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



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 the part I, you’ve learned 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.

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 $$y=1+j0$$).

We’ll continue to learn how to match Type #3 and Type #4 impedance below.

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.

Matching Type #3 impedance: r < 1, g < 1, x > 0 or b < 0.

Type #3 impedance is located within the shaded area in the Smith chart.

Fig. 1  Type #3 impedance in the Smith chart

The process to match a Type #3 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 $$z=r+jx$$ and admittance $$y=g+jb$$ in the Smith chart, point X.

Fig. 3  Read Type #3 impedance in the Smith chart

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

$$z=r+jx=0.4+j1.1$$

$$y=g+jb=0.29-j0.8$$

Fig. 4  Type #3 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.

There are 4 options to satisfy this first rule and the first 2 options add a capacitor $$C_p$$ in shunt to move the susceptance $$b$$ along the $$g=0.29$$ circle until meeting $$r=1$$ circle at points O1-S1 & O2-S1. We call them Option #1 and Option #2 respectively.

The other 2 options add a capacitor $$C_s$$ in series to move the reactance $$x$$ along the $$r=0.4$$ circle until meeting $$g=1$$ circle at points O3-S1 & O4-S1. We cal them Option #3 and Option #4 respectively.

Fig. 5  Type #3 impedance matching Step 1, Option #1, #2, #3, and #4

We can fairly accurately read the locations of all 4 points in the Smith chart:

Point O1-S1:

$$z=r+jx=1+j1.56$$

$$y=g+jb=0.29-j0.46$$

The susceptance of the added capacitor $$C_p$$ is $$-0.46-(-0.8)=0.34$$

Point O2-S1:

$$z=r+jx=1-j1.56$$

$$y=g+jb=0.29+j0.46$$

The susceptance of the added capacitor $$C_p$$ is $$0.46-(-0.8)=1.26$$

Point O3-S1:

$$z=r+jx=0.4+j0.49$$

$$y=g+jb=1-j1.23$$

The reactance of the added capacitor $$C_s$$ is $$0.49-1.1=-0.61$$

Point O4-S1:

$$z=r+jx=0.4-j0.49$$

$$y=g+jb=1+j1.23$$

The reactance of the added capacitor $$C_s$$ is $$-0.49-1.1=-1.59$$

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 a capacitor $$C_s$$, Option #1, or an inductor $$L_s$$, Option #2, in series respectively.

And move both points O3-S1 & O4-S1 along the $$g=1$$ circle to the origin $$z=1$$ by simply adding a capacitor $$C_p$$, Option #3, or an inductor $$L_p$$, Option #4, in shunt respectively.

Fig. 6  Type #3 impedance matching, Step 1 & Step 2

And the values of added elements are:

Option #1:  $$C_s=0-1.56=-1.56 … impedance$$

Option #2:  $$L_s=0-(-1.56)=1.56 … impedance$$

Option #3:  $$C_p=0-(-1.23)=1.23 … admittance$$

Option #4:  $$L_p=0-1.23=-1.23… admittance$$

You can determine which option is the best for your application after a few lab tests.

Question: Match this Type #3 impedance $$z=0.35+j0.65$$ using Smith chart.

Ans.

Fig. 7  Type #3 impedance matching, Step 1 & Step 2

Option #1:   $$C_p=0.71$$     $$C_s=-0.75$$

Option #2:   $$C_p=1.67$$   $$L_s=0.75$$

Option #3:   $$C_s=-0.17$$     $$C_p=1.36$$

Option #4:   $$C_s=-1.13$$   $$L_p=-1.36$$

Matching Type #4 impedance: r < 1, g < 1, x < 0 or b > 0.

Type #3 impedance is located within the shaded area in the Smith chart.

Fig. 8  Type #4 impedance in the Smith chart

The process to match a Type #4 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. 9  Normalize the impedance

3. Locate the impedance $$z=r+jx$$ and admittance $$y=g+jb$$ in the Smith chart, point X.

Fig. 10  Read Type #4 impedance in the Smith chart

As showed in Fig. 10, we can simultaneously read the impedance $$z$$ and admittance $$y$$ of point X:

$$z=r+jx=0.5-j1.0$$

$$y=g+jb=0.4+j0.8$$

Fig. 11  Type #4 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.

There are 4 options to satisfy this first rule and the first 2 options add an inductor in shunt to move the susceptance $$b$$ along the $$g=0.4$$ circle until meeting $$r=1$$ circle at points O1-S1 & O2-S1. We call them Option #1 and Option #2 respectively.

The other 2 options add an inductor in series to move the reactance $$x$$ along the $$r=0.5$$ circle until meeting $$g=1$$ circle at points O3-S1 & O4-S1. We cal them Option #3 and Option #4 respectively.

Fig. 12  Type #4 impedance matching Step 1, Option #1, #2, #3, and #4

We can fairly accurately read the locations of all 4 points in the Smith chart:

Point O1-S1:

$$z=r+jx=1+j1.22$$

$$y=g+jb=0.4-j0.49$$

The susceptance of the added inductor $$L_p$$ is $$-0.49-0.8=-1.29$$

Point O2-S1:

$$z=r+jx=1-j1.22$$

$$y=g+jb=0.4+j0.49$$

The susceptance of the added inductor $$L_p$$ is $$0.49-0.8=-0.31$$

Point O3-S1:

$$z=r+jx=0.5+j0.5$$

$$y=g+jb=1.0-j1.0$$

The reactance of the added inductor $$L_s$$ is $$0.5-(-1.0)=1.5$$

Point O4-S1:

$$z=r+jx=0.5-j0.5$$

$$y=g+jb=1+j1.0$$

The reactance of the added inductor $$L_s$$ is $$-0.5-(-1.0)=0.5$$

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 a capacitor $$C_s$$, Option #1, or an inductor $$L_s$$, Option #2, in series respectively.

And move both points O3-S1 & O4-S1 along the $$g=1$$ circle to the origin $$z=1$$ by simply adding a capacitor $$C_p$$, Option #3, or an inductor $$L_p$$, Option #4, in shunt respectively.

Fig. 13  Type #4 impedance matching, Step 1 & Step 2

And the values of added elements are:

Option #1:  $$C_s=0-1.22=-1.22 … impedance$$

Option #2:  $$L_s=0-(-1.22)=1.22 … impedance$$

Option #3:  $$C_p=0-(-1.0)=1.0 … admittance$$

Option #4:  $$L_p=0-1.0=-1.0… admittance$$

You can determine which option is the best for your application after a few lab tests.

Question: Match this Type #4 impedance $$z=0.3-j0.6$$ using Smith chart.

Ans.

Fig. 14  Type #4 impedance matching, Step 1 & Step 2

Option #1:   $$L_p=-1.81$$     $$C_s=-0.71$$

Option #2:   $$L_p=-0.86$$   $$L_s=0.71$$

Option #3:   $$L_s=1.06$$     $$C_p=1.53$$

Option #4:   $$L_s=0.14$$   $$L_p=-1.53$$

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In the next article, you’ll learn how to match any type of impedance by using both Smith chart and a proprietary spreadsheet so you can get answers within a fraction of second without knowing those sophisticate formulas.

However, if you feel need to know more basics of Smith chart or impedance matching, you should start with this article “Smith Charts-Basics, Parameters, Equations, and Plots.” and follow the instructions to go over all articles.

‘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.’