Technical Article

# Mathematical Construction and Properties of the Smith Chart

July 23, 2015 by Trevor Gamblin

## Smith Charts are an extremely useful tool for engineers and designers concerned with RF circuits. This article covers the mathematics behind creating the chart and its physical interpretation.

Smith Charts are an extremely useful tool for engineers and designers concerned with RF circuits. This article covers the mathematics behind creating the chart and its physical interpretation.

Intermediate

#### Introduction

The Smith Chart has been in use since the 1930s as a method to solve various RF design problems - notably impedance matching with series and shunt components - and it provides a convenient way to find these solutions without the use of a calculator. In order to understand the construction of the chart, you'll need to understand high school algebra and the basics of complex numbers, as well as have a basic understanding of impedance in electronic circuits. That said, even if you don't fully understand the derivation below, you can still use the chart to help you with your own design. By taking the standard reflection coefficient formula and manipulating it so that it provides us with the equations for circles of various radii, we'll be able to construct the basic Smith Chart. That's all the Smith Chart really is: a collection of circles, each one centered in a different place in (or outside) the plot, and each one representing either constant resistance or constant reactance.

#### Deriving the Smith Chart

Once we get past the derivation, there will be a few simplified images showing how those equations can be used and combined to get the final product. Let's get started by writing the equation for the reflection coefficient of a load impedance, given a source impedance:

$$\Gamma=\frac{Z_{source}-Z_{load}}{Z_{source}+Z_{load}}$$

The reflection coefficient is just the ratio of the complex amplitude of a reflected wave to the amplitude of the incident wave. This is the main equation we'll be using, but there will be some quick transformations to it. First, we'll want to simplify it a little by normalizing the equation with respect to Zload, dividing each term on the right side:

$$\Gamma = \frac{\frac{Z_{source}}{Z_{load}}-\frac{Z_{load}}{Z_{load}}}{\frac{Z_{source}}{Z_{load}}+\frac{Z_{load}}{Z_{load}}}$$

$$\Gamma = \frac{Z_{O}-1}{Z_{O}+1}$$

Where,

$$Z_{O} = \frac{Z_{source}}{Z_{load}}$$

At this point, recall that Zo, being an impedance of complex value, can be represented in the form R + jX. Since the reflection coefficient (which is currently in polar form) can also be represented in rectangular coordinates (we'll use A + jB for it), the above formula can be transformed into this:

$$A + jB = \frac{R + jX -1}{R + jX + 1}$$

Great! At this point we've got the equation in the form we need to start constructing the Smith Chart. The next step - solving for the real and imaginary parts of the equation - is probably the most difficult part of the entire derivation, and even then you only need to understand the concept of complex conjugates to do it. Let's go ahead and split it into real and imaginary components, first by multiplying by the complex conjugate (it helps if you separate the existing real and imaginary parts using brackets as shown below):

$$A + jB = \frac{(R-1)+jX}{(R+1)+jX} \cdot \frac{(R+1)-jX}{(R+1)-jX}$$

$$A+jB = \frac{R^2-1+X^2+2jX}{(R+1)^2+X^2}$$

At this point we can separate the real and imaginary components. After that, there will be two final simplifications to do before we'll have the equations to draw the Smith Chart. Here are the separated real and imaginary parts (we'll call them Equations 1 and 2):

$$A=\frac{R^2-1+X^2}{(R+1)^2+X^2} \text{ (Equation 1)}$$

$$B=\frac{2X}{(R+1)^2+X^2} \text{ (Equation 2)}$$

Finally, you will want to do just a little more algebra (tedious, I know). Solving the real component, A, for X2, you will get Equation 3:

$$X^2=\frac{A(R+1)^2-R^2+1}{1-A} \text{ (Equation 3)}$$

You can substitute this into Equation 2 to get the first of our two final equations, which allows us to determine the circles of constant resistance (Equation 4):

$$(A-\frac{R}{R+1})^2 + B^2 = (\frac{1}{R+1})^2 \text{ (Equation 4)}$$

Does that look familiar? It's a circle, with a radius of $$\frac{1}{R+1}$$ and a center of $$(\frac{R}{R+1},\; 0)$$. By varying the value of R in this equation, you can draw each of the circles in the Smith Chart.

Similarly, solving for R (I used Equation 2) will get you solutions that look like this:

$$R=\frac{\sqrt{-BX(BX-2)}-B}{B}$$

which, when substituted and simplified into Equation 1, will get you this result (Equation 5):

$$(A-1)^2+(B-\frac{1}{X})^2=(\frac{1}{X})^2 \text{ (Equation 5)}$$

Just like the previous result, this is a circle with radius $$\frac{1}{X}$$, but this time there are two sets of circles (more on that in a bit), with centers at ($$1$$, $$1/X$$). These are circles (they appear as arcs on the diagram) of constant reactance. Now you should see how the standard Smith Chart is drawn; it consists of constant resistance circles graphed together with the constant reactance arcs. Below you'll find some simplified images of both equations graphed separately and combined. But first, let's talk about how to interpret the Smith Chart and its physical relevance.

There is quite a bit of information to obtain from analyzing the equations we've derived. Here are just a few things of note:

1. At infinite R and X, both types of circles converge to the same location (typically shown on a Smith Chart at the far right or far left side of the diagram). This is at the point (1, 0).
2. Setting R = 0 will result in a circle centered at (0, 0) on your chart with a radius of 1, which is the "boundary" of the chart.
3. Approaching X = 0 results in an infinite radius; this is represented by a line crossing the center of the chart. How do we interpret this? This is often called the real axis. In terms of reactances, lines above the real axis in the chart (the positive arcs from the second derived equation) represent inductive reactances, while those below (negative arcs) represent capacitive reactances.
4. What happens if R < 0? The standard Smith Chart doesn't provide much detail about this, but situations with R lying outside the boundary suggest oscillation in any would-be circuit (which is pretty handy to know).
5. Based on the knowledge we now have on resistance and reactance on the chart, we know that every point represents a series combination of resistance and reactance (R + jX). This'll help us when we want to do some plotting.

#### Using the Chart for Impedance Manipulation

So how does one use the chart? In order to plot an impedance for impedance matching purposes, best practice is to find the relevant constant resistance circle (the one that corresponds to the real part of your impedance), then move along its arc until you find its intersection with the corresponding reactance value.

For example, assume you have an impedance of Z = 0.3 - 0.6j. First find the 0.3 constant resistance circle. Since your impedance has a negative complex value, this represents a capacitive impedance in the theoretical series network, and you'll move counter-clockwise along the 0.3 resistance circle to find where it hits the -0.6 reactance arc (if it were a positive complex value, it would represent an inductor and you would move clockwise). You can continue to do this as an easy way to perform impedance matching for your circuit, with the Smith Chart as a very intuitive physical aid. You only have to follow these steps:

1. Knowing the load impedance value, find it on the Smith Chart and use it as your starting location.
2. If you know the impedance that the source would like to "see", you can add series components (shunts will be mentioned below) by adding and subtracting reactance values until you have the desired impedance.

It's important to note two things:

1. You may often find in practice that the numbers in your chart are small in comparison to the component values you are seeking. Normalization again comes into play here; it is often most convenient to normalize whatever impedance (e.g. Z = 200 + j400) you are working with by dividing it by the value that makes it easiest for you to plot (often this is the value of the real component, but use your best judgment). That way, you can work in a less crowded area of the Smith Chart, which brings me to the next point:
2. It's easy when working with a Smith Chart to be caught dealing with impedance values closer to the point (1, 0), in which case you will have some difficulty with errors in your calculated values. That's why it is best to normalize your impedance values when working with the chart, allowing yourself to work with wider arcs when matching impedances and ensuring continued accuracy as you add components in series.

#### One Final Note - Admittance and Immittance Charts

Up until now there's been no mention of admittance in the Smith Chart. If you are unfamiliar, admittance Y is the reciprocal of impedance, or $$Y=\frac{1}{Z}$$. The terms corresponding to resistance and reactance are called conductance and susceptance, respectively. It is actually surprisingly easy to plot the equivalent chart for admittance - all you have to do is mirror the Chart horizontally. This is an important step too, as by flipping it over, you now have a chart that will assist you in dealing with shunt components rather than those in series.

The process of plotting admittance is essentially reversed - where adding an inductor to a series circuit would move the impedance value clockwise along a constant resistance circle, a shunt inductor would move it counter-clockwise along a constant admittance circle; shunt capacitors similarly move your values clockwise on an admittance chart, where a series capacitor would be counter-clockwise.

Combining both chart types gets you what is called an immittance chart, which (once you add a few other details that aren't covered here) becomes even more useful than the standard Smith Chart, although it'll certainly look more intimidating to someone who is unfamiliar with how it is created.

Hopefully this article has provided you with a solid overview of how the Smith Chart is constructed and functions. Be sure to comment below about any concerns or questions you might have.