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Table of Contents Cover image Front matter Copyright Preface Dedication Acknowledgements Chapter 1 Circuit Analysis Chapter 2 Basic Building Blocks Chapter 3 Dynamic Range Chapter 4 Component Technology Chapter 5 Power Supplies Chapter 6 The Power Amplifier Chapter 7 The Pre Amplifier Appendix Index Front matter Valve Amplifiers Valve Amplifiers Fourth Edition Morgan Jones AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD • PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKY.

Chapter 1 Circuit Analysis

Chapter 2 Basic Building BlocksChapter 3 Dynamic Range

Chapter 4 Component TechnologyChapter 5 Power Supplies

Chapter 6 The Power AmplifierChapter 7 The Pre-AmplifierAppendix

Index

SINGAPORE • SYDNEY • TOKYO

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11 12 13 14 15 10 9 8 7 6 5 4 3 2 1

Almost 40 years ago the author bought his first valve amplifier; it cost him

£3, and represented many weeks’ pocket money Whilst his pocket moneyhas increased, so have his aspirations, and the DIY need was born

Although there were many sources of information on circuit design, theelectronics works gave scant regard to audio design, whilst the Hi-Fi booksbarely scratched the surface of the theory The author, therefore, spent muchtime in libraries trying to link this information together to form a basis foraudio design This book is the result of those years of effort and aims topresent thermionic theory in an accessible form without getting too boggeddown in maths Primarily, it is a book for practical people armed with acalculator or computer, a power drill and a (temperature-controlled)soldering iron

The author started a B.Sc in Acoustical Engineering, but left after a year tojoin BBC Engineering as a Technical Assistant, where he received excellenttuition in electronics and rose to the giddy heights of a Senior Engineerbefore being made redundant by BBC cuts He has also served time inHigher Education, and although developing the UK’s first B.Sc (Hons.)Media Technology course and watching students blossom into graduateswith successful careers was immensely satisfying, education is achieved byclass contact – not by committees and paper chases

Early on, he became a member of the Audio Engineering Society, and hasdesigned and constructed many valve pre-amplifiers and power amplifiers,loudspeakers, pick-up arms and a pair of electrostatic headphones

It is now 18 years since work began on the 1st edition of Valve Amplifiers,

yet much has changed in this obsolete technology since then

The relentless infestation of homes by computers has meant that test andmeasurement has become both cheaper and more easily integrated, eitherbecause it directly uses the processing power of a computer, or because itborrows from the technology needed to make them Thus, the Fast FourierTransform has become a tool for all to use, from industrial designer to keenamateur – enabling spectrum analysis via a £100 sound card that was theprovince of world class companies only 20 years ago As a happyconsequence, this edition benefits from detailed measurements limited

primarily by the author’s time Computer modelling is now freely available– exemplified by Duncan Munro’s PSUD2 linear power supply freeware.The spread of Internet trading has made the market for valves truly global.Exotica such as Loctals, European ‘Special Quality’ valves, and finalgeneration Soviet bloc valves are now all readily available worldwide toany Luddite with the patience to access the Internet – we no longer need to

be constrained to conservative (but expensive) choices of traditional audiovalves Even better, many of the 1950s engineering books that you thoughthad gone forever are available from the second-hand book sellers on theInternet

Paradoxically, although digital electronics has improved the supply ofvalves, other analogue components are dying Capacitors are the worstaffected by the lack of raw materials; polycarbonate disappeared in 2001,and silvered-mica capacitors and polystyrene are both endangered species.Controls have succumbed to the ubiquitous digital encoder, so mechanicalswitch ranges have contracted and potentiometers face a similar Darwinianfate It is particularly galling to discover a use for Zeners just as majorsemiconductor manufacturers stop making them

Despite, or perhaps because of, these problems, valves and vinyl have

become design icons, both in television adverts and the bits in between Therelentless hype from manufacturers of audio servers that favourconvenience over sound quality has forced manufacturers of CD players to

justify their products on sound quality ( and convenience, because although

nobody mentions it, a CD player is unable to wipe your entire music library

at the drop of an operating system) CD and vinyl are now the only reliablesources of quality audio – which is perhaps a step forward from the 1980swhen it was FM radio and vinyl

Note for the MP3 generation: That shiny 120 mm disc was invented for storing music (such as Beethoven’s 9th Symphony) at far higher quality than a compressed download Try it some time – you might even like it.

The author would like to dedicate this book to the dwindling band of BBCengineers, particularly at BBC Southampton, and also to those at BBCWood Norton, of which he has many colourful memories

Special thanks are due to Euan McKenzie who undertook the onerous task

of proofreading at long distance on short notice and in record time to anappropriately low uncertainty

Thermionic design cannot proceed in a vacuum, so the author is grateful forthe perceptive insights and insults freely offered by Stuart Yaniger over therecent years

An annual celebration of awe and wonder has been the European TriodeFestival This delightfully civilised bacchanalia has humbled the authorwith splendid works of art and engineering whilst at the same timereassuring him that he was not alone Thank you, Christian, for first inviting

me, and even more thanks to subsequent organisers for successfullymaintaining the momentum

Finally, the author would like to thank those readers who took the time andtrouble to breach the publishing citadel and give the author hugely usefulfeedback

Chapter 1 Circuit Analysis

In order to look at the interesting business of designing and building valveamplifiers, we need some knowledge of electronics funmentals.Unfortunately, fundamentals are not terribly interesting, and to cover themfully 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 complexnumbers and vectors, whilst the omission of differential calculus is aparticularly poignant loss All that is left is ordinary algebra, and although

there are lots of equations, they are timid, miserable creatures and quite

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 small change in the associated value, such as Δ Vgk

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 aballoon, rub it briskly against a jumper and then place the rubbed faceagainst 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 forcebetween it and the wall (Although the wall was initially uncharged, placingthe balloon on the wall induced a charge.)

Charged objects come in two forms: negative and positive Unlike chargesattract, and like charges repel Electrons are negative and positrons arepositive, but whilst electrons are stable in our universe, positrons encounter

an electron almost immediately after production, resulting in mutualannihilation and a pair of 511 keV gamma rays

If we don’t have ready access to positrons, how can we have a positivelycharged 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 wouldsay 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 havesaid that the second object was more positively charged than the first It’sjust a matter of which way you look at it

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

The balloon would, therefore, stick to the jumper Or would it? Certainly itwill be attracted to the jumper, but what happens when we place the two incontact? The balloon does not stick This is because the fibres of the jumperwere able to touch the whole of the charged area on the balloon, and theelectrons were so attracted to the jumper that they moved back onto thejumper, 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 approximately6,240,000,000,000,000,000 electrons had passed, but much handier

Simply being able to say that a large number of electrons had flowed past agiven 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 ofelectrons since the dawn of time, but we do want to know about electronflow at any given instant Thus, we could define the flow as the number ofcoulombs of charge that flowed past a point in one second This is stillrather 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 isusually stated in the following form

One coulomb is the charge moved by one ampere flowing for one second.

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 atcapacitors (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 thismovement This force is known as the electro motive force (EMF) Currentcontinues to flow whilst this force is applied, and it flows from a higherpotential 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.

One joule of work is done if a force of one newton moves one metre from its point of application.

This very physical interpretation of work can be understood easily once werealise that it means that one joule of work would be done by moving onekilogramme a distance of one metre in one second Since charge is directlyrelated 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 the motion of electrons, the force is called

the Electro-Motive Force, and it is measured in volts.

If one joule of work is done moving one coulomb of charge, then the system is said to have a potential difference of one volt (V).

The concept of work is important because work can be done only by theexpenditure of energy, which is, therefore, also expressed in joules

In our specialised sense, doing work means moving charge (electrons) tomake currents flow

Batteries and Lamps

If we want to make a current flow, we need a circuit A circuit is exactly

that a loop or path through which a current can flow, from its starting point

all the way round the circuit, to return to its starting point Break the circuit,

and the current ceases to flow

The simplest circuit that we might imagine is a battery connected to an

incandescent lamp via a switch We open the switch to stop the current flow

(open circuit) and close it to light the lamp Meanwhile, our helpful friend

(who has been watching all this) leans over and drops a thick piece of

copper across the battery terminals, causing a short circuit.

The lamp goes out Why?

Ohm’s Law

To answer the last question, we need some property that defines how much

current flows That property is resistance, so we need another definition,

units and a symbol

If a potential difference of one volt is applied across a resistance, resulting in a current of one ampere, then the resistance has a value

of one ohm (Ω).

This is actually a simplified statement of Ohm’s law, rather than a strictdefinition of resistance, but we don't need to worry too much about that

We can rearrange the previous equation to make I or R the subject.

These are incredibly powerful equations and should be committed tomemory

The circuit shown in Figure 1.1 is switched on, and a current of 0.25 Aflows What is the resistance of the lamp?

Figure 1.1 Use of Ohm’s law to determine the resistance of a hot lamp.

Now this might seem like a trivial example, since we could easily havemeasured the resistance of the lamp to 3½ significant figures using ourshiny, new, digital multimeter But could we? The hot resistance of anincandescent lamp is very different from its cold resistance; in the exampleabove, the cold resistance was 80 Ω

We could now work the other way and ask how much current would flowthrough an 80 Ω resistor connected to 240 V

Incidentally, this is why incandescent lamps are most likely to fail atswitch-on The high initial current that flows before the filament haswarmed up and increased its resistance stresses the weakest parts of thefilament, they become so hot that they vaporise, and the lamp blows

Power

In the previous example, we looked at an incandescent lamp and rated it bythe current that flowed through it when connected to a 240 V battery But

we all know that lamps are rated in watts, so there must be some connection

between the two

One watt (W) of power is expended if one joule of work is done in one second.

This may not seem to be the most useful of definitions, and, indeed, it isnot, but by combining it with some earlier equations:

Kirchhoff’s Laws

There are two of these: a current law and a voltage law They are both verysimple and, at the same time, very powerful

The current law states:

The algebraic sum of the currents flowing into, and out of, a node is equal to zero.

What it says in a more relaxed form is that what goes in, comes out If wehave 10 A going into a node, or junction, then that much current must alsoleave that junction – it might not all come out on one wire, but it must allcome out A conservation of current, if you like (see Figure 1.2)

Figure 1.2 Currents at a node (Kirchhoff’s current law).

From the point of view of the node, the currents leaving the node areflowing in the opposite direction to the current flowing into the node, so we

must give them a minus sign before plugging them into the equation.

This may have seemed pedantic, since it was obvious from the diagram thatthe incoming currents equalled the outgoing currents, but you may need tofind a current when you do not even know the direction in which it isflowing Using this convention forces the correct answer!

It is vital to make sure that your signs are correct

The voltage law states:

The algebraic sum of the EMFs and potential differences acting around any loop is equal to zero.

This law draws a very definite distinction between EMFs and potential

differences EMFs are the sources of electrical energy (such as batteries), whereas potential differences are the voltages dropped across components.

Another way of stating the law is to say that the algebraic sum of the EMFsmust equal the algebraic sum of the potential drops around the loop Again,you could consider this to be a conservation of voltage (see Figure 1.3)

Figure 1.3 Summation of potentials within a loop (Kirchhoff’s voltage law).

Resistors in Series and Parallel

If we had a network of resistors, we might want to know what the total

resistance was between terminals A and B (see Figure 1.4)

Figure 1.4 Series/parallel resistor network.

We have three resistors: R1 is in parallel with R2, and this combination is in

series with R3

As with all problems, the thing to do is to break it down into its simplestparts If we had some means of determining the value of resistors in series,

we could use it to calculate the value of R3 in series with the combination of

R1 and R2, but as we do not yet know the value of the parallel combination,

we must find this first This question of order is most important, and we

will return to it later

If the two resistors (or any other component, for that matter) are in parallel,then they must have the same voltage drop across them Ohm’s law mighttherefore, be a useful starting point

Using Kirchhoff’s current law, we can state that:

So:

Dividing by V:

The reciprocal of the total parallel resistance is equal to the sum of the reciprocals of the individual resistors.

For the special case of only two resistors, we can derive the equation:

This is often known as ‘product over sum’, and whilst it is useful for mentalarithmetic, it is slow to use on a calculator (more keystrokes)

Now that we have cracked the parallel problem, we need to crack the seriesproblem

First, we will simplify the circuit We can now calculate the total resistance

of the parallel combination and replace it with one resistor of that value –

an equivalent resistor (see Figure 1.5).

Figure 1.5 Simplification of Fig 1.4 using an equivalent resistor.

Using the voltage law, the sum of the potentials across the resistors must beequal to the driving EMF:

Using Ohm’s law:

But if we are trying to create an equivalent resistor, whose value is equal tothe combination, we could say:

Hence:

The total resistance of a combination of series resistors is equal to the sum of their individual resistances.

Using the parallel and series equations, we are now able to calculate the

total resistance of any network (see Figure 1.6).

Figure 1.6

Now this may look horrendous, but it is not a problem if we attack itlogically The hardest part of the problem is not wielding the equations ornumbers, but where to start

We want to know the resistance looking into the terminals A and B, but we

do not have any rules for finding this directly, so we must look for a pointwhere we can apply our rules We can apply only one rule at a time, so we

look for a combination of components made up only of series or parallel

components

In this example, we find that between node A and node D there are only

parallel components We can calculate the value of an equivalent resistorand substitute it back into the circuit:

We redraw the circuit (see Figure 1.7).

Figure 1.7

Looking again, we find that now the only combinations made up of series

or parallel components are between node A and node C, but we have a

choice – either the series combination of the 2 Ω and 4 Ω, or the parallelcombination of the 3 Ω and 6 Ω The one to go for is the seriescombination This is because it will result in a single resistor that will then

be in parallel with the 3 Ω and 6 Ω resistors We can cope with the threeparallel resistors later:

We redraw the circuit (see Figure 1.8)

• The critical stage is choosing the starting point.

• The starting point is generally as far away from the terminals as it ispossible to be

• The starting point is made up of a combination of only series or parallel

reduces confusion and errors – do it!

Potential Dividers

Figure 1.9 shows a potential divider This could be made up of two discreteresistors, or it could be the moving wiper of a volume control As before,

we will suppose that a current I flows through the two resistors We want to

know the ratio of the output voltage to the input voltage (see Figure 1.9)

Figure 1.9 Potential divider.

Hence:

This is a very important result and, used intelligently, can solve virtuallyanything

Equivalent Circuits

We have looked at networks of resistors and calculated equivalent

resistances Now we will extend the idea to equivalent circuits This is a

tremendously powerful concept for circuit analysis.

It should be noted that this is not the only method, but it is usually thequickest and kills 99% of all known problems Other methods includeKirchhoff’s laws combined with lots of simultaneous equations and thesuperposition theorem These methods may be found in standard texts, butthey tend to be cumbersome, so we will not discuss them here

The Thévenin Equivalent Circuit

When we looked at the potential divider, we were able to calculate the ratio

of output voltage to input voltage If we were now to connect a batteryacross the input terminals, we could calculate the output voltage Using ourearlier tools, we could also calculate the total resistance looking into theoutput terminals As before, we could then redraw the circuit, and the result

is known as the Thévenin equivalent circuit If two black boxes were made,

one containing the original circuit and the other the Thévenin equivalentcircuit, you would not be able to tell from the output terminals which waswhich The concept is simple to use and can break down complex networksquickly and efficiently (see Figure 1.10)

Figure 1.10 A ‘black box’ network and its Thévenin equivalent circuit.

This is a simple example to demonstrate the concept First, we find the

equivalent resistance, often known as the output resistance Now, in the

world of equivalent circuits, batteries are perfect voltage sources; they have

zero internal resistance and look like a short circuit when we consider their

resistance Therefore, we can ignore the battery, or replace it with a piece ofwire whilst we calculate the resistance of the total circuit:

Next, we need to find the output voltage We will use the potential dividerequation:

still a short circuit, so we will throw it away (see Figure 1.12)

Figure 1.13

Using the potential divider rule:

Using ‘product over sum’:

Looking at the right-hand side, we can perform a similar operation to theright of the dashed line

First, we find the resistance of the parallel combination of the 36 Ω and 12

Ω resistors, which is 9 Ω We now have a potential divider, whose outputresistance is 6 Ω and the Thévenin voltage is 2 V

Now we redraw the circuit (see Figure 1.14)

Figure 1.14

We can make a few observations at this point First, we have three batteries

in series, why not combine them into one battery? There is no reason why

we should not do this provided that we take note of their polarities.Similarly, we can combine some, or all, of the resistors (see Figure 1.15)

Figure 1.15

The problem now is trivial, and a simple application of Ohm’s law willsolve it We have a total resistance of 10 Ω and a 5 V battery, so thecurrent must be 0.5 A

Useful points to note:

• Look for components that are irrelevant, such as resistors directlyacross battery terminals.

• Look for potential dividers on the outputs of batteries and ‘Thévenise’them Keep on doing so until you meet the next battery

• Work from battery terminals outwards

• Keep calm, and try to work neatly – it will save mistakes later

Although it is possible to solve most problems using a Thévenin equivalentcircuit, sometimes a Norton equivalent is more convenient

The Norton Equivalent Circuit

The Thévenin equivalent circuit was a perfect voltage source in series with

a resistance, whereas the Norton equivalent circuit is a perfect current source in parallel with a resistance (see Figure 1.16).

Figure 1.16 The Norton equivalent circuit.

We can easily convert from a Norton source to a Thévenin source, or viceversa, because the resistor has the same value in both cases We find thevalue of the current source by short circuiting the output of the Théveninsource and calculating the resulting current – this is the Norton current

To convert from a Norton source to a Thévenin source, we leave the sourceopen circuit and calculate the voltage developed across the Norton resistor –this is the Thévenin voltage

For the vast majority of problems, the Thévenin equivalent will be quicker,mostly because we become used to thinking in terms of voltages that caneasily be measured by a meter or viewed on an oscilloscope Occasionally, aproblem will arise that is intractable using Thévenin, and converting to aNorton equivalent causes the problem to solve itself Norton problems

usually involve the summation of a number of currents, when the only othersolution would be to resort to Kirchhoff and simultaneous equations.

Units and Multipliers

All the calculations up to this point have been arranged to use convenientvalues of voltage, current and resistance In the real world, we will not be sofortunate, and to avoid having to use scientific notation, which takes longer

to write and is virtually unpronounceable, we will prefix our units withmultipliers [1]

Note that the case of the prefix is important; there is a large difference

between 1 mΩ and 1 MΩ Electronics uses a very wide range of values;small-signal pentodes have anode to grid capacitances measured in fF(F=farad, the unit of capacitance), and petabyte data stores are alreadycommon Despite this, for day-to-day electronics use, we only need to usepico to mega

Electronics engineers commonly abbreviate further, and you will often hear

a 22 pF (picofarad) capacitor referred to as 22 ‘puff’, whilst the ‘ohm’ iscommonly dropped for resistors, and 470 kΩ (kiloohm) would bepronounced as ‘four-seventy-kay’

A rather more awkward abbreviation that arose before high-resolutionprinters became common (early printers could not print ‘μ’), is the

abbreviation of μ (micro) to m This abbreviation persists, particularly inthe US text, and you will occasionally see a 10 mF capacitor specified,although the context makes it clear that what is actually meant is 10 μF Forthis reason, true 10 mF capacitors are invariably specified as 10,000 μF.Unless an equation states otherwise, assume that it uses the base physicalunits, so an equation involving capacitance and time constants would expectyou to express capacitance in farads and time in seconds Thus, 75μs=75×10 −6 s, and the value of capacitance determined by an equation

an equation using real-world units such as mA or MΩ, in which case the

equation or its accompanying text will always explain this break from

convention

The Decibel

The human ear spans a vast dynamic range from the near silence heard in

an empty recording studio to the deafening noise of a nearby pneumaticdrill If we were to plot this range linearly on a graph, the quieter soundswould hardly be seen, whereas the difference between the noise of the drilland that of a jet engine would be given a disproportionate amount of room

on the graph What we need is a graph that gives an equal weighting to

relative changes in the level of both quiet and loud sounds By definition,

this implies a logarithmic scale on the graph, but electronics engineers went

one better and invented a logarithmic ratio known as the decibel ( dB) that

was promptly hijacked by the acoustical engineers (The fundamental unit isthe bel, but this is inconveniently large, so the decibel is more commonlyused.)

The dB is not an absolute quantity It is a ratio, and it has one formula for

use with currents and voltages and another for powers:

The reason for this is that P∝ V2 or I2, and with logarithms, multiplying thelogarithm by 2 is the same as squaring the original number Using adifferent formula to calculate dBs when using powers ensures that theresulting dBs are equivalent, irrespective of whether they were derived frompowers or voltages

This might seem complicated when all we wanted to do was to describe thedifference in two signal levels, but the dB is a very handy unit.

Useful common dB values are:

For example, if we had two cascaded amplifiers, one with a voltage gain of

0.5 and the other with a voltage gain of 10, then by multiplying the

individual gains, the combined voltage gain would be 5 Alternatively, wecould find the gain in dB by saying that one amplifier had −6 dB of gain

whilst the other had 20 dB, and adding the gains in dB to give a total gain

of 14 dB

When designing amplifiers, we will not often use the above example, as

absolute voltages are often more convenient, but we frequently need dBs todescribe filter and equalisation curves

Alternating Current (AC)

All the previous techniques have used direct current ( DC), where the

current is constant and flows in one direction only Listening to DC is not

very interesting, so we now need to look at alternating currents ( AC).

All of the previous techniques of circuit analysis can be applied equallywell to AC signals

The Sine Wave

The sine wave is the simplest possible alternating signal, and its equation is:

where

v=the instantaneous value at time t

Vpeak=the peak value

ω=angular frequency in radians/second ( ω=2 πf)

t=the time in seconds

θ=a constant phase angle.

These mathematical concepts are shown on the diagram (see Figure 1.17)

Figure 1.17

Equations involving changing quantities use a convention Upper caseletters denote DC, or constant values, whereas lower case letters denote theinstantaneous AC, or changing, value It is a form of shorthand to avoidhaving to specify separately that a quantity is AC or DC It would be nice tosay that this convention is rigidly applied, but it is often neglected, and thecontext of the symbols usually makes it clear whether the quantity is AC orDC

In electronics, the word ‘peak’ (pk) has a very precise meaning and, whenused to describe an AC waveform, it means the voltage from zero volts tothe peak voltage reached, either positive or negative Peak to peak (pk–pk)means the voltage from positive peak to negative peak, and for a

symmetrical waveform, Vpk–pk=2 V pk

Although electronics engineers habitually use ω to describe frequency, they

do so only because calculus requires that they work in radians Since ω=2

πf, we can rewrite the equation as:

If we now inspect this equation, we see that apart from time t, we could

vary other constants before we allow time to change and determine the

waveform We can change Vpeak, and this will change the amplitude of the sine wave, or we can change f, and this will change the frequency The inverse of frequency is period, which is the time taken for one full cycle of

the waveform to occur:

If we listen to a sound that is a sine wave, and change the amplitude, thiswill make the sound louder or softer, whereas varying frequency changes

the pitch If we vary θ (phase), it will sound the same if we are listening to

it, and unless we have an external reference, the sine wave will look exactlythe same viewed on an oscilloscope Phase becomes significant if we

compare one sine wave with another sine wave of the same frequency or a

harmonic of that frequency Attempting to compare phase betweenwaveforms of unrelated frequencies is meaningless

Now that we have described sine waves, we can look at them as they wouldappear on the screen of an oscilloscope See Figure 1.18

Figure 1.18

Sine waves A and B are identical in amplitude, frequency and phase Sinewave C has lower amplitude, but frequency and phase are the same Sinewave D has the same amplitude, but double the frequency Sine wave E has

identical amplitude and frequency to A and B, but the phase θ has been

changed

Sine wave F has had its polarity inverted Although, for a sine wave, we

cannot see the difference between a 180° phase change and a polarityinversion, for asymmetric waveforms there is a distinct difference Weshould, therefore, be very careful if we say that two waveforms are 180° out

of phase with each other that we do not actually mean that one is invertedwith respect to the other

The sawtooth waveform G has been inverted to produce the waveform H,

and it can be seen that this is completely different from a 180° phasechange (Strictly, the UK term ‘phase splitter’ is entirely incorrect for thisreason, and the US description ‘phase inverter’ is much better, but fallsshort of the technically correct description ‘polarity inverter’ used bynobody.)

The Transformer

When the electric light was introduced as an alternative to the gas mantle,there was a great debate as to whether the distribution system should be AC

or DC The outcome was settled by the enormous advantage of the

transformer, which could step up, or step down, the voltage of an AC

supply The DC supply could not be manipulated in this way, and evolutiontook its course

A transformer is essentially a pair of electrically insulated windings that aremagnetically coupled to each other, usually on an iron core They vary fromthe size of a fingernail to that of a large house, depending on power ratingand operating frequency, with high frequency transformers being smaller.The symbol for a transformer is modified depending on the core material.Solid lines indicate a laminated iron core and dotted lines denote a dustcore, whilst an air core has no lines (see Figure 1.19)

Figure 1.19 Transformer symbols.

The perfect transformer changes one AC voltage to another, moreconvenient voltage with no losses whatsoever; all of the power at the input

is transferred to the output:

Having made this statement, we can now derive some useful equations:Rearranging:

The new constant n is very important and is the ratio between the number of

turns on the input winding and the number of turns on the output winding ofthe transformer Habitually, when we talk about transformers, the input

winding is known as the primary and the output winding is the secondary.

Occasionally, an audio transformer may have a winding known as a tertiary

winding, which usually refers to a winding used for feedback or monitoring,but it is more usual to refer to multiple primaries and secondaries

When the perfect transformer steps voltage down, perhaps from 240 V to

12 V, the current ratio is stepped up, and each ampere of primary current isdue to 20 A drawn from the secondary This implies that the resistance ofthe load on the secondary is different from that seen looking into theprimary If we substitute Ohm’s law into the conservation of powerequation:

The transformer changes resistances by the square of the turns ratio This

will become very significant when we use audio transformers that mustmatch loudspeakers to output valves

As an example, an output transformer with a primary to secondary turnsratio of 31.6:1 would allow the output valves to see the 8 Ω loudspeaker as

an 8 kΩ load, whereas the loudspeaker sees the Thévenin output resistance

of the output valves stepped down by an identical amount

The concept of looking into a device in one direction, and seeing one thing,whilst looking in the opposite direction, and seeing another, is verypowerful, and we will use it frequently when we investigate simpleamplifier stages

Practical transformers are not perfect, and we will investigate theirimperfections in greater detail in Chapter 4

Capacitors, Inductors and Reactance

Previously, when we analysed circuits, they were composed purely ofresistors and voltage or current sources

We now need to introduce two new components: capacitors and inductors Capacitors have the symbol C, and the unit of capacitance is the farad ( F).

1 F is an extremely large capacitance, and more common values range from

a few pF to tens of thousands of μF Inductors have the symbol L, and the unit of inductance is the henry ( H) The henry is quite a large unit, and

common values range from a few μH to tens of H Although the henry, andparticularly the farad, is rather large for our very specialised use, its sizederives from the fundamental requirement for a coherent system of units; acoherent system allows units (such as the farad) to be derived from baseunits (such as the ampere) with an absolute minimum of scaling factors.The simplest capacitor is made of a pair of separated plates, whereas aninductor is a coil of wire, and this physical construction is reflected in theirgraphical symbols (see Figure 1.20)

Figure 1.20 Inductor and capacitor symbols.

Resistors had resistance, whereas capacitors and inductors have reactance.

Reactance is the AC equivalent of resistance – it is still measured in ohms

and is given the symbol X We will often have circuits where there is a

combination of inductors and capacitors, so it is normal to add a subscript

to denote which reactance is which:

Looking at these equations, we find that reactance changes with frequencyand with the value of the component We can plot these relationships on agraph (see Figure 1.21)

Figure 1.21 Reactance of inductor and capacitor against frequency.

An inductor has a reactance of zero at zero frequency More intuitively, it is

a short circuit at DC As we increase frequency, its reactance rises

A capacitor has infinite reactance at zero frequency It is open circuit at DC

As frequency rises, reactance falls