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January 2016 • The PCB Design Magazine 43 quiet power HoW To DESIgN A PDN FoR THE WoRST-CASE SCENARIo Figure 7: Flat impedance profile with a single second-order notch at 1 MHz frequency. Figure 8: Step responses of circuits from Figure 7. Figure 9: Worst-case transient peak-to-peak noise with 1A 100ps rise time step excitations. Q=10. note the logarithmic horizontal scale. the impedance, one would expect 100 mVpp worst-case transient noise. Instead, based on the reverse pulse technique, we get 120, 234, 346 and 453 mVpp worst case values. The big- gest hit is the initial factor-of-two increase; as we showed [1] , this happens because instead of a flat impedance starting at DC with the target impedance value, we start with zero (or very low) impedance and then continue with a flat target impedance at higher frequencies. When we have just one dominant peak, reaching the target impedance at the peak, but having very low impedance at DC and at high frequencies, we create a the bandpass filter. This produces the worst-case noise when we re- petitively hit this peak with a 50% duty cycle square wave. The bandpass filter picks out the fundamental harmonic from the square wave, creating a 4/PI times higher response. As the number of resonant peaks increases in the impedance profile, the worst-case noise goes up. In the example shown here, the peaks are fairly well separated on the frequency scale, interacting only mildly. The small interaction reduces somewhat the worst-case peak noise from the pathological worst case of 120, 240, 360, 480 mVpp values that we get when the peak responses do not interact. Next, we look at a single disturbance in a flat impedance profile. We use the same 100 mOhm target impedance as before and drive a deep second-order notch into it with three different Q values: 1, 3 and 10. Figure 7 shows the imped- ance profiles, Figure 8 shows the step responses. Note that all three responses reach a 1 mOhm minimum impedance at 1 MHz. Interestingly, for a single disturbance in the impedance profile with the same maximum and minimum values, the worst-case transient noise does not depend on the Q of the notch. When we calculate the worst-case noise with the re- verse pulse technique, we get 290 mVpp for all three cases. Figure 9 shows the actual worst-case time-domain response for the Q=10 case.