(upper), unwrapped phase (middle), and overcompensated (lower); for an inductor (right) and a capacitor (left).
The trajectories are distorted by a frequency dependent phase shift due to the delay of a portion of transmission line between the VNA reference port and the parasitic. For the inductive case, the elements are series elements, and the normal Smith chart, or impedance chart, is shown. The marker readout gives the resistance, in Ω; the reactance, in Ω; and the equivalent inductance. It is clear that the apparent value of the reactive element is not constant. Most VNAs include the equivalent inductance or capacitance value associated with the reactance for the marker position and frequency.
For the capacitive example, since it is a shunt impedance, an inverse Smith chart or admittance chart is used. The value displayed for the real part of the admittance is the conductance, in milli‐Siemans (or mS, inverse Ω), and for the imaginary part is the susceptance, also in milli‐Siemans. The reactive part is converted to an equivalent shunt capacitance or inductance, determined by the sign of the imaginary part of the admittance. Again, it is clear that the apparent value of the shunt reactive element is not constant. In fact, both trajectories show an attribute of having a resonance, since they cross the real axis. However the fact that the magnitude of reflection is not a minimum at the crossing indicates that this is not a true resonant structure, but rather a device whose phase response is distorted by a length or delay of a transmission line between the measurement plane and the discrete impedance or admittance.
It is reasonably simple to investigate the effect of removing the delay, by using two markers, spaced in frequency. By reading the value of the imaginary element of each marker while adding in electrical delay, the phase shift of the delay line can be removed, and the resulting underlying element characteristics are revealed. When both marker readings show the same value for the reactive element, then the proper delay has been removed, as shown in the middle portion of Figure 2.36. In this case, the left plots give a capacitance of 1 pf in shunt with 100 Ω, and the right plots show an inductance of 3 nH in series with a resistance of 25 Ω.
The lower traces show the same measurement, but with even more electrical delay removed from the response. Electrical delay is a common scaling function in VNAs that provides a linear phase shift versus frequency for any particular trace. A related function is port extension, which also provides a phase shift but that shift is associated with the port of the analyzer, rather than with just the particular trace. With electrical delay scaling, only the trace that is active has the delay applied, and different traces of the same parameter can have different delays. With port extension, all traces that are associated with a particular port, for example, S11 and S21 with port 1, will have their phase response modified by the port extension. Electrical delay applies the same phase shift regardless of the parameter type, but port extensions properly account for a two‐times phase shift for reflection parameters in contrast with a one‐times phase shift for transmission parameters. Therefore, it is perhaps better to use port extension to accommodate changes in reference plane and reserve electrical delay for when one wants to remove the linear phase shift of a particular parameter.
The delay or port extension is adjusted until the trace rotation is minimized, while still maintaining a trajectory that follows a clockwise rotation. Foster demonstrated that all real devices should have phase that increases with frequency causing the clockwise rotation, so the proper amount of delay to be removed can often be determined by looking at the rotation direction of the trace trajectory. This is demonstrated in the lower traces of Figure 2.36, where an additional 10% of the delay from the middle traces has been removed, making the response overcompensated and causing a reactive element value at the two markers to be different.
2.4.1.2 Impedance Transformation
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 ZL, 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 ZL.
2.4.2 Transforming S‐Parameters to Other Impedances
While it is most common to define S‐parameters in a 50 Ω impedance, or 75 Ω for cable‐television applications, situations arise where it is necessary to define an S‐parameter matrix in other than 50 Ω, or to have it defined with 50 Ω on one port and with a different impedance on another port. This requirement occurs for matching circuits, or impedance transformers, as well as the use of waveguide adapters where it is common practice to define the terminal impedance as 1 Ω. Unfortunately, while S‐parameter definitions don't prohibit different impedances on different ports, the most common data files for S‐parameters, the so‐called Touchstone™ or S2P files, provide for only a single impedance in their definition. (Recently a second revision of the S2P file format has been defined that allows different impedances on different ports, but it has not yet been widely implemented.) Thus, it is often necessary to transform S‐parameters from one reference impedance to another. If the complete S‐parameter matrix is available, then a matrix transformation (Tippet and Speciale 1982) can be used to convert the impedance of by applying
(2.17)