Solving for unknown time
Sometimes it is necessary to determine the
length of time that a reactive circuit will take to reach a
predetermined value. This is especially true in cases where
we're designing an RC or L/R circuit to perform a precise
timing function. To calculate this, we need to modify our
"Universal time constant formula." The original formula
looks like this:
However, we want to solve for time, not the
amount of change. To do this, we algebraically manipulate
the formula so that time is all by itself on one side of the
equal sign, with all the rest on the other side:
The ln designation just to the right
of the time constant term is the natural logarithm
function: the exact reverse of taking the power of e.
In fact, the two functions (powers of e and natural
logarithms) can be related as such:
If ex = a, then ln a = x.
If ex = a, then the natural
logarithm of a will give you x: the power that e must
be was raised to in order to produce a.
Let's see how this all works on a real
example circuit. Taking the same resistor-capacitor circuit
from the beginning of the chapter, we can work "backwards"
from previously determined values of voltage to find how
long it took to get there.
The time constant is still the same amount:
1 second (10 kΩ times 100 �F), and the starting/final values
remain unchanged as well (EC = 0 volts starting
and 15 volts final). According to our chart at the beginning
of the chapter, the capacitor would be charged to 12.970
volts at the end of 2 seconds. Let's plug 12.970 volts in as
the "Change" for our new formula and see if we arrive at an
answer of 2 seconds:
Indeed, we end up with a value of 2 seconds
for the time it takes to go from 0 to 12.970 volts across
the capacitor. This variation of the universal time constant
formula will work for all capacitive and inductive circuits,
both "charging" and "discharging," provided the proper
values of time constant, Start, Final, and Change are
properly determined beforehand. Remember, the most important
step in solving these problems is the initial set-up. After
that, it's just a lot of button-pushing on your calculator!
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REVIEW:
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To determine the time it takes for an RC
or L/R circuit to reach a certain value of voltage or
current, you'll have to modify the universal time constant
formula to solve for time instead of change.
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The mathematical function for reversing an
exponent of "e" is the natural logarithm (ln), provided on
any scientific calculator.
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