Electrical Engineering 1 1 Basic Concepts

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1 Basic Concepts 1.1 Electric Circuits and Current Flow The student will be able to:

- List the most important effects caused by the electric current - draw a simple wiring diagram with a battery and a bulb, and show the direction of the current flow - give a definition of the unit of the current (ampère) .

To explain some commonly-known effects, we start with an experiment (see fig. 1.1-a): Consider a simple circuit consisting of commonly-known electrical elements, a bulb, a battery, and a switch. Elements are sometimes also called devices. Wires interconnect the elements and a compass is installed close to the wires. After closing the switch we can observe that

- the filament (incandescent ~) is heated and gives light - the compass needle is deflected - a chemical reaction occurs inside the battery.

We usually say that the electric current (elektrischer Strom) is responsible for all the effects shown. It only exists if a closed circuit – simply called circuit (Stromkreis) – exists. An electric circuit or electric network is an interconnection of electrical elements linked together in a closed path so that an electric current may flow continuously. We do not have sense organs for an electric current; we can only observe its effects:

- generation of heat (thermal energy) - generation of magnetic phenomena - generation of chemical reactions (changes) inside the material through which a current flows.

As these effects of the electric current only occur as long as the circuit is closed, we model the current as a movement of “elements” which circulate in the circuit. When a current flows through a metallic conductor no material is being transported. We conclude that only elements of the atoms, electrons, not the atoms themselves are moving. We know that a battery has a positive pole (+) and a negative pole (-). If we interchange the battery terminals, the needle is deflected in the opposite direction. From this result we define a reference direction (Richtungssinn) for the current: the current is flowing from the positive terminal of the battery through the bulb to the negative terminal, generally:

- outside the battery: from plus to minus - inside the battery: from minus to plus.

The structure of the introductory experiment can be described clearly using a circuit diagram (Schaltplan, see fig. 1.1-b). We use graphical symbols (Schaltzeichen) for the elements connected (e.g. battery (Batterie), clamps (Klemmen), wires (Drähte), leads ((Verbindungs-~)Leitungen), lines (Leitungen),

fig. 1.1-a: experiment to show the physical effects caused by an electric current

+-

I

battery bulb

lead, wire, line

detachable clamp

switch

fig. 1.1-b: circuit diagram including symbols and reference direction of the current

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resistors, transistors etc.). The physical quantity (physikalische Größe) used to describe an electric current is simply called current (elektrische Stromstärke, elektrischer Strom). For all physical quantities standardized symbols are used; for the current the symbol I has been chosen. In the circuit diagram the direction of the current is shown using an arrow on the line, or close to it on either side, and the symbol I is written by the arrow. Generally, each of the phenomena caused by a current could be used to define it. Today, the unit for the current is defined based on the magnetic effect. Experiment: Two thin metallic bands are arranged in parallel. They are connected to a battery B using a switch S. Observation: When the switch is closed, a circuit is built and a current will flow. A force will act on the bands due to the magnetic field built by a current-carrying conductor (see EMF). 1.2 SI-Units To understand the behaviour of an electrical element (e.g. the battery, the bulb etc.), we need to consider certain quantities associated with it, such as current and voltage. These quantities (and others), must be carefully defined. This can be done only if we have a standard system of units so that when a quantity is described by measuring it, we can all agree on what the measurement means. The system of units we employ today is the international system of units (le Système Internationale d’Unités), which is normally referred to as the SI standard system, adopted in 1960 by the General Conference on Weights and Measures. The SI system is composed of six basic units, in addition the unit for the amount of substance has been introduced. The SI units are very precisely defined in terms of permanent and reproducible quantities.

SI unit SI quantity name symbolLength meter m Mass kilogram kg Time second s (electric) current ampère A thermodynamic temperature kelvin K amount of substance mole mol luminous intensity candela cd

+ -

S open

S closed

S

B

FF

I

The unit introduced for the physical quantity current I is 1 ampere (1 A). Definition (1 ampere): One ampere is the magnitude of a constant electric current flowing through parallel conductors arranged in vacuum, having infinite length and negligible circular cross-section, and being separated at a distance d = 1m if the measured forces per unit length of the conductors is F = 2 * 10-7 N

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The derived units for other physical quantities are obtained by combining the base units. Some derived units are given as an example in the following table together with their formulas in terms of the fundamental (or base) units.

unit (derived) quantities name symbol formulaacceleration m/s² velocity m/s1 force Newton 1N 1kgm/s2

energy, work Joule 1J 1Nm electric charge Coulomb 1C 1As power Watt 1W 1J/s voltage Volt 1V 1J/C

The advantage of the SI system is that it incorporates a decimal system for relating larger or smaller quantities to the basic units. The powers of 10 are represented by standard prefixes:

factor name literal Factor name Literal10-18 atto a 1018 Exa E 10-15 femto f 1015 Peta P 10-12 piko p 1012 Tera T 10-9 nano n 109 Giga G 10-6 mikro µ 106 Mega M 10-3 milli m 103 kilo k 10-2 centi c 102 hekto h 10-1 deci d 101 Deka da

Example for the common use of a prefix. The centimetre (cm), which is 0.01 m. The decimal multiplier must always accompany the units and is never written by itself. Examples: 2,500 m = 2.5 * 103 m = 2.5 km; 12.5 µA = 12.5 . 10-6 A = 1.25 . 10-5 A = 0.0000125 A Exercises:

- If a microprocessor can perform 4 million instructions per second, how many nanoseconds are required to execute an operation requiring 5 instructions? A: 1,250 ns

- The fastest recorded speed run by a human being for 10 m is approximately 39.8 km/h. If this speed were sustained over 400 m, how long would it take to cover that distance? A: 36.2 s

1.3 Charges and Atomic Model We are familiar with the gravitational force of attraction between bodies, which is responsible for holding us on earth. There are also much stronger forces which are both attractive and repulsive. One of the most important of these forces is electrical and is caused by the presence of electrical charges (elektrische Ladungen). We explain the existence of electrical forces (elektrische Kräfte) of both attraction (Anziehung) and repulsion (Abstoßung) by postulating that there are two kinds of charges (Ladungsarten), positive (positiv) and negative (negativ), and that unlike (ungleichnamige) charges attract and like (gleichnamige) charges repel. From a simple experiment we know that two types of charges exist, marked by a positive (+) or a negative (-) sign. Our experiences with electricity have their roots in the observation that, for example, a piece of glass and a piece of resin (or rubber) attract one another if they are first rubbed together and then

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separated.1 Also, if a second piece of glass is rubbed with another piece of resin, the two pieces of glass or resin repel one another, while each glass piece attracts each peace of resin. These electrical phenomena of attraction and repulsion are understood in terms of electric charge. When glass and resin, for example, are rubbed together, a small amount of charge is transferred from one to the other, causing each material to become non-neutral or charged (geladen). Materials behaving on electrification like glass are said to be positively charged (positiv geladen). A rubber rod rubbed with a silk, and materials behaving like resin are said to be negatively charged (negativ geladen). Charges of the same sign repel each other, while charges of opposite sign attract (fig. 1-2a). The forces exerted by charges are called Coulomb forces2: For example, consider two charged spheres carrying charges Q1 and Q2, separated at a distance r. The force exerted on one charge by the other varies directly as the product of the charges (the sign of the charge is included in its value) and inversely as the square of the distance between them (Coulomb’s law): The constant of proportionality, k, is approximately 9·109 Nm²C-2 when the other parameters are given in SI units and the surrounding medium is free space (vacuum). If two spheres carrying opposite charges touch each other, the electric forces vanish immediately. The charges are balanced (charge balance, Ladungsausgleich), we say they are discharged (entladen). The coulomb forces also occur in vacuum. To describe this phenomenon, we use a model called the field model (Feldmodell): Each charge changes the state of the space surrounding the charge, so that forces are exerted on other charges. This state (due to a charge) is called electric field (elektrisches Feld, see EMF for further discussion). To describe both, the magnitude and the direction of the force exerted on a charge, we use a vector quantity, called electric field strength (elektrische Feldstärke) Ē .

Definition (electric field strength E): The electric field strength E (at a certain point in space) is the force F exerted on a positive charge (at that point) devided by the charge Q.

Q

FE (1.2)

So, the direction of E shows in the direction of F if Q is positive. The force onto a negative charge is directed in the opposite direction (fig. 1-2b). An electric field is also responsible for the movement (Bewegung) of charges in a conductor. It is set up from the positive and negative charges of the battery poles, and is built in the conductor between. 1 Thales of Miletus (640-540 B.C.) wrote that a piece of amber rubbed in silk attracts pieces of straw. 2 Charles A. Coulomb (1736-1806)

2

29

02

21 1094

1;

C

Nmk

r

QQkF

(1.1)

fig. 1-2 a: coulomb forces on charges

fig. 1-2 b: forces on charges in a homogeneous electric field

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Now, we want to know where the positive and negative charges exist. They are properties of the atoms. To describe their structure we use the Bohr Sommerfeld atomic model: 3 In this classical model atoms (Atome) are built from protons (Protonen) and neutrons (Neutronen), arranged in a so-called nucleus (Kern), surrounded by electrons (Elektronen) circulating around the central nucleus in an electron shell (Elektronenschale). Most of the mass of the atom is concentrated in the nucleus, the mass of an electron is roughly 1/2000 of the mass of a proton or neutron. The charge is quantized, which means that a smallest possible charge exists, called elementary charge (Elementarladung) e = 1.602 . 10-19 coulombs (C). Electrons carry a negative elementary charge, protons carry a positive elementary charge. Each charge is an integral multiple of the elementary charge. Neutrons do not possess a charge. The definition of which of the charges is defined positive or negative was chosen arbitrarily.

elementary particle rest mass / kg charge

proton 1.673 . 10-27 +e

neutron 1.675 . 10-27 0

electron 0.911 . 10-30 -e

The coulomb forces between protons and electrons are responsible for the attraction of electrons and protons in atoms. These forces lead to the centripetal force keeping the electrons on orbit. Strong attractive forces in the nucleus (van der Waals forces) keep the nucleus together. Generally, in an atom the number of electrons and the number of protons is equal. This electron doesn’t cause electric forces, it is electrically neutral (neutral), uncharged. In the Bohr Sommerfeld atomic model electrons circulate on definite shells (called K, L, M, etc.). The maximum number of electrons possible on the shells depend on the shell number. If we number the shell beginning with n = 1 for the inner shell the maximum number of electrons on the n-th shell is 2 n². The outermost shell where electrons can be found is responsible for both the electrical and chemical behaviour of the atom. These electrons are called valency electrons (Valenzelektronen). Not in all cases, the inner shells are fully occupied. Example:

element symbol total

K

number of L

electronsM

N

O

P

hydrogen H 1 1 helium He 2 1 1 oxygen O 8 2 6 neon Ne 10 2 8 aluminum Al 13 2 8 3 copper Cu 29 2 8 18 1 silver Ag 47 2 8 18 18 1 gold Au 79 2 8 18 32 18 1

A more comprehensive description can be given using the orbital model of quantum physics. But, it is not really necessary to be able to understand the basic technical concepts. Each electron can be attributed a certain energy (Energie). The lowest possible energy is called normal energy level or ground state (Grundzustand). By supplying energy an electron can be excited (angeregt; excited state, Anregung or angeregter Zustand). During transition from a state of higher energy to a lower

3 Niels Bohr (1885-1962), Arnold Sommerfeld (1868-1951)

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level the atom can radiate light. The frequency (or wavelength) of the light emitted is interrelated with the difference in energy. Practical aspect (example): A L(ight) E(mitting) D(iode) is a semiconductor device which is able to emit light if a current is supplied to the diode. Due to the current (energy) the electrons are excited to a higher energetic level. When falling back to the ground state, each electron will emit a photon of wavelength = c h / W

c velocity of light in vacuum: c 2.998 108 m/s h Planck constant h 6.64 * 10-34 Js 4.16 * 10-15 eV s ( 1eV = 1.602 * 10-19 J) W energy difference of the charge

For a GaAsP diode an energy difference of W 1.92 eV must be supplied. So, the wavelength of the light emitted will be = c h / W 2.998 108 m/s * 4.16 * 10-15 eV s / 1.92 eV = 6.5 * 10-7 m = 650 nm If more energy is supplied, e.g. by heat, an atom can be ionised. There are two possibilities: One or more electrons can be removed (example for H(ydrogen): W 2.2 * 10-18 J). A positive ion

(Ion) is created, which acts like a positive charge. One or more electrons are added. A negative ion will be created, acting like a negative charge. Normally, ions try to recombine (rekombinieren). Elements (atoms, molecules, elementary particles) which cause electric effects are called [charge] carriers (Ladungsträger), e.g. electrons, ions, protons. 1.4 Conductor and Insulator - Semi-conductor The movement of charges differs between different material. The main questions which have to be answered are:

- How many free carriers (per volume) can we find in a given material? - How good can they move in the material?

Depending on the amount of carriers and their mobility the material is divided into three groups: conductors, semi-conductors and insulators. A material which is well-suited to conduct an electric current is called conductor (Leiter). In conducting material many freely movable carriers are available, such as metals (Ag (silver), Cu (copper), Al (aluminium), Au (gold)). The atoms of pure metals (high-grade) are arranged to form a periodic three-dimensional lattice (Gitter), e.g. the crystal structure for copper (cu), an arrangement which is called face-centered cubic. At room-temperature the atoms are ionised. The atomic residues oscillate around their position of rest. The oscillation amplitude increases with higher temperature. The valency electrons can move quasi freely between these residues (diameter of atoms: 0.1 nm, diameter of nucleus 1 fm); so, they are called free electrons (freie Elektronen). Normally, they cannot leave the metal. Only if the work of emission (Austrittsarbeit) is supplied can the attractive forces between electrons and nucleus be overcome, and the electrons will leave the metal (e.g. oscilloscope, see EMI). First, the electrons move randomly (ungeordnet) in the lattice and this motion cannot be observed from outside the material. An electric current will only flow when a privileged direction is superimposed by an external force (electric field). To become a good conductor a metal must be of high quality (nearly pure, e.g. E-Cu has a purity of more than 99.98 %). Even very low pollution reduces the movability of the free electrons significantly, and the conduction properties of the material are significantly reduced.

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Example: We want to estimate how many free electrons we can expect in 1 cm³ of copper (Cu): material properties and physical constants: relative atomic mass 63.54 (the mass of 1 mole of copper; 63.54 g) density = 8.93 g cm-3 constant of Avogadro: N = 6.022 . 1023 mol-1 (number of atoms per mole) 1 valency electron per atom electrical properties one free electron per atom (at room temperature) solution: number of atoms (nuclei) in 1 g of Cu: nn = 6.022 . 1023 1/mol / 63.54 g/mol 9.48 *. 1021 g-1

number of free carriers in 1 cm³ of Cu: ne = nn = 9.48 . 1021 g-1 ( 9.48 . 1021 g-1 * 8.93 g/cm³) 8.5 * 1022 cm-3.

Roughly 1023 free electrons are available in cm³ of copper. Exercise: Find the number of free electrons in silver (which is the best electrical conductor). A second group of conductors is built from acids, alcalines and saline solutions. Positive and negative ions are the carriers in these liquids. In case of a current flow mass will be transported (charging and discharging of accumulators, galvanization techniques etc.). A material where no free charges occur is called an insulator (Isolator). Only vacuum is an ideal insulator. However, in technical insulators the number of free carriers is very small, gases (molecular structure), glass, porcelain, silk, paper, rubber, etc. Also many liquids are insulators, like water or oil. The latter is used if good cooling properties are needed, e.g. in electrical machines. If a large amount of energy is supplied to these materials (heating), free charges can be built and the insulating properties vanish. Usually, when a larger number of charges is generated in an insulation material, the insulating properties vanish, and the material very often will be destroyed. In semiconducting material (semiconductors, Halbleiter) at room temperature only a small number of free (mobile) charges exist (typically: 1010 .. 1014 cm-3); they behave more like insulators. If energy is supplied (e.g. the temperature is increased), the number of charges can be increased so that the material behaves more like a conductor. The number of free carriers can also be increased by adding foreign atoms (impurities) called doping (dotieren). Examples:

Semiconductor material Application chemical elements:

carbon (C) selenium (Se) germanium (Ge) silicon (Si)

resistor rectifier diode, transistor diode, transistor

compounds: gallium arsenide (GaAs) indium antimonide (InSb) oxide of zinc

luminescent diode, LED Hall element varistor

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1.5 Charge and Current Current is the movement of carriers. It can be the movement of only positive carriers, or of only negative carriers, or of both at the same time. In metals we only have a movement of electrons, which are negative carriers. Positive carriers, free protons, do not occur in technical applications. Moving positive charges can only be ions (in liquids or gases) or holes (in semiconductors). In liquids and semiconductors positive and negative carriers can move at the same time. Henry Rowland4 proved experimentally that the magnetic effect of the electric current results from the movement of carriers. He showed that positive carriers moving in the direction of the reference direction cause the same magnetic effect as a current flowing in this direction. If instead, negative carriers are moving opposite to the reference direction, we will find the same result. Thus, we need a definition for the direction which is independent from the type of carrier.

Definition (current reference direction, Strombezugsrichtung):

A current flowing in a conductor from cross-section 1 to cross-section 2 is defined positive, if positive carriers move from 1 to 2 or negative carriers move from 2 to 1. Fig. 1-3 a shows two conductors, connected between the poles of a battery. Positive carriers only can move in conductor 1, and negative carriers can only move in conductor 2. The moving directions for the carriers is different, however, the current reference direction is independent. In the special case of a metallic conductor, the carriers (electrons) move opposite to the current reference direction.

4 Henry A. Rowland (1848-1901)

batt

ery

+

batt

ery

-

1

2

2

1

++ +

+ ++

--

----

conductor 1

conductor 2

I

fig. 1-3 a: current reference direction and movement direction of carriers

+-

I

movement of electrons

current reference direction

fig. 1-3 b: current reference direction and direction of electrons in metallic conductors

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In the next step, we want to find the relationship between current and the quantity of electricity (Ladungs-menge) Q. We are looking at a certain cross-section A of a given conductor (see fig. 1-4) and counting the number of carriers passing the cross-section during a time-interval t.

Current is the rate of movement of electric charge. Certain amounts of positive charge QR

+ and negative charge QR- are moving to the right across a cross-

section of the cylinder, and similar quantities, QL+ and QL

-, are moving to the left. The net positive charge moving to the right QR is QR = QR

+- QR

-- QL

++ QL

-

since negative charge moving in one direction is equivalent to positive charge moving in the opposite direction. If we observe the charge crossing an area of the cylinder in a certain time interval, let’s say t, then the current directed to the right is defined as the net positive charge transferred to the right per unit of time:

t

Q

t

Qi R

(1.3a)

This rate of movement may not be constant but may vary with time. Hence the general definition of current is

dt

dqi (1.3b)

The unit of current is the ampere (A): ;1

1

1A

s

C

t

qi 5

)(1

)(1)(1

econds

oulombCmpèreA

Charge is the quantity of electricity responsible for electric phenomena. Current (Stromstärke, Strom) is the time rate of flow of electric charge past a given point. Note: Lower case letters are used, such as q, to denote a variable that is a function of time, q(t). We use an upper case letter, such as Q, to represent a constant. As a matter of vocabulary, we say that a current exists in or through an element. In circuit theory, current is specified as the movement of positive charges, called conventional current (technische Stromrichtung). This convention was developed by Benjamin Franklin. Today, we know that charge flow in metal conductors results from electrons. Nevertheless, we will conceive current as the flow of positive charge, according to accepted convention. Although we know that current flow in metallic conductors results from electron motion, the conventional current flow, which is universally adopted, represents the movement of positive charges.

5 read [x] as “the unit of (the physical quantity) x is”. Example: [F] = 1 N (the unit of (the physical quantity) force is 1 N(ewton)

A++ +

+ ++-

- --

- -I

fig. 1-4: positive and negative carriers flow through cross-section A

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Current flow along a lead or through an element will be specified by two indicators: an arrow (Pfeil), which establishes a current reference direction (Strombezugspfeil) and a value (Wert) which quantifies the current flow in the reference direction: The current i1 is the flow of electric charge from terminal A to terminal B. If i2 is the flow of electric charge from terminal b to terminal A, it must be the same size but have different direction. Therefore, i2 is the negative of i1 12 ii Note: Without a reference direction a current’s numerical value alone is not enough to specify the current. Several types of currents are frequently encountered: If a current is constant, we represent it by the constant I. A constant current is called direct current (or dc, Gleichstrom). A time-varying current

)(tii can take many forms, such as a ramp, a sinusoid, an exponential, etc. The sinusoidal current is called an alternating current (or ac, Wechselstrom). Examples:

a dc current: Ii solar cell, electronic circuits a ramp with slope c: sttci 0, saw tooth generator

a sinusoid (sine current): 0),sin(ˆ ttii household

an exponential: 0,0 teIi at switching

i1i2A A

B B

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1.5.1 Current, Current Density and Charge Density The current was defined to be the rate of change of charge per unit time crossing an area of a conductor. The density of charges (electrons) in metals is about n 1023 free electrons/cm³. When connecting a voltage source to a conductor (of given length l and cross-section A) free electrons move through the conductor at a certain speed v. The total charge in the given volume V = Al is neVQ (1.4 a) passing the cross-section in the time t = l/v. From this we find the current flow through the conductor

neAvt

lneA

t

neV

t

QIi

(1.4 b)

As already mentioned the current reference direction shows in the direction of moving positive charges which means opposite to the direction of the moving electrons. Example #5: Given a current I = 1 A, flowing through a copper wire (cross-section: A = 1.5 mm², l = 1 m), find the speed of the electrons. Solution:

s

mm

mmCmm

A

neA

iv 0416.0

²5.110602.1³10

11920

Up to very high currents electrons move slowly. Another important quantity is the current-density (Stromdichte) J defined to be the current per cross-section.

nevA

neAv

A

IJ (1.5 a)

Example #5 (continued): (given: I = 2 A, A = 1.5 mm²) Find the current-density in the wire. Solution:

²

33.1²5.1

2

mm

A

mm

AJ

Practical values: J 10 A/mm² (typical values: 1-2 A/mm²). Equation (1.6a) holds in homogeneous media only. If the material is inhomogeneous the current-density may vary over the cross-section and J will be found by differentiating I with respect to the cross-section:

dA

dIJ (1.5 b).

It should be clear that the speed of the electrons may vary over the cross-section if the conductor is arbitrarily shaped. If v varies, so does J.

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1.5.2 Current and Magnetic Forces Electric currents produce forces as do stationary electric charges. The forces produced by currents (movement of charge) are called magnetic forces, and they possess many of the properties of those produced by ordinary bar magnets. Magnetic field lines surrounding a current-carrying conductor are cylindrical. The direction of the magnetic field lines is given by the right-hand rule: if the thumb of the right hand is pointed in the direction of the current, the fingers will point in the direction of the magnetic field. The needle of a compass that is placed in this field will align with the field lines. Example #6: Given two wires arranged in parallel: I1 = I2 = 1 A, l = 10 m, d = 2.5 mm, find the forces between the wires. Solution:

mNm

mAA

A

NF 8.0

³105.2

1011102

27

Usually, in a household the forces acting between parallel wires are negligibly small: per unit length (1 m) the values lie in the range of some mN. If in the case of a short-circuit the maximum possible current occurs, (65 A) the forces will still be small enough. 1.5.3 Charge and Current If the charge q is known, the current can be found using equation (1.2). Conversely, if the current is known, the charge can also be calculated. The total charge qT entering a general circuit element with a current flowing from the left towards the right terminal in any time range, let’s say between t = t0 and t = t1, can be found by integrating equation (1.2)

1

0

)()()( 01

t

t

T ditqtqq (1.7a)

This can be rewritten as follows. If the total charge q0 at time t0 is given, we find the charge at t1 by integrating i(t) in the time range t0 t t1 and adding the value of q0.

01

1

0

)()( qditqt

t

where

0

)()( 00

t

ditqq (1.7b)

27

21

102A

Nk

d

lIIkF

(1.6)

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Example #1: For the current defined piecewise as follows:

st

stss

t

stA

t

ti

30

31

101

00

)(

a) Sketch the graph of the function i(t) b) Determine the total charge transported in the time range 0 t 4s. Example #2: For the current given as a graph a) find the function i(t) mathematically. b) Determine the function for the charge q(t), if q(0) = 0 C. c) Sketch q(t). Example #3: If the total charge which has entered a terminal of an element is

stemCmC

sts

tmC

st

tq

st 224

202

00

)(

)2/(2

a) find the current i(t) in the time range –1s t 3s b) sketch the function i(t) and q(t). Example #4: For example #1 find the total charge transported.

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1.6 Work and Voltage Since charges exert forces on other charges, energy must be expended in moving a charge in the vicinity of other charges. Thus charges produce a type of force field. The unit of energy is the joule (J), defined as the energy extended in the exertion of a force of one Newton in moving an object through a distance of one meter (1 J = 1N m). For example, consider moving a charge q from a point a to another point b along a chosen path in the presence of another charge Q. Over certain portions of the path, the force exerted by Q may oppose the movement, while over other portions the force may be in the direction of q as to aid the movement of q. Hence the net energy Wab expended in moving the charge q from a to b may be positive or negative. This is similar to gravitational potential energy. For example, lifting a mass of m = 100 g (chocolate) to a height h = 1m above the earth requires work on our part against the gravitational force (F = m g 100 g 10 ms-2 = 1 kg m s-2 =: 1 N(ewton)). We say that points above the earth are at a higher gravitational potential with respect to the earth as our reference. The change of (potential) energy stored in the mass is

hmgsFW (1.8 a) Similarly, if the movement of a charge Q requires work, then we say that the voltage of point b with respect to point a is the work per unit charge required to move the charge from point a to point b:

Q

WU (1.8 b)

Alternatively, we say that there is a potential difference between points a and b. 1.6.1 Voltage The basic variables in an electrical circuit are current and voltage. These variables describe the flow of charge through the elements of a circuit and the energy required to cause charge to flow. The notation we use to describe the voltage consists of two parts:

- a value (perhaps represented by a variable name, e.g. v, u) and - an assigned direction (voltage reference direction).

There are two ways to label the voltage across an element. The voltage uAB is the proportional to the work needed to move a positive charge from terminal A to terminal B. Conversely, uBA is the work needed to move a positive charge in the opposite direction (from B to A). The voltages uBA and uAB are similar but different. They have the same magnitude but opposite directions:

ABBA uu

Voltage across a circuit element

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Sometimes we read uAB as the voltage (or potential difference) at terminal A with respect to terminal B. Alternatively, we sometimes say that uAB is the voltage drop (Spannungsfall) from terminal A to terminal B. When considering uAB , terminal A is called “positive terminal” (or + terminal, terminal at higher potential) and terminal B is called “negative terminal” (or – terminal, terminal at lower potential). The voltage across an element is defined to be the work (energy) required to move a unit positive charge (+1 C) from the negative terminal to the positive terminal. The unit of the voltage is the volt, V. Some practical values:

telecommunication 1 µV electronic circuits (TTL) 1 .. 40 V (5V) accumulators in cars 12 V (24 V) household 230 V (US: 110 V) energy distribution 10 .. 30 kV 110 .. 400 kV (1100 kV) (bolt of) lightning 10 .. 50 MV

1.6.2 Energy and Power In transferring charge through an element, work is being done (or energy is being supplied). Let us consider now the rate at which energy is being delivered to or by a cir- cuit element. The power and energy delivered to an element are of great importance. For example, the output of an electric light bulb can be expressed in terms of power. We know that a 100-watt bulb delivers more light than a 25-watt bulb. If the voltage across the element is u and a charge q is moved through the element from the positive to the negative terminal, the energy absorbed by the element, say w, is given by quw

If the time involved is t, then the rate at which the work is being done, is given by

t

qu

t

w

(1.9 a)

If t gets smaller and smaller, we finally find

iudt

dqup

dt

dw

t

w

t

)(lim0

(1.9 b)

Power is simply the product of the voltage across an element times the current through the element. Power (Leistung) is the time rate of expending or absorbing energy. Power has the unit of watts, W. (p6=w/t=uq/t=V C/s = V As / s = VA =: W) The quantities u and i are generally functions of time. Therefore p is a time-varying quantity. It is sometimes called the instantaneous power (Augenblickswert der Leistung) because its value is the power at the instant of time at which u and i are measured. Two circuit variables correspond to each element of a circuit: a voltage and a current. There are two different ways to arrange the directions of this current and voltage. If the current reference direction arrow and the voltage reference direction arrow point in the same (opposite) direction(s) which means that the 6 x read “unit of x”; example: U = “unit of voltage” = 1V

Work and Power in a mechanical systems (forklift)

16

arrows are in parallel (anti-parallel), the current and voltage so defined are said to satisfy the passive sign convention (Verbraucherpfeilsystem) (active sign convention (Erzeugerpfeilsystem). In the passive (sign) convention, the voltage indicates the work required to move a positive charge in the direction indicated by the current. Accordingly, the power p = ui is the power absorbed (aufgenommene Leistung) by the element (also called “the power dissipated by the element” or the “power delivered to the element). The power absorbed can be either positive or negative. This depends on the values of the element voltage and current. In the active (sign) convention, the voltage indicates the work required to move a positive charge in the direction opposite to the direction indicated by the current. Accordingly, the power is the power supplied by the element (also called “the power delivered by the element”). Again, the power delivered by an element can either be positive or negative, depending on the values of the element voltage and current. Example: (to illustrate the concept of power delivered to or absorbed by an element) Char- ging a discharged automobile battery The power absorbed by an element and the power delivered by the same element are related by power absorbed = - power delivered Summary:

{u}7 {i} {p}

> 0 < 0

> 0 < 0

> 0: absorption power absorbed (or “dissipated”)

passive sign convention

> 0 < 0

< 0 > 0

< 0: supply power supplied (or “delivered”)

> 0 < 0

> 0 < 0

> 0: supply

active sign convention

> 0 < 0

< 0 > 0

< 0: absorption

When the element voltage and current adhere to the passive convention, the energy absorbed by an element can be determined from equation 1.9 pdtdw On integrating, we get

t

dpw )( (1.10 a)

If the element only receives power for t t0 and we let t0 = 0, then we have

t

dpw0

)( (1.10 b)

7 x magnitude of x; example: if U = 1V, then U = „magnitude of U“ = 1

Electrical Engineering 1 1 Basic Concepts

EE1-WS1213-Ise 17/17

Example #7: a) Given U = 4 V, I = 10 A, a1) passive convention a2) active convention Find the power absorbed (or delivered) and the energy absorbed (or delivered) over a 10 s interval. b) Given U = - 4 V, I = 10 A, b1) passive convention b2) active convention Find the power absorbed and the energy absorbed (or delivered) over a 10 s interval. c) Given u = 8 e-t, i = 2.5 e-t, active convention. Find the power delivered and the energy supplied by this

element over the first second of operation. Assume that both u and i are zero for t 0. 1.7 Passive and Active Elements Electric circuits consist of interconnections of circuit elements. Circuit elements, e.g. a battery, a bulb, a switch, wiring, etc., may be represented by a model of their behaviour described in terms of the terminal current and voltage of each element. We may classify circuit elements in two categories, passive and active, by determining whether they absorb energy or supply energy. A circuit element is said to be passive (passiv) or (active (aktiv)), if it cannot (can) deliver more energy than has previously been supplied to it by the rest of the circuit. That is, at each t the net energy absorbed by a passive (supplied by an active) element up to t must be nonnegative (negative):

0)()()(:

0)()()(:

tt

tt

duidptwwtwactive

duidptwwtwpassive

(1.10 a,b)

If (1.10 a) does not hold at each time t, the element is active. Examples of active elements are batteries, generators, electronic power supplies, etc. The feature of an electric circuit component is that its behaviour is described in terms of a voltage-current relation. The “voltage-current” characteristic may be obtained experimentally or from physical principles. Although no device is exactle linear for all values of current, we can often assume a range of linear operation. Example: voltage-current relationship (with assumed range of linearity) for an incandescent lamp a diode