In Figure 2 of my last column, you can see a simple PDN impedance-profile approximation. I showed that the piece-wise-linear Bode plots of various PDN components can create peaking at some interim frequencies. In general, this kind of peaking occurs when complex impedances are connected in parallel, and at any given frequency one of them is capacitive (with negative phase) while another is inductive (with positive phase).

In this column, I will look at this peaking phenomenon in more detail. I need to talk about this because, unfortunately, even today, you can find articles, Web postings, books and some simple CAD tools giving the wrong answer to this problem.

To start, let's look at the impedance profile of a single bypass capacitor. Any real-life capacitor will have some series resistance and series inductance associated with the capacitor plates and terminals. If the parallel DC and AC leakage is neglected, which is usually an acceptable simplification for PDN applications, a series C-R-L equivalent circuit is produced, as shown on the left in Figure 1. To understand how to properly calculate the antiresonance between parallel-connected capacitors, a model with frequency-independent C, R and L is sufficient. To emphasize the fact that the antiresonance has a lot to do with the complex nature of the impedances, we plot both the magnitude and the phase of the capacitor's impedance.

Figure 1: C-R-L equivalent circuit of a bypass capacitor (on the left) and impedance magnitude and phase of a capacitor with C = 100 uF, R = 0.003 Ohm, L = 2 nH (on the right).

This capacitor has a series resonance frequency (SRF) of 356 kHz. Below SRF the impedance is capacitive: the impedance magnitude slopes downward as frequency increases and the phase is negative. At SRF the capacitive and inductive reactances cancel, the phase is zero and the impedance magnitude equals R. Above SRF the impedance magnitude slopes upward as frequency increases and the phase is positive.

As a next step, we add a second (different) capacitor with C = 1uF R = 0.01 Ohm L = 1nH in parallel to the first one. Figure 2 shows the schematics and the impedances of the two capacitors separately and in parallel. To make sure the image is not overcrowded, here we show only the magnitudes of impedances.

Figure 2: Impedance magnitude of Capacitor 1 (100 uF 0.003 Ohm 2 nH) and Capacitor 2 (1 uF 0.015 Ohm 1 nH) and their parallel equivalent. The thin lines show the impedance magnitudes of the two individual capacitors; the heavy blue line is the impedance magnitude of their parallel equivalent.