The time dependence of voltage across a discharging capacitor is given by the exponential decay. The voltage on the capacitor at time t= 0 sec is 24 volt. If a 220 +/- 22uF capacitor and 1 Mega ohm (5% tolerance) resistor are used in the RC circuit, what would be the value of the time constant (T=RC) and the associated error? What would be the voltage V after t = 100 sec? Propagate the errors and find the percent error in the voltage V.
C = q/V
q = C V
i = -dq/dt = -C dV/dt
but V = i r
so
i = -C r di/dt
i + C r di/dt = 0
V/r + C r (1/r) dV/dt) = 0
V + r C dV/dt = 0
dV/dt = -1/rC V
dV/V = -1/rC dt
ln V = - (1/rC) t
V = Vi e^-(1/RC)t
let T = RC
then
V = Vi e^-t/T
if RC = 10^6 * 220*10^-6 = 220 = T
Vi = 24
V = 24 e^-t/220
if t = 100
V = 24 e^-(100/220) = 24 e^-.454
= 24*.635 = 15.2
You can do the change for new C and R and get error in V
A. T = RC = 1*10^6 * 220*10^-6 = 220 s.
tolerance = 10 + 5 = +- 15 %.
T = 0.85*220 = 187 s, min.
T = 1.15*220 = 253 s, max.
B. t/T = 100/220 = 0.45.
V = 24/e^0.45 = 15.2 volts.
C. t/T = 100/253 = 0.395, min.
V = 24/e^0.395 = 16.2 volts, max.
t/T = 100/187 = 0.535, max.
V = 24/e^0.535 =
To calculate the time constant (T=RC) and associated error, we can use the given values for the capacitor and resistor.
1. Time Constant (T=RC):
The time constant is the product of the resistance (R) and the capacitance (C) in an RC circuit. So, we can calculate it as follows:
T = R * C
Given:
Resistance, R = 1 Mega ohm (5% tolerance) = 1,000,000 ohm
Capacitance, C = 220 +/- 22uF
The time constant is given by:
T = (1,000,000 ohm) * (220 * 10^-6 F)
Calculating the value:
T = 0.22 seconds
2. Associated Error:
To determine the error in the time constant, we can calculate the maximum and minimum values.
For the capacitance:
Max C = 220 + (0.05 * 220) uF = 231 uF
Min C = 220 - (0.05 * 220) uF = 209 uF
Calculating the maximum time constant:
Max T = (1,000,000 ohm) * (231 * 10^-6 F) = 0.231 seconds
Calculating the minimum time constant:
Min T = (1,000,000 ohm) * (209 * 10^-6 F) = 0.209 seconds
Therefore, the associated error in the time constant is given by:
Error = Max T - Min T
Error = 0.231 - 0.209
Error = 0.022 seconds
3. Voltage at t = 100 sec:
The voltage on the capacitor at time t = 0 sec is given as 24 volts. The voltage across a discharging capacitor follows exponential decay, given by the equation:
V(t) = V0 * e^(-t/T)
Plugging in the values:
V(100) = 24 * e^(-100/0.22)
Calculating the voltage at t = 100 sec:
V(100) ≈ 1.459 volts
4. Percent Error:
To propagate the errors and find the percent error in the voltage V, we need to calculate the maximum and minimum values.
Calculating the maximum voltage:
Max V = 24 * e^(-100/(0.209 + 0.022))
Calculating the minimum voltage:
Min V = 24 * e^(-100/(0.231 - 0.022))
The percent error is given by:
Percent Error = (Max V - Min V) / Max V * 100
Calculating the percent error:
Percent Error = (Max V - Min V) / Max V * 100
Please note that for the precise calculation, the specific values of the maximum and minimum voltage need to be substituted into the above equations.