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December 2015 • The PCB Design Magazine 47 going through our circuit. The voltage as a re- sult of the 1A swept-frequency sine-wave excita- tion gives us the complex impedance. Figure 2 shows the impedance magnitude and phase at node Z out in the frequency range of 100Hz and 1GHz. The sweep was logarithmic with 100 fre- quency points per decade. The impedance profile shows three peaks with almost the same peak value, all slightly above 100 mOhm. The series loss values are very low, 1–6 mOhm, resulting in deep val- leys in between the impedance peaks. The an- tiresonance frequencies spread across almost three decades of frequencies. While this im- pedance profile would be very rare in practice, and would most likely be the result of either careless design or lack of any systematic de- sign whatsoever, we cannot rule out either the possibility that this could represent an ac- tual circuit. Making use of the three distinct peaks separated by deep valleys, Sandler uses a semi-heuristic approach to find what is called a rogue wave: it defines three repetitive bursts hitting the peak impedances one after the oth- er, leaving the timing adjustment to an opti- mizer to find the biggest noise. The result is 750 mVpp for a series of 2A cur- rent step, which is equivalent to 375 mVpp/A. Compared to a perfectly flat impedance profile matching the largest peak, 126 mOhm, the op- timization from Sandler's paper predicts a worst case of almost exactly three times of that value. The question is: is this really the worst case, or is it possible to find a different sequence of current steps that would produce an even big- ger transient noise? We can turn to the reverse pulse technique to get the answer. The reverse pulse technique starts with the step response of the circuit. Since the basic as- sumption is that the PDN is linear and time invariant (LTI), it does not matter whether we look at the response for a positive or negative going current step excitation; they are mirror images of each other. Figure 3 shows the step response for a positive going current step. With- out restricting generality, we assume that the DC voltage on the supply rail is zero and there- fore most of the transient response will be neg- Figure 2: Impedance magnitude and phase from the circuit shown in Figure 1. Note that both axes are logarithmic; in particular, the frequency scale is logarithmic to clearly show the resonance peaks separated by three orders of magnitude. quiet power SYSTEMATIC ESTIMATIoN oF WoRST-CASE PDN NoISE