I am going through an online electronics tutorial. They use the following problem to demonstrate the use of the universal time constant chart.

"An RC circuit is to be designed in which a capacitor (C) must charge to 20 percent (0.20) of the maximum charging voltage in 100 microseconds (0.0001 second). Because of other considerations, the resistor (R) must have a value of 20,000 ohms. What value of capacitance is needed?"

The answer using the universal TC chart is RC = 455 micro farads and capacitance is 0.023 micro farads. I thought I would put it through the mathematical formula for determining percentage and I get an entirely different answer. The following working is for RC value:
Vpc = 100(1-e^-t/RC)
RC = -t/ln(1-Vpc/100)
RC = -0.0001/ln(1-0.2/100)
RC ≈ 0.05 secs

 

100 is the wrong thing to plug into that formula.

 

Your right. Its actually 1. Which gives the correct answer of 448 micro seconds.

 

I am going through an online electronics tutorial. They use the following problem to demonstrate the use of the universal time constant chart.

"An RC circuit is to be designed in which a capacitor (C) must charge to 20 percent (0.20) of the maximum charging voltage in 100 microseconds (0.0001 second). Because of other considerations, the resistor (R) must have a value of 20,000 ohms. What value of capacitance is needed?"

The answer using the universal TC chart is RC = 455 micro farads and capacitance is 0.023 micro farads. I thought I would put it through the mathematical formula for determining percentage and I get an entirely different answer. The following working is for RC value:
Vpc = 100(1-e^-t/RC)
RC = -t/ln(1-Vpc/100)
RC = -0.0001/ln(1-0.2/100)
RC ≈ 0.05 secs

Here is where tracking units would have saved the day.

Assuming Vpc is Voltage as a percentage, then the units on that 100 should have been percent (100%). As you noted, 20% = 0.20.

So when you did the math and had 1 - 0.2/100 that should have been 1 - 0.2/100% and, noting that dividing a pure number (0.2) by a percentage (100%) would not yield a pure number, which you need in order to be able to subtract it from 1, would have put up red warning flags.

You have other units issues. You say that RC = 455 microfarads, but RC has units of time, not capacitance.

You also have RC = -0.0001/ln(something). Since ln(anything) is dimensionless, you are claiming that RC is dimensionless. Yet on the next line it magically has units of seconds.

Track your units properly and the answer will have the proper units naturally -- and if it doesn't then you KNOW you have made a mistake.

\(
V_{pc} \; = \; 100% \(1 \; - \; e^{ \( 1 \; - \; \frac{-t}{RC} \)} \)
RC \; = \; \frac{-t}{\ln\( 1 \; - \; \frac{V_{pc}}{100%}\)}
RC \; = \; -\(100 \: \mu s\) {\ln\( 1 \; - \; \frac{20%}{100%}\)}
RC \; = \; 448 \: \mu s
\)

 

RC Time Constant. The time required to charge a capacitor
to 63% (actually 63.2%) of maximum voltage, or to discharge it
to 37% (actually 36.8%) of its final voltage is known as the
TIME CONSTANT (TC) of the circuit. The charge and discharge
curves of a capacitor are shown in figure 36. Note that the
charge curve is like the curve in figure 34, graph 1), on page
53, and the discharge curve like the curve in figure 34, graph B.
FIGURE 36. RC TIME CONSTANT.
The value of the time constant in seconds is equal to the
product of the circuit resistance in Ohms and the circuit.
capacitance in farads. The value of one time constant is
expressed mathematically as t=RC. Some forms of this formula
used in calculating RC time constants are:
t(in seconds)
= R (in Ohms) x C(in farads)
t(in seconds)
= R (in megohms) x C(in microfarads)
t(in microseconds)
= R (in Ohms) x C(in microfarads)
t(in microseconds)
= R (in megohms) x C(in picofarads)

 

You don't need to specify the units in a bajillion different equations.

The time constant for an RC circuit is simply the product of the resistance and the capacitance. Resistance has units of voltage per unit current and capacitance has units of charge per unit voltage. Since current has units of charge per unit time, the product of the two has units of time. The specific units will naturally work themselves out if you just track the units properly.