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发表于 2007-1-25 15:34:15
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Introduction
Finding the –3 dB bandwidth of an arbitrary linear network can be a difficult problem in
general. Consider, for example, the standard recipe for computing bandwidth:
1) Derive the input-output transfer function (using node equations, for example)
2) Set s = jw;
3) Find the magnitude of the resulting expression;
4) Set the magnitude = 1/ of the “midband” value; and
5) Solve for w
It doesn't take a great deal of insight to recognize that explicit computation (by hand) of
the –3 dB bandwidth using this method is generally impractical for all but the simplest
systems. In particular, the order of the denominator polynomial obtained in step 1 above is
equal to the number of poles (natural frequencies), which in turn equals the number of
degrees of freedom (measured, say, by the number of initial conditions one may independently
specify), which in turn equals the number of independent energy storage elements
(e.g., L or C), which in turn can be as large as the number of energy storage elements
(phew!). Thus, a network with n capacitors might require the equivalent of finding the
roots of an nth-order polynomial. If n exceeds just four, no algebraic closed form solution
exists. Even if n = 2, it might be labor-intensive to obtain the final numerical result.
Now, machine computation is cheap and getting cheaper all the time, so perhaps the analysis
of networks doesn’t present much of a problem. However, we are interested in developing
design insight so that if a simulator tells us that there is a problem, we have some idea
of what to do about it. We therefore seek methods that are reasonably simple to apply, yet
conveys the desired insight, even if it yields answers that might be approximate. Simulators
can then be used to provide final quantitative verification.
Two such approximate methods are open- and short-circuit time constants. The former
provides an estimate of the high-frequency rolloff while the latter yields an estimate of the
low-frequency rolloff point. These methods are valuable because they identify which elements
are responsible for the bandwidth limitation. This information alone is often sufficient
to suggest what modifications should be tried next.
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