What is the voltage across a capacitor after a time of two constants when a. charging from zero voltage and b. discharging from a fully charged condition?
U₀ is the max charge of capacitor.
Discharging
U=U₀exp(-t/T)=
=U₀exp(-2T/T)=
=U₀exp(-2)=0.135U₀
Charging
U=U₀{1-exp(-t/T)}=
= U₀{1-exp(-2)}= U₀(1-0.135)=0.865U₀
To calculate the voltage across a capacitor after a certain time, we need to consider the charging and discharging behavior of the capacitor.
a. Charging from zero voltage:
When a capacitor is charging from zero voltage, we can use the equation:
Vc = V0 * (1 - e^(-t / RC))
Where:
Vc is the voltage across the capacitor at time t,
V0 is the maximum voltage the capacitor can charge to (usually the applied voltage),
t is the time elapsed, and
RC is the time constant (the product of resistance R and capacitance C).
b. Discharging from a fully charged condition:
When a capacitor is discharging from a fully charged condition, we can use the equation:
Vc = V0 * e^(-t / RC)
Where the terms have the same meaning as in the previous equation.
In both cases, it's important to note that the time constant (RC) is a characteristic of the specific capacitor and the resistance in the circuit.
To calculate the voltage across a capacitor after a certain time, you will need to know the capacitive reactance (Xc), which is given by the formula Xc = 1/(2πfC), where f is the frequency of the signal and C is the capacitance.
a. Charging from zero voltage:
When a capacitor is being charged from zero voltage, it follows an exponential charging curve. The voltage across the capacitor can be determined using the charging equation Vc(t) = Vc(0)(1 - e^(-t/RC)), where Vc(t) is the voltage at time t, Vc(0) is the initial voltage (zero in this case), t is the time, R is the resistance, and C is the capacitance.
b. Discharging from a fully charged condition:
When a capacitor is discharged from a fully charged condition, it also follows an exponential curve. The voltage across the capacitor can be calculated using the discharging equation Vc(t) = Vc(0)e^(-t/RC), where Vc(t) is the voltage at time t, Vc(0) is the initial voltage (fully charged condition in this case), t is the time, R is the resistance, and C is the capacitance.
By substituting the appropriate values for the time, resistance, and capacitance into the respective equations, you can calculate the voltage across the capacitor in each scenario.