Parallel resonance

Howard Johnson, PhD - February 2, 2012

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 define the damping constant, Q, as the ratio of energy stored within the system divided by energy lost per radian of oscillation. The higher the damping constant is, 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. This phenomenon 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.

Now consider a computer system. On a graph of power-supply impedance versus frequency, the highest peaks—the sharp resonances—draw your attention. With a step excitation, however, the peak response depends almost entirely on the values of capacitance and inductance, not the damping factor.

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 its inductive and capacitive asymptotes: j2πfL and 1/j2πfC, respectively (Figure 2). The place at which these two straight lines cross is the circuit impedance, Z_C. In response to a single step input, the initial perturbation does not exceed the current times the impedance.

My point? A huge resonance in the power system is sometimes OK, provided that you stimulate it only once.