Appendix G.Transformed frequency domain measurements using spice

306 TRANSFORMED FREQUENCY DOMAIN MEASUREMENTS USING SPICE

First of all, the SPICE S-parameter simulation can be expanded to include unequal input and output impedance levels. This is useful in analyzing impedance steps and transformers with two different resistance levels. Second, rather than using ac steady state voltage sources, a time domain pulse can be used. The automatic network analyzer is made to simulate a time domain reflectometer by mathematically producing an “impulse” that can be replicated in SPICE approximately by using the PULSE function or more accurately by the piecewise linear (PWL) transient function. This SPICE replica of the network analyzer “impulse” can be used to show how various circuit elements react to this “impulse.” Simple formulas are given that can be used to obtain the value of the inductance, capacitance, or impedance step directly from time domain “impulse” data.

G.2 FREQUENCY DOMAIN S-PARAMETERS

In [4] the S parameters were obtained for a given circuit from SPICE for a twoport circuit in which the input and output resistances were both 50 . However, the conversion circuit used to obtain the S parameters can be modified so that the two ports of the circuit are at different impedance levels (Fig. G.1). This is done by the addition of an ideal transformer whose turns ratio is

where R01 and R02 are the characteristic impedance levels of the input and output sides, respectively. The input impedance, Z1, looking into the transformer from the port-1 side is

which is equal to R01 when ZL D R02. The voltage drop caused by the independent current source of value 1/R01 is

The voltage at node 11 in Fig. G.1 is numerically the same as S11, since

For the output side, the secondary current in the transformer is I2 D I1/n. The portion of current going into the transformer primary from the current source side is

308 TRANSFORMED FREQUENCY DOMAIN MEASUREMENTS USING SPICE

The voltage at node 2 is

The voltage across R21 with the marked polarity is V 21 D 2V2/n which, upon replacing V2 with the value above, gives a value for V 21 that is numerically equal to S21:

The right-hand side is obtained using Eqs. (G.1) and (G.2). For a matched load when ZL D R02, S21 D 1 as expected. Finding the S22 and S12 for a circuit is achieved by direct analogy. A suggested SPICE listing for finding the S parameters with unequal source and load impedance levels is shown in Section G.6.

G.3 TIME DOMAIN REFLECTOMETRY ANALYSIS

Time domain reflectometer (TDR) measurements from an automatic network analyzer can be directly compared with a theoretical circuit model in the time domain in SPICE. All that is required is to make the two modifications described in Section G.7. Of course, a near-ideal impulse can be implemented in SPICE, but the point of view taken here is to calculate time domain data that would actually be measured using the time domain feature on a network analyzer. Two actual time domain “impulses” were measured on a network analyzer under the conditions in which the maximum frequency was 18 and 26 GHz, respectively. A third “impulse” with a maximum frequency of 50 GHz was calculated from Hewlett Packard’s Microwave Development System (MDS) program. This was justified on the basis that (1) the program and the network analyzer use the same algorithm for the chirp-Z transform and (2) the measured and calculated 18 GHz “impulses” are nearly identical. Circuit responses to “impulses” with different maximum frequency content will of course differ, so the time domain response must be always coupled to the frequencies used in producing the “impulse.”

Two options for representing the network analyzer “impulse” are illustrated in Fig. G.2. The approximate impulse is modeled in SPICE as a PULSE, while the more accurate approach is achieved by using the piecewise linear (PWL) SPICE function. In the present case, the PWL function uses 77 different x, y coordinate pairs to represent the 26 GHz “impulse.” The result is indistinguishable from the measured “impulse” within the resolution of the graph in Fig. G.2. A similar fit was made for the 18 and 50 GHz “impulses.” The piecewise linear fit of the network analyzer “impulses” for use in SPICE are found in Section G.7. The approximate trapezoidal pulse for a 50 source impedance has the advantage of providing results similar to the PWL approximation with a lot less data entry. However, to closely represent the actual time domain response of the network analyzer, the piecewise linear approach should be used.

Capacitance, pF

FIGURE G.5 Maximum and minimum values of the time domain response for a shunt capacitance using the piecewise linear approximation to the measured “impulse.”

extremum) is monotonic, and hence it gives a unique value for the capacitance. The series inductance response is a mirror image to the shunt capacitance about the t D 0 line. Thus the first maximum of t is monotonically related to the value of the series inductance. It is this first maximum of j tj that can be used to find the value of the two types of discontinuities.

The time domain not only allows determination of the discontinuity type and size but also the discontinuity position in time. Here again there is some ambiguity in the part of the “impulse” response curve that should be used to find the position. For shunt capacitances, a position midway between the lower and upper part of the curve could be used, that is, where t D 0. A comparison of this choice with the theoretical distance is provided in Fig. G.6, where it is seen that this method is accurate if the discontinuity is very small. However, for larger discontinuities a better choice would be at the location of the first extremum of j tj.

The minimum t of the shunt capacitor, the maximum t of the series inductor, the maximum t of the series capacitance, the minimum t of the shunt inductance, and the peak t (either negative or positive) of the impedance step can each be described by a simple formula that could be readily stored in a handheld calculator. These formulas are listed below. The parameters for these expressions are in Table G.1 for the 18 GHz, 26 GHz, and 50 GHz “impulses.”

Measurements of a given series-mounted chip capacitor on a LCR meter at, say 1 MHz, would not necessarily give accurate correlation with a time domain model of a simple series capacitance as modeled by Eq. (G.10) or direct measurements on a network analyzer. However, a better fit with the time domain data can be obtained by using a model of a high-frequency capacitor as described in Chapter 2.

G.5 MULTIPLE DISCONTINUITIES

The formulas above are correct when there is only one significant discontinuity in the circuit that is being measured. When there are multiple discontinuities, the SPICE analysis will also display the results that would be expected in a real time domain reflectometer measurement. The gating error in measuring a discontinuity in the presence of other discontinuities is analyzed in [6]. That analysis uses a rectangular gating function with a depth of 40 dB rather than the chirp-Z transform in the network analyzer software. Four sources of error are identified. The first is out-of-gate attenuation associated with incomplete suppression of reflections outside the gating function. The second is truncation error where the gate is made too narrow to pick up all of the response due to the discontinuity in question. The third is masking error where the transmission coefficients of previous discontinuities reduces the signal getting to the discontinuity under investigation. The fourth is multi-reflection aliasing error that occurs when the circuit has commensurate line lengths, and the residual reflection of one discontinuity adds to or subtracts from the reflection of the discontinuity under investigation.

When two discontinuities are sufficiently separated in time, each can be analyzed separately, though the accuracy of predicting the later discontinuities from the formulas in Table G.1 are less accurate than when there is only one discontinuity. Three impedance steps are set up such that if they were alone, they would have had a t of 0.3, 0.2, and 0.1, respectively (Fig. G.7), as is done by [6]. However, when the three steps are put in one circuit suitably separated from one another, the SPICE analysis shows that t D 0.3 (0% error), t D 0.181 (9.6% error), and t D 0.087 to 0.076 (12.9% to 24% error depending on whether line lengths are commensurate). The estimated errors in [6] are 0%, 12%, and 20%, respectively, which correlate well with the SPICE results given the approximations that are employed.

In addition the technique described here can be used in a wide variety of circuits to model two elements that are too close together to be resolved separately in time. Adjustment of a theoretical time domain model to make its response match that of the measured time domain measurements gives a method to extract an equivalent circuit in the time domain.