Capacitors

Capacitors are electronic components which absorb and release electrical energy, they operate very differently to resistors which simply dissipate energy in the form of heat primarily.

They are a fundamental building block for the vast majority of electronic circuits with many applications, almost any electronic device you take apart will be full of them!

Capacitor Operation:

Structurally capacitors are simple components, they consist of two parallel plates separated by a small distance. Between the plates is an insulative material such as plastic or ceramic or even air and are commonly referred to as the dielectric material of the capacitor. Yes, this means any two conductors separated by a distance have capacitance (air is the inductive material)! And yes you are certainly allowed to have your mind explode at this point.

When a power source is connected to a capacitor electrons flow onto one of the plates from the power source and electrons leave the other plate to return to the source resulting in the situation shown below:

With time you can see electrons spreading out over the left plate due to the electric field from the power source and electrons leaving the right plate to return to the power source. This increase in electrons on one plate and reduction on the other side creates a surplus of electrons on the left plate (negative charge) and a surplus of protons on the other plate due to all the electrons leaving the plate leaving the protons within their atoms in the right plate (positive charge).

The net result is that the capacitor creates an electric field between the plates which points in the opposite direction to the power source, i.e. the electric field within the capacitor opposes the electric field of the power source and due to this over time less and less current flows due to a lower net electric field towards the plates which continues to reduce in time gradually ending in zero current flow.

This is a key difference to the humble resistor, a capacitor absorbs the energy provided by the voltage source and stores it in an internal electric field which acts to oppose the power source. This can be thought of as capacitor action.

Another way to think of it, is that it's actually quite similar to a Van Der Graph Generator where the electrons are trying to spread across the surface of the sphere to minimize the electric field within the sphere where in the case of a capacitive circuit the electrons continue to spread out over the surface of the plate until the electrons no longer experience an electric force (the net electric fields cancel out).

We can say that the capacitor is becoming charged as electrons enter one plate and leave the other.

Capacitance:

The property of capacitance denotes a capacitor's ability to store charge, a capacitor with more capacitance allows more charge to be stored across its plates for an applied voltage.

A capacitor with larger plates allows more electrons to spread across the surface as the surface area is larger.

When the plates are brought closer together there will be more attraction between the electrons on the left plate and the protons on the right plate causing more electrons to accumulate which increases the capacitance.

Capacitance also depends on the dielectric material. Ceramic, mica, glass, plastics are great dielectric materials.

We can combine these observations into a mathematical relationship relating capacitance to geometry and the dielectric as shown below:

Where C is the capacitance in F, A is the area of the plates in m^2, d is the distance between the plates in m, ε is the permittivity of the dielectric material,  εr is the relative permittivity and ε0 (epsilon naught) is the permittivity of free space, aka electric constant (where ε0 ≈ 8.854×10−12 F⋅m−1)

The self capacitance of a given conductor is given by:

Which agrees with the definition that a capacitor with higher capacitance allows more charge per volt. The unit of capacitance is the Farad (F)

Practical capacitance values range from pF (very small capacitors) to mF (very large) to the extreme case of Super Capacitors! Which have capacitance in F.

Voltage/Current Relationship:

From the formula for self capacitance we can derive the voltage/current relationship for capacitors by recalling that current is equal to the change in charge over time and taking the derivative with respect to time of both sides results in:

Note: The current varies with time and is proportional to the rate of change of voltage.

This reflects capacitor operation in that initially electrons experience the electric field from the power source dispersing over the surface of the wire and plate and slowing down with time due to the opposing electric field from the electrons on the plate with the electric field within the wire approaching zero over time resulting in the current asymptotic to zero and the voltage slowly ramping up to the voltage of the supply.

Energy stored in a capacitor:

We can also quantify the energy (Work) stored by the capacitor within the electric field with the equation below:

Capacitors in Series:

This is where capacitors also differ to resistors in that capacitance reduces in series.

For the simplest case of two capacitors connected together with the same capacitance then each will drop a voltage equal to half the supply voltage. We can think of the effective distance between the left most and right most plates as being larger than the case of a single capacitor which acts to reduce the overall capacitance.

Mathematically the reciprocals are added together (as is the case for parallel resistors):

Capacitors in Parallel:

Capacitance simply adds in parallel.

The reason for this is simply that the effective length of the plates has increased, each capacitor charges to the same voltage of the source hence for the case of two capacitors with the same value we have an overall effective doubling of capacitance.

Capacitor Applications:

There are countless applications for capacitors in electronic circuits! Some of the most commonly encountered applications are:

  • Filter circuits
  • Smoothing of AC power
  • Oscillators
  • AC coupling
  • Decoupling
  • Many more!

We will learn more about each of the above applications throughout this course.

Examples:

1. Determine the capacitance of a capacitor with paper dielectric of relative permittivity 3.85 with a plate distance of 1um and area of 1mm^2:

2. What is the energy stored in a capacitor with a voltage of 5V and capacitance of 1uF?

3. What is the total capacitance for the circuit below?

In the circuit above we have capacitors C1 and C4 in series with C2 in parallel with C3 hence we can calculate the total capacitance with the equation shown below:
Recall that capacitance adds in parallel and so C2||C3 in simply C2+C3 and series parts C1 and C4 adds like parallel resistance so we can rewrite this as:

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