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Design Ideas |
An RC circuit charges or
discharges to within 36.8% of the total excursion in one time constant, t, where t=RC. However, 36.8%
is an awkward figure, and you usually need a calculator to achieve reasonable
accuracy. You need a shortcut for these "back-of-the-envelope"
calculations you often have to do. The "practical time constant,"
tP,
is one such constant. The practical time constant is the time to charge or
discharge to 50% of the total excursion:
tP=0.7RC.
The exact value is:
–ln(1/2=0.6931…).
Thus, it takes 2tP to charge to within one-fourth of the total excursion, 3tP to charge to within one-eighth of the total excursion, and NtP to charge to within 1/(2N) of the total excursion. In short, just divide by two for each practical time constant. Figure 1 shows a practical example for analyzing a 16-bit data-acquisition circuit.
R is the equivalent resistance looking back toward the source, and C is the total load capacitance, including parasitics. Assume that R is approximately 1 k[ohm] and C is approximately 12 pF. To calculate whether the RC time constant limits the dynamic performance and whether C can charge from 0 to 5V to within 1/2LSB in 10 µsec, try the following "conference-room algorithm" (so called because you can use it to approximate an answer on your feet, usually without a calculator):
1. Calculate t:
t=(1 k[ohm])(12 pF)=12 nsec.
2. Multiply by 0.7:
tP=(0.7)(12 nsec)=8.4 nsec.
3. Calculate the number of tP intervals:
1/2(16-bit LSB)=1/(217),
or 17 practical time constants.
4. Multiply tP by the number of intervals:
(17)(8.4)=143 nsec.
The calculations show that the circuit is fast enough after all.
The long method is to solve the exponential expression for t:
The 141.4-nsec result differs from that predicted by the practical-time-constant method by approximately 1%. (DI #1942)
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