# Time constant and capacitors

• Hannah7h
In summary, the rate at which a capacitor charges and discharges is determined by resistance and capacitance in a circuit. The time constant, equal to RC, is the time it takes for the charge on a discharging capacitor to decrease to 37% of the initial charge (Qo). This is also equivalent to the time it takes for the charge on a charging capacitor to increase by 63% of Qo. This relationship is derived from solving the differential equation for an RC circuit, which results in exponential expressions for charge, current, and voltage. The constant 'e-1' in this solution is where the 63% value comes from. In summary, the time constant is the time it takes for the charge to reach

#### Hannah7h

So the rate at which a capacitor charges and discharges is dependent on resistance in a circuit and the magnitude of capacitance of the capacitor? So the time constant is equal to RC. So using this equation where Q=Qoe-t/RC ,time constant is the time taken (when the capacitor is discharging) for charge on a capacitor (Q) to decrease to 37% of Qo ( i.e. charge on the capacitor when it is is fully charged). But I've been reading around and what I don't get is how the time constant is also equal to the time taken for the charge (Q) on a charging capacitor to increase by 63% of Qo . If this makes any sense, would be good if someone could maybe explain it mathematically as well i.e. how 'e' is involved?

Hannah7h said:
i.e. how 'e' is involved?
Have you studied calculus?

cnh1995 said:
Have you studied calculus?

Um haven't studied it in much detail, but I could give it a go

Hannah7h said:
Um haven't studied it in much detail, but I could give it a go
Well, KVL equation for an RC circuit is a differential equation. Solving that differential equation, you get exponential expressions for charge, current and voltage.

cnh1995 said:
Well, KVL equation for an RC circuit is a differential equation. Solving that differential equation, you get exponential expressions for charge, current and voltage.

Ok so... how does this relate to the 63% of Qo?

Hannah7h said:
time constant is the time taken (when the capacitor is discharging) for charge on a capacitor (Q) to decrease to 37% of Qo
Or in other words, time taken to lose 63% of Q0.
Hannah7h said:
on a charging capacitor to increase by 63% of Qo
You can see that the time taken to gain a charge of 0.63Qo (while charging) is equal to the time taken to lose the same charge of 0.63Q0(while discharging). It's obvious, isn't it?

Hannah7h said:
Ok so... how does this relate to the 63% of Qo?
If you are asking where 63% comes from, it comes from the constant 'e-1' in the solution to the differential equation of the RC circuit.

the time constant is the time taken to get to within 37% of the final value.
In decay the final value is 0%!
In growth the final value is 100%

cnh1995 said:
If you are asking where 63% comes from, it comes from the constant 'e-1' in the solution to the differential equation of the RC circuit.

Ok that was pretty obvious now I look at it, thank you for explaining it otherwise I probably wouldn't have got there lol

lychette said:
the time constant is the time taken to get to within 37% of the final value.
In decay the final value is 0%!
In growth the final value is 100%

Yep this makes sense, thanks!

## 1. What is a time constant?

A time constant is a measure of how quickly a capacitor charges or discharges. It is equal to the product of the capacitance and the resistance in an electrical circuit.

## 2. How is the time constant calculated?

The time constant (τ) is calculated by dividing the capacitance (C) by the resistance (R), or τ = C/R. It is measured in seconds.

## 3. What is the relationship between time constant and charging/discharging of a capacitor?

The time constant determines the rate at which a capacitor charges or discharges. A larger time constant means a slower charging or discharging process, while a smaller time constant results in a faster process.

## 4. How does the time constant affect the behavior of a capacitor in a circuit?

The time constant affects the behavior of a capacitor by determining the amount of time it takes for the capacitor to reach a certain level of charge or discharge. It also determines the shape of the charging/discharging curve, with a larger time constant resulting in a more gradual curve and a smaller time constant resulting in a steeper curve.

## 5. How can the time constant be used in circuit analysis?

The time constant can be used in circuit analysis to predict the behavior of a capacitor in a circuit over time. It can also be used to calculate the voltage or current at different points in the charging/discharging process.