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December 2015 • The PCB Design Magazine 53 move the frequency around the middle peak in five values: 300 kHz, 900 kHz, 1020 kHz, 1100 kHz and 5000 kHz. Figure 10 shows the schematics and wave- forms for the three repetition frequencies ex- actly matching the three peak frequencies. We see that in this case the transient noise is higher, actually 4/p times higher, than the product of the impedance peak magnitude and current magnitude. This is because the high- Q peak picks out the fundamental harmonic and greatly attenuates the harmonics. In the Fourier series of a square wave with 50% duty cycle, the fundamental-frequency sine wave has a 4/p times higher magnitude than that of the square wave. Finally the table in Figure 11 summa- rizes the characteristic noise signature num- bers we obtained. Note that the reverse pulse technique yielded the highest noise, and it is proven to be the absolute worst case. It is also true that the rogue-wave optimization could provide the same (correct) answer, however, in a multi-resonance case like this example, without operator guidance it could take a lot of computing resources and eventually it may not converge. The first entry in the table is calculated as twice the peak deviation of the step re - sponse (from the last row in Figure 7) minus the steady state response. This is the peak-to- peak noise as a result of a single rising edge followed by a single falling edge with a large time separation in between. This estimate is 85.7% smaller than the true maximum. The second entry equals the biggest peak in Figure Figure 9b: Response waveforms. Note that the square-wave excitation frequencies map out the middle resonance. The biggest response comes from the 1020 kHz square wave, matching the middle resonance frequency in the impedance profile. quiet power SYSTEMATIC ESTIMATIoN oF WoRST-CASE PDN NoISE