This entry is part 19 of 24 in the series Electronics DC Revision

Capacitance

What is Capacitance?

Before I answer that question, let me start by clearing up something which has the potential to be very confusing… Capacitance has been given the designation C, which, as you will remember from the article on electricity, is also the symbol used for the unit of coulombs. I have no idea why, I can only think that the naming committee absolutely hated electronics students! To further confuse the issue, the actual unit of capacitance is the farad, having the symbol F.

That naming conflict shouldn’t be a problem in general use, but may be confusing when I talk about how the basic calculations are derived. Where there is the potential for misinterpretation I will try to remember to qualify the symbol/designation with it’s full name.

OK. With that out of the way. What is capacitance?

Capacitance (designation C) is the ability to store an electric charge (Q), which is a measure of coulombs (C) per volt (V) in the unit of farads (symbol F).

\({Capacitance} = \frac{Charge}{Volt} \\\) \(C = \frac{Q}{V} \\\)
Rearranging the equation we get:
\(Q = V.C \\\)
This is important for understanding why stored charge capacity is higher for the same capacitor experiencing a higher voltage.

Capacitors

As you might expect. A capacitor is a device which exhibits the property of capacitance. It works by storing charge on two differentially charged plates separated by a dielectric layer.

What is a dielectric?

A dielectric is an insulator in the sense that when placed in an electric field electric charges do not flow as they do in a conductor because it has no free electrons. However, it exhibits the property of being polarised by the electric field.

The orbit of the cloud of electrons around the nucleus is distorted slightly (elongated if you will) by the electric field. Now instead of there being an evenly neutral charge balance all around the atom, the side where the bulk of the cloud exists will be slightly more negatively charged than the side where the nucleus (and hence protons) are closest to the edge (which will be positively charged). Thus the material becomes polarised by the electric field.

All insulators are dielectric to some degree. The ease with which an insulator can be polarised is given by it’s coefficient of Relative Permeability (µ_r). This constant is a multiplier that is used to scale the constant for the Permittivity of Free Space (µ₀) to find it’s absolute electrostatic permeability. Some materials such as ceramics can have a relative permeability (µ₀) value as high as 7500, while others like mica have a µ₀ value of only 5 (others much less than that).

So although all insulators are dielectrics, the term is usually only used when it’s electrostatic permeability property is of key importance.

Capacitor Construction

The circuit schematic symbol for capacitors gives a big clue as to their construction:

Two parallel lines separated by a gap. A real world capacitor is simply two conductive plates separated by a gap that is usually filled with some sort of dielectric material (though air gap’d capacitors are a real thing too).

The value of capacitance is determined by:

The area of the plates.
The distance between the plates.
The permittivity of the dielectric between the plates.

If you want to get technical then:

\(C = \frac{A.\mu _r. \mu _0}{d} \\\)
Where:

C = Capacitance in farads (F)

A = Area of the plate in meters (m²)

µ_r = Relative Permeability of the dielectric (a unit less number)

µ₀ = Permittivity of free space in farads per meter (^F/_m)

d = Distance between the plates in meters (m)

Capacitance is everywhere and affects almost every electronic component and circuit. It even affects electrical wires run parallel to one another!

How Capacitors Work

We begin with a fully discharged capacitor in a circuit with a battery, but with the switch open so no current is flowing.

The net charge on each plate is neutral. Each positive charge is balanced by a negative charge.

Next the switch is thrown resulting in negative charge flowing in to the bottom plate to fill any available free slots. Meanwhile free electrons from the top plate flow towards the positive terminal of the battery.

Resulting in a positively charged top plate and a negatively charged bottom plate.

Meanwhile the charges generate an electric field normal to the plate that they are on and saturate the neutrally charged dielectric between the plates. Note that no electrons cross the boundary between the plates and the dielectric, only the electric field.

The strength of this field is proportional to:

Amount of the charge on the plates. The more charge, the stronger the field;
Distance between the plates. The larger the gap, the weaker the field;
Permeability of the dielectric used. The easier it allows a field, the stronger the field can be.

Although no current can flow in the dielectric. The orbits of it’s electron clouds are distorted by the attraction of the positive and negatively charged plates.

This results in the dielectric itself becoming polarised in the opposite polarity to the adjacent plate.

The now polarised dielectric generates it’s own internal electric field in opposition to the field generated by the plates. Again note that no electrons cross the boundary between dielectric and plate.

Despite the fact that no electrons have ever crossed the boundary between the plates and the dielectric. The net effect is that the opposing charge forces across the boundary neutralise part of the charge in the plates.

As a result the plates can take on more charge from the external circuit than would otherwise have been possible.

Once again it should be emphasised that at no time in any part of these proceedings did any electrons cross over between either plate and the dielectric. Electrons only came in from the external circuit, and those leaving were solely sourced from the plate itself.

After some predetermined time period, that we will talk about later, there is no spare capacity left to take on, or relinquish, any further electrons and current flow ceases.

The switch can now be opened.

However, the charge remains upon the capacitor plates because there is no completed circuit for it to drain away.

In theory this charge would remain for ever. However in practice all real world capacitors have a very small internal leakage current which results in a very slow self discharge.

Charging & Discharging a Capacitor

What is not immediately obvious from the diagrams and descriptions above is that the rate of charge and discharge is not linear. It is in fact exponential. As the charge on the plates of the capacitor tend towards full, rate of flow decreases. Similarly as the state of charge tends towards zero the discharge rate slows down.

The reason for this is that voltage across the capacitor is directly related to the amount of charge stored. Remember from earlier that C = ^Q/_V or put another way V = ^Q/_C so as the level of charge drops, so too does the voltage. Conversely as the level of charge increases, so too does the voltage.

How does this translate to charging and discharging rates (assuming charge/discharge into a constant resistance load)?

While Charging

Let V_S be the supply voltage and V_L be the voltage across the load and let V_C be the voltage across the capacitor, then from Ohms law we can substitute I.R for V_L and substituting ^Q/_C for V, we get:
\(V_S = V_L + V_C = IR + \frac{Q}{C} \\\)

As charging progresses, Q increases and so V_C decreases, meaning that V_R increases and hence IR decreases.

While Discharging

\(V_L = V_C = \frac{Q}{C} \\\) \(I = \frac{V_L}{R_L} \)

As the capacitor discharges, Q decreases and so does V_C, which means that so does V_L and hence the current also decreases.

Basic Circuit

The basic circuit to charge, or discharge, a capacitor is just a resistor connected in series.

The time constant (τ [lower case tau]) for this circuit is just R.C

Note that this is not the same thing as the time to fully charge or discharge the capacitor.

The time constant (τ) is used in a few scenarios, one of which is in calculating the charge / discharge times of a capacitor, and another is in calculating the cutoff frequency (f_c).

\(\tau = RC = \frac{1}{2\pi f_c} \\\)
Rearranging this we get:
\(f_c = \frac{1}{2\pi RC} = \frac{1}{2\pi \tau} \)

With regards to charging the capacitor. The time constant is the time to charge using a constant voltage from completely discharged, up to 63.2% of the charging voltage. Similarly to discharge to 36.8% of some initial voltage towards zero. The values of 63.2% and 36.8% are approximations of 1 – e^-1 and e^-1 respectively (where e is Euler’s constant and is the base for natural logarithms) since the charge / discharge function is exponential.

To demonstrate this, the SPICE analysis below was performed using a 100R resistor in series with a 1µF capacitor. This gives an RC time constant of 100 x 1 x 10^-6 = 0.1mS

The red line is showing charging voltage, and the blue line shows charging current.

Visual inspection of the graph shows that at 0.1mS the voltage is around 3V (red line) which is ³/₅ * 100 = 60% which without using a cursor supports the 63.2% prediction. Similarly by visual inspection at 0.1mS the blue line is somewhere near to 2V which is ²/₅ * 100 = 40% which again is close enough to the predicted 36.8%.