. & 10.  Normalized admittance $$y=g+jb$$.

Both $$g$$ and $$b$$ are derived from $$r$$ and $$x$$ by formulas.

11. & 12.  Impedance $$Z=R+jX$$, use this number to calculate $$r(a)$$ and $$x(a)$$ if this is what you only are able to provide.

13. & 14.  Normalized impedance $$z(a)=r(a)+jx(a)$$ via $$Z$$.

$$z(a)=Z/Z_0=(R+jX)/Z_0$$

If $$Z_0=50$$ ohms then,

$$z(a)=(R+jX)/50$$

4 methods to provide normalized impedance:

Frequency $$F$$, characteristic impedance $$Z_0$$, and normalized impedance $$z=r+jx$$ are the only data needed for all other worksheets in this spreadsheet to calculate the matching component values.

You have 4 alternative ways to provide $$z$$.

Method #1.   The default and most popular way: You provide $$S_{11}$$ data, $$|S_{11}|$$ and $$\angle S_{11}$$.

$$Γ$$ and $$z$$ will be calculated automatically based on the $$S_{11}$$ data.

Fig. 4   Data entry Method #1

Method #2.    You provide Reflection Coefficient $$Γ$$ data, both $$Γ_r$$ and $$Γ_i$$, to replace both $$Γ$$ equations, and $$z$$ will be calculated automatically based on the $$Γ$$ data. All $$S_{11}$$ data are ignored.

Fig. 5   Data entry Method #2

Method #3.    You provide directly the normalized impedance $$z$$, both $$r$$ and $$x$$, to replace both $$z$$ equations. All $$S_{11}$$ and $$Γ$$ data are ignored.

Fig. 6   Data entry Method #3

Method #4.    You provide pre-normalized impedance $$Z$$, both $$R$$ and $$X$$ into spaces 9 & 10, to get $$r(a)$$ and $$x(a)$$. You then copy and paste both data to spaces $$r$$ and $$x$$. All $$S_{11}$$ and $$Γ$$ data are ignored.

Fig. 7   Data entry Method #4

NOTE: Unless you provide the $$z$$ data via the default method #1, all other methods #2 through #4 will disturb the formats of the “Data entry row”, and you should reset this row as soon as possible by copying from the whole “Backup row” data after you are done with the task.

Fig. 8   The “Data entry row” after a few operations

Fig. 9   Copy the “Backup row” to reset the “Data entry row”

DO NOT modify the “Backup row”.

Now we have entered the normalized impedance $$z$$ using one of those 4 methods in the “Start” sheet, we are ready to see the results. Go ahead to visit Part II, the matching circuits and their values, as well as all plots on the Smith chart.You will be amazed by what you see there.

‘Note: This is an article written by an RF engineer who has worked in this field for over 40 years. Visit ABOUT to see what you can learn from this blog.’