Welcome, my name is Bart Smolders. And in this web lecture we're going to talk about the Smith chart, which is a very useful tool in microwave engineering and antennas. The objective of this lecture is to provide you some history and use cases for the Smith chart. We're going to use it to visualize the reflection coefficient and the complex impedance and we're going to show some examples. Now the Smith chart looks like this. It consists of many circles, curves, which are part of circles, many numbers on the Smith chart. Now, it's originates from the periods during the II World War, so its in fact 80 years old already. And in that period there were no computers, so it was very useful to have a kind of approach, a tool, to plot the complex reflection coefficient and to transform that directly to complex impedances, complex admittances, VSWR etc. Phillip Smith invented the Smith chart, which is the results that we were going to discuss today. Now, the use of the Smith chart is to make a graphical representation of the complex reflection coefficients and the complex impedance. And it's very useful when you are designing a matching or a tuning circuit, as we will see in one of the follow-up web lectures. The Smith chart might look rather complicated at first sight, but in fact it's not, because we're only plotting the complex reflection coefficient in the complex domain. Now, this reflection coefficient gamma was already derived in one of the previous web lectures. It consists of a real part and an imaginary part an because of that we can plot the real part along the horizontal axis in the complex domain and the imaginary part along the vertical axis. Now the absolute value of gamma is always smaller than one, which means that all values of gamma fall within a unit circle with radius one. Now, as an example we could plot the reflection coefficient in the complex domain and could look like this. But in fact, what would be interesting is to know the relation between the reflection coefficient of a circuit and the input impedance. Now this was already derived in one of the previous web lectures and this relation was given in this form. Where Zin is the input impedance of a particular circuit at hand, and Z0 is the reference impedance, which is usually 50 ohms. Now, if you would plot the impedance values related to a particular reflection coefficients, you would see these kind of curves which consists of circles and parts of circles. Now what is it what we see? Now the normalized input impedance, normalized to the reference impedance consists of a real part and an imaginary part as well. Now the real parts if we would plot it in the complex domain of the reflection coefficient. So on the Smith chart, if we could keep this real part constant would mean that we would walk along one of these circles as indicated here in the Smith chart. So it very the imaginary part, we would walk along the particular circle at hands with a constant real part of the input impendence. Similarly, we can keep the imaginary part constants and then we would move along these curves which in fact are parts of circles which extends the unit circle of the complex reflection coefficient. In all cases, we're going to consider in this web lecture we will assume that the reference impedance Z0 is equal to 50 ohm. Which means that if the input impedance is exactly equal to the reference impedance, we would be at the center of the Smith chart, in the middle. Now let's take a closer look at some examples. One of the most simple examples could be a short circuits where the input impendence equal to 0. The associated reflection coefficient would be equal to -1. So that means that this particular point would pop up at the left side of the Smith chart along the horizontal axis. In case of an open circuit, the input impedance would be infinite and the reflection coefficient will be +1. So we would pop up at the right side of the Smith chart as indicated here along again, the horizontal axis. In case of a perfect matched circuits with an input impedance of 50 ohm equal to the reference impedance 50 ohm. We would find that reflection coefficient is 0 and we will be at the center of the Smith chart as indicated here. When we have a more complex impedance, so for example 50- j100 ohm, we can easily find this point on the Smith chart, how? Well, best is to 1st consider the real part. So in this case the normalized value is 1 and we find the circle with a real part which is equal to 1 starting in the middle of the Smith chart as indicated here by the red box. So we have to move along this circle. The reactants, the normalized reactance in this case is equal to -j2, which means that we need to find the curve related to a reactance of -j2. And we need to move along this curve as well. Well, at the intersection of both curves, we find the particular point on Smith chart as indicated here. Up to now, we looked into the Smith chart by considering the input impedance and the relation to the reflection coefficient. Of course, it's also possible to plot admittance is instead of impedances. So we would have circles corresponding to a constant conductance or we would have curves corresponding to constant susceptances. By combining both the impedances and the admittance is we get a rather complicated Smith chart as indicated here. Where the red curves correspond to impedance is real and imaginary parts, and the blue curves correspond to admittances. To summarize this web lecture, the Smith chart is very useful tool to visualize the reflection coefficient and the input impedance or input admittance. It is used nowadays in many cut tools, design tools and measurement equipment like a vector network analyzers. And we're going to use the Smith chart for the design of matching circuits. Hope to see you back next time.
