Electronics.
"The study of the design of circuits using transistors and microchips, and with the behaviour and movement of electrons in a semiconductor, conductor, vacuum, or gas."
Table of Contents:
1. Capacitors and Capacitance.
2. Inductors.
Foreword.
This chapter originally was placed in the end of the electromagnetism chapter, but as i wrote it I soon realised it deserved its own chapter. It is important to remember that the Electronics chapter is nothing more than an annex of Electromagnetism - a real-world, macroscopic application of the fundamental laws set out in the Electromagnetism chapter.
The electronics chapter primarily consists of various electronic components with interesting mathmatical and physical properties.
1. Capacitors and Capacitance.
Capacitors are a type of electrical component that can be connected in circuits. Their purpose is to store electrical energy by producing a potential difference between two very close plates. Its electric symbol is identical to a cell but with equal length lines. They also sometimes use curved lines rather than straight ones as shown - this typically represents a cylindrical capacitor but the fun of those is contained in a later section.
All capacitors have a unique and inherent value of capacitance. Capacitance is a measure of how much electrical energy they can store and an understanding can be easily inferred from the equation that defines it:
C = Q/V.
This applies when a conductor (materials which allow easy flow of electric current through them) is isolated from other bodies and given a charge Q, that causes the potential difference, with the other plate, to be raised to V.
By rewriting the equation as:
Q = CV
It becomes clear that the enclosed charge, Q, and the potential difference, V, are in direct proportion. The constant of proportionality is C.
From this we can produce a neater definition of capacitance:
"The amount of charge a conductor will hold at a certain potential."
- ELECTRICITY AND MAGNETISM 3rd Edition by W.J. Duffin.
The unit of capacitance is the farad (F), after the physicist Michael Faraday who developed a method to measuring capacitance. Like the Tesla, the farad is a huge unit of measurement so for practical uses, nano (10^-9) and even pico (10^-12) farads are used to describe the capacitance of a capacitor.
Charging and Discharging.
Capacitors are charged when they are connected to a power source – either a battery or single cell. The process of charging capacitors with electrical energy is very simple – the potential difference across the terminals of the battery causes a current to flow around the circuit.
However, the flow of electrons into one of the conducting plates makes it negative and once it is negative, the other plate becomes positively charged by a process called electromagnetic induction.
When the plate is neutral it is very easy for current to flow into it but as it becomes more and more negative, it becomes increasingly more difficult for electrons to flow onto the plate because opposite charges repel.
For a constant current, the rate of charging for a capacitor is shown by the graph on the left.
As shown in the graph, when the capacitor charges, the potential difference between the two plates tends toward the voltage from the battery - the maximum charge that it can hold is the voltage from the battery but squeezing in the last few electrons takes huge energy
A capacitor
will discharge if its terminals become connected and if a wire is run between
the two plates, a current will flow until the two plates have
neutral charge. By using inverse logic, we can see that the reverse process occurs when discharging.
Because like charges repel, it is easiest to get a current to flow when the plates have lots of charge (when there is a high potential difference between the plates) and hardest at the end of the discharge process, when the plates have almost identical charge. This exponential effect is shown in the graph above right.
It is however
possible to force a capacitor to charge and discharge linearly – something
which is vital for certain circuits such as voltage-controlled oscillators
(VCOs). This can be done by adding a variable resistor to the circuit
connecting the two plates of the capacitor. When charging electrical energy
onto the plates, the resistor can be initially set to have very high resistance
and as the voltage between the plates rises, the resistance falls to offset the
difficulty experienced when adding charge to the plates. The opposite effect
works when discharging – the resistance is initially set very high and falls as
voltage falls.
Ideal Capacitors.
Ideal electrical components are the practical physicists worst enemy. They are all entirely theoretical but can be used to either give good approximations (such as ideal voltmeters having infinite resistance) or to model perfect scenarios. Before an understanding of ideal capacitors can be made however, an understanding of Gauss' law of electric fields must be achieved. Luckily, you're spoilt for choice. The differential form of this equation can be found in the Maxwell's Equation section of the Electromagnetism chapter whilst reading the Maxwell's Equation section in the Maths in Physics chapter will prepare you for the integral form.
The lazy reader (seriously? I worked hard on those sections!) can find an overview of Guass' Law here:
"The electric flux (divergence across the surface area of an object) as an electric field passes through any closed surface is proportional to the charge contained within the object. This electric flux depends on both the magnitude and sign of the enclosed charge and the constant of proportionality is the permittivity of free space."
Returning to ideal capacitors, we can formulate a a very simple definition that comes in two parts:
- "All the flux of E (electric flux) leaving one of the conductors (the plates of the capacitor) must end up on the other conductor." - ELECTRICITY AND MAGNETISM 3rd Edition' by W.J. Duffin:
This means when the conductor is storing electrical energy and it discharges (by allowing the charges on the two conducting surfaces to equal) no other electrical conductor influences this equalling of the charges. In real life this is practically unachievable, any other components in the circuit leach energy from the system so the current flow is never perfect.
Gauss' law shows that the only way to create or destroy electric flux entering or leaving an object is if the object has some internal charge – known as enclosed charge. The electric flux strength produced is directly proportional to the size of this enclosed charge.
- "The charges on the two conductors [the plates of the capacitor] must be equal and opposite– as though one plate was charged from the other." - ELECTRICITY AND MAGNETISM 3rd Edition' by W.J. Duffin:
This second definition is also not achievable in experiments as any external electric fields, however slight, will affect the distribution of charges on the two conducting surfaces.
A phenomena called the edge effect also limits this assumption to the realms of theoretical physics - as discussed in more depth later.
Calculating and Measuring Capacitance.
In theory, you could experimentally determine the value of C for any capacitor by imagining transferring a charge from one plate to another and then calculating the resultant potential difference. To do this we assume that we are working with an ideal capacitor – one composed of two conductors that hold opposite and equal charge when the conductor stores energy and one which discharges perfectly such that all of the electric flux leaving one conducting surface ends on the other.
If we take these assumptions, it is possible to construct an equation to find capacitance, but before I could understand that I had to research Gauss' law of electric fields in a slightly different form:
2. Inductors.
Inductors are very similar to conductors - with one small difference. Both store energy from electrical circuits in the form of potential difference across a field. However, conductors store it in the form of an electric field whilst inductors use magnetic fields to store energy.
Practical Set Up.
The internal workings of inductors are far simpler than conductors - they are composed of a single, tightly-wound coil wrapped around a ferromagnetic core (although some inductors don't have cores at all).
The easiest way to understand inductors is by seeing them in action in a circuit:
Picture a parallel circuit as shown left, with a battery connected in parallel with an inductor and a component, like a lamp. When the current first flows, the inductor has incredibly high resistance (more on that later) and so the electrons flow round the path of least resistance - which is the branch connecting the lamp initially.
Externally, as we watch the circuit, this causes the lamp to turn on with a steady, continued power output - as seen in a constant light intensity. It is important to note here that the majority of the current flowing out the battery ends up going through the branch with the lamp but some still flows through the inductor under very high resistance.
However, one key property of inductors is that as the current flows through the inductor, the resistance of the inductor falls (very elegant explanation to come!).