correction, the effect of mismatch can be substantially reduced (to the level of the calibration standards quality) if it is stable, but cable instability limits the repeatability of the cable mismatch and often is the dominant error in a return‐loss measurement.
For transmission measurements, the effects of mismatch do directly affect calibration, though it is reduced to a small level by the quality of the calibration standards. Often, the major instability in a cable is the phase response versus frequency. Even if the amplitude of the cable is stable, if the phase response changes, the VNA error correction will become corrupted because of phase shift of the cable mismatch error. Methods for determining the quality of cable and the effects of flexure will be described in Chapter 9.
2.4 Measurements Derived from S‐Parameters
S‐parameter measurements provide substantial information about the qualities of a DUT. In many cases, the transformation and formatting of these parameters is necessary to more readily understand the intrinsic attributes of the DUT. Some of these transformations are graphical in nature, such as plotting on a Smith chart; some are formatting such a group delay and SWR; and some are functional transformation such as time‐domain transforms. Some of the more important transformations are discussed next, with emphasis on some particularly interesting results.
2.4.1 The Smith Chart
The Smith chart is a visualization tool that every RF engineer should strive to master. It provides a compact form for describing the match characteristics of a DUT, as well as being a useful tool for moving the match point of a device to a more desired value. Invented by Philip Smith (1944), it maps the normalized complex value of a termination impedance onto a circular‐based chart, from which the impedance effects of adding lengths of transmission line onto the termination impedance are easily computed. The original intention for the use of a Smith chart was for the computing of impedances presented to a generator as lengths of transmission line were added to a load and was intended particularly for the use of telephone line impedance matching. Adding a length of transmission lines changes the apparent termination impedance, ZT, according to
(2.14)
where α and β are the real and imaginary propagation constants, and z is the distance from the load. This computation was tedious, in part because the argument of the hyperbolic tangent is complex, so a nomographic approach was desirable. A Smith chart solves by this mapping impedance to reflection coefficient (Γ), and plotting the return loss on a polar plot, as
(2.15)
The genius of the Smith chart is recognizing that rotating an impedance value through a length of transmission line is the same as rotating the phase of the reflection coefficient value on the chart. The Smith chart maps the impedance onto the polar reflection coefficient plot, but with the graticule lines marked with circles of constant resistance and circles of constant reactance. As such, any return loss value can be plotted, and the equivalent resistance and reactance can be determined immediately. To see the effect of adding some Z0 transmission line, the impedance is simply rotated on the polar plot by the phase shift of the transmission line. If the line is lossy, the return loss is modified by the line loss (two times the one‐way loss of the line), and from this new position, the resistance and the reactance are directly read.
2.4.1.1 Series and Shunt Elements
The original intent of the Smith chart was to show S11 at a fixed frequency and use the chart to derive the change in impedance due to a change in distance from the generator. But the use of the Smith chart in VNAs differs from the original intent in that the display shows return loss or S11 as a function of frequency, and the phase rotation displayed is due to a phase shift in a transmission line or device caused by the increase in frequency. Various characteristics, such as capacitance, inductance, loss, and delay, can be directly inferred from the Smith chart trajectory displayed on a VNA, and it is often more informative than just the LogMag plot or the Phase plot individually. In many instances, the Smith chart is useful for determining the principal component characteristics of the DUT. Since by most designs, the DUT should ideally be matched, the deviation from matched conditions is due to some parasitic series or shunt element. Series elements show up in a Smith chart trajectory as following a contour of constant resistances. Shunt elements are not intuitively deduced from a Smith chart but can be deduced from an admittance chart (also called an inverse Smith chart), which follows the same conformal mapping of a Smith chart (impedance chart) but with the inverse of impedance (admittance) displayed as lines of constant conductance or susceptance.
For high‐frequency measurements, shunt capacitance or series inductance is almost always the parasitic values that must be dealt with. Note that the parasitic effect of a series capacitance or a shunt inductance actually diminishes with frequency, with the capacitor becoming a short, and the inductor an open, and these elements typically cause only low‐frequency degradation.
Figure 2.35 shows both an impedance plot (left) and admittance plot (right), each with two circuit elements: a 40 Ω load with a shunt capacitance (dark trace, left), and a 60 Ω load with a series inductance.
Figure 2.35 Impedance and admittance Smith charts.
Note that on the impedance chart, the highlighted marker values for the series inductive circuit display constant resistance and inductance, even as the frequency varies. The impedance plot of the shunt RC circuit (dark trace) does not show either constant resistance or constant capacitance. However, the same trace on the admittance chart (right) does show constant conductance and constant capacitance, where the series inductive LR circuit does not show constant value. From these charts, it is clear that the impedance chart allows determination of series elements easily as their trajectory follows constant resistance circles, and the admittance chart allows determination of shunt elements as their trajectory follows constant conductance circles.
In evaluating practical responses for input impedance, it is often the case the series or shunt element is at the end of some short, or long, transmission line that wraps the response around the Smith chart. In this case, for useful information to be discerned, it is necessary to remove excess delay from the measurement. However, it is sometimes difficult to know the exact delay that must be removed. In such a case, it is possible to use two marker readouts of the VNA to attempt to determine the value of a parasitic element by means of unwrapping the phase of the response until the underlying value is determined. Figure 2.36 shows a set of responses where the DUT characteristic is delayed by a short length of transmission line, as might be found in a printed circuit board (PCB) fixture or on‐wafer probe, followed by a series resistance and inductance (left, upper) or a shunt capacitance and conductance (right, upper). These are the two most common cases of parasitic characteristics.