# Time taken to charge the capacitor

Which equation can be used to calculate the time taken to charge the capacitor at the given amount of current and voltage at a constant capacitance?

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en.wikipedia.org/wiki/Capacitance –  kenny Nov 20 '11 at 12:06
The problem is overconstrained - the current and voltage can't both be given, as one of them will vary (whichever is less "enforced" ie less approximates an ideal source) during the charging of the cap. –  Chris Stratton Nov 20 '11 at 18:35

If you want a "simple" equation, and it seems that you do, you could start with definition of current.

First, let's start with the farad. It is usually expanded as $F=\frac {As}{V}$.

Now let's write that with symbols for capacitance, current, voltage and time:

$C=\frac {It}{U}$

Since we have constant current and voltage and we need time, we'll divide the equation with current and multiply with voltage so that we can get time.

That gives us $\frac{UC}{I}=t$.

If this is just a school problem, then we have a solution.

In real life things will work differently. As the capacitor charges, the voltage on the capacitor will drop resulting in drop of current and the time will therefore be longer.

Here's an example: Let's assume that at the beginning, the capacitor is discharged. First we have the voltage on the resistor which is $U_r=Ri$. Then we have voltage on the capacitor which is $U_c=\frac{1}{C} \int {i \mbox{ }dt}$.

So we know that $E=Ri+\frac{1}{C} \int {i \mbox{ }dt}$. To solve this, we need to turn it into differential equation.

$(E=Ri+\frac{1}{C} \int {i \mbox{ }dt}) / \frac{d}{dt}$

Since $E$ is constant, it will turn into zero. The integration and differentiation will cancel each-other out and we'll get:

$R\frac{di}{dt}+\frac{i}{C}=0$ Next we divide everything with $R$ and get
$\frac{di}{dt}+\frac{i}{RC}=0$

After that we move the $\frac{1}{RC}i$ to the other side and multiply everything with $dt$ and divide everything with $i$ and we get:

$\frac{di}{i}=-\frac{1}{RC}dt$

Now we integrate everything and get $\int {\frac{di}{i}} = -\int {\frac{1}{RC}dt}$ As a result, we get:

$\ln{i}=-\frac{t}{RC}+C_1$

Now to get rid of the logarithm, we raise everything to $e$
$i=C_1 e^{-\frac{t}{RC}}$

Now we have the general solution and we need to determine the constants. So first we look at what's happening when the time is equal to zero:

$i=C_1 e^{-\frac{0}{RC}} = C_1$.

We also know that the initial current is $i_{(0)}=\frac{E}{R}$. From that we can determine that $C_1=\frac{E}{R}$.

The complete equation for the current is:
$i_{(t)}=\frac{E}{R} e^{-\frac{t}{RC}}$

This is a classical capacitor charging equation and it is available on many sources on the Internet.

The $RC$ is also called the time constant, so $\tau=RC$. It is usually considered that five time constants are enough to charge a capacitor.

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A possibly more descriptive version of your last sentence: after five time constants, the capacitor is charged to 1-e^-5 = .993 of fully charged (at the given voltage), which is usually considered charged. –  Jefromi Nov 20 '11 at 18:18

The basic equations are

time constant = R C = 1/(2 Pi fc)
i = dq/dt = C du/dt


Hope it helps!

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For this circuit:

When the everything starts out at 0 V and then the input is changed to Vin at time t=0:

$V_{out} = V_{in}(1 - e^{-RCt})$

When R is in Ohms and C in Farads, then t is in seconds.

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There are TWO cases, as Chris indicated.

Case 1 is where you charge a capacitor from a constant voltage source with resistance and capacitance known. (Resistance is any circuit resistance plus capacitor internal resistance plus any added resistance. This is the case covered by eg Andreja Ko & Olin.
You get

$V_{out} = V_{in}(1 - e^{-RCt})$

{stolen from Olin}, which you can rearrange for t.

Case 2 is charging at constant current.
More current charges quicker.
More capacitance takes longer.
Charging to a higher voltage takes longer

SO:

• t = V x C / I
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The capacitor (C) in the circuit diagram is being charged from a supply voltage (Vs) with the current passing through a resistor (R).

The voltage across the capacitor (Vc) is initially zero but it increases as the capacitor charges.
The capacitor is fully charged when Vc = Vs.
The charging current (I) is determined by the voltage across the resistor (Vs - Vc):

 Charging current,  I = (Vs - Vc) / R   (note that Vc is increasing)


At first Vc = 0V so the initial current,

 Io = Vs / R


Vc increases as soon as charge (Q) starts to build up (Vc = Q/C),
This reduces the voltage across the resistor and therefore reduces the charging current.
This means that the rate of charging becomes progressively slower.

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