where Γ<sub>L</sub> is the load reflection coefficient, ζ<sub>L</sub> = Z<sub>L</sub>/Z<sub>0</sub> is the normalized load value, L is the length of the line segment (same as the len parameter), and β = 2π/λ<sub>g</sub> = 2π√(ε<sub>eff</sub>)/λ<sub>0</sub>. Note that both input impedance and input reflection coefficient are functions of the product βL or the product 2π√(ε<sub>eff</sub>)(f.L)/c, where c is the speed of light. For a given fixed value of ε<sub>eff</sub> (same as the eeff parameter), both Z<sub>in</sub> and Γ<sub>in</sub> are functions of the product (fL) rather than functions of f or L individually. This property has a practical application. If you fix the length of the transmission line segment and vary the frequency, you will get a circular plot on the Smith chart that is equivalent to changing the line segment length at a fixed frequency. For example, in the Smith chart of the previous step, L = 37.5mm was fixed. S11 at 2GHz falls at the origin of the Smith chart.
Now let's interpret the same circular curve at a fixed frequency of f = 2GHz, assuming that L now varies instead. The origin of the Smith chart obviously corresponds to L = 37.5mm. The intercept points point that corresponded to a frequency of 1GHz in the circle with the positive vertical axis above Smith chart now corresponds to L = 75*would represent a line length of (11GHz × 37.5GHz/2GHz5mm) = 56.25mm, while the intercept points with the negative vertical axes corresponds to L = 75*(2.5GHz/2GHz) = 9318.75mm. To verify these resultsSimilarly, you can change the length of the T-Line segment first point that used to 56.25mm and then correspond to 93.75GHz, and run a network analysis over the 1-3GHz frequency rangeof 4GHz now would represent a line length of (4GHz × 37. The Smith chart results for the two cases are shown in the figures below. The data points corresponding to f 5mm)/2GHz = 2GHz has been marked in both figures, which are identical to the previous results75mm.
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The two Smith charts below show the one-to-one correspondence between the frequency points at a fixed length of 37.5mm and the length points at a fixed frequency of 2GHz.
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