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Chapter 26: Smith Charts

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Chapter 26: Smith Charts Chapter 26: Smith Charts Chapter Learning Objectives: After completing this chapter the student will be able to: Plot an impedance or admittance on a Smith chart. Use a Smith chart to determine the input impedance of a terminated transmission line. Use a Smith chart to design a single-stub tuner to match a load impedance to a transmission line’s characteristic impedance. You can watch the video associated with this chapter at the following link: Historical Perspective: Phillip H. Smith (1905-1987) was a an American electrical engineer who worked for Bell Telephone Laboratories. He is best known for inventing the Smith chart, which is a very useful graphical tool for working with transmission lines and RF circuits. Smith Chart image © 2015 IEEE. Used with permission. 1 26.1 Introduction to the Smith Chart The Smith chart is a graphical method for solving problems involving transmission lines and radio frequency (RF) loads such as antennas. It was invented in 1936, and for more than 50 years, it was the most important tool in the RF engineer’s toolkit. Since the 1980s, computer tools have become much more powerful, and pencil-and-paper Smith charts are used less frequently, but they are still incredibly useful for visualizing the problems associated with transmission lines and loads. Many computer tools and digital measurement devices still use Smith charts to convey information to their users. Smith charts can be used to plot impedances, admittances, reflection coefficients, and many other quantities that are beyond the scope of this discussion such as scattering parameters, noise figures, and measures of stability. We know that the impedance of a transmission line (the ratio of the voltage divided by the current) varies from point to point along the line, and that the impedance repeats itself every half-wavelength. We will take advantage of this repetition by plotting the impedance on a circle, where the circumference represents a half wavelength. In that way, one revolution around the circle will return you to the same point, and moving a half wavelength along the line will return you to the same impedance. In order to allow one chart to handle any possible transmission line, we will “normalize” all the impedances in the system by dividing them by the characteristic impedance of the transmission line, Zc. We will use a lower-case z to represent impedances that have been normalized in this manner. (Equation 26.1) Since the load impedance can have both a real part (due to resistance) and an imaginary part (due to capacitance and/or inductance), we can represent it as follows: (Equation 26.2) Normalizing both the real and imaginary components of the impedance gives: (Equation 26.3) Where the normalization is performed as indicated by Equation 26.1: 2 (Equation 26.4) (Equation 26.5) Recall from chapter 25 that we derived the following equation for the input impedance of a transmission line: (Copy of Equation 25.10) If we normalize this equation by dividing by ZC and then divide both the numerator and denominator by ZC to normalize the elements inside the fraction, we find: (Equation 26.6) Recalling from chapter 24 the expression for reflection coefficient in terms of impedances: (Copy of Equation 24.25) Since the load impedance can be complex, the reflection coefficient can also be complex. We will break it into the real part r and the imaginary part i. We will also divide both the numerator and denominator of Equation 24.25 by ZC to normalize the impedances: (Equation 26.7) Performing a little algebra on Equation 26.7, we can rearrange it to solve for zL: (Equation 26.8) Separating both zL and G into real and imaginary parts gives: (Equation 26.8) 3 We need to separate the right side into purely real and purely imaginary components. To do so, we need to eliminate the imaginary component in the denominator. Multiplying both the numerator and denominator of the right side by the complex conjugate of the denominator gives: (Equation 26.9) Multiplying the right side out gives: (Equation 26.10) We can now simplify the right side. Notice that the denominator has become completely real, so the right side can be separated into a real and an imaginary component: (Equation 26.11) Simplifying further gives: (Equation 26.12) Separating the real and imaginary components gives: (Equation 26.13) For this equation to be true, the real components on both sides must be equal to each other: (Equation 26.14) After many steps of algebra, this equation can be rewritten as follows: (Equation 26.15) This equation is in the form of a circle, where the center of the circle is at (r/(r+1), 0) and the radius of the circle is 1/(r+1). The “x” and “y” variables are the real part and imaginary part of the reflection coefficient, r and i. There are actually a number of circles, each of which has a 4 different center along the x-axis and each of which has a different radius. These are known as “r-circles,” since they are defined by the parameter r (the normalized real load impedance). Figure 26.1 shows a few of these r-circles. r=1 r=2 r=3 r=4 r=0 =0.2 r r=1.5 r=0.6 r=0.4 Figure 26.1. A series of r-circles. We can also write an equation that equates the imaginary components of Equation 26.13. (Equation 26.16) Once again, this equation can be manipulated into the following form: (Equation 26.17) This is again the equation of a series of circles, whose centers are at (1, 1/x) and whose radius is 1/x. These circles are known as x-circles, since their characteristics are determined by the parameter x, which is the imaginary component of the normalized load impedance. Recall that imaginary impedance can be positive (for an inductive load) or negative (for a capacitive load). If x>0, the circle will be above the x-axis, while if x<0, the circle will be below the x-axis. Figure 26.2 shows a series of x-circles with both positive and negative values of x. 5 x=1 x=2 x=3 x=4 x=-4 x=-3 x=-2 x=-1 Figure 26.2. A series of x-circles. Positive x values are above the x-axis, negative values are below. If we combine the r-circles and the x-circles into one plot, we obtain Figure 26.3. We will restrict our attention to the inside of the r=0 circle, since anything outside that circle will correspond to negative real impedance, which is not physically possible. Notice that this means we will only capture a portion of each x-circle, making them look more like arcs than circles. Figure 26.3. A combination of r-circles and x-circles shown inside the circle r=0. This superposed image of r-circles and x-circles is also known as a Smith chart. Professional versions of the Smith chart are widely available, as shown in Figure 26.4. 6 Figure 26.4. A Professional Quality Smith chart (© 2015 IEEE, used with permission) Every intersection on this chart corresponds to a particular normalized load impedance, and you can interpolate between points if you have a very precise impedance to be plotted. Example 26.1: Plot the following points on the Smith chart in Figure 26.4. a) zL=0+j0 b) zL=∞ c) zL=1+j0 d) zL=0.5+j0.5 e) zL=1.5-j1.5 f) zL=2+j1 7 26.2 Calculating Admittances on the Smith Chart You will recall from chapter 25 that several of the impedance-matching strategies involved placing loads in parallel, and the math for parallel loads is made easier when we work with admittances rather than impedances. We can actually find the admittance that corresponds to a given impedance directly using the Smith chart. Rewriting Equation 26.7 in terms of admittances gives: (Equation 26.18) Multiplying numerator and denominator by YLYC produces: (Equation 26.19) Dividing numerator and denominator by Yc to normalize all admittances gives: (Equation 26.20) As seen in Equation 26.20, we can factor a -1 term out of the numerator, which is equal to ej. Doing so yields a form that is identical to the expression in terms of impedance: (Equation 26.21) Since the Smith chart is actually drawn on the complex plane, Equation 26.21 says that we can find an admittance from an impedance (or vice versa) by moving radians (180°) around a circle at a constant radius. Example 26.2: Use Figure 26.4 to calculate the admittance that corresponds to: a) zL=0.5+j0.5 b) zL=1.5-j1.5 c) zL=2+j1 8 26.3 Calculating Input Impedance Using a Smith Chart We can also use a Smith chart to determine the impedance at any point along the line, including the input impedance at the generator (the opposite end from the load). Recall from chapter 25 that we have the following expression for the impedance at any point z: (Copy of Equation 25.2) Since the input impedance is the impedance at z=-L, we can write: (Equation 26.22) If we divide both sides of the equation by Zc and then divide the numerator and denominator of the right side by ejkL, we find: (Equation 26.23) We can combine the angle from the load with the phase shift introduced by moving from the load to the input to obtain: (Equation 26.24) Where: (Equation 26.25) More generally, if you want to move a distance z’ away from the load, this angle is: This means that if you know the impedance at any point along the line (such as at the load), you can simply rotate around the Smith chart an angle of 2kz’ to find it at a different location along the line.
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