# Smith Chart Explanation

I'm working on designing some hopefully simple RF (912 MHz) attenuators. I need a few different levels of attenuation but each different attenuator can be fixed.

I have prototyped one using just off the shelf resistors in a T-pad configuration that gives me a decent level of attenuation (pretty flat 19dB) when I perform an S21 measurement using my network analyzer.

However, the smith chart is all over the place when I measure S11.

Now I should mention that my prototype is very shotty. Basically I took apart a coaxial cable and hand soldered in some close-to-calculated carbon 5% resistors in between the two SMA connector ends.

My questions are as follows: What is a Smith chart and how do I use it to improve my attenuator+cable design? Is this a feasible method for creating basic fixed RF attenuators given that they don't need to be super accurate and only need to function over a very specific range of frequencies (905-920 ish MHz)?

As always, thanks for your help.

EDIT:

This is the SC of my taken apart coaxial cable WITHOUT the attenuator

This is the SC of my cable WITH the attenuator in the middle of it

Here are the log plots of attenuation over the frequency range I am interested in: First no attenuator:

Second with attenuator:

Also, another question struck me. If I'm just trying to reduce signal power at the output, does it matter where/how the loss happens? So I know poor impedance matching as indicated by my charts means a higher VSWR... but doesn't that just help attenuation? Thanks again.

• Can you give us a picture of the smitch chart(not really important if this is what I think it is) and can you give us a graph of S11 as log mag plot? Aug 1 '12 at 15:33
• Are the resistors carbon film or carbon composition? Carbon film are not suitable for UHF work as they are formed by cutting a spiral track in a carbon film cylinder, so have very substantial inductance. Carbon composition have a solid carbon body and may be suitable for UHF work depending on other factors. Aug 1 '12 at 15:37
• A Smith chart is a brilliant graphical means of either working out what impedance you need to match between two known impedances or an indication of what you will end up with if you add a complex impedance to another one. You can get automated versions of the SC but they are relatively easy toi use once understood. Aug 1 '12 at 15:37
• @RussellMcMahon I agree, I am wondering the magnitude of the reflected signal received before I comment. Aug 1 '12 at 16:06
• But now you've basically combined so many questions together, that it's hard for us to give you a good set of answers in this format...Maybe you could break out some of your follow-up questions into new questions for the site? Aug 1 '12 at 23:35

A Smith Chart is not so much an attenuator design aid
as a means of evaluating and adjusting a design.

So - see attenuator articles below and then Smith chart articles.

Carbon resistors may be carbon film or carbon composition?

• Carbon film are not suitable for UHF work as they are formed by cutting a spiral track in a carbon film cylinder, so have very substantial inductance.

• Carbon composition have a solid carbon body and may be suitable for UHF work depending on other factors.

UHF Attenuators:

RF attenuator basic tutorial

Attenuator design tutorial - looks good.

Interest - Commercial products

Wikipedia

What is a Smith Chart?

Wikipedia gives a better than average concise summary:
From here

• The Smith chart, invented by Phillip H. Smith (1905–1987),1[2] is a graphical aid or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assist in solving problems with transmission lines and matching circuits.[3] Use of the Smith chart utility has grown steadily over the years and it is still widely used today, not only as a problem solving aid, but as a graphical demonstrator of how many RF parameters behave at one or more frequencies, an alternative to using tabular information. The Smith chart can be used to represent many parameters including impedances, admittances, reflection coefficients, scattering parameters, noise figure circles, constant gain contours and regions for unconditional stability, including mechanical vibrations analysis.[4][5] The Smith chart is most frequently used at or within the unity radius region. However, the remainder is still mathematically relevant, being used, for example, in oscillator design and stability analysis.[6]

Somewhat gentle introduction - 27 page powerpoint intro - still gets deep quite quickly BUT a Smith Chart can be very very useful with almost no maths or numerics involved.

Superb Smith Chart resource - essentially an index of indexes - breaks subject up into sections and provides many references for each.

Another good list of references

Smith Chart tutorial from Maxim - reasonable "dense" but looks understandable.

You'll understand this once you've read it :-)

Free software based Smith Chart

Freeware Smith Chart software

Sim Smith - Java based

Many Smith Chart related pages

To give a very quick explanation of the Smith chart, it rests on one simple idea:

The reflection coefficient ($\Gamma$ or $S_{11}$) of a termination on a transmission line is related to the impedance of the termination (Z) by

$\Gamma=\frac{Z-Z_0}{Z+Z_0}$

Where $Z_0$ is the characteristic impedance of the line. All of these variables are complex numbers.

The Smith chart is a graphical means of calculating this relationship.

Basically you plot the reflection coefficient on the chart in polar coordinates: the distance of the point from the center of the chart is the magnitude of the reflection coefficient, and the angle from the x-axis is the argument of the reflection coefficient. Then the lines on the chart enable you to read off the load impedance. Often the chart is normalized to a characteristic impedance of 1 Ohm, so you'd multiply the read load impedance by your actual Z0 (often 50 Ohms) to get the physical load impedance.

Conversely you could plot your load impedance value by reference to the lines drawn on the chart, and read off the reflection coefficient by using a ruler to measure the distance from the chart's center and to locate the angle from the scale around the outer edge.

It's useful to be able to quickly switch between reflection coefficient and load impedance because certain circuit adjustments have an effect that's more easily calculated in one form or the other.

For example, adding a series resistor adds a fixed value to the real part of the load impedance. Or adding a series inductor adds a frequency-dependent value to the imaginary component of the load impedance. On the other hand, moving back along the transmission line to a point further away from the load adds a frequency-dependent value to the phase of the reflection coefficient.

The curves drawn on the chart posted by Russell show examples of these kinds of transformations.

I should add that there's an alternative form of the Smith chart, called an admittance Smith chart, that looks the same but mirrored across the y-axis. This allows calculating the relationship between admittance and reflection instead of impedance. It's useful, for example, if you are adjusting your load by placing a parallel element instead of a series element.

Russel has given an extensive list of links to grasp the concept of the Smith Chart.

I'll try to give a short summary of what the Smith Chart does with the example. I am also a student and the concept was new for me.

The answer is 100% based on the perfect article by Maxim Integrated referred by Russel (URL).

Theory

1) Set-up: transmission line and load

2) Well-known formula for the coefficient of reflection:
$${\Gamma _L} \equiv \frac{{{V_{relf}}}}{{{V_{inc}}}} = \frac{{{Z_L} - {Z_0}}}{{{Z_L} + {Z_0}}} \equiv {\Gamma _r} + j \cdot {\Gamma _i}$$
3) Let's normalize the load impedance by Z0 and denote the real part as r and imaginary part as x:
$$z \equiv \frac{{{Z_L}}}{{{Z_0}}} \equiv r + j \cdot x$$ 4) Now, using lengthy but straightforward mathematical manipulations, described in the article, you may show that: $${({\Gamma _r} - \frac{r}{{r + 1}})^2} + {\Gamma _i}^2 = {(\frac{1}{{r + 1}})^2}$$
and
$${({\Gamma _r} - 1)^2} + {({\Gamma _i} - \frac{1}{x})^2} = {(\frac{1}{x})^2}$$

As you might remember from school these are the equations of two circles for coordinates ${\Gamma _r}$ and ${\Gamma _i}$. This forms the beauty of the Smith Chart: you may find complex impedance of the load knowing real and imaginary parts of the reflection coefficient (${\Gamma _r}$ and ${\Gamma _i}$) by intersecting corresponding circles in the Smith Chart.

Example (again borrowed from the article)

Find the complex impedance of the point Z2 on the Smith Chart below

Solution:

Find the corresponding circles for r and x. Corresponding values are located at the horizontal axis (r) and at the big circle around the Smith Cart (x)(marked with green arrows): r=1.5, x=-2 (we added the minus sign because the point is located in the bottom half plane).
Remember to multiply by Z0.

$$Z2 \equiv {Z_L} = {Z_0} \cdot z = {Z_0} \cdot (r + j \cdot x) = {Z_0} \cdot r + j \cdot {Z_0} \cdot x = 50 \cdot 1.5 + 50 \cdot j \cdot ( - 2)\,\Omega = 75 - j \cdot 100\,\Omega$$