lecture notes 1 final

7/28/2019 Lecture Notes 1 Final

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DIGITAL AND ANALOG CIRCUITS

1.Basic elements and components in electric circuits

Faculty of Information Technologies CTU in Prague

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Lecture Notes 11. Basic elements and components in electric citcuits1.1 Introduction - the basic difference between analog and digital technology1.2 Electrical circuits and their parameters

1.3 Shining a light as an example

1.3.1. Electric current

1.3.2 Electric voltage

1.3.3. Which way does the current flow?

1.3.4 Resistance

1.3.5. Ohms equations

1.3.6 Electrical laws

1.3.6.1 DC circuit equations and laws

a) Ohms and Joules laws

b) Kirhoffs circuit laws1,4 Electrical elements

1.4.1 Elements versus components

1.4.2 Electrical resistance

1.4.3 Electrical capacitance

1.4 .4 Electrical inductance


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1. Basic elements and components in electric citcuits

1.1. Introduction - the basic difference between analog and digital

technologies

First, we should explain the basic difference between analog and digital technologies.

In analog technology, a wave is recorded or used in its original form. So, for example,

in an analog tape recorder, a signal is taken straight from the microphone and laid onto tape.

The wave from the microphone is an analog wave, and therefore the wave on the tape is

analog as well. That wave on the tape can be read, amplified and sent to a speaker to produce

the sound.

In digital technology, the analog wave is sampled at some interval, and then turnedinto numbers that are stored in the digital device. For example, on a CD, the sampling rate is

44000 samples per second. So on a CD, there are 44000 numbers stored per second of music.

To hear the music, the numbers are turned into a voltage wave that approximates the original

signal.

The two big advantages of digital technology are:

The recording does not degrade over time. As long as the numbers can be read, youwill always get exactly the same wave.

Groups of numbers can often be compressed by finding patterns in them. It is alsoeasy to use special computers called digital signal processors (DSPs) to process and

modify streams of numbers

Electricity does not differentiate between the circuits used to power it. Electricity is

simply available for use in an analog circuit or a digital circuit. It uses the path that it is given

and makes no decision about the better of the two. Deciding which is the better alternative

leads to a discussion of the inherent differences between the two circuits.

a) Function

1. Before the differences between analog circuits and digital circuits can be established,the term "circuits" must be defined. Circuits are closed paths that can be used to

describe many situations. In electronics, an electric current flows through a series of

electronic components which are arranged and connected together to create a closed

path or circuit. Circuits can be composed in any number of ways. The most popular

way is through a printed circuit board or PCB. The PCB serves as the foundation for

the electrical components to be assembled upon.

b) What Is The Primary Difference Between Analog And Digital

Circuits?

2. The way an analog circuit processes a signal is different from a digital circuit. Theoperations supported on an analog circuit can be duplicated on a digital circuit, but

there are differences between how these operations are completed. Noise, precisionand design issues are the three main differences between analog and digital circuits.


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The Issue Of Noise

3. Analog circuits have inherent noise or disturbances caused for any number of reasons.

These disturbances are always significant because each translate as an actual change in

the signal and even loss of information. Digital circuits do not have the same problems

with noise. They are designed in a manner that can eliminate noise. Digital circuits

recreate signal at specific points in the circuit thereby eliminating noise issues.

Consistency in the data identifies digital circuits as more stable.

The Issue Of Precision

4. Precision is a reference to the random fluctuations of electrical current in electricalconductors. Analog circuits are made up of components that have physical limitations.These limitations can cause random fluctuations which cause similar problems to that

of noise--loss of information, for example. In digital circuits more precision can be

had by increasing the number of digits representing the signal. Digital operating

systems perform without precision being affected.

The Issue Of Design

5. Compare analog circuits to digital circuits of the same output capacity and you willfind it is a physically larger system. Digital circuits are much smaller and use

integrated circuits, or chips, to do the same job faster and easier than most analog

circuits. Smaller means easier to manufacture. Smaller also means easier and simpler

to place with in a larger system. Analog circuits are usually hand built, more

expensive to make, and require more space once they are in place .

Now when the basic differences between analog and digital circuits are quite clear, we can

turn our attention to electrical circuits, mention basic circuit elements and parameters.

And study circuit analysis as well.

1.2 Electrical circuits and their parameters

Lets start with basic terminology.

An electrical networkis an interconnection ofelectrical elements such as resistors, inductors,

capacitors,various non-linear elements like diodes, transistors, etc, transmission lines,

voltage sources, current sources, and switches.

An electrical circuit is a network that has a closed loop, giving a return path for the current.

A network is a connection of two or more components, and may not necessarily be a circuit.


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Electrical networks that consist only of sources (voltage or current), linear lumped elements

(resistors, capacitors, inductors), and lineardistributed elements (transmission lines) can be

analyzed by algebraic and transform methods to determine DC response, AC response, and

transient response.

Note: Circuits with lumped elements, or briefly lumped circuits have a finite number of

circuit elements, circuit quantities are only functions of time, they are independent on the

space distribution of their elements. As we can find, their mathematical models are linear or

non-linear differential or algebraic equations with constant or variable perameters.

If the wave nature of the physical phenomena cannot be ommited, it is necessary to

use the methods of electromagnetic field analysis. However there is an exception to this rule

with transmission lines and windings of electrical machines. However the mutual distance

between the conductors is much less than their length, so the wave nature of the

electromagnetic field is apparent on the lenghtwise direction only. In this case, the properties

of the existing electromagnetic field can also be described by voltages and currents. Because

of the continuous distribution of the field, a continuous model with an infinite number ofinfinitesimally small elements must be used. Therefore we call this case a circuit with

distributed elementsor briefly a distributed circuit.

A network that also contains active electronic components is known as an electronic circuit.

Such networks aregenerally nonlinearand require more complex design and analysis tools.

It might be useful ta to start with a concrete example

1.3 Shining a light as an example

Have you ever taken an electric torch to pieces to find out how it works? Look at the diagram

below which shows the arrangement of parts inside one kind of torch:


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Fig.1.1.: Structure of an electric torch

Why did the designer choose this particular combination of materials? The metal parts

of the torch must conduct electric current if the torch is to function, but they must also be

able to stand up to physical forces. The spring holding the cells in place should stay springy,

while the parts of the switch must make good electrical contact and be undamaged by

repeated use.

The lamp and reflector make up an optical system, often intended to focus the light

into a narrow beam. The plastic casing is an electrical insulator. Its shape and colour are

important in making the torch attractive and easy to handle and use.

A torch is a simple product, but a lot of thought is needed to make sure that it will

work well. Can you think of other things which the designer should consider?

A different way of describing the torch is by using a circuit diagram in which the

parts of the torch are represented by symbols (see Fig. 1.2):

Fig.1.2.:Circuit diagram of an electric torch

There are two electric cells ('batteries'), a switch and a lamp (the torch bulb). The lines

in the diagram represent the metal conductors which connect the system together.

A circuit is a closed conducting path. In the torch, closing the switch completes the

circuit and allows current to flow. Torches sometimes fail when the metal parts of the switch

do not make proper contact, or when the lamp filament is 'blown'. In either case, the circuit is

incomplete.

1.3.1. Electric current

An electric current is a flow of charged particles, an organized flow of charges. As a

physical qantity, it is defined as the time rate of a positive charge passing through a certain

area. Inside a copper wire, current is carried by small negatively-charged particles, called

electrons. The electrons drift in random directions until a current starts to flow. When this

happens, electrons start to move in the same direction. The size of the current depends on the

number of electrons passing per second. If a charge of value dq [C] passes through the

considered area within the time inteval dt [s], the current is:

dt

dq

i = (1.1)


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Current is represented by the symbol i, and is measured inamperes, or 'amps', A. One

ampere is a flow of 6.24 x 1018 electrons per second past any point in a wire. That's more than

six million million million electrons passing per second. This is a lot of electrons, but

electrons are very small and each carries a very tiny charge.

Current is a scalar quantity whose positive direction is identified with the direction ofmotion of positive charges.We schall indicate it in schemes by an arrow. When current is a

time function i(t) exhibiting both positive and negative values, the actual direction is identical

with the denoted direction if the instantaneous value of the current is positive, and it is

opposite, if the value is negative.

In electronic circuits, currents are most often measured in milliamps, mA,

that is, thousandths of an amp.

1.3.2. Electric voltage

In the torch circuit, what causes the current to flow? The answer is that the cells

provide a 'push' which makes the current flow round the circuit. When the cells are new,

enough current flows to light the lamp brightly. On the other hand, if the cells have been used

for some time, they may be 'flat' and the lamp glows dimly or not at all.

Each cell provides a push, called its potential difference, orvoltage. This is represented by

the symbol U (often V(in the USA) orE), and is measured in Volts,V. Voltage is an integral

quantity that is the rate of work effects of the electric field.The voltage between two points A

and B along a certain path s [m] is equal to the ratio of the work A[J] that is done by the

forces of the electric field, and the moved charge q[C], ie.:

qAuAB = (1.2)

Voltage is a scalar quantity whose directionis identified with the directions of the forces

acting on a positive charge. We indicate it by an arrow . Concerning the actual and reference

directions of voltages, the same considerations as for furrents are valid.

Typically, each cell provides 1.5 V. Two cells connected one after another, inseries, provide

3 V, while three cells would provide 4.5 V (see Fig.1.3):

Fig 1.3.: Cells connected in series


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Which arrangement would make the lamp glow most brightly? Lamps are designed to

work with a particular voltage, but, other things being equal, the bigger the voltage, the

brighter the lamp.

Strictly speaking,a battery consists of two or more cells. These can be connected in series, as

is usual in a torch circuit, but it is also possible to connect the cells in parallel, like this in

Fig.1.4.:

Fig. 1.4: Cells connected in parallel

A single cell can provide a little current for a long time, or a big current for a short

time. Connecting the cells in series increases the voltage, but does not affect the useful life of

the cells. On the other hand, if the cells are connected in parallel, the voltage stays at 1.5 V,

but the life of the battery is doubled.

A torch lamp which uses 300 mA from C-size alkaline cells should operate

for more than 20 hours before the cells are exhausted.

1.3.3. Which way does the current flow?

One terminal of a cell or battery ispositive, while the other is negative. It is convenient

to think of current as flowing from positive to negative. This is calledconventional current.

Current arrows in circuit diagrams always point in the conventional direction. However, you

should be aware that this is the direction of flow for a positively-chargedparticle.

In a copper wire, the charge carriers are electrons. Electrons are negatively-charged

and therefore flow from negative to positive. This means that electron flow is opposite indirection to conventional current.

Current flow in electronic systems often involves charge carriers of both types. For

example, in transistors, current can be carried by electrons and also by holes, which behave as

positive charge carriers.

When the behaviour of a circuit is analyzed, what matters is the amount of charge which is

being transferred. The effectof the current can be accurately predicted without knowing about

whether the charge carriers are positively or negatively charged.

A cell provides a steady voltage, so that current flow is always in the same direction. This is

called direct current, ord.c. In contrast, the domestic mains provides a constantly changing

voltage which reverses in polarity 50 times every second. This gives rise to alternating

current, ora.c., in which the charge carriers move backwards and forwards in the circuit.


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1.3.4 Resistance

If a thick copper wire was connected from the positive terminal of a battery directly to

the negative terminal, you would get a very large current for a very short time. In a torch, this

does not happen.

Part of the torch circuit limits, or resists, the flow of current. Most of the circuit

consists of thick metal conductors which allow current to flow easily. These parts, including

the spring, switch plates and lamp connections, have a low resistance. The lamp filament, on

the other hand, is made up of very thin wire. It conducts much less easily than the rest of the

circuit and has a higher resistance.

The flow of current through the filament causes it to heat up and glow white hot. In

air, the filament would quickly oxidise. This is prevented by removing all the air inside theglass of the lamp and replacing it with a non-reactive gas.

The resistance,R , of the filament is measured in ohms,. If the battery voltage is 3 V (2 C-

size cells in series) and the lamp current is 300 mA, 0.3 A, what is the resistance of the

filament?

This is calculated from:

where R is resistance, V is the voltage across the lamp, and I is

current. (Although it may not appear logical, the symbol I is

always used for current. Cis used for capacitance.)

In this case, 10 is the resistance of the lamp filament once it has heated up. Itsresistance is less when cold and there will be a surge of current, more than 300 mA, when thetorch is first switched on.

Resistance values in electronic circuits vary from a few ohms, , to values in kilohms,k, (thousands of ohms) and megohms, M, (millions of ohms). Electronic componentsdesigned to have particular resistance values are calledresistors.

1.3.5 Ohm's equationsThe relationship between current, voltage and resistance was

discovered by Georg Simon Ohm (in the picture). He made his own wires and

was able to show that the size of an electric current depended upon their lengthand thickness. The current was reduced by increasing the length of the wire, or

by making it thinner. Current was increased if a shorter thicker wire was used.

In addition, larger currents were observed when the voltage across the wire

was increased.

From experiments like these, Ohm found that, at constant temperature, the ratio

voltage/current was constant for any particular wire, that is:

(1.3)

Ohm's Lawstates that, at constant temperature, the electric current flowing in a conducting

material is directly proportional to the applied voltage, and inversely proportional to theresistance.

I

UR =


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Rearranging the formula gives two additional equations:

I

UR = (1.4)

These simple equations are fundamental to electronics and, once you have learned

to use them effectively, you will find that they are the key to a whole range of circuit

problems. You are going to need these equations, so learn them now.

1.3.6 Electrical laws

A number of electrical laws apply to all electrical networks. These include:

Kirchhoff's current law: The sum of all currents entering a node is equal to the sum ofall currents leaving the node.

Kirchhoff's voltage law: The directed sum of the electrical potential differencesaround a loop must be zero.

Ohm's law: The voltage across a resistor is equal to the product of the resistance andthe current flowing through it (at constant temperature).

Norton's theorem: Any network of voltage and/or current sources and resistors iselectrically equivalent to an ideal current source in parallel with a single resistor.

Northon s Theorem will be mentioned later.

Thevenin's theorem: Any network of voltage and/or current sources and resistors iselectrically equivalent to a single voltage source in series with a single resistor.

Thevenin s Theorem will be mentioned later.

Other more complex laws may be needed if the network contains nonlinear or reactive

components. Non-linear self-regenerative heterodyning systems can be approximated.

Applying these laws results in a set of simultaneous equations that can be solved either by

hand or by a computer.

1.3.6.1 DC circuit equations and laws

A) Ohm's and Joule's Laws

Joule's first law, also known as theJoule effect, is aphysical law expressing the relationship

between the heat generated by the current flowing through a conductor. It is named for James

Prescott Joule who studied the phenomenon in the 1840s. It is expressed as:

tRIQ 2= (1.5)

where Q is the heat generated by a constant currentIflowing through a conductor of electrical

resistanceR, for a time t. When current, resistance and time are expressed in amperes, ohms,

and seconds, respectively, the unit ofQ is the joule. Joule's first law is sometimes called the

JouleLenz law since it was later independently discovered by Heinrich Lenz. The heating

effect of conductors carrying currents is known as Joule heating.

RIU .=


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The principle of conservation of electric charge implies that:

At any node (junction) in an electrical circuit, the sum of currents flowing into that

node is equal to the sum of currents flowing out of that node.

or

The algebraic sum of currents in a network of conductors meeting at a point is zero.

(Assuming that current entering the junction is taken as positive and current leaving

the junction is taken as negative).

Recalling that current is a signed (positive or negative) quantity reflecting direction towards

or away from a node, this principle can be stated as:

=

=n

1k

k 0I (1.7)

Where n is the total number of branches with currents flowing towards or away from the node.

The law is based on the conservation of charge whereby the charge (measured in

coulombs) is the product of the current (in amperes) and the time (which is measured in

seconds).

Fig. 1.6.: The sum of all the voltages around the loop is

equal to zero. v1 + v2 + v3 + v4 = 0

Another is Kirchhoff's voltage law (KVL).

Kirchhoff's voltage law is also known as Kirchhoff's

second law, a closed circuit law, and Kirchhoff's loop law. The algebraic sum of the voltage

(potential) differences in any loop must equal zero.(This circuit is a closed circuit) (see the

example in Fig.1.6) . Any complex circuit can be divided into many closed circuits. This law

means that in the circuit there is an electric cell and electric resistance. The electric cell gives

the charge a electromotive force, and then the electric resistance dissipates this force. But in

electric resistance if the direction is opposite of the current's direction, this electric resistance

adds to the electromotive force. This Kirchhoff's second law is based on potential energypreservation law.

. The principle of conservation of energy implies that:

The directed sum of the electrical potential differences (voltage) around any closed

circuit must be zero.

or

More simply, the sum of the emfs in any closed loop is equivalent to the sum of the

potential drops in that loop.

or

The algebraic sum of the products of the resistances of the conductors and thecurrents in them in a closed loop is equal to the total emf available in that loop.


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KVL, similarly to KCL, it can be stated as:

=

=n

1k

k 0U (1.8)

Here, n is the total number of voltages measured.

1.4 Electrical elements

The concept ofelectrical elements is used in the analysis of electrical networks. Any

electrical network can be modeled by decomposing it down to multiple, interconnected

electrical elements in a schematic diagram or circuit diagram. Each electrical element affects

the voltage in the network or current through the network in a particular way. By analyzing

the way a network is affected by its individual elements, it is possible to estimate how a real

network will behave on a macro scale.

1.4.1 Elements versus components

There is a distinction between real, physical electrical or electronic components and

the ideal electrical elements by which they are represented.

Electrical elements do not exist physically, and are assumed to have idealproperties according to a lumped element model.

Conversely,electrical components do exist, have less than ideal properties, theirvalues always have a degree of uncertainty, they always include some degree of

nonlinearity and typically require a combination of multiple electrical elementsto approximate their functions.

Circuit analysis using electric elements is useful for understanding many practical electrical

networks using components.

The elements

The four fundamental circuit variables are current, I; voltage, V, charge, Q; and

magnetic flux, . Only 5 elements, produced by manipulating these four variables, are

required to represent any component or network in a linear system:

Two sources:o Current source, measured in amperes - produces a current in a conductor.

Affects charge according to the relation dQ = Idt.

o Voltage source, measured in volts - produces a potential difference betweentwo points. Affects magnetic flux according to the relation d = Vdt.

in this relationship does not necessarily represent anything physically meaningful.

In the case of the current generator, Q, the time integral of current, represents the

quantity of electric charge physically delivered by the generator. Here is the time

integral of voltage but whether or not that represents a physical quantity depends onthe nature of the voltage source. For a voltage generated by magnetic induction it is


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meaningful, but for an electrochemical source, or a voltage that is the output of

another circuit, no physical meaning is attached to it.

Three passive (linear)elements:o Resistance R, measured in ohms - produces a voltage proportional to the

current flowing through the element. Relates voltage and current according to

the relation dV=R . dI.

o Capacitance C, measured in farads - produces a current proportional to the rateof change of voltage across the element. Relates charge and voltage according

to the relation dQ = C. dV.

o Inductance L, measured in henries - produces a voltage proportional to the rateof change of current through the element. Relates flux and current according to

the relation d =L . dI.

Four abstract active elements:o

Voltage Controlled Voltage Source (VCVS) Generates a voltage based onanother voltage with respect to a specified gain. (has infinite input impedance

and zero output impedance).

o Voltage Controlled Current Source (VCCS) Generates a current based on avoltage with respect to a specified gain, used to model Field Effect Transistors

and vacuum tubes (has infinite input impedance and infinite output

impedance).

o Current Controlled Voltage Source(CCVS) Generates a voltage based on aninput current with respect to a specified gain. (has zero input impedance and

zero output impedance).

o Current Controlled Current Source (CCCS) Generates a current based on aninput current and a specified gain. Used to model Bipolar Junction Transistors.(Has zero input impedance and infinite output impedance).

1.4.2 Electr ical resistanceReistanceR is constant for the given temperature and material. Therefore, the resistance of an

object can be defined as the ratio of voltage to current, in accordance with Ohm's law:

I

UR = (1.9)

- Resistance of a conductor

DC resistance

By a derivation of Ohm's law, the resistanceR of a conductor of uniform cross section can be

computed as

A

lR = (1.10)


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where

l is the length of the conductor, measured in metres [m]

A is the cross-sectional area of the current flow, measured in square metres [m]

(Greek: rho) is the electrical resistivity (also called specific electrical resistance)

of the material, measured in ohm-metres ( m). Resistivity is a measure of the

material's ability to oppose electric current.

For practical reasons, any connections to a real conductor will almost certainly mean

the current density is not totally uniform. However, this formula still provides a good

approximation for long thin conductors such as wires.

Resistor

Resistor

Three resistors

Type Passive

Electronic symbol

(Europe)

(US)

Fig.1.7.: Resistors and their symbols

A resistor is a two-terminal electronic component that produces a voltage across its terminals

that is proportional to the electric current passing through it in accordance with Ohm's law:

V=IR (1.11)

Resistors are elements of electrical networks and electronic circuits and are ever-

present in most electronic equipment. Practical resistors can be made of various compounds

and films, as well as resistance wire (wire made of a high-resistivity alloy, such as

nickel/chrome).

The primary characteristics of a resistor are the resistance (measured in ohms), the

tolerance, maximum working voltage and the power rating. Other characteristics include

temperature coefficient, noise, and inductance. Less well-known is critical resistance, the

value below which power dissipation limits the maximum permitted current flow, and above

which the limit is applied voltage. Critical resistance is determined by the design, materialsand dimensions of the resistor.


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Resistors can be integrated into hybrid and printed circuits, as well as integrated

circuits. Size, and position of leads (or terminals) are relevant to equipment designers;

resistors must be physically large enough not to overheat when dissipating their power. The

ohm (symbol: ) is the SI unit of electrical resistance, named after Georg Simon Ohm.

Commonly used multiples and submultiples in electrical and electronic usage are the

milliohm (1x103), kilohm (1x103), and megohm (1x106).

The behavior of an ideal resistor is dictated by the relationship specified in Ohm's law:

RIU .= (1.12)

Ohm's law states that the voltage (V) across a resistor is proportional to the current (I)

through it where the constant of proportionality is the resistance (R).

Equivalently, Ohm's law can be stated as:

IR

U= (1.13)

This formulation of Ohm's law states that, when a voltage (V) is maintained across a

resistance (R), a current (I) will flow through the resistance.

This formulation is often used in practice. For example, if V is 12 volts and R is 400 ohms, a

current of 12 / 400 = 0.03 amperes will flow through the resistance R.

A two-terminal is a resistor if its characteristics

u=f(i) or i=g(u) (1.14)

is a curve in a u, i plane passing through the first and third quadrants only (see Fig.1.8). Thus,

for the associated voltage currents references, the polarity of the two variables is always the

same. This so called volt-ampere (VA) characteristics specifiies the set of all possible valuesthat the pair of variables u(t) and i(t) may take at any time t. The VA characteristics of a linear

resistor is a

stright line

through the

origin and

can be

expressed

by Ohms

law, where.

Fig.1.8.: VA characteristics of a linear resistor, nonlinear symmetrical and nonlinear

asymmetrical resistors

R=u/I is the resistance of the resistor in Ohms[]. Using these perameters, the instantaneous

power of a linear resistor can be expressed in the following form:

2

Riiup == . (1.15)


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In resisors with nonlinear symmetrical VA characteristics (see Fig. 1.8) a change of voltage

polarity evokes the same polarity change if the current without changing its magnitude. Thus,

the interchange of the terminals has no effect. In the resistors with the assymmetrical VA

characteristics shown in Fig. 1.8), the voltage polarity change evokes both, the polarity and

the magnitude change of the current. Therefore, it is important how the terminals are

connected to the circuits.

Non-linear resistors may also be characterized by resistance and conductance.

However these parameters arent unique and they must me strictly specified.The static

resistance R and the static conductance can be defined for any operating point of the VA

chatacteristics by the ratio of its coordinates, i.e:

i

if

i

uR

)(==

u

ug

u

iG

)(== (1.16)

In many applications, the operation of non-linear resistors is specified by the small

variations of voltage and current in the close vicinity of the so called fixed operating point

that is adjusted by the constant DC component of the signal. It is useful to define for it thedifferencial resistance and the differencial conductance by increments of voltage and current:

di

du

i

uiR

0id ==

lim)(

du

di

u

iuG

0ud ==

lim)( (1.17)

These parameters depend on the position of the fixed operating point, and they are

related with the slope of the tangent line to the VA characteristics at the respective point. At

the origin(i=0, u=0) the definitions of static parameters (1.16) give uncertain values. At this

point, the characteristics can be aproximated by the tangent line so that at the origin the static

and differencial parameters are identical.

1.4.3 Capacitance

In electromagnetism and electronics, capacitance is the ability of a body to hold an

electrical charge. Capacitance is also a measure of the amount of electrical energy stored (or

separated) for a given electric potential. A common form of energy storage device is a

parallel-plate capacitor. In a parallel plate capacitor, capacitance is directly proportional to the

surface area of the conductor plates and inversely proportional to the separation distance

between the plates. If the charges on the plates are +Q and Q, and V gives the voltage

between the plates, then the capacitance is given by

UQC= (1.18)

The SI unit of capacitance is the farad; 1 farad is 1 coulomb per volt.

The energy (measured in joules) stored in a capacitor is equal to the workdone to charge it.

Consider a capacitance C, holding a charge +q on one plate and q on the other. Moving a

small element of charge dq from one plate to the other against the potential difference V= q/C

requires the work dW:

dqC

q

dW = (1.19)


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where Wis the work measured in joules, q is the charge measured in coulombs and Cis the

capacitance, measured in farads.

The energy stored in a capacitance is found by integrating this equation. Starting with an

uncharged capacitance (q = 0) and moving charge from one plate to the other until the plates

have charge +Q and Q requires the workW:

stored

22Q

0

ingch WCU2

1

C

Q

2

1dq

C

qW ==== arg (1.12)

Capacitors

The capacitance of the majority of capacitors used in electronic circuits is several orders of

magnitude smaller than the farad. The most common subunits of capacitance in use today are

the millifarad (mF), microfarad (F), the nanofarad (nF) and the picofarad (pF).

The capacitance can be calculated if the geometry of the conductors and the dielectric

properties of the insulator between the conductors are known. For example, the capacitance of

a parallel-plate capacitor constructed of two parallel plates both of area A separated by a

distance dis approximately equal to the following:

d

AC 0r= (1.21)

where

Cis the capacitance;

A is the area of overlap of the two plates;

r is the relative static permittivity (sometimes called the dielectric constant) of the

material between the plates (for a vacuum, r= 1);

0 is the electric constant (0 8.8541012 F m1); and

dis the separation between the plates.

Capacitance is proportional to the area of overlap and inversely proportional to the

separation between conducting sheets. The closer the sheets are to each other, the greater the

capacitance. The equation is a good approximation if d is small compared to the otherdimensions of the plates so the field in the capacitor over most of its area is uniform, and the

so-calledfringing fieldaround the periphery provides a small contribution.

Combining the SI equation for capacitance with the above equation for the energy stored in a

capacitance, for a flat-plate capacitor the energy stored is:

. 20r2

stored Ud

A

2

1CU

2

1W == (1.22)

where Wis the energy, in joules; Cis the capacitance, in farads; and Vis the voltage, in volts.


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18

A capacitor is the basic passive two-terminal in which the delivered electrical energy is stored

in the form of energy of magnetic field. The basic characteristics of this element is the relation

between its charge q [C]and voltage u(t) [V], i.e.:

u= f(q) (1.23)

This characteristics is independent of the waveforms q(t)

and u(t) and we call it the volt-coulomb (VC) characteristics.Generally , any two terminal is a

capacitor, if its VC characteristics is a curve passing through the first and third quadrants of

the u, q, -plane only. An example of a linear capacitor is given in Fig 1.9.

Fig.1.9.: VC characteristics of a linear and a non-linear capacitors.

Capacitors may be classified according to the forms od their VC charachteristics. The

characteristics of a linear capacitor in Fig. 1.9 is a straightline through the origin. Its

mathematical description is:

uCq .= (1.24)

Where C is capacitance [F]. The inverse value is the elastance:

S= 1/C [F-1] (1.25)

For non-linear capacitors the same definition of capacitance may be used, giving here the

static capacitance as a function of voltage, i.e.:

u

uf

u

quC

)()( == (1.26)

For the case of excitation of the capacitor by relativelly small signals it is necessary to define

the differencial capacitance:

du

dquC =)( (1.27)

The magnitude is also a function of voltage, and it is related with the slope of the tangent lineto the VC characteristics at the respective operating point.


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Forc the linear capacitor the fundamental relation between voltage u(t) and current i(t) is:

dt

tduCti

)()( = (1.28)

Very often the inverse relations are also needed. First we can find the charge by integration of

the equation (1.28).The integration constant can be determined by utilizing our knowledge of

the linked flux at a certain time constant. 0t . Mostly we choose 0t0 = . As the initial point to

start observing the circuit phenomena.

For a linear capacitor , the voltage can be expressed directly by its current and capacitance.

Choosing 0t0 = , we obtain:

+=t

0

di

C

10utu )()()( (1.29)

Fig.1.10.:Hydrodynamical analogy

Energy stored in a capacitor is given by the work necessary to create its electric field.

Assuming a zero charge at t=0 the energy at time t is:

)()(

)()()()()(

)()(

tCu2

1

C

tq

2

1dq

C

qdqqudiutAtW 2

2tq

0

t

0

tq

0

e ====== (1.30)

As can be deduced, the state variables of a capacitor are voltage and charge, which are ,

therefore always continuous functions.Conversely, the current of a capacitor can be

discontinuous since we suppose that no magnetic field is created in this element.

1.4.4. Inductance

In 1824, Oersted discovered that current passing though a coil created a magnetic field

capable of shifting a compass needle. Seven years later, Faraday and Henry discovered just

the opposite. They noticed that a moving magnetic field would induce current in an electrical

conductor. This process of generating electrical current in a conductor by placing the

conductor in a changing magnetic field is called electromagnetic induction or just induction. It

is called induction because the current is said to be induced in the conductor by the magnetic

field.


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20

Lenz's Law

Soon after Faraday proposed his law of induction, Heinrich Lenz developed a rule for

determining the direction of the induced current in a loop. Basically, Lenz's law states that

an induced current has a direction such that its magnetic field opposes the change inmagnetic field that induced the current. This means that the current induced in a conductor

will oppose the change in current that is causing the flux to change.

Faraday also noticed that the rate at which the magnetic field changed also had an effect on

the amount of current or voltage that was induced. Faraday's Law for an uncoiled conductor

states that the amount of induced voltage is proportional to the rate of change of flux lines

cutting the conductor. Faraday's Law for a straight wire is shown below.

dt

dU

= (1.31)

where:

U = the induced voltage in volts

d/dt = the rate of change of magnetic flux in webers/second

Induction is measured in unit of enries (H) which reflects this dependence on the rate of

change of the magnetic field. One henry is the amount of inductance that is required to

generate one volt of induced voltage when the current is changing at the rate of one ampere

per second. Note that current is used in the definition rather than magnetic field. This is

because current can be used to generate the magnetic field and is easier to measure and

control than magnetic flux.induction occurs in an electrical circuit and affects the flow ofelectricity it is called inductance, L. Self-inductance, or simply inductance, is the property of

a circuit whereby a change in current causes a change in voltage in the same circuit. When

one circuit induces current flow in a second nearby circuit, it is known as mutual-

inductance. When an AC current is flowing through a piece of wire in a circuit, an

electromagnetic field is produced that is constantly growing and shrinking and changing

direction due to the constantly changing current in the wire. This changing magnetic field will

induce electrical current in another wire or circuit that is brought close to the wire in the

primary circuit. The current in the second wire will also be AC and in fact will look very

similar to the current flowing in the first wire. An electrical transformer uses inductance to

change the voltage of electricity into a more useful level. In nondestructive testing, inductance

is used to generate eddy currents in the test piece. It should be noted that since it is the

changing magnetic field that is responsible for inductance, it is only present in AC circuits.

High frequency AC will result in greater inductive reactance since the magnetic field is

changing more rapidly.The property of self-inductance is a particular form of electromagnetic

induction. Self inductance is defined as the induction of a voltage in a current-carrying wire

when the current in the wire itself is changing. In the case of self-inductance, the magnetic

field created by a changing current in the circuit itself induces a voltage in the same circuit.

Therefore, the voltage is self-induced.

The term inductor is used to describe a circuit element possessing the property of

inductance and a coil of wire is a very common inductor. In circuit diagrams, a coil or wire isusually used to indicate an inductive component. Taking a closer look at a coil will help


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21

understand the reason that a voltage is induced in a wire carrying a changing current. The

alternating current running through the coil creates a magnetic field in and around the coil that

is increasing and decreasing as the current changes. The magnetic field forms concentric

loops that surround the wire and join to form larger loops that surround the coil as shown in

Fig.1.11. below. When the current increases in one loop the expanding magnetic field will cut

across some or all of the neighboring loops of wire, inducing a voltage in these loops. This

causes a voltage to be induced in the coil when the current is changing.

Fig.1.11.:Magnetic field of an inductor

By studying this image of a coil, it can be seen that the number of turns in the coil will

have an effect on the amount of voltage that is induced into the circuit. Increasing the number

of turns or the rate of change of magnetic flux increases the amount of induced voltage.

Therefore, Faraday's Law must be modified for a coil of wire and becomes the following.

dt

dNU

= (1.32)

where:

U = induced voltage in volts

N = number of turns in the coil

d/dt = rate of change of magnetic flux in

webers/second

Inductance is the property in an electrical circuit where a change in the electric

current through that circuit induces an electromotive force (EMF) that opposes the change in

current.


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The coil in the fig.1.11 simulates an inductor.The main issue is how the magnetic field

lines go across the inductor (lines with arrows). There is some magnetic field at the top

bottom of the coil too.

The current I going through the inductor

generates a magnetic field which is perpendicular

to I.

The magnetic field H is given by the loops that

surround the current I. The direction of the

magnetic field is given by the arrows around the

loops. If the current was to flow in the opposite

direction the magnetic field arrows would be

reversed. It is the magnetic field which contains the current through the coil which by the

principle called self-induction will induce a voltage U. More specifically speaking, the

voltage U across the inductor L is given by:

U = /T (1.33)

which reads - the

voltage U is caused by the change in flux over the correspondent change in time, but since

the change in flux is given by the inductance L and the change in current across the coil I,

the voltage U becomes:

U = L*I/T (electrical definition for inductance) (1.34)

On the other hand the physical definition of inductance L is given by:

L = N2* A/l (physical definition for inductance) (1.35)

where stands for the relative ease with which current flows through the inductor or

Permeability of the medium. N stands for the number of turns in the coil, A stands for its

cross-sectional area, and the length of the coil is given by l. Hence this formula tells us that

the more number of turns the larger the inductance (i.e.: current can be contained better), also

the larger the cross-sectional area the larger the inductance (since there is more flux of current

that can be contained) and the longer the coil the smaller the inductance (since more current

can be lost through the turns). L is also proportional to , since the better the permeabilitycurrent will flow with more ease.

Inductance and Energy.

By containing the current via the magnetic field the inductor is capable of storing

energy. A transformer will certainly remind us of the ability of storing energy associated with

inductors.Whereas for a capacitor the energy stored depends on the voltage across it, for the

inductor the energy stored depends on the current being held, such that:

W = 1/2*L* I2 (1.36)


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23

where W stands for the energy on the inductor.

Faraday also noticed that the rate at which the magnetic field changed also had an

effect on the amount of current or voltage that was induced. Faraday's Law for an uncoiled

conductor states that the amount of induced voltage is proportional to the rate of change of

flux lines cutting the conductor. Faraday's Law for a straight wire is shown below.

dt

dU

= (1.37)

Where:

U = the induced voltage in volts

d/dt = the rate of change of magnetic flux in webers/second

The SI unit of inductance is the henry (H), named after American scientist andmagnetic researcher Joseph Henry. 1 H = 1 Wb/A. The quantitative definition of the (self-)

inductance of a wire loop in SI units (webers per ampere, known as henries) is

i

NL

= (1.38)

whereL is the inductance, denotes the magnetic flux through the area spanned by the loop,

Nis the number of wire turns, and i is the current in amperes.

InductorsAn inductor or a reactor is a passive electrical component that can store energy in a

magnetic field created by the electric current passing through it. An inductor's ability to store

magnetic energy is measured by its inductance, in units of henries. Typically an inductor is a

conducting wire shaped as a coil, the loops helping to create a strong magnetic field inside the

coil due to Ampere's Law. Due to the time-varying magnetic field inside the coil, a voltage is

induced, according to Faraday's law of electromagnetic induction, which by Lenz's Law

opposes the change in current that created it. Inductors are one of the basic electronic

components used in electronics where current and voltage change with time, due to the ability

of inductors to delay and reshape alternating currents. In everyday speak inductors are

sometimes called chokes, but this refers to only a particular type and purpose of inductor.

.

Fig. 1.12:Various types of inductors


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24

Inductance (L) (measured in henries) is an effect resulting from the magnetic field that

forms around a current-carrying conductor which tends to resist changes in the current.

Electric current through the conductor creates a magnetic flux proportional to the current, and

a change in this current creates a corresponding change in magnetic flux which, in turn, by

Faraday's Law generates an electromotive force (EMF) that opposes this change in current.

Inductance is a measure of the amount of EMF generated per unit change in current. For

example, an inductor with an inductance of 1 henry produces an EMF of 1 volt when the

current through the inductor changes at the rate of 1 ampere per second. The number of loops,

the size of each loop, and the material it is wrapped around all affect the inductance. For

example, the magnetic flux linking these turns can be increased by coiling the conductor

around a material with a high permeability such as iron. This can increase the inductance by

2000 times, although less so at high frequencies.

An inductor is the basic passive two-terminal in which the delivered electrical energyis stored in the form of energy of the magnetic field. The basic characteristics of this element

is the relation between its linked magnetic flux [ ]Fc and current [ ]Ai , i.e.:

)(ifc = )( cgi = (1.39)

This charaacteristics is independent on waveforms c and i(t), and we call it ampere-

weber (Awb) characteristics, Gennerally , any two terminal is a inductor if its Awb

characteristics is a curve passing through the first and third quadrants of the c , i plane only.

Examples of such charactteristics, together with the associated reference directions of voltage

and current and the used specific symbols are shown in Fig 1.13.

Fig. 1.13. examples for Awb characteristics of a linear and non-linear inductors

Inductors may be classified according to the forms of their Awb characteristics. The

charascteristics of a linear inductor in Fig 1.34 on left is a stright line through the origin. Its

mathematical description is as follows:

iLc .= (1.40)


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25

Where L is the inductance in henries [H]. Its inverse value is the inverse inductance1HL1 = / For non linear inductors (Fig. 1.13 on the right), the same definition of

inductance can be used. However here it defines the so called static inductance that is a

function of current, i.e.:

i

iiL c

)()(

= (1.41)

Similarly as for resistors, the differential inductance can also be defined for inductors by the

relation:

di

id

itL cc

0id

)(lim)(

==

(1.42)

It is also a function of the current, and it is related with the slope of the tangent line to theAwb characteristics at the respective operating point.

For the linear inductor of inductance L, the relation between voltage and current is as

follows:

dt

tdiLtu

)()( = (1.43)

Very often the inverse relations alre also needed. First we can find the linked flux by

integration of the equation above.

The current of a linear conductor can be expressed directly by its inductance and

voltage at the chosen time constant. Especially for 0t0 = we have:

+=+=t

0

t

0

du0iduL

10iti )()()()()( (1.44)

Types of Inductors

Although the most common type of inductor is the bar coil type which has been

already presented, there is also surface mount inductors, Toroids (ring-shaped core) , thin film

inductors and tTransformers (which are actually a combination inductor elements and will be

dealt with in AC Electronics). The choice of a particular kind of inductor depends on the

space availability, frequency range of operation, and certainly power requirements.

Bar-Coil Surface Mount Thin Film Toroid Type


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The surface mount type inductors are very small in size and therefore deserve to be

considered when space becomes and issue. The thin film inductors are fabricated by several

processing steps similar to the fabrication of transistors and diodes (They are very small in

size too).

Magnitude of the current (voltage) by the resistor it depends only on resistivity, by

the capacitor and inductor itdepends on not only on capacitance (inductance), but also

on frequency!!!

Known thesis, that on capacitor current leads voltage by /2, while on inductor

current lags voltage by /2 is the consequence of differentiation (integration) and

therefore it is valid only for sin function !!!

AC RLC circuits will be discussed later.


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lecture notes 1 final

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