Capacitance of a Capacitor

Capacitors are short term charge-stores, a bit like an electrical spring.  They are used widely in electronic circuits.  It consists of two metal plates separated by a layer of insulating material called a dielectric.  The symbol for a capacitor is shown below:

There are two types of capacitor, electrolytic and non-electrolytic.  We won’t worry at the moment what these terms mean, other than to say:

• Electrolytic capacitors hold much more charge;

• Electrolytic capacitors have to be connected with the correct polarity, otherwise they can explode.

If we pump electrons onto the negative plate, electrons are repelled from the negative plate.  Since positives do not move, a positive charge is induced.  The higher the potential difference, the more charge is crowded onto the negative plate and the more electrons repelled from the positive plate.  Therefore charge is stored.  The plates have a certain capacitance.

Capacitance is defined as:

The charge required to cause unit potential difference in a conductor.

Capacitance is measured in units called farads (F) of which the definition is:

1 Farad is the capacitance of a conductor, which has potential difference of 1 volt when it carries a charge of 1 coulomb.

So we can write from this definition:

            Capacitance (F) = Charge (C)

                                       Voltage (V)

In code, this is written:

[Q - charge in coulombs (C); C – capacitance in farads (F); V - potential difference in volts (V)]  

A 1 farad capacitor is actually a very big capacitor indeed so instead we use microfarads (mF) where 1 mF = 1 × 10^-6 F.  Smaller capacitors are measured in nanofarads (nF), 1 × 10^-9 F, or picofarads (pF), 1 × 10^-12 F.  A working voltage is also given.  If the capacitor exceeds this voltage, the insulating layer will break down and the component shorts out.  The working voltage can be as low as 16 volts, or as high as 1000 V.  

The voltage rises as we charge up a capacitor, and falls as the capacitor discharges.  The current falls from a high value as the capacitor charges up, and falls as it discharges.

If we connect a capacitor in series with a bulb:

• If connected to a d.c. circuit, the bulb flashes, then goes out.

• In an a.c. circuit, the bulb remains on.

We can say that a capacitor blocks d.c., but allows a.c. to flow.

Measuring Capacitance

This circuit can be used to measure the value of a capacitor:

• The reed switch is operated from a 400 Hz supply.

•  It operates on the forward half cycle, to charge up the capacitor.

• No current flows on the reverse half cycle so the reed switch flies back to discharge the capacitor.

We can use I = Q/t to work out the charge going onto the plates.  We also know that f = 1/t, so we can combine the two relationships to give I = Qf, Þ Q = I/f

Since C = Q/V, we can now write C = I/fV

Energy in a Capacitor

When we charge up a capacitor, we make a certain amount of charge move through a certain voltage.  We are doing a job of work on the charge to build up the electric field in the capacitor.  Thus we can get the capacitor to do a job of useful work.

  We know that:

1.      Energy = charge × voltage

2.      Q = CV.

This second relationship tells us that the charge – voltage graph is a straight line:

The capacitor is charged with charge Q to a voltage V.  Suppose we discharged the capacitor by a tiny amount of charge, dQ.  The resulting tiny energy loss (dW) can be worked out from the first equation:

dW = V × dQ           

This is the same as the area of the pink rectangle on the graph.

If we discharge the capacitor completely, we can see that:

Energy loss = area of all the little rectangles

 = area of triangle below the graph

 = ½ QV

  By substitution of Q = CV, we can go on to write:  

E = ½ CV²

Graphical Representation and Quantitative Treatment of Capacitor Discharge

Many electronic circuits use the charge and discharge of a capacitor.  If we discharge a capacitor, we find that the charge decreases by the same fraction for each time interval.  If it takes time t for the charge to decay to 50 % of its original level, we find that the charge after another t seconds is 25 % of the original (50 % of 50 %).  This time interval is called the half-life of the decay.  The decay curve against time is called an exponential decay.

The voltage, current, and charge all decay exponentially during the capacitor discharge.

We can plot a graph using a circuit like this:

We can note the voltage and current at time intervals and plot the data, which gives us the exponential graph shown on the next page.  We should note the following about the graph:

• Its shape is unaffected by the voltage.

• The half life of the decay is independent of the voltage.

• The current  follows exactly the same pattern as I = V/R.

• The charge is represented by the voltage, as Q = CV.

The graph is asymptotic, i.e. in theory the capacitor does not completely discharge.  In practice, it does.

The graph is described by the relationship:

Q = Q₀ e ^–t/RC

[Q – charge (C); Q0 – charge at the start; e – exponential number (2.718…); t – time (s); C – capacitance (F); R – resistance (W).]

For voltage and current, the equation becomes:

• V= V₀ e ^–t/RC

• I = I₀ e ^–t/RC

The product RC (capacitance × resistance) which we see in the formula is called the time constant.  The units for the time constant are seconds.  We can go back to base units to show that ohms × farads are seconds.  So if we discharge the capacitor for RC seconds, we can easily find out the fraction of charge left:

            V= V₀ e ^–RC/RC = V₀ e ^–1 = 0.37 V₀

So after RC seconds the voltage is 37 % of the original.  This is used widely by electronic engineers.  To increase the time taken for a discharge we can:

• Increase the resistance.

• Increase the capacitance.

We can link the half-life to the capacitance.  At the half life:

• Q = Q₀/2

• t = t_1/2

Q₀/2 = Q₀ e ^{– t}_1/2^/RC

Þ ½ = e ^{– t}_1/2^/RC

Þ 2^-1 = e ^{– t}_1/2^/RC

Þ e ^{+ t}_1/2^/RC = 2

Þ log_e (2) = t_1/2/RC  [In text books you may see the natural logarithm written as ‘ln’]

Þ t_1/2  = log_e (2) × RC = 0.693 × RC

The half-life is 69 % of the time constant.

Worked Example