Davis W.A.Radio frequency circuit design.2001

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.

*n = SQRT(R02/R01)

*Input side

*.

*.

*.

*Output side

G.7 IMPULSE RESPONSE SPICE NET LIST MODIFICATION

Time domain analysis with SPICE requires replacing the AC statement with a TRAN statement similar to the following:

.TRAN .01ps 250ps 0 .2ps

In addition the VIN statement should be replaced with one containing either the PULSE or the PWL transient function. The pulse statement has the form PULSE(initial volt, pulse volt, delay time, rise time, fall time, pulse width, period). The measured “impulse” from the time domain measurements from the network analyzer is approximated by the following PULSE statements. The 18 GHz pulse, with base width of 94.019 ps, is approximated as follows:

VIN 10 11 PULSE(0 1 0 39.482p 39.491p 15.05p)

The 26 GHz pulse, with a base width of 64.150 ps, is approximated as follows:

VIN 10 11 PULSE(0 1 0 26.395p 28.477p 9.278p)

Alternately, the more accurate piecewise linear fit could be used. The 18 GHz “impulse” is approximated using 81 points:

VIN 10 11 PWL(0ps 0, 1.5ps 0.0037, 3.0ps 0.0089, 4.5ps 0.0156,

316 TRANSFORMED FREQUENCY DOMAIN MEASUREMENTS USING SPICE

+6.0ps 0.0237, 7.5ps 0.0336, 9.0ps 0.0452, 10.5ps 0.0589,

+12.0ps 0.0745, 13.5ps 0.0922, 15.0ps 0.1121, 16.5ps 0.1342,

+18.0ps 0.1584, 19.5ps 0.1848, 21.0ps 0.2134, 22.5ps 0.2441,

+24.0ps 0.2767, 25.5ps 0.3111, 27.0ps 0.3472, 28.5ps 0.3848,

+30.0ps 0.4236, 31.5ps 0.4634, 33.0ps 0.5038, 34.5ps 0.5448,

+36.0ps 0.5858, 37.5ps 0.6267, 39.0ps 0.6671, 40.5ps 0.7065,

+42.0ps 0.7448, 43.5ps 0.7815, 45.0ps 0.8162, 46.5ps 0.8489,

+48.0ps 0.8789, 49.5ps 0.9062, 51.0ps 0.9304, 52.5ps 0.9513,

+54.0ps 0.9687, 55.5ps 0.9824, 57.0ps 0.9922, 58.5ps 0.9983,

+60.0ps 1.0002, 61.5ps 0.9982, 63.0ps 0.9922, 64.5ps 0.9823,

+66.0ps 0.9686, 67.5ps 0.9512, 69.0ps 0.9303, 70.5ps 0.9061,

+72.0ps 0.8788, 73.5ps 0.8487, 75.0ps 0.8161, 76.5ps 0.7813,

+78.0ps 0.7446, 79.5ps 0.7064, 81.0ps 0.6669, 82.5ps 0.6266,

+84.0ps 0.5857, 85.5ps 0.5446, 87.0ps 0.5037, 88.5ps 0.4633,

+90.0ps 0.4235, 91.5ps 0.3847, 93.0ps 0.3471, 94.5ps 0.3111,

+96.0ps 0.2767, 97.5ps 0.2441, 99.0ps 0.2135, 100.5ps 0.1849,

+102.0ps 0.1585, 103.5ps 0.1342, 105.0ps 0.1122, 106.5ps 0.0923,

+108.0ps 0.0746, 109.5ps 0.0591, 111.0ps 0.0455, 112.5ps 0.0338,

+114.0ps 0.0240, 115.5ps 0.0158, 117.0ps 0.0092, 118.5ps 0.0040,

+120.0ps 0)

The piecewise linear fit for the 26 GHz “impulse” is approximated using 77 points:

VIN 10 11 PWL(0ps 0,1ps .005, 2ps .015, 3ps .0267, 4ps .0402,

+5ps .0556, 6ps .0731, 7ps .0925, 8ps .1140, 9ps .1375,

+10ps .1632, 11ps .1909, 12ps .2204, 13ps .2519, 14ps .2850,

+15ps .3198, 16ps .3560, 17ps .3933, 18ps .4318, 19ps .4709,

+20ps .5106, 21ps .5505, 22ps .5904, 23ps .6299, 24ps .6688,

+25ps .7067, 26ps .7433, 27ps .7784, 28ps .8116, 29ps .8427,

+30ps .8713, 31ps .8972, 32ps .9202, 33ps .9401, 34ps .9566,

+35ps .9697, 36ps .9792, 37ps .9850, 38ps .9871, 39ps .9855,

+40ps .9801, 41ps .9710, 42ps .9584, 43ps .9423, 44ps .9227,

+45ps .9001, 46ps .8745, 47ps .8462, 48ps .8155, 49ps .7825,

+50ps .7477, 51ps .7112, 52ps .6734, 53ps .6346, 54ps .5952,

+55ps .5553, 56ps .5154, 57ps .4756, 58ps .4363, 59ps .3979,

+60ps .3604, 61ps .3240, 62ps .2891, 63ps .2557, 64ps .2240,

+65ps .1942, 66ps .1663, 67ps .1404, 68ps .1166, 69ps .0948,

+70ps .0751,71ps .0575, 72ps .0418, 73ps .0279, 74ps .0160,

+75ps .0058, 76ps 0)

The piecewise linear fit for the 50 GHz “impulse” is approximated using 46 points:

VIN 10 11 PWL( 0ps -4.530E-03, 1ps -611.4E-06, 2ps 6.941E-03,

+3ps 0.019, 4ps 0.037, 5ps 0.061, 6ps 0.091, 7ps 0.130,

+8ps 0.175, 9ps 0.229, 10s 0.289, 11ps 0.355, 12ps 0.425,

+13ps 0.500, 14ps 0.576, 15ps 0.651, 16ps 0.724, 17ps 0.792,

+18ps 0.854, 19ps 0.906, 20ps 0.948, 21ps 0.979, 22ps 0.996,

+23ps 1.000, 24ps 0.990, 25ps 0.967, 26ps 0.931, 27ps 0.884,

318 TRANSFORMED FREQUENCY DOMAIN MEASUREMENTS USING SPICE

+28ps 0.828, 29ps 0.763, 30ps 0.693, 31ps 0.618, 32ps 0.542,

+33ps 0.467, 34ps 0.394, 35ps 0.325, 36ps 0.262, 37ps 0.204,

+38ps 0.155, 39ps 0.112, 40ps 0.077, 41ps 0.049, 42ps 0.028,

+43ps 0.013, 44ps 3.141E-03, 45ps -2.711E-03)

ACKNOWLEDGMENT

The authors wish to acknowledge the help of Terry Jamison in the analysis of the impulse response data.

REFERENCES

1.M. E. Hines and H. E. Stinehelfer Sr, “Time-Domain Oscillographic Microwave Network Analysis Using Frequency-Domain Data,” IEEE Trans. Microwave Theory Tech., Vol. MTT-22, pp. 276–282, 1974.

2.T. B. Mills, “S-Parameters in Spice,” RF Design, pp. 45–48, 1989.

3.K. B. Kumar and T. Wong, “Methods to Obtain Z, Y, H, G and S Parameters from the SPICE Program,” IEEE Circuits Devices, Vol. 4, pp. 30–31, 1988.

4.R. Goyal, “S-Parameter Output from the SPICE Program,”IEEE Circuits Devices, Vol. 4, pp. 28–30, 1988.

5.C. E. Smith, “Frequency Domain Analysis of RF and Microwave Circuits Using SPICE,” IEEE Trans. Microwave Theory Tech., Vol. MTT-42, pp. 1904–1909, 1994.

6.Ke Lu and T. J. Brazil, “A Systematic Error Analysis of HP8510 Time-Domain Gating Techniques with Experimental Verification,” 1993 IEEE MTT-S Microwave Theory Tech. Symp. Digest, pp. 1259–1262, 1993.