Electronics and circuits leacture notes

They are:  the independent voltage source  the independent current source  the dependent voltage source  the dependent current source Passive Circuit Elements Passive circuit eleme

48520 Electronics and

7 5

CIRCUIT ELEMENTS AND TYPES OF CIRCUITS 1A.4

ACTIVE CIRCUIT ELEMENTS 1A.4

PASSIVE CIRCUIT ELEMENTS 1A.4

TYPES OF CIRCUITS 1A.4

INDEPENDENT SOURCES 1A.5

THE INDEPENDENT VOLTAGE SOURCE 1A.5

THE INDEPENDENT CURRENT SOURCE 1A.7

THE RESISTOR AND OHM’S LAW 1A.8

THE SHORT-CIRCUIT 1A.11

THE OPEN-CIRCUIT 1A.12

CONDUCTANCE 1A.12

PRACTICAL RESISTORS 1A.13

PREFERRED VALUES AND THE DECADE PROGRESSION 1A.14

THE ‘E’SERIES VALUES 1A.14

MARKING CODES 1A.16

KIRCHHOFF’S CURRENT LAW 1A.19

KIRCHHOFF’S VOLTAGE LAW 1A.23

COMBINING RESISTORS 1A.26

SERIES RESISTORS 1A.26

PARALLEL RESISTORS 1A.27

COMBINING INDEPENDENT SOURCES 1A.30

COMBINING INDEPENDENT VOLTAGE SOURCES IN SERIES 1A.30

COMBINING INDEPENDENT CURRENT SOURCES IN PARALLEL 1A.32

THE VOLTAGE DIVIDER RULE 1A.34

THE CURRENT DIVIDER RULE 1A.36

DEPENDENT SOURCES 1A.38

THE DEPENDENT VOLTAGE SOURCE 1A.38

THE DEPENDENT CURRENT SOURCE 1A.40

POWER 1A.42

POWER ABSORBED IN A RESISTOR 1A.48

AMPLIFIERS 1A.49

UNITS OF GAIN 1A.50

AMPLIFIER POWER SUPPLIES 1A.52

SATURATION 1A.53

CIRCUIT MODEL 1A.54

THE OPERATIONAL AMPLIFIER 1A.55

FEEDBACK 1A.56

CIRCUIT MODEL 1A.57

THE IDEAL OP-AMP 1A.58

OP-AMP FABRICATION AND PACKAGING 1A.60

NEGATIVE FEEDBACK 1A.61

NEGATIVE FEEDBACK IN ELECTRONICS 1A.62

AN AMPLIFIER WITH NEGATIVE FEEDBACK 1A.63

THE NONINVERTING AMPLIFIER WITH AN IDEAL OP-AMP 1A.69

INPUT RESISTANCE OF THE NONINVERTING AMPLIFIER 1A.71

EQUIVALENT CIRCUIT OF THE NONINVERTING AMPLIFIER 1A.71

THE BUFFER 1A.74

THE INVERTING AMPLIFIER 1A.75

INPUT RESISTANCE OF THE INVERTING AMPLIFIER 1A.77

EQUIVALENT CIRCUIT OF THE INVERTING AMPLIFIER 1A.77

CIRCUITS WITH RESISTORS AND INDEPENDENT CURRENT SOURCES ONLY 1B.5

NODAL ANALYSIS USING BRANCH ELEMENT STAMPS 1B.8

CIRCUITS WITH VOLTAGE SOURCES 1B.11

CIRCUITS WITH DEPENDENT SOURCES 1B.13

SUMMARY OF NODAL ANALYSIS 1B.16

MESH ANALYSIS 1B.19

PLANAR CIRCUITS 1B.19

PATHS,LOOPS AND MESHES 1B.20

MESH CURRENT 1B.21

MESH ANALYSIS METHODOLOGY 1B.22

CIRCUITS WITH RESISTORS AND INDEPENDENT VOLTAGE SOURCES ONLY 1B.23

CIRCUITS WITH CURRENT SOURCES 1B.25

CIRCUITS WITH DEPENDENT SOURCES 1B.27

SUMMARY OF MESH ANALYSIS 1B.29

SUMMARY 1B.30

EXERCISES 1B.31

GUSTAV ROBERT KIRCHHOFF (1824-1887) 1B.34

L ECTURE 2A – C IRCUIT A NALYSIS T ECHNIQUES

INTRODUCTION 2A.1

LINEARITY 2A.2

SUPERPOSITION 2A.3

SUPERPOSITION THEOREM 2A.5

SOURCE TRANSFORMATIONS 2A.9

PRACTICAL VOLTAGE SOURCES 2A.9

PRACTICAL CURRENT SOURCES 2A.11

PRACTICAL SOURCE EQUIVALENCE 2A.13

MAXIMUM POWER TRANSFER THEOREM 2A.15

THÉVENIN’S AND NORTON’S THEOREM 2A.19

SUMMARY OF FINDING THÉVENIN EQUIVALENT CIRCUITS 2A.27

SUMMARY 2A.31

L ECTURE 2B – L INEAR O P -A MP A PPLICATIONS

NEGATIVE IMPEDANCE CONVERTER 2B.11

VOLTAGE-TO-CURRENT CONVERTER 2B.13

CAPACITOR v-iRELATIONSHIPS 3A.4

ENERGY STORED IN A CAPACITOR 3A.6

SUMMARY OF IMPORTANT CAPACITOR CHARACTERISTICS 3A.9

THE INDUCTOR 3A.10

INDUCTOR v-iRELATIONSHIPS 3A.13

ENERGY STORED IN AN INDUCTOR 3A.18

SUMMARY OF IMPORTANT INDUCTOR CHARACTERISTICS 3A.21

PRACTICAL CAPACITORS AND INDUCTORS 3A.22

INTRODUCTION 3B.1

THE SILICON JUNCTION DIODE 3B.2

THE FORWARD-BIAS REGION 3B.3

THE REVERSE-BIAS REGION 3B.5

THE IDEAL DIODE MODEL 3B.14

THE CONSTANT VOLTAGE DROP MODEL 3B.16

THE PIECE-WISE LINEAR MODEL 3B.18

THE SMALL SIGNAL MODEL 3B.20

BASIC DIODE CIRCUITS 3B.24

HALF-WAVE RECTIFIER 3B.24

FULL-WAVE RECTIFIER 3B.26

LIMITER CIRCUITS 3B.27

SUMMARY 3B.29

L ECTURE 4A – S OURCE -F REE RC AND RLC IRCUITS

INTRODUCTION 4A.1

DIFFERENTIAL OPERATORS 4A.2

PROPERTIES OF DIFFERENTIAL OPERATORS 4A.4

THE CHARACTERISTIC EQUATION 4A.8

THE SIMPLE RCCIRCUIT 4A.12

PROPERTIES OF THE EXPONENTIAL RESPONSE 4A.15

SINGLE TIME CONSTANT RCCIRCUITS 4A.18

THE SIMPLE RLCIRCUIT 4A.20

SINGLE TIME CONSTANT RLCIRCUITS 4A.23

SUMMARY 4A.27

EXERCISES 4A.28

L ECTURE 4B – N ONLINEAR O P -A MP A PPLICATIONS

INTRODUCTION 4B.1

THE COMPARATOR 4B.2

PRECISION RECTIFIERS 4B.6

THE SUPERDIODE 4B.7

PRECISION INVERTING HALF-WAVE RECTIFIER 4B.9

PRECISION FULL-WAVE RECTIFIER 4B.14

SINGLE-SUPPLY HALF-WAVE AND FULL-WAVE RECTIFIER 4B.16

THE UNIT-STEP FORCING FUNCTION 5A.2

THE DRIVEN RCCIRCUIT 5A.6

THE FORCED AND THE NATURAL RESPONSE 5A.10

FINDING A PARTICULAR SOLUTION USING THE INVERSE DIFFERENTIAL

OPERATOR 5A.12

FINDING A PARTICULAR SOLUTION BY INSPECTION 5A.14

FINDING A PARTICULAR SOLUTION USING AN INTEGRATING FACTOR 5A.15

STEP-RESPONSE OF RCCIRCUITS 5A.18

ANALYSIS PROCEDURE FOR SINGLE TIME CONSTANT RCCIRCUITS 5A.28

RLCIRCUITS 5A.29

ANALYSIS PROCEDURE FOR SINGLE TIME CONSTANT RLCIRCUITS 5A.31

SUMMARY 5A.32

EXERCISES 5A.33

LEONHARD EULER (1707-1783) 5A.37

L ECTURE 5B –O P -A MP I MPERFECTIONS

INTRODUCTION 5B.1

DCIMPERFECTIONS 5B.2

OFFSET VOLTAGE 5B.3

INPUT BIAS CURRENTS 5B.4

FINITE OPEN-LOOP GAIN 5B.7

NONINVERTING AMPLIFIER 5B.7

INVERTING AMPLIFIER 5B.8

PERCENT GAIN ERROR 5B.10

FINITE BANDWIDTH 5B.11

OUTPUT VOLTAGE SATURATION 5B.12

OUTPUT CURRENT LIMITS 5B.13

SLEW RATE 5B.14

FULL-POWER BANDWIDTH 5B.15

SUMMARY 5B.16

EXERCISES 5B.18

INTRODUCTION 6A.1

SINUSOIDAL SIGNALS 6A.3

SINUSOIDAL STEADY-STATE RESPONSE 6A.5

THE COMPLEX FORCING FUNCTION 6A.11

THE PHASOR 6A.17

FORMALISATION OF THE RELATIONSHIP BETWEEN PHASOR AND SINUSOID 6A.20

GRAPHICAL ILLUSTRATION OF THE RELATIONSHIP BETWEEN A PHASOR

AND ITS CORRESPONDING SINUSOID 6A.21

PHASOR RELATIONSHIPS FOR R, L AND C 6A.22

PHASOR RELATIONSHIPS FOR A RESISTOR 6A.22

PHASOR RELATIONSHIPS FOR AN INDUCTOR 6A.24

PHASOR RELATIONSHIPS FOR A CAPACITOR 6A.26

SUMMARY OF PHASOR RELATIONSHIPS FOR R, L AND C 6A.28

ANALYSIS USING PHASOR RELATIONSHIPS 6A.29

IMPEDANCE 6A.30

ADMITTANCE 6A.35

SUMMARY 6A.37

EXERCISES 6A.38

JOSEPH FOURIER (1768-1830) 6A.43

L ECTURE 6B – C IRCUIT S IMULATION

TIME-DOMAIN (TRANSIENT)SIMULATIONS 6B.7

ACSWEEP /NOISE SIMULATIONS 6B.10

EXERCISES 6B.15

L ECTURE 7A – S INUSOIDAL S TEADY -S TATE R ESPONSE

INTRODUCTION 7A.1

ANALYSIS USING PHASORS 7A.2

NODAL ANALYSIS 7A.3

MESH ANALYSIS 7A.5

SUPERPOSITION 7A.7

THÉVENIN’S THEOREM 7A.8

NORTON’S THEOREM 7A.10

PHASOR DIAGRAMS 7A.12

POWER IN THE SINUSOIDAL STEADY-STATE 7A.20

INSTANTANEOUS POWER 7A.20

AVERAGE POWER 7A.21

ROOT-MEAN-SQUARE (RMS)VALUES 7A.23

RMSVALUE OF A SINUSOID 7A.24

PHASORS AND RMSVALUES 7A.25

AVERAGE POWER USING RMSVALUES 7A.26

APPARENT POWER 7A.26

POWER FACTOR 7A.26

COMPLEX POWER 7A.27

REACTIVE POWER 7A.28

SUMMARY OF POWER IN ACCIRCUITS 7A.30

SUMMARY 7A.33

EXERCISES 7A.34

OLIVER HEAVISIDE (1850-1925) 7A.40

L ECTURE 7B – A MPLIFIER C HARACTERISTICS

ACCOUPLING AND DIRECT COUPLING 7B.17

HALF-POWER FREQUENCIES AND BANDWIDTH 7B.19

LINEAR WAVEFORM DISTORTION 7B.20

INTRODUCTION 8A.1

FREQUENCY RESPONSE FUNCTION 8A.2

DETERMINING THE FREQUENCY RESPONSE FROM CIRCUIT ANALYSIS 8A.3

MAGNITUDE RESPONSES 8A.5

PHASE RESPONSES 8A.9

DETERMINING THE FREQUENCY RESPONSE FROM EXPERIMENT 8A.11

BODE PLOTS 8A.12

APPROXIMATING BODE PLOTS USING FREQUENCY RESPONSE FACTORS 8A.13

SUMMARY 8A.14

EXERCISES 8A.15

L ECTURE 8B – F IRST -O RDER O P -A MP F ILTERS

INTRODUCTION 8B.1

BILINEAR FREQUENCY RESPONSES 8B.2

FREQUENCY RESPONSE REPRESENTATION 8B.4

MAGNITUDE RESPONSES 8B.6

PHASE RESPONSES 8B.10

SUMMARY OF BILINEAR FREQUENCY RESPONSES 8B.14

FREQUENCY AND MAGNITUDE SCALING 8B.15

FREQUENCY SCALING (DENORMALISING) 8B.16

L ECTURE 9A – S ECOND -O RDER S TEP R ESPONSE

INTRODUCTION 9A.1

SOLUTION OF THE HOMOGENEOUS LINEAR DIFFERENTIAL EQUATION 9A.2

DISTINCT REAL ROOTS 9A.3

REPEATED REAL ROOTS 9A.4

ONLY REAL ROOTS 9A.5

DISTINCT COMPLEX ROOTS 9A.7

REPEATED COMPLEX ROOTS 9A.9

THE SOURCE-FREE PARALLEL RLCCIRCUIT 9A.10

THE OVERDAMPED PARALLEL RLCCIRCUIT 9A.13

THE CRITICALLY DAMPED PARALLEL RLCCIRCUIT 9A.17

THE UNDERDAMPED PARALLEL RLCCIRCUIT 9A.21

RESPONSE COMPARISON 9A.25

THE SOURCE-FREE SERIES RLCCIRCUIT 9A.26

COMPLETE RESPONSE OF THE RLCCIRCUIT 9A.27

FORCED RESPONSE 9A.28

NATURAL RESPONSE 9A.29

CASE I–OVERDAMPED 9A.31

CASE II–CRITICALLY DAMPED 9A.33

CASE III–UNDERDAMPED 9A.34

MAXIMUM VALUE AND PEAK TIME 9A.36

SUMMARY 9A.38

EXERCISES 9A.39

WILLIAM THOMSON (LORD KELVIN)(1824-1907) 9A.43

L ECTURE 9B – W AVEFORM G ENERATION

INTRODUCTION 9B.1

OPEN-LOOP COMPARATOR 9B.2

COMPARATOR WITH HYSTERESIS (SCHMITT TRIGGER) 9B.3

ASTABLE MULTIVIBRATOR (SCHMITT TRIGGER CLOCK) 9B.5

WAVEFORM GENERATOR 9B.8

SUMMARY 9B.11

INTRODUCTION 10A.1

RESONANCE 10A.2

PARALLEL RESONANCE 10A.4

PHASOR DIAGRAM OF THE PARALLEL RLCCIRCUIT 10A.7

QUALITY FACTOR 10A.8

SECOND-ORDER CIRCUIT RELATIONS 10A.11

BANDWIDTH 10A.12

SERIES RESONANCE 10A.16

OTHER RESONANT FORMS 10A.18

THE SECOND-ORDER LOWPASS FREQUENCY RESPONSE 10A.24

PEAK FREQUENCY 10A.26

BANDWIDTH 10A.28

CASE I–RELATIVE PEAK > 2 (Q0  11 2 ) 10A.29

CASE II–RELATIVE PEAK < 2 (1 2 Q0  11 2) 10A.31

CASE III–NO RELATIVE PEAK (Q0 1 2) 10A.32

BODE PLOTS 10A.33

THE SECOND-ORDER HIGHPASS FREQUENCY RESPONSE 10A.34

PEAK FREQUENCY 10A.36

BANDWIDTH 10A.36

STANDARD FORMS OF SECOND-ORDER FREQUENCY RESPONSES 10A.37

SUMMARY 10A.38

EXERCISES 10A.39

JAMES CLERK MAXWELL (1831-1879) 10A.43

L ECTURE 10B – S ECOND -O RDER O P -A MP F ILTERS

INTRODUCTION 10B.1

FILTER DESIGN PARAMETERS 10B.1

THE LOWPASS BIQUAD CIRCUIT 10B.3

THE UNIVERSAL BIQUAD CIRCUIT 10B.8

APPROXIMATING THE IDEAL LOWPASS FILTER 10B.10

THE BUTTERWORTH LOWPASS FILTER 10B.12

SUMMARY 10B.15

EXERCISES 10B.16

L ECTURE 11A – C OMPLEX F REQUENCY

INTRODUCTION 11A.1

COMPLEX FREQUENCY 11A.2

THE DAMPED SINUSOIDAL FORCING FUNCTION 11A.6

GENERALIZED IMPEDANCE AND ADMITTANCE 11A.10

FREQUENCY RESPONSE AS A FUNCTION OF  11A.13

FREQUENCY RESPONSE AS A FUNCTION OF  11A.17

THE COMPLEX-FREQUENCY PLANE 11A.22

VISUALIZATION OF THE FREQUENCY RESPONSE FROM A POLE-ZERO PLOT 11A.28

L ECTURE 11B – S PECIALTY A MPLIFIERS

PROGRAMMABLE GAIN AMPLIFIERS 11B.9

PGADESIGN ISSUES 11B.10

TRANSFER FUNCTIONS 12A.2

CHARACTERISTIC EQUATION 12A.3

POLE-ZERO PLOT 12A.3

TRANSFER FUNCTION FORM 12A.4

RELATIONSHIP TO DIFFERENTIAL EQUATION 12A.5

CIRCUIT ABSTRACTION 12A.6

FORCED RESPONSE 12A.7

FREQUENCY RESPONSE 12A.11

NATURAL RESPONSE 12A.14

COMPLETE RESPONSE 12A.19

SUMMARY 12A.22

EXERCISES 12A.24

PIERRE SIMON DE LAPLACE (1749-1827) 12A.27

L ECTURE 12B – S ENSOR S IGNAL C ONDITIONING

INTRODUCTION 12B.1

SENSORS 12B.2

PROCESS CONTROL SYSTEMS 12B.3

PROGRAMMABLE LOGIC CONTROLLERS 12B.4

SMART TRANSDUCERS 12B.5

PROGRAMMABLE AUTOMATION CONTROLLERS 12B.6

BRIDGE CIRCUITS 12B.7

BRIDGE DESIGN ISSUES 12B.11

AMPLIFYING AND LINEARIZING BRIDGE OUTPUTS 12B.12

DRIVING REMOTE BRIDGES 12B.15

INTEGRATED BRIDGE TRANSDUCERS 12B.18

STRAIN,FORCE,PRESSURE AND FLOW MEASUREMENTS 12B.19

HIGH IMPEDANCE SENSORS 12B.21

TEMPERATURE SENSORS 12B.22

SUMMARY 12B.24

EXERCISES 12B.25

INTRODUCTION 13A.1

DIFFERENTIAL EQUATIONS OF PHYSICAL SYSTEMS 13A.2

LINEAR APPROXIMATIONS OF PHYSICAL SYSTEMS 13A.4

THE TRANSFER FUNCTION 13A.7

BLOCK DIAGRAMS 13A.8

Lecture 1A – Basic Laws & Op-Amp Amplifiers

Current Voltage Circuit elements and types of circuits Independent sources

The resistor and Ohm’s Law Practical resistors Kirchhoff’s Current Law

(KCL) Kirchhoff’s Voltage Law (KVL) Combining resistors Combining

independent sources The voltage divider rule The current divider rule

Dependent sources Power Amplifiers The operational amplifier Negative

feedback The noninverting amplifier The inverting amplifier

Introduction

Electric circuit theory and electromagnetic theory are the two fundamental

theories upon which all branches of electrical engineering are built Many

branches of electrical engineering, such as power, electric machines, control,

electronics, communications, and instrumentation, are based on electric circuit

theory Circuit theory is also valuable to students specializing in other branches

of the physical sciences because circuits are a good model for the study of

energy systems in general, and because of the applied mathematics, physics,

and topology involved

Electronic circuits are used extensively in the modern world – society in its

present form could not exist without them! They are used in communication

systems (such as televisions, telephones, and the Internet), digital systems (such

as personal computers, embedded microcontrollers, smart phones), and

industrial systems (such as robotic and process control systems) The study of

electronics is therefore critical to electrical engineering and related professions

One goal in this subject is to learn various analytical techniques and computer

software applications for describing the behaviour of electric circuits Another

goal is to study various uses and applications of electronic circuits

We will start by revising some basic concepts, such as KVL, KCL and Ohm’s

Law We will then introduce the concept of the electronic amplifier, and then

study a device called an operational amplifier (op-amp for short), which has

been used as the building block for modern analog electronic circuitry since its

invention in the 1960’s

Current

Charge in motion represents a current The current present in a discrete path,

such as a metallic wire, has both a magnitude and a direction associated with it – it is a measure of the rate at which charge is moving past a given reference

point in a specified direction Current is symbolised by i and thus:

i

Figure 1A.1

The arrow does not indicate the “actual” direction of charge flow, but is simply part of a convention that allows us to talk about the current in an unambiguous manner

The use of terms such as “a current flows through the resistor” is a tautology and should not be used, since this is saying a “a charge flow flows through the resistor” The correct way to describe such a situation is “there is a current in the resistor”

A current which is constant is termed a direct current, or simply DC Examples

of direct currents are those that exist in circuits with a chemical battery as the source A sinusoidal current is often referred to as alternating current, or AC1 Alternating current is found in the normal household electricity supply

Current defined as

the rate of change of

charge moving past

Voltage

A voltage exists between two points in a circuit when energy is required to

move a charge between the two points The unit of voltage is the volt (V) and

is equivalent to JC-1 In a circuit, voltage is represented by a pair of +/- signs:

Circuit Elements and Types of Circuits

A circuit element is an idealised mathematical model of a two-terminal

electrical device that is completely characterised by its voltage-current relationship Although ideal circuit elements are not “off-the-shelf” circuit components, their importance lies in the fact that they can be interconnected (on paper or on a computer) to approximate actual circuits that are composed

of nonideal elements and assorted electrical components – thus allowing for the analysis of such circuits

Circuit elements can be categorised as either active or passive

Active Circuit Elements

Active circuit elements can deliver a non-zero average power indefinitely

There are four types of active circuit element, and all of them are termed an

ideal source They are:

 the independent voltage source

 the independent current source

 the dependent voltage source

 the dependent current source

Passive Circuit Elements

Passive circuit elements cannot deliver a non-zero average power indefinitely

Some passive elements are capable of storing energy, and therefore delivering power back into a circuit at some later time, but they cannot do so indefinitely

There are three types of passive circuit element They are:

 the resistor

 the inductor

 the capacitor

Types of Circuits

The interconnection of two or more circuit elements forms an electrical

network If the network contains at least one closed path, it is also an electrical

Ideal circuit

elements are used

to model real circuit

Independent Sources

Independent sources are ideal circuit elements that possess a voltage or current

value that is independent of the behaviour of the circuits to which they belong

The Independent Voltage Source

An independent voltage source is characterised by a terminal voltage which is

completely independent of the current through it The representation of an

independent voltage source is shown below:

vs

Figure 1A.3

If the value of the voltage source is constant, that is, does not change with time,

then we can also represent it as an ideal battery:

Vs

Vs

Figure 1A.4

Although a “real” battery is not ideal, there are many circumstances under

which an ideal battery is a very good approximation

Independent voltage source defined

An ideal battery is equivalent to an independent voltage source that has a constant value

function of time In this case we represent the voltage symbolically as v t

A few typical voltage waveforms are shown below The waveforms in (a) and (b) are typical-looking amplitude modulation (AM) and frequency modulation (FM) signals, respectively Both types of signals are used in consumer radio communications The sinusoid shown in (c) has a wide variety of uses; for example, this is the shape of ordinary household voltage A “pulse train”, such

as that in (d), can be used to drive DC motors at a variable speed

t v

(c)

t v

(a)

t v

(d)

t v

The Independent Current Source

An independent current source establishes a current which is independent of

the voltage across it The representation of an independent current source is

In other words, an ideal current source is a device that, when connected to

anything, will always push i out of terminal 1 and pull s i into terminal 2 s

Since the current produced by a source is in general a function of time, then the

most general representation of an ideal current source is as shown below:

AS 1102 IEC 60617

"intuitive"

Figure 1A.8

Independent current source defined

The most general representation of an ideal independent current source

The Resistor and Ohm’s Law

In 1827 the German physicist George Ohm published a pamphlet entitled “The Galvanic Circuit Investigated Mathematically” It contained one of the first efforts to measure currents and voltages and to describe and relate them mathematically One result was a statement of the fundamental relationship we now call Ohm’s Law

Consider a uniform cylinder of conducting material, to which a voltage has been connected The voltage will cause charge to flow, i.e a current:

This is Ohm’s Law The unit of resistance (volts per ampere) is referred to as

the ohm, and is denoted by the capital Greek letter omega, Ω

A simple resistor

Ohm’s Law

The ideal resistor relationship is a straight line through the origin:

Even though resistance is defined as Rv i , it should be noted that R is a

purely geometric property, and depends only on the conductor shape and the

material used in the construction For example, it can be shown for a uniform

resistor that the resistance is given by:

A

l

(1A.3)

where l is the length of the resistor, and A is the cross-sectional area The

resistivity,  , is a constant of the conducting material used to make the

resistor

The circuit symbol for the resistor is shown below, together with the direction

of current and polarity of voltage that make Ohm’s Law algebraically correct:

i

AS 1102 IEC 60617

The resistance of a uniform resistor

The circuit symbol for the resistor

Consider the circuit shown below

10

1    

R

v i

and:

mA10A01.01000

10

2      

R

v i

Note that i2 i1, as expected

Example

Consider the circuit shown below

50  3cos( ) A v t ( )

t Ri t v

The Short-Circuit

Consider a resistor whose value is zero ohms An equivalent representation of

such a resistance, called a short-circuit, is shown below:

arbitrarycircuit

circuit

Figure 1A.11

By Ohm’s Law:

V 0

(1A.4)

Thus, no matter what finite value i t has, v t will be zero Hence, we see that

a zero-ohm resistor is equivalent to an ideal voltage source whose value is zero

volts, provided that the current through it is finite

The short-circuit

The Open-Circuit

Consider a resistor having infinite resistance An equivalent representation of

such a resistance, called an open-circuit, is shown below:

arbitrarycircuit

arbitrarycircuit

v i

(1A.5)

Thus, no matter what finite value v t has, i t will be zero Thus, we may

conclude that an infinite resistance is equivalent to an ideal current source

whose value is zero amperes, provided that the voltage across it is finite

Practical Resistors

There are many different types of resistor construction Some are shown below:

with heat sink

array

chip - thick film chip - thin film

chip array

Figure 1A.13 – Some types of resistors

The “through-hole” resistors are used by hobbyists and for prototyping real

designs Their material and construction dictate several of their properties, such

as accuracy, stability and pulse handling capability

The wire wound resistors are made for accuracy, stability and high power

applications The array is used where space is a premium and is normally used

in digital logic designs where the use of “pull-up” resistors is required

Modern electronics utilises “surface-mount” components There are two

varieties of surface-mount chip resistor – thick film and thin film

Some types of resistors

Fundamental standardization practices require the selection of preferred values

within the ranges available Standard values may at first sight seem to be strangely numbered There is, however, a beautiful logic behind them, dictated

by the tolerance ranges available

The decade progression of preferred values is based on preferred numbers

generated by a geometric progression, repeated in succeeding decades In 1963, the International Electrotechnical Commission (IEC) standardized the preferred number series for resistors and capacitors (standard IEC 60063) It is based on the fact that we can linearly space values along a logarithmic scale so a percentage change of a value results in a linear change on the logarithmic scale

For example, if 6 values per decade are desired, the common ratio is

468.110

6  The six rounded-off values become 100, 150, 220, 330, 470, 680

The ‘E’ Series Values

The IEC set the number of values for resistors (and capacitors) per decade based on their tolerance These tolerances are 0.5%, 1%, 2%, 5%, 10%, 20% and 40% and are respectively known as the E192, E96, E48, E24, E12, E6 and E3 series, the number indicating the quantity of values per decade in that series For example, if resistors have a tolerance of 5%, a series of 24 values can be assigned to a single decade multiple (e.g 100 to 999) knowing that the possible extreme values of each resistor overlap the extreme values of adjacent resistors in the same series

Any of the numbers in a series can be applied to any decade multiple set Thus, for instance, multiplying 220 by each decade multiple (0.1, 1, 10 100, 1000 etc.) produces values of 22, 220, 2 200, 22 000, 220 000 etc

The ‘E’ series of preferred resistor and capacitor values according to IEC 60063 are reproduced in Table 1A.1

The IEC also defines how manufacturers should mark the values of resistors and capacitors in the standard called IEC 60062 The colours used on fixed leaded resistors are shown below:

0 1 red 2

Figure 1A.14 – Colour code marking of leaded resistors

The resistance colour code consists of three or four colour bands and is

followed by a band representing the tolerance The temperature coefficient band, if provided, is to the right of the tolerance band and is usually a wide band positioned on the end cap

IEC labelling for

leaded resistors

The resistance colour code includes the first two or three significant figures of

the resistance value (in ohms), followed by a multiplier This is a factor by

which the significant-figure value must be multiplied to find the actual

resistance value (i.e the number of zeros to be added after the significant

figures)

Whether two or three significant figures are represented depends on the

tolerance: ±5% and wider require two band; ±2% and tighter requires three

bands The significant figures refer to the first two or three digits of the

resistance value of the standard series of values in a decade, in accordance with

IEC 60063 as indicated in the relevant data sheets and shown in Table 1A.1

The colours used and their basic numerical meanings are recognized

internationally for any colour coding used in electronics, not just resistors, but

some capacitors, diodes, cabling and other items

The colours are easy to remember: Black is the absence of any colour, and

therefore represents the absence of any quantity, 0 White (light) is made up of

all colours, and so represents the largest number, 9 In between, we have the

colours of the rainbow: red, orange, yellow, green, blue and violet These take

up the numbers from 2 to 7 A colour in between black and red would be

brown, which has the number 1 A colour intermediate to violet and white is

grey, which represents the number 8

The resistor colour code explained

When resistors are labelled in diagrams, such as schematics, IEC 60062 calls for the significant figures to be printed as such, but the decimal point is replaced with the SI prefix of the multiplier Examples of such labelling are shown below:

Resistor Value IEC Labelling

Note how the decimal point is expressed, that the ohm symbol is shown as an

R, and that 1000 is shown as a capital K The use of a letter instead of a decimal point solves a printing problem – the decimal point in a number may not always be printed clearly, and the alternative display method is intended to help misinterpretation of component values in circuit diagrams and parts lists

In circuit diagrams and constructional charts, a resistor’s numerical identity, or

designator, is usually prefixed by ‘R’ For example, R15 simply means resistor

Kirchhoff’s Current Law

A connection of two or more elements is called a node An example of a node

is depicted in the partial circuit shown below:

Even if the figure is redrawn to make it appear that there may be more than one

node, as in the figure below, the connection of the six elements actually

constitutes only one node

Kirchhoff’s Current Law (KCL) is essentially the law of conservation of electric charge If currents directed out of a node are positive in sense, and currents directed into a node are negative in sense (or vice versa), then KCL can be stated as follows:

KCL: At any node of a circuit, the currents algebraically sum to zero

1      

Note that even if one of the elements – the one which carries i – is a short-3

KCL defined

The directions of i1, i2, i and the polarity of v were chosen arbitrarily (the 3

directions of the 13 A and 2 A sources are given) By KCL (at either of the two

nodes), we have:

02

11611

116

236

1132

1132

v v v

v v v

v v v

Having solved for v, we can now find that:

A61

61

1  v  

2

62

3

63

i3  v  

Just as KCL applies to any node of a circuit, so must KCL hold for any closed region, i.e to satisfy the physical law of conservation of charge, the total current leaving (or entering) a region must be zero

1 i i i

i   

For Region 3:

4 5

2 i i

i  

You may now ask, “Since there is no current from point a to point b (or vice

versa) why is the connection (a short-circuit) between the points there?” If the connection between the two points is removed, two separate circuits result The voltages and currents within each individual circuit remain the same as before

Having the connection present constrains points a and b to be the same node,

and hence be at the same voltage It also indicates that the two separate portions are physically connected (even though there is no current between

Kirchhoff’s Voltage Law

Starting at any node in a circuit, we form a loop by traversing through elements

(open-circuits included!) and returning to the starting node, never encountering

any other node more than once

For example, the paths fabef and fdcef are loops:

f

Figure 1A.19

Loop defined

energy If voltages across elements traversed from – to + are positive in sense, and voltages across elements that are traversed from + to – are negative in sense (or vice versa), then KVL can be stated as follows:

KVL: Around any loop in a circuit, the voltages algebraically sum to zero

KVL can also be stated as: In traversing a loop, the sum of the voltage rises equals the sum of the voltage drops

Example

In the circuit shown in Figure 1A.18, we select a traversal from – to + to be

positive in sense Then KVL around the loop abcefa gives:

0

6 8 3 2

The polarities of v1, v2, v and the direction of i were chosen arbitrarily (the 3

polarities of the 10 V and 34 V sources are given) Applying KVL we get:

034

10v1  v2 v3 Thus:

24

3 2

2412

246

42

246

42

i i i

i i

V8244

V4222

3 2 1

i v i v

Combining Resistors

Relatively complicated resistor combinations can be replaced by a single equivalent resistor whenever we are not specifically interested in the current, voltage or power associated with any of the individual resistors

Series Resistors

Consider the series combination of N resistors shown in (a) below:

v1

arbitrary circuit

R i R v

2 1

(1A.12)

and then compare this result with the simple equation applying to the equivalent circuit shown in Figure 1A.20b:

i R

Thus, the value of the equivalent resistance for N series resistances is: