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Smith Chart 5 Smith Chart After completing this section, students should be able to do the following. • Describe Smith chart as a polar plot of reflection coefficient. • Estimate the reflection coefficient from the Smith Chart. • Read the reflection coefficient off the Smith Chart • Mark the point on the Smith Chart given reflection coefficient. • Calculate load impedance if reflection coefficient and transmission-line impedance are given • Calculate reflection coefficient if load impedance and transmission-line impedance are given • Explain how was Smith Chart developed • Read normalized impedance and reflection coefficient on Smith Chart given a random point on the Smith Chart • Given impedance find the normalized position of the impedance on the Smith Chart. • Describe the reason for introducing admittance • Write impedance and admittance of an inductor and capacitor. • Explain why is the susceptance of an inductor negative and reactance is positive. • Explain why is the admittance Smith Chart rotated 180 degrees. • Distinguish between load impedance and normalized load impedance. De- scribe impedances on the Smith Chart as normalized impedances • Given impedance, read admittance on combo Y/Z chart • Given a random point on Y/Z chart, find admittance, impedance and the reflection coefficient. • Explain electrical length • Calculate the input impedance and input reflection coefficient • Describe input reflection coefficient in terms of load reflection coefficient. 88 Smith Chart 5.1 Smith Chart Smith Chart is a handy tool that we use to visualize impedances and reflection coefficients. Lumped element and transmission line impedance matching would be challenging to understand without Smith Charts. Simulation software such as ADS and measurement equipment, such as Network Analyzers, use Smith Chart to represent simulated or measured data. Smith Chart first looks like Black Magic, but it is a straightforward and useful tool that will help us better understand impedance/admittance transformations and transmission lines. Definition 16. Smith Chart is a polar plot of the reflection coefficient. An example of a Smith Chart is given in Figure 40 Figure 40: Smith Chart In essence, Smith Chart is a unit circle centered at the origin with a radius of 1. Smith Chart is used to represent the reflection coefficient graphically in polar coordinates. The reflection coefficient's real and imaginary axis (the Cartesian coordinates) is not shown on the actual Smith Chart. However, the center of the Smith Chart is where the origin of the coordinate system would be. We usually represent the reflection coefficient in polar coordinates, with a magnitude and an angle. Magnitude is the distance between the point and the origin, and the angle is measured from the x-axis. An example location of several reflection coefficients is given in Figure 41. If you do not see why the points are positioned as shown, review the polar representation of complex numbers. Figure 42, 43 circle and line represent all points on the Smith Chart that have constant magnitude or angle of the reflection coefficient. It is challenging to 89 Smith Chart Figure 41: Examples of location of reflection coefficient on the Smith Chart. measure impedances directly at high frequencies, as it is difficult to measure (or sometimes even define) voltage and current. To measure impedances, engineers use Network Analyzer shown in Figure 44. Example 18. We can estimate the reflection coefficient by looking at the posi- tion of a point on the Smith Chart. We know that the radius of the circle is 1. So, if the point is half-way between the center and the perimeter, we know the magnitude of the reflection coefficient is 0.5. We can also estimate the angle of the reflection coefficient. For example, if the magnitude is in the first quadrant, we know that the angle is between 00 and 900. If the point is on the positive y-axis, we know that the angle is equal to 900. Observe the simulation below and see if you can estimate the reflection coefficient’s magnitude and angle by looking at the point's position. The simulation below shows only positive angles. In practice, if the point is below the x-axis, we use negative angles to describe the angle of the reflection coefficient. For example, if the point is at the negative y-axis, we would say that the reflection coefficient’s angle is −900. Geogebra link: https: // tube. geogebra. org/ m/ u9ehhbaj 90 Smith Chart Figure 42: Points of constant magnitude of the reflection coefficient. jΓj=0.5, jΓj=1 Figure 43: Points of constant phase of the reflection coefficient. < Γ = 45◦, < Γ = 120◦ 91 Smith Chart Figure 44: HP8510 Network Analyzer in Microwave Laboratory measures impedances up to 26.5GHz. This piece of equipment is on permanent loan curtesy of Defense Micro Electronic Activity (DMEA), Sacramento 92 Impedance and admittance circles on the Smith Chart 5.2 Impedance and admittance circles on the Smith Chart Reflection Coefficient and Impedance Reflection coefficient and impedance are related through Equation 178. We can find an impedance that corresponds to the reflection coefficient, Equation 179. Every point on the Smith Chart represents one reflection coefficient Γ and one impedance ZL. Z − Z Γ = L ◦ (178) ZL + Z◦ 1 + Γ Z = Z (179) L 0 1 − Γ All impedances on the Smith Chart are normalized to the transmission line impedance Z0. The normalized impedance is denoted in Equation 180 with lowercase zL. ZL 1 + Γ zL = = (180) Z0 1 − Γ Derivation of Impedance and Admittance Circles on the Smith Chart Impedance and reflection coefficient are complex numbers. The normalized impedance has a real and imaginary part zL = rL + jxL, and the reflection coefficient can also be shown in Cartesian coordinates as Γ = Γr + jΓi. We can now substitute these equations into Equation 181. 1 + Γr + jΓi rL + jxL = (181) 1 − (Γr + jΓi) We can equate the real and imaginary parts on the left and right side of Equation 181 to get the equations of constant rL and xL. 93 Impedance and admittance circles on the Smith Chart 2 2 rL 2 1 Γr − + Γi = (182) 1 + rL 1 + rL 2 2 2 1 1 (Γr − 1) + Γi − = (183) xL xL These are equations of a circle, with the constant resistance circle's center at r 1 ( L ; 0) and radius ; and the constant reactance imaginary circle cen- 1 + rL 1 + rL 1 1 ter at (1; ) and radius of . xL xL Figures 45- 46 show circles on the Smith Chart that represent constant (nor- malized) reactances, and resistances. Figure 45: All points on the circle have the constant real part of the impedance (resistance). Normalized resistance circles. 1 Admittance circles can be similarly derived using the fact that YL = and ZL the Equation 180 94 Impedance and admittance circles on the Smith Chart Figure 46: All points on the circle have the constant imaginary part of the impedance (reactance). Normalized reactance circles. 95 Impedance and Admittance on Smith Chart 5.3 Impedance and Admittance on Smith Chart Brief review of impedance and admittance Impedance's Z = R + jXreal part is called resistance R, and the imaginary part is called reactance X. It is easier to add impedances when elements are in series; see Figure 47. To find the total impedance, we add resistances and reactances separately. The real part of an admittance Y = G + jB is called conductance G, and the imaginary part is called susceptance B. It is easier to add admittances when elements are in parallel; see Figure 47. When adding two admittances, we add conductances and susceptances separately. Figure 47: It is easier to use admittance when the circuit elements are in parallel and impedance when the circuit elements are in series. Smith Chart in Figure 48 has impedance circles, and impedance coordinates on it. We can use this Smith Chart to read off the values for the impedance, and reflection coefficient. In the next section, we will learn to use impedance/ad- mittance (Z/Y) Smith Chart, where both impedance and admittance circles are shown. We will use the Z/Y Smith Chart when we add impedances or admit- tances in parallel or series, which is useful in impedance matching that we will talk about in the next chapter. 96 Impedance and Admittance on Smith Chart Figure 48: On Smith Chart impedances are shown in red, and admittances in green. 97 Impedance and Admittance on Smith Chart Impedance on the Smith Chart Figure 49, shows how to find the location of normalized impedance zL = 1+j1 on the Smith Chart. zL = 1+j1 is at a point where the circle of constant resistance rL = 1 crosses the circle of constant reactance xL = 1. Figure 50 shows how to find the reflection coefficient if normalized load impedance zL = 1 + j1 is given. Measure the distance between the origin and the point using the scale ”Reflection Coefficient E or I" on the Smith Chart's bottom to find the reflection coefficient's magnitude. To find the reflection coefficient's angle, we read the scale "Angle of Reflection Coefficient" on the Smith Chart's perimeter, shown o in green. The reflection coefficient is therefore 0:5ej62 , which is close to the o actual value 0:5ej64 . If we use a ruler and compass, and a nicely sharpened pencil, we will get exactly the right answer. Try it out! Figure 49: Red circle that represents all points on Smith Chart with normalized resistance r = 1 and blue circle that represents all points on Smith Chart with normalized reactance x = 1 cross at point where ZL = 1 + j1. 98 Impedance and Admittance on Smith Chart Figure 50: To find the magnitude of the reflection coefficient from impedance ZL = 1 + j1, we measure the distance between the origin and the point using the scale ”Reflection Coefficient E or I" on the bottom of the Smith Chart.
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