Impedance Matching by Using Smith Chart Associated with A Proprietary Spreadsheet, Part II-The Results

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In Part I, we have learned how to provide the normalized impedance \(z\) using one of the 4 available methods in the “Start” sheet, and we are ready to see and discuss the results in this article.

At the end of this article, after you have learned the whole process, you will be provided a link to download this great and unique program.

These articles will greatly help you understand impedance matching and Smith chart:

Smith Charts-Basics, Parameters, Equations, and Plots.

Understanding Smith Chart by Learning Examples and Questions.

RF Circuits Impedance Matching Basics.

Impedance Matching-Using Lump Elements, Formulas, and Conversions-Part I.

Impedance Matching-Using Lump Elements, Formulas, and Conversions-Part II.

Impedance Matching by Using Smith Chart – A Step-by-Step Guide, Part I.

Impedance Matching by Using Smith Chart – A Step-by-Step Guide, Part II.

 

4 Types of Impedance

Based on the values of r, g, x, and b, we can roughly categorize the impedance into 4 different types:

  • Type #1: r ≥ 1, x any value.
  • Type #2: g ≥ 1, b any value.
  • Type #3: r < 1, g < 1, x > 0 or b < 0.
  • Type #4: r < 1, g < 1, x < 0 or b > 0.

Fig. 1   Four types of impedance in the Smith chart

 

Theoretically all these 4 types of impedance can be perfectly matched to 50Ω (or any other values) by using only 2 lumped elements, inductors and capacitors, if not considering the limited component values we are able to get as well as their tolerances.

The impedance of a capacitor \(C\) at frequency \(F\) is \(-1/2πFC\).

The impedance of an inductor \(L\) at frequency \(F\) is \(2πFL\).

The admittance of a capacitor \(C\) at frequency \(F\) is \(2πFC\).

The admittance of an inductor \(L\) at frequency \(F\) is \(-1/2πFL\).

 

How to read the worksheet

All values of \(Z0\) (ohm), \(F\) (MHz), \(r\), and \(x\) are originated from the “Start” sheet as soon as you provide them using anyone of the 4 methods.

And you are done with your part, it’s time to find the answers you are looking for.

 

Let’s define the matching components first:

\(C_p\): normalized admittance of the capacitor in shunt (parallel).

The real capacitor value is:

\(C_{p} (pF)=C_p \times 10^6\div (2πFZ_0) …. (1)\)

 

\(L_p\): normalized admittance of the inductor in shunt (parallel).

The real inductor value is:

\(L_{p} (nH)=Z_0 \times 10^3\div (2πF(-L_p)) …. (2)\)

 

\(C_s\): normalized impedance of the capacitor in series.

The real capacitor value is:

\(C_{s} (pF)=10^6\div (2πF(-C_s)Z_0) …. (3)\)

 

\(L_s\): normalized impedance of the inductor in series.

The real inductor value is:

\(L_{s} (nH)={L_s \times Z_0 \times 10^3}\div (2πF) …. (4)\)

 

Matching Type #1 impedance:

Fig. 2   \(r\) value in the “Start” sheet, Type #1 impedance

 

If the resistance \(r\), real part of the normalized impedance \(z\), is greater or equal to 1 as we find in the “Start” sheet, then it’s a Type #1 impedance and we would find the matching results in the “Type #1” worksheet.

There are 2 impedance matching options using 2 lossless lumped elements for this particular Type #1 impedance.

The Option #1 matching circuit includes a capacitor \(C_p\) in shunt with the to-be-matched impedance and then an inductor \(L_s\) in series.

Fig. 3   Type #1 impedance matching, Option #1

 

In Figure 3, \(Z_L\) is the normalized to-be-matched Type #1 impedance \(z\) and,

\(C_p=0.755\) is normalized admittance of the capacitor in shunt.

And, by equation (1),

\(C_{p} (pF)=0.83\)

 

\(L_s=0.780\) is normalized impedance of the inductor in series.

And, by equation (4),

\(L_{s} (nH)=2.14\)

 

As showed in the Smith chart of Fig. 3, \(C_p\) moves the point \(X\) to touch the \(r=1\) circle along the circle of \(g_1=0.622\), and \(L_s\) further moves it to the origin, which is the final goal.

 

 

The Option #2 matching circuit includes an inductor \(L_p\) in shunt with the to-be-matched impedance and then add a capacitor \(C_s\) in series.

 

Fig. 4   Type #1 impedance matching, Option #2

 

\(L_p=-0.215\) is normalized admittance of the inductor in shunt.

And, by equation (2),

\(L_{p} (nH)=12.78\)

 

\(C_s=-0.780\) is normalized impedance of the capacitor in series.

And, by equation (3),

\(C_{s} (pF)=1.41\)

 

As showed in the Smith chart of Fig. 4, \(L_p\) moves the point \(X\) to touch the \(r=1\) circle along the circle of \(g_1=0.622\), and \(C_s\) further moves it to the origin, which is the final goal.

 

<<<<<<<<<<>>>>>>>>>>

 

Matching Type #2 impedance:

Fig. 5   \(g\) value in the “Start” sheet, Type #2 impedance

 

If the conductance \(g\), real part of the normalized admittance \(y\), is greater or equal to 1 as we find in the “Start” sheet, then it’s a Type #2 impedance and we would find the matching results in the “Type #2” worksheet.

There are 2 impedance matching options using 2 lossless lumped elements for this particular Type #2 impedance.

The Option #1 matching circuit includes a capacitor \(C_s\) in series with the to-be-matched impedance and then an inductor \(L_p\) in shunt.

 

Fig. 6   Type #2 impedance matching, Option #1

 

In Figure 6, \(C_s=-0.215\) is normalized impedance of the capacitor in series.

And, by equation (3),

\(C_{s} (pF)=5.11\)

 

\(L_p=-0.780\) is normalized admittance of the inductor in shunt.

And, by equation (2),

\(L_{p} (nH)=3.52\)

 

As showed in the Smith chart of Fig. 6, \(C_s\) moves the point \(X\) to touch the \(g=1\) circle along the circle of \(r_1=0.622\), and \(L_p\) further moves it to the origin, which is the final goal.

 

The Option #2 matching circuit includes an inductor \(L_s\) in series with the to-be-matched impedance and then add a capacitor \(C_p\) in shunt.

Fig. 7   Type #2 impedance matching, Option #2

 

\(L_s=0.755\) is normalized impedance of the inductor in series.

And, by equation (4),

\(L_{s} (nH)=2.07\)

 

\(C_p=0.780\) is normalized admittance of the capacitor in shunt.

And, by equation (1),

\(C_{p} (pF)=0.86\)

 

As showed in the Smith chart of Fig. 7, \(L_s\) moves the point \(X\) to touch the \(g=1\) circle along the circle of \(r_1=0.622\), and \(C_p\) further moves it to the origin, which is the final goal.

 

<<<<<<<<<<>>>>>>>>>>

 

Matching Type #3 impedance:

Fig. 8   \(z\) & \(y\) values in the “Start” sheet, Type #3 impedance

 

If both the resistance \(r\), real part of the normalized impedance \(z\), and the conductance \(g\), real part of the normalized admittance \(y\), is less than 1, and also \(x\), the imaginary part of \(z\), is positive as we find in the “Start” sheet, then it’s a Type #3 impedance and we would find the matching results in the “Type #3” worksheet.

There are 4 impedance matching options using 2 lossless lumped elements for this particular Type #3 impedance.

 

The Option #1 matching circuit includes a capacitor \(C_p\) in shunt with the to-be-matched impedance and then a capacitor \(C_s\) in series.

Fig. 9   Type #3 impedance matching, Option #1

 

In Figure 6, \(C_p=0.57\) is normalized admittance of the capacitor in shunt.

And, by equation (1),

\(C_{p} (pF)=0.63\)

 

\(C_s=-1.036\) is normalized impedance of the capacitor in series.

And, by equation (3),

\(C_{s} (pF)=1.06\)

 

As showed in the Smith chart of Fig. 9, \(C_p\) moves the point \(X\) to touch the \(r=1\) circle along the circle of \(g_1=0.483\), and \(C_s\) further moves it to the origin, which is the final goal.

 

The Option #2 matching circuit includes a capacitor \(C_p\) in shunt with the to-be-matched impedance and then add an inductor \(L_s\) in series.

 

Fig. 10   Type #3 impedance matching, Option #2

 

\(C_p=1.569\) is normalized admittance of the capacitor in shunt.

And, by equation (1),

\(C_{p} (pF)=1.72\)

 

\(L_s=1.036\) is normalized impedance of the inductor in series.

And, by equation (4),

\(L_{s} (nH)=2.84\)

 

As showed in the Smith chart of Fig. 10, \(C_p\) moves the point \(X\) to touch the \(r=1\) circle along the circle of \(g_1=0.483\), and \(L_s\) further moves it to the origin, which is the final goal.

 

The Option #3 matching circuit includes a capacitor \(C_s\) in series with the to-be-matched impedance and then a capacitor \(C_p\) in shunt.

Fig. 11   Type #3 impedance matching, Option #3

 

In Figure 11, \(C_s=-0.3\) is normalized impedance of the capacitor in series.

And, by equation (3),

\(C_{s} (pF)=3.66\)

 

\(C_p=1.362\) is normalized admittance of the capacitor in shunt.

And, by equation (1),

\(C_{p} (pF)=1.49\)

 

As showed in the Smith chart of Fig. 11, \(C_s\) moves the point \(X\) to touch the \(g=1\) circle along the circle of \(r_1=0.35\), and \(C_p\) further moves it to the origin, which is the final goal.

 

The Option #4 matching circuit includes a capacitor \(C_s\) in series with the to-be-matched impedance and then add an inductor \(L_p\) in shunt.

Fig. 12   Type #3 impedance matching, Option #4

 

\(C_s=-1.254\) is normalized impedance of the capacitor in series.

And, by equation (3),

\(C_{s} (pF)=0.88\)

 

\(L_p=-1.362\) is normalized admittance of the inductor in shunt.

And, by equation (2),

\(L_{p} (nH)=2.02\)

 

As showed in the Smith chart of Fig. 12, \(C_s\) moves the point \(X\) to touch the \(g=1\) circle along the circle of \(r_1=0.35\), and \(L_p\) further moves it to the origin, which is the final goal.

 

<<<<<<<<<<>>>>>>>>>>

 

Matching Type #4 impedance:

Fig. 13   \(z\) & \(y\) values in the “Start” sheet, Type #4 impedance

 

If both the resistance \(r\), real part of the normalized impedance \(z\), and the conductance \(g\), real part of the normalized admittance \(y\), is less than 1, and also \(x\), the imaginary part of \(z\), is negative as we find in the “Start” sheet, then it’s a Type #4 impedance and we would find the matching results in the “Type #4” worksheet.

There are 4 impedance matching options using 2 lossless lumped elements for this particular Type #4 impedance.

 

The Option #1 matching circuit includes an inductor \(L_p\) in shunt with the to-be-matched impedance and then a capacitor \(C_s\) in series.

Fig. 14   Type #4 impedance matching, Option #1

 

In Figure 14, \(L_p=-1.183\) is normalized admittance of the inductor in shunt.

And, by equation (2),

\(L_{p} (nH)=2.32\)

 

\(C_s=-1.43\) is normalized impedance of the capacitor in series.

And, by equation (3),

\(C_{s} (pF)=0.77\)

 

As showed in the Smith chart of Fig. 14, \(L_p\) moves the point \(X\) to touch the \(r=1\) circle along the circle of \(g_1=0.328\), and \(C_s\) further moves it to the origin, which is the final goal.

 

The Option #2 matching circuit includes an inductor \(L_p\) in shunt with the to-be-matched impedance and then add an inductor \(L_s\) in series.

Fig. 15   Type #4 impedance matching, Option #2

 

\(L_p=-0.244\) is normalized admittance of the inductor in shunt.

And, by equation (2),

\(L_{p} (nH)=11.23\)

 

\(L_s=1.43\) is normalized impedance of the inductor in series.

And, by equation (4),

\(L_{s} (nH)=3.92\)

 

As showed in the Smith chart of Fig. 15, \(L_p\) moves the point \(X\) to touch the \(r=1\) circle along the circle of \(g_1=0.328\), and \(L_s\) further moves it to the origin, which is the final goal.

 

The Option #3 matching circuit includes an inductor \(L_s\) in series with the to-be-matched impedance and then a capacitor \(C_p\) in shunt.

Fig. 16   Type #4 impedance matching, Option #3

 

In Figure 16, \(L_s=1.655\) is normalized impedance of the inductor in series.

And, by equation (4),

\(L_{s} (nH)=4.54\)

 

\(C_p=0.938\) is normalized admittance of the capacitor in shunt.

And, by equation (1),

\(C_{p} (pF)=1.03\)

 

As showed in the Smith chart of Fig. 16, \(L_s\) moves the point \(X\) to touch the \(g=1\) circle along the circle of \(r_1=0.532\), and \(C_p\) further moves it to the origin, which is the final goal.

 

The Option #4 matching circuit includes an inductor \(L_s\) in series with the to-be-matched impedance and then add an inductor \(L_p\) in shunt.

Fig. 17   Type #4 impedance matching, Option #4

 

\(L_s=0.657\) is normalized impedance of the inductor in series.

And, by equation (4),

\(L_{s} (nH)=1.8\)

 

\(L_p=-0.938\) is normalized admittance of the inductor in shunt.

And, by equation (2),

\(L_{p} (nH)=2.92\)

 

As showed in the Smith chart of Fig. 17, \(L_s\) moves the point \(X\) to touch the \(g=1\) circle along the circle of \(r_1=0.532\), and \(L_p\) further moves it to the origin, which is the final goal.

 

With this 2-part articles, Impedance Matching by Using Smith Chart Associated with A Proprietary Spreadsheet, you would be able to get the impedance matching results to basically any types of impedance instantly.

You are only responsible for data entry and then, based on the “Results” provided by this program, you start working on the bench and slightly modify the circuits to accommodate all component and other design tolerances to get you satisfactory final answers.

This program will save you a lot of time and hassles.

You can download it Here.

 

If you feel the Smith chart is too difficult to understand or for any reason you don’t like to use it, but you also would like to have a plain program to instantly provide you the impedance matching circuits, then you should continue to visit this article: Impedance Matching Using a Proprietary Spreadsheet.

 

 

‘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.’

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