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

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We have learned the impedance matching for all 4 Types of impedance using formulas & their conversions, and it’s recommended to visit these 2 articles before you continue to read further here:

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

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

 

In this article you’ll learn the step-by-step guide to match Type #1 and Type #2 impedance by simply using Smith chart without knowing those formulas. The only thing you need to do is enter S11 or to-be-matched impedance and you’ll get the approximate result by following all steps.

However, if you are vague to the Smith chart then you should STOP here and go back to learn the Smith Chart Basics first. Only after finishing reading the sequence and knowing all basics, you then can use this skill effectively.

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 into 50Ω by using only 2 lumped elements, inductors and capacitors, if not considering the limited amount of component values we are able to get as well as their tolerances.

However, I would not suggest to use lumped elements for impedance matching over 5 GHz because it’s hard to find small enough elements, value and size, and the path length will play a main role for the matching process.

The very basic RULES of impedance matching are:

  1. Add a lossless element, capacitor or inductor, to get the real part of either impedance or admittance to be 1.
  2. Add the second lossless element to tune out the remaining imaginary part, reactance or susceptance, so the resultant impedance or admittance is a real number 1 (\(z=1+j0\) or \(y=1+j0\)).

 

Matching Type #1 impedance: r ≥ 1, x any value.

Type #1 impedance is located within the area of \(r=1\) circle.

 

The process to match a Type #1 impedance into 50Ω:

1. If return loss \(S_{11}\) or reflection coefficient \(Γ\) is given by datasheets, then refer to this article Smith Charts-Basics, Parameters, Equations, and Plots to learn how to convert the number to impedance.

2. Normalize the given impedance. If the impedance is \(Z=R+jX\), then the normalized impedance is \(z=Z/50=r+jx\).

Fig. 2  Normalize the impedance

 

3. Locate the impedance \(z=r+jx\) in the Smith chart, point X.

Fig. 3  Type #1 impedance in the Smith chart

 

As showed in Fig. 3, we can simultaneously read the impedance \(z\) and admittance \(y\) of point X:

\(z=r+jx=2.0+j1.0\)

\(y=g+jb=0.4-j0.2\)

Fig. 4  Type #1 impedance \(z\) & \(y\)

 

Follow the first basic rule of impedance matching, add a lossless element, capacitor or inductor, to get the real part of either impedance or admittance to be 1.

The only options to satisfy this first rule is add a capacitor, Option #1, or an inductor, Option #2, in shunt to move the admittance \(y\) along the \(g=0.4\) circle until meeting \(r=1\) circle at points O1-S1 & O2-S1.

 

Fig. 5  Type #1 impedance matching, Step 1

 

We can fairly accurately read the locations of both points in the Smith chart:

Point O1-S1:

\(z=r+jx=1-j1.22\)

\(y=g+jb=0.4+j0.49\)

The admittance of the added capacitor \(C_p\) is \(0.49-(-0.2)=0.69\)

 

Point O2-S1:

\(z=r+jx=1+j1.22\)

\(y=g+jb=0.4-j0.49\)

The admittance of the added inductor \(L_p\) is \(-0.49-(-0.2)=-0.29\)

 

We then follow the 2nd rule and move both points O1-S1 & O2-S1 along the \(r=1\) circle to the origin \(z=1\) by simply adding an inductor, Option #1, or a capacitor, Option #2, in series respectively.

 

 

Fig. 6  Type #1 impedance matching, Step 1 & Step 2

And the impedance values of added elements are:

Option #1:  \(L_s=0-(-1.22)=1.22\)

Option #2:  \(C_s=0-1.22=-1.22\)

 

Question: Match this Type #1 impedance \(z=1.4-j0.6\) using Smith chart.

Ans.

Option #1:   \(C_p=0.23\)     \(L_s=0.81\)

Option #2:   \(L_p=-0.75\)   \(C_s=-0.81\)

 

 

 

 

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

 

Matching Type #2 impedance: g ≥ 1, b any value.

Type #2 impedance is located within the area of \(g=1\) circle.

 

The process to match a Type #2 impedance into 50Ω:

1. If return loss \(S_{11}\) or reflection coefficient \(Γ\) is given by datasheets, then refer to this article Smith Charts-Basics, Parameters, Equations, and Plots to learn how to convert the number to impedance.

2. Normalize the given impedance. If the impedance is \(Z=R+jX\), then the normalized impedance is \(z=Z/50=r+jx\).

Fig. 7  Normalize the impedance

 

3. Locate the impedance \(z=r+jx\) in the Smith chart, point X.

 

Fig. 8  Type #2 impedance in the Smith chart

 

As showed in Fig. 8, we can simultaneously read the impedance \(z\) and admittance \(y\) of point X:

\(z=r+jx=0.4-j0.2\)

\(y=g+jb=2.0+j1.0\)

Fig. 9  Type #2 impedance \(z\) & \(y\)

 

Follow the first basic rule of impedance matching, add a lossless element, capacitor or inductor, to get the real part of either impedance or admittance to be 1.

The only options to satisfy this first rule is add a capacitor, Option #1, or an inductor, Option #2, in series to move the impedance \(z\) along the \(r=0.4\) circle until meeting \(g=1\) circle at points O1-S1 & O2-S1.

 

Fig. 10  Type #2 impedance matching, Step 1

 

We can fairly accurately read the locations of both points in the Smith chart:

Point O1-S1:

\(z=r+jx=0.4-j0.49\)

\(y=g+jb=1.0+j1.22\)

The impedance of the added capacitor \(C_s\) is \(-0.49-(-0.2)=-0.29\)

 

Point O2-S1:

\(z=r+jx=0.4+j0.49\)

\(y=g+jb=1.0-j1.22\)

The impedance of the added inductor \(L_s\) is \(0.49-(-0.2)=0.69\)

 

We then follow the 2nd rule and move both points O1-S1 & O2-S1 along the \(g=1\) circle to the origin \(z=1\) by simply adding an inductor, Option #1, or a capacitor, Option #2, in shunt respectively.

 

 

Fig. 11  Type #2 impedance matching, Step 1 & Step 2

 

And the admittance values of added elements are:

Option #1:  \(L_p=0-1.22=-1.22\)

Option #2:  \(C_p=0-(-1.22)=1.22\)

 

Question: Match this Type #2 impedance \(z=0.5+j0.3\) using Smith chart.

Ans.

Option #1:   \(C_s=-0.8\)     \(L_p=-1.0\)

Option #2:   \(L_s=0.2\)     \(C_p=1.0\)

 

 

 

 

We’ll continue to discuss Type #3 and Type #4 impedance matching in the next article.

 

 

 

 

‘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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