Parallel Resonance

A child's swing says a lot about resonance.

A child sits on a swing, feet dangling, perfectly at rest. Give him a gentle push. The child moves forward to a maximum height, reverses course under the influence of gravity, and then swings back and forth. The height of the child's excursions depends on the energy, E1, supplied by your initial push. Damping forces, such as air resistance and the child's foot-dragging, rob energy from each cycle. These damping forces control the ride's duration but have little to do with the size of the initial excursion. Mathematicians think of the damping in terms of a quality factor, Q, defined as the ratio of energy stored within the system divided by energy lost per radian of oscillation. The higher the quality factor, the lower the rate of energy loss, and the longer the ride.

If you push the swing repeatedly in sync with its natural movement, the oscillations grow. They keep growing until the amount of lost energy during each cycle, which varies with oscillation size, balances the fixed amount supplied by each push (or the child falls off). The phenomenon of a growing response is called resonance.

Figure 1 illustrates an electrical circuit that resonates. This circuit might represent part of a power system, perhaps the interaction between the total effective series inductance of a bypass capacitor array, L, and the bulk capacitance of a power-and-ground-plane pair, C. Resistance R represents the various damping factors throughout the system. A step-current waveform excites the circuit. Note that the size of the first excursion varies only modestly, going from 0.75 to 0.95 as the damping constant ranges a full order of magnitude—from two to 20. Like a swing after one push, the damping constant determines the rate of decay but has little to do with the size of the first perturbation.

In the frequency domain, the response looks different (Figure 2). A sinusoidal waveform repeats endlessly, bringing the system to a full and complete resonant balance. The peak response to a sinusoidal excitation varies in almost direct proportion to the damping constant. In a practical power system, stimulating the system (i.e., drawing a pulse of current) repetitively at the exact frequency of resonance can develop a huge response, exceeding the maximum operating voltage limits of your logic.

That is why, in a computer power system, we look at a graph showing the power-supply impedance magnitude versus frequency. The highest peaks in this graph—the sharp resonances—draw your attention and raise your fears.

What I want you to remember is that with only a single step excitation, the peak response depends almost entirely on the values of capacitance and inductance, not the quality factor.

To determine the peak response to a single step of current, a circuit theorist looks at the value of circuit impedance, defined as:

You can determine the circuit impedance for any frequency-response impedance graph from the inductive and capacitive asymptotes: j2πfL and 1/j2πfC, respectively (Figure 2). The place at which these two straight lines cross is the circuit impedance, ZC.

The rule for a parallel resonant circuit as shown in Figure 1, is that the response to a single, isolated step input never exceeds the product of the step current and the circuit impedance.

My point? Even though a huge resonance in the power system impedance looks scary, if you plan to stimulate it just one time (e.g., at startup), and wait for the resonance to die away before stimulating again, the result might not be too bad.