Since R cannot be negative for antennas or passive devices, we will restrict R to be greater than or equal to zero. look at the set of curves defined by zL = R + iY, where Y is held constant and R varies from 0 to infinity. The zL=6 resistance circle has been added in green, and zL=2 resistance circle is in black. In Figure 3, the zL=0.1 resistance circle has been added in purple. We'll now add several values for the constant resistance, as shown in Figure 3: The real part of the load impedance is constant along each of these curves. These circles are called constant resistance curves. We've left the resistance circle of 1.0 in red on the Smith Chart. A few points along the circle are plotted. In Figure 2, the black ring represents the set of all impedances where the real part of z2 equals 0.3. Suppose we want to know what the curve z2 = 0.3 + i*Y looks like on the Smith Chart. Several points are plotted along this curve, z1 = 1, z1 = 1 + i*2, and zL = 1 - i*4. The black curve is a constant resistance circle: this is where all values of z1 = 1 + i*Y will lie on.
In Figure 1, the outer blue ring represents the boundary of the smith chart. and any possible value for Y that you could think of, what is the resulting curve? The answer is shown in Figure 1: What would the curve corresponding to equation look like if we plotted it on the Smith Chart for all values of Y? That is, if we plotted z1 = 1 + 0*i, and z1 = 1 + 10*i, z1 = 1 - 5*i, z1 = 1. Now, suppose we have the normalized load impedance given by: Constant Resistance Circlesįor a given normalized load impedance zL, we can determine and plot it on the Smith Chart. It is just a convention that is used everywhere. To make the Smith Chart more general and independent of the characteristic impedance Z0 of the transmission line, we will normalize the load impedance ZL by Z0 for all future plots:Įquation doesn't affect the reflection coefficient tow. Along this curve, all of the power is reflected by the load impedance.
The outter ring of the Smith Chart is where the magnitude of is equal to 1. That is, this is the only point on the smith chart where no power is reflected by the load impedance. The center of the Smith Chart is the point where the reflection coefficient is zero. In Figure 2, plotting the set of all values for the complex reflection coefficient, along the real and imaginary axis.
The complex reflection coefficient, or, must have a magnitude between 0 and 1.Īs such, the set of all possible values for must lie within the unit circle: Recall that the complex reflection coefficient () for an impedance ZL attached to a transmission line with characteristic impedance Z0 is given byįor this tutorial, we will assume Z0 is 50 Ohms, which is often, but not always the case. The Smith Chart displays the complex reflection coefficient, in polar form, for an arbitrary impedance (we'll call the impedance ZL or the load impedance).įor a primer on complex math, click here. This section of the antenna theory site will present an intro to the Smith Chart basics. But for now, just admire the Smith Chart and its curvy elegance. In fact, we are going to learn an even more complicated version of the Smith Chart known as the immitance Smith Chart, which is twice as complicated, but also twice as useful. We will build up the Smith Chart from scratch, so that you can understand exactly what all of the lines mean. A larger version is shown here.įigure 1 should look a little intimidating, as it appears to be lines going everywhere. With modern computers, the Smith Chart is no longer used to the simplify the calculation of transmission line equatons however, their value in visualizing the impedance of an antenna or a transmission line has not decreased. See, for instance, the input impedance equation for a load attached to a transmission line of length L and characteristic impedance Z0.
Smith Charts were originally developed around 1940 by Phillip Smith as a useful tool for making the equations involved in transmission lines easier to manipulate. The Smith Chart is used to display a real antenna's impedance when measured on a Vector Network Analyzer (VNA). Smith Charts are also extremely helpful for impedance matching, as we will see. Smith Charts can be used to increase understanding of transmission lines and how they behave from an impedance viewpoint. The Smith Chart is a fantastic tool for visualizing the impedance of a transmission line and antenna system as a function of frequency. SMITH CHART, SOLUTIONS OF PROBLEMS USING SMITH CHART
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