Circuit Analysis

Publisher Summary

This chapter illustrates Circuit Analysis. Electrons are charged particles. Charged objects are attracted to other charged particles or objects. Charged objects come in two forms—negative and positive. Unlike charges attract, and like charges repel. Electrons are negative and positrons are positive, but while electrons are stable in the universe, positrons encounter an electron almost immediately after production, resulting in mutual annihilation and a pair of 511 keV gamma rays. An electron is very small, and does not have much of a charge, so one needs a more practical unit for defining charge. That practical unit is the coulomb (C). One could say that 1 C of charge had flowed between one point and another, which would be equivalent to saying that approximately 6,240,000,000,000,000,000 electrons had passed, but much handier. Simply being able to say that a large number of electrons had flowed past a given point is not in it very helpful. One might say that a billion cars have traveled down a particular section of motorway since it was built, but if he/she were planning a journey down that motorway, he or she would want to know the flow of cars per hour through that section.

In order to look at the interesting business of designing and building valve amplifiers, we need some knowledge of electronics funmentals. Unfortunately, fundamentals are not terribly interesting, and to cover them fully would consume the entire book. Ruthless pruning is, therefore, necessary to condense what is needed in one chapter.

It is thus with deep sorrow that the author has had to forsaken complex numbers and vectors, whilst the omission of differential calculus is a particularly poignant loss. All that is left is ordinary algebra, and although there are lots of equations, they are timid, miserable creatures and quite defenceless.

If you are comfortable with basic electronic terms and techniques, then please feel free to go directly to Chapter 2, where valves appear.

Mathematical Symbols

Unavoidably, a number of mathematical symbols are used, some of which you may have forgotten, or perhaps not previously met:

a≡b a is totally equivalent to b

a=b a equals b

a≈b a is approximately equal to b

a∝b a is proportional to b

a≠b a is not equal to b

a>b a is greater than b

a<b a is less than b

a≥b a is greater than, or equal to, b

a≤b a is less than, or equal to, b

As with the = and ≠ symbols, the four preceding symbols can have a slash through them to negate their meaning (a ∋ b, a is not less than b).

√a the number which when multiplied by itself is equal to a (square root)

an a multiplied by itself n times. a4=a×a×a×a (a to the power n)

± plus or minus

∞ infinity

° degree, either of temperature (°C), or of an angle (360° in a circle)

parallel, either parallel lines, or an electrical parallel connection

Δ a small change in the associated value, such as ΔVgk.

Electrons and Definitions

Electrons are charged particles. Charged objects are attracted to other charged particles or objects. A practical demonstration of this is to take a balloon, rub it briskly against a jumper and then place the rubbed face against a wall. Let it go. The balloon remains stuck to the wall. This is because we have charged the balloon, and so there is an attractive force between it and the wall. (Although the wall was initially uncharged, placing the balloon on the wall induced a charge.)

Charged objects come in two forms: negative and positive. Unlike charges attract, and like charges repel. Electrons are negative and positrons are positive, but whilst electrons are stable in our universe, positrons encounter an electron almost immediately after production, resulting in mutual annihilation and a pair of 511 keV gamma rays.

If we don’t have ready access to positrons, how can we have a positively charged object? Suppose we had an object that was negatively charged, because it had 2,000 electrons clustered on its surface. If we had another, similar, object that only had 1,000 electrons on its surface, then we would say that the first object was more negatively charged than the second, but as we can’t count how many electrons we have, we might just as easily have said that the second object was more positively charged than the first. It’s just a matter of which way you look at it.

To charge our balloon, we had to do some work and use energy. We had to overcome friction when rubbing the balloon against the woollen jumper. In the process, electrons were moved from one surface to the other. Therefore, one object (the balloon) has acquired an excess of electrons and is negatively charged, whilst the other object (woollen jumper) has lost the same number of electrons and is positively charged.

The balloon would, therefore, stick to the jumper. Or would it? Certainly it will be attracted to the jumper, but what happens when we place the two in contact? The balloon does not stick. This is because the fibres of the jumper were able to touch the whole of the charged area on the balloon, and the electrons were so attracted to the jumper that they moved back onto the jumper, thus neutralising the charge.

At this point, we can discard vague talk of balloons and jumpers because we have just observed electron flow.

An electron is very small, and doesn’t have much of a charge, so we need a more practical unit for defining charge. That practical unit is the coulomb (C). We could now say that 1 C of charge had flowed between one point and another, which would be equivalent to saying that approximately 6,240,000,000,000,000,000 electrons had passed, but much handier.

Simply being able to say that a large number of electrons had flowed past a given point is not in itself very helpful. We might say that a billion cars have travelled down a particular section of motorway since it was built, but if you were planning a journey down that motorway, you would want to know the flow of cars per hour through that section.

Similarly in electronics, we are not concerned with the total flow of electrons since the dawn of time, but we do want to know about electron flow at any given instant. Thus, we could define the flow as the number of coulombs of charge that flowed past a point in one second. This is still rather long-winded, and we will abbreviate yet further.

We will call the flow of electrons current, and as the coulomb/second is unwieldy, it will be redefined as a new unit, the ampere (A). Because the ampere is such a useful unit, the definition linking current and charge is usually stated in the following form.

This is still rather unwieldy, so symbols are assigned to the various units: charge has symbol Q, current I and time t.

This is a very useful equation, and we will meet it again when we look at capacitors (which store charge).

Meanwhile, current has been flowing, but why did it flow? If we are going to move electrons from one place to another, we need a force to cause this movement. This force is known as the electro motive force (EMF). Current continues to flow whilst this force is applied, and it flows from a higher potential to a lower potential.

If two points are at the same potential, no current can flow between them. What is important is the potential difference (pd).

A potential difference causes a current to flow between two points. As this is a new property, we need a unit, a symbol and a definition to describe it. We mentioned work being done in charging the balloon, and in its very precise and physical sense, this is how we can define potential difference, but first, we must define work.

This very physical interpretation of work can be understood easily once we realise that it means that one joule of work would be done by moving one kilogramme a distance of one metre in one second. Since charge is directly related to the mass of electrons moved, the physical definition of work can be modified to define the force that causes the movement of charge.

Unsurprisingly, because it causes...