Parallel resonance
A scene that involves pushing a child on a swing helps explain the concept of resonance.
Howard Johnson, PhD -- EDN, 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,
ZC. In response to a single step
input, the initial perturbation does not
exceed the current times the impedance.
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,
ZC. 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.
Talkback
-
Howie, in a related topic, could you talk about optimizing transformer bandwidth, ie controlling self resonance at the top end vs enough inductance to avoid shunting the source impedance at lower frequencies, etc..
cheers,
Steve
Steve Ungstad - 2012-5-2 15:37:00 PST -
Damping factor and "Q" seem to be reciprocally related. The higher the "Q" the less energy is lost, which in control systems can mean overshoot and even oscillation. While a filter with very high "Q" will be quite selective, it is also likely to ring, since it does not lose energy quickly. While it is possible to have both high "Q" and adequate damping, it is certainly not simple.
William Ketel - 2012-2-2 19:00:25 PST -
"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."
The first statement of the Q-factor is correctly stated, however I've not read of the Q-factor being synonymously associated with damping. In fact quite the opposite : An overdamped system (high damping constant) has high energy loss and low Q. I normally associate the damping factor with the decay constant.
Sandeep Kasargod - 2012-2-2 09:42:19 PST
