Time Constant and Energy Stored in Capacitors

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Time Constant and Energy Stored in Capacitors

Capacitors discharge exponentially. That means that their charge falls away in a similar way to radioactive material decay. In radioactivity you have a half-life, in capacitance you have a 'time constant'.

The rate of removal of charge is proportional to the amount of charge remaining.

Time Constant and Energy diagram

As time steps forward in equal intervals, T (called the time constant), the charge drops by the same proportion each time. It turns out that each for interval, T, the charge or current drops to about 0.37 (37%) of its initial value. (Note: For the mathematicians amongst you, this number can be calculated using 1/e, where e is the exponential constant with a value of 2.718.)

We can calculate the time constant, T using the equation:

T = RC

Where:

T = time constant

R = resistance in the circuit (Ω)

C = capacitance of the circuit (F)

So the factor that governs how quickly the charge drops is a combination of the capacitance of the capacitor and the resistance it is discharging through.

In practice it takes 0.69 x RC (ln2 x RC) for the charge to be half its original value. In this time the discharge current also drops to half its original value too.

To calculate the charge left, Q, on a capacitor after time, t, you need to use the equation:

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Where:

Q0 = initial charge on the capacitor

Q = charge on the capacitor at any time

t = time

RC = time constant

Likewise the current or voltage at any time can be found using:

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As all of these relationships are exponential, natural log graphs can be drawn to obtain values for the time constant. For instance:

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(Remeber for y = mx + c

m gives the gradient of the graph

c is the intercept on the y axis when x = 0)

Time Constant and Energy diagram

The potential difference across the plates of a capacitor is directly proportional to the charge stored on the plates. This gives a straight line through the origin on a voltage-charge graph. The area under this graph gives the energy stored in a capacitor.

Time Constant and Energy diagram

As the area under the graph is a triangle,

area = ½ base x height.

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Note: the energy used by the cell to charge the capacitor, W = QV, but the energy stored on the capacitor = 1/2 QV. So half the energy is lost in the circuit as heat energy as the capacitor is changed.

As capacitors are able to store energy, they can be used in back-up systems in electrical devices, such as computers.

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