Читать книгу Handbook of Microwave Component Measurements - Joel P. Dunsmore - Страница 127

One aspect of rotation on the Smith chart that is often misunderstood is that the rotation around the center of the chart for a transmission line delay occurs only if the transmission line impedance matches the reference impedance of the Smith chart. Consider a case of a termination consisting of a 25 Ω resistor to ground, shunted by a 3 pF capacitor, and evaluated from DC to 10 GHz. The impedance trajectory is shown in the light trace in Figure 2.37, which shows a small deviation from 25 Ω due to the shunt capacitance. The darker trace shows the same impedance, but at the end of a transmission line that has 180° of phase shift at 10 GHz. The value of the impedance trajectory centers on 50 Ω, and the value of the trace at 180° phase shift matches that exactly of zero phase shift. At the frequency where the phase shifts 90° due to the transmission line (5.4 GHz) plus the slight phase shift of the DUT, the impedance is nearly 100 Ω. This is a well‐known aspect of ¼ wave (or 90°, or λ/4) transmission line transformers. If impedance of the line is Z₀, then the impedance at the end of a ¼ wave section is

(2.16)

Figure 2.37 An impedance value rotated by 180° 50 Ω line.

One consequence of this is that the maximum deviation of impedance due to a transmission line depends completely on the impedance of the transmission line. Figure 2.38 shows the Smith chart trajectories for the same termination, but this time with a 12.5 Ω line, a 25 Ω line, and a 100 Ω line before the termination. Of course, at 180°, no transformation of impedance takes place, and the impedance value at the end of the line is identical to that at the 0° phase shift. It is interesting to note that the smallest deviation of impedance is for the case where the line matches the impedance of the termination, rather than matching the system impedance, as in Figure 2.37.

Figure 2.38 25 Ω termination proceeded by half‐wavelength segments of 12.5, 25, and 100 Ω lines.

The other important aspect to note is that when the transmission line is of greater impedance than Z_L, the resulting impedance will transform to a higher value, while when the transmission line is of lower impedance, the resulting impedance will be lower than Z_L.