BYST Circuit -F2003: Introduction 7 CPE220 Electric Circuit Analysis Chapter 1: Introduction.

BYSTBYSTCircuit -F2003: IntroductionCircuit -F2003: Introduction 1

CPE220 Electric Circuit CPE220 Electric Circuit AnalysisAnalysis

Chapter 1: Introduction


Systems of UnitsSystems of Units1.11.1

Chapter 1Chapter 1

Electrical circuit analysis deals with measurable quantities. Hence, we must give both a number and a unit. This requires a standard language that virtually all professionals can understand. Such an international measurement language is the International System of Units (SI). The SI is built upon seven basic units: the meter, kilogram, second, ampere, kelvin, mole, and candela as shown in Table 1.1. Units for other quantities such as volume, force, energy, etc., are derived from these seven base units.


Base Quantity

Table 1.1 SI base units

Name Symbol

length meter m

mass kilogram kg

time second s

electric current ampere A

thermodynamic kelvin K

temperature

amount of substance mole mol

luminous intensity candela cd

Prefixes are often used to emphasize the significant figures when the magnitudes of the measurable quantities are too large or too small. Common prefixes and their corresponding factors are shown in Table 1.2.


Factor Prefix Symbol

10-24 yocto y

10-21 zepto z

10-18 atto a

10-15 femto f

10-12 pico p

10-9 nano n

10-6 micro

10-3 milli m

10-2 centi c

10-1 deci d


Factor Prefix Symbol

10 deka da

102 hecto h

103 kilo k

106 mega M

109 giga G

1012 tera T

1015 peta P

1018 exa E

1021 zetta Z

1024 yotta Y


Fundamental QuantitiesFundamental Quantities

To study and analyze electric circuits, we need to determine the movement of the electrically charged particles, "q" or simply charges. The concept of charge is the underlying principle for explaining all electrical phenomena. The movement of charge can be described by the "through"

1.21.2

We encounter so many electronic devices in our daily life such as radio, television, mobile phone, etc.. These electronic devices are formed by many electric circuits which are the interconnection of electrical elements. Hence, it is important to know the properties of the involved components as well as the components are connected to form the circuit.


1.2.1 Charge

quantity called "current" and the "across" quantity called "voltage".

Charge, q, is defined as an electrical property of the atomic particles of which matter consists. From basic physics, atoms are fundamental building blocks of all matter. Each atom is composed of electrons, protons, and neutrons.

In the SI system, charge is measured in coulombs (C). The charge on an electron is -1.602 x 10-19 C. Hence, in 1 C of charge, there are 1/(1.602 x 10-19) = 6.24 x 1018 electrons. The law of conservation of charge states that charge can neither be created nor destroyed, only transferred. In other words,


When you rub a comb with a woolen cloth, a negative (corresponding to an electron) charge is produced on the comb and a positive (corresponding to a proton) charge is produced on the cloth. (Benjamin Franklin defined the charge on the comb as negative). The comb acquires its negative charge because some of the electrons on the cloth are rubbed off onto the comb.

the algebraic sum of the electric charges in a system does not change.

Charge in motion results in energy transfer. Of special interest in the case where the motion is confined to a definite closed path. In this case, current "flows".


1.2.2 Current

Current, i, is the rate at which charge is transferred or "flows" through an electronic device. Mathematically, the current is defined as:

dq(t)dt

i(t) = (1.1)

where i = the current in amperes (A)q = the charge in coulombs (C)

That is, we define the current, i, flowing in an electronic device as the amount of charge passing through that device per unit of time. The unit of current is the ampere (A) and

1 ampere is 1 coulomb per second.


It is convenient to think of current as the flowing motion of positive charge although, physically, current flow in metallic conductors results from negative charge (electron) motion. That is, the charge carriers are negative and move in the opposite direction.

Figure 1.1 Charge/Current Relationship.

From Fig. 1.1, the charge and the current flow through the conductor, passing from one side of the imaginary surface to the other.

+ + + ++ + + +++

i(t) i(t)


During the time interval from ta to tb the charge passing through the imaginary surface can be determined by Eq. 1.2 as following:

b

a

t

t

q(t) i(t)dt (1.2)

Current defined by Eq. 1.1 demonstrates that current is a time-varying function. That is, it is not a constant-valued function. Practically, there are so many types of current. However, the most common types of current in electric circuit are shown in Fig. 1.2 which are: direct current (dc), alternating (sinusoidal) current (ac), exponential current, damped sinusoidal current, and transient current.


i

t

0 5 10 15 20-1

-0.5

0

0.5

1

0 5 10 15 20-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20-3

-2

-1

0

1

2

3

4

5

0 5 10 15 200

0.2

0.4

0.6

0.8

1

Direct Current (dc)

Alternating/Sinusoidal Current (ac)

Exponential Current

Damped SinusoidalCurrent

Transient Current

Figure 1.2 Types of current: Direct Current (dc), Alternating/Sinusoidal Current (ac), Exponential Current, Damped Sinusoidal Current and Transient Current.


The two most common types of current used with electronic devices or apparatus are direct current and alternating current. From Fig. 1.2, a direct current (dc) is a current that is constant in time. By convention the symbol "I" is used to represent such a constant current. An alternating current (ac), on the other hand, is a current that varies sinusodially with time as shown in Fig. 1.3.

Acos()

i(t)

Tp

t

Figure 1.3 A sinusoidal (alternating) current.


=

Tp = the fundamental period (sec)1f

= the phase

i(t) = Acos(t + ) (1.3) t

Where

= the angular frequency (rad/sec)

= 2ff = the frequency (cycles/sec)

Such current is usually called a "sinusoidal current" and can be represented as a cosine function:

The sinusoidal current described by Eq. 1.3 has these following properties:

1. Periodical: i.e.

i(t + Tp) = i(t) (1.4)


2. Distinction: sinusoidal currents with different frequencies are themselves distinct.

3. The rate of oscillation of the current will be increased if the frequency "f" is increased.

In defining current, both the direction of the arrow and the value are required as illustrated in Fig. 1.4.

Figure 1.4 Two different methods of labeling the same current.

(a) (b)

A negative current of -3 A flowing in one direction as shown in Fig. 1.4(b) is the same as a current of +3 A flowing in the opposite direction.


As previously mentioned, the direction of current flow is conventionally taken as the direction of positive charge movement. Hence, the direction of the arrow and the positive value of current indicate the direction of a net positive charge. For example, in Fig. 1.4(a), the direction of the arrow and the value 3 A indicate that a net positive charge of 3 C/s is moving to the right or that a net negative charge of -3 C/s is moving to the left each second.

To define the current i1(t) having only either the value of the current or the arrow is an in complete, improper, and incorrect definition as shown in Fig. 1.5(a) and (b), respectively. Fig. 1.5(c) is the correct definition of a current.


Figure 1.5 (a,b) Incomplete, improper, and incorrect definitions of a current. © the correct definition of i1(t).

Example 1.1: DC current.

Find the charge passing through a point in a conductor if the current passing through the conductor has the waveform indicated.

i

t

2 A

Solution:

The current does not change with time. Hence, the charge 2 C is passing through the point each second. Ans.


Example 1.2: For the given current, which is given in graphical form, find the charge passing through a given point in the time interval 0-4 s.

Solution:

For the time interval 0-4 s, the charge passing through a given point on the conductor that carries the current is:

t (sec)

10 A

10 2 43

i(t)

4

0

q(t) i(t)dt1 2

0 1

10tdt 10dt 3 4

2 3

( 10t 30)dt (10t 30)dt


Ans.

= 5 + 10 -25 +35 C

= 25 C

1.2.3 Voltage

Charges in motion yield an energy transfer. The separation of charge creates an electric force, which was recognized by the 18th century Italian physicist Allessandro Antonio Volta. This force is known as "voltage" or "potential difference". Voltage is defined as the energy required to move a unit charge through an element, measured in volts (V). In the other words, voltage is the energy per unit charge that is created by the charge separation.


In Fig. 1.6, the current I flowing through a circuit element from point "a" to point "b" requires an energy to move the charges. Voltage between points "a" and "b" is defined as:

a b

circuitelement

i i. . . . . .

Figure 1.6 Current i flowing from point a to point b induces the voltage across point a and b.

dwdq

vab = (1.5)

where vab = the voltage between a and b,

w = the energy in joules (J),

q = the charge in coulombs (C).

+ -v


1 volt is 1 joule per coulomb.

Traditionally, vab has two meanings:

Voltage polarity indicates the relative potential between two points. Usually, the plus "+" sign is assigned to a higher potential point and minus "-" sign is assigned to a lower potential point.

The potential at point a is assumed higher than the potential at point b. vab is the potential at point a with respect to point b.

vab = -vba

Hence,

(1.6)


Actual direction and polarity will be governed by the sign of the value. Hence, during the analysis, direction and polarity can be arbitrarily assigned on circuit diagram. In Fig. 1.7 (a), the “+” sign is placed at the point “A” but the numerical

Figure 1.7 Point B is 5 V positive with respect to point A in (a) and (b). Point A is 5 V positive with respect to point B in (c) and (d).


value of vAB is -5 V. That is, conventionally, point “A” is -5 V positive with respect to point “B” or, actually, point “B” is 5 V positive with respect to point “A” as illustrated in Fig. 1.7(b). Similarly, the representation of vAB in Fig. 1.7(c) and (d) have the same meaning which is point “A” is 5 V positive with respect to point “B”. Therefore, the definition of voltage requires both the numerical value of voltage and the “plus-minus” sign pair.

Like electric current, a constant voltage is called a “dc voltage” and is represented by “V”, whereas a sinusoidally time-varying voltage is called an “ac voltage” and is represented by “v”. A dc voltage is usually produced by a battery and an ac voltage is produced by an electric generator.


1.2.4 Power and Energy

Two other important quantities that are frequently used in describing electrical system are “power, p” and “energy, w”.

Power, p, is defined as the rate at which energy, w, is expended or absorbed, measured in watts (W).

That is, the relationship between the power and the energy can be expressed as:

dwdt

p = (1.7)

where p = the power in watts (W),

w = the energy in joules (J),

t = the time in seconds (sec).

PowerPower


We can rewrite Eq. 1.7 as:

(1.8)dwdq

p =dqdt

. = vi

Thus, the power absorbed or supplied by an element is the product of the voltage across the element and the current through it. If the numerical value of power is positive (+), power is being delivered to or absorbed by the element. On the other hand, if the numerical value of power is negative (-), power is being supplied by the element.

Current direction and voltage polarity determine the sign of power. The passive sign convention is satisfied when the

Passive Sign ConventionPassive Sign Convention


current enters through the positive(plus-marked) terminal of an element asshown in Fig. 1.8.

circuitelementv

–

+i

i

. . .

Figure 1.8 The passive sign convention.

When this passive sign convention is being used:

If v i > 0, then the circuit element is absorbing power. If v i < 0, then the circuit element is supplying power.


In general,

Power absorbed = -Power supplied

Power Conservation TheoremPower Conservation Theorem

The sum of the powers absorbed by all the elements in a circuit equals zero.

Example 1.3

Verify the power conservation theorem for the following circuit.


Solution:

p1 = -(300)(2) = -600 W

p2 = (240)(2) = 480 W

p3 = (60)(1.5) = 90 W

p4 = (60)(0.5) = 30 W

Σpi = -600 +480 + 90 + 30 W

= 0 W

supplied supplied

absorbed absorbed

absorbed absorbed

absorbed absorbed

+300 V –

+60 V –

+60 V –

+ 240 V –

2 A

2 A 1.5 A

0.5 A

3 41

2

Ans.


EnergyEnergy

The power p in Eq. 1.8 is a time-varying and is called the “instantaneous power”. The energy absorbed or supplied by an element from time t0 to time t is determined by:

0

t

t

w(t) p(t)dt (1.9)

0

t

t

v(t)i(t)dt

That is, energy is the capacity to do work, measured in joules (J).


A simple model for many circuits consists of one source and one load. The source produces the power and the load consumes the power.

There is a voltage rise at the source and a voltage drop at the load (when proceeding in the direction of the reference current).

Source Load V

+

i

–

Voltage drop in the direction of the current: power consumed.

Voltage rise in the direction of the current: power supplied.

The source get its energy through this connection.

Source-Load CircuitSource-Load Circuit


Source-Load Circuit: Hydraulic AnalSource-Load Circuit: Hydraulic Analogog

At the source (pump) there is a pressure rise in the direction of the flow.

At the turbine (load) there is a pressure drop in the direction the flow.

Pump Turbine

High pressure

Low pressureThe pump get its energy through this connection.


Voltage and Current SourcesVoltage and Current Sources1.31.3

Thus far voltage, current, power and energy have been defined. In this section an element which is the basic building block of a circuit will be discussed. In general, the basic circuit elements can be classified by the relationship of the current through the element to the voltage across the element. For example, the voltage across an element can be linearly proportional to the current through it (a resistor) or the voltage across an element can be proportional to the derivative (a inductor) or the integral (a capacitor) of the current with respect to time.

There are two types of basic electrical elements: passive elements and active elements. An passive element is one that


never supplies energy while an active element is capable of generating energy. In this section, only the active elements will be discussed.

The most important active elements are voltage and current sources which generally deliver power to the circuit connected to them. The voltage and current sources can be classified as either independent or dependent sources.

An ideal independent source is an active element that can supply a specified voltage or current that is completely independent of a current or voltage elsewhere in the circuit.


An ideal dependent source, on the other hand, is an active element in which the source voltage or current depends on another voltage or current in the circuit.

1.3.1 Independent Voltage Sources

An ideal dependent source in Fig. 1.9a will deliver to the circuit whatever current is necessary to maintain its terminal voltage vs. The current arrow labeled "i" in Fig.

vs

ivs vs

i

Figure 1.9 Circuit symbol of the independent voltage source.

(a) (b) (c)


The lower-case vs indicates a time-varying terminal voltage of the independent voltage source. As mentioned earlier, the most two common current types are direct current (dc) and alternating current (ac). Hence, if vs is constant in time, such the independent voltage source is termed "an independent dc voltage source" represented by either of the symbols shown in Fig. 1.10a and b.

1.9b indicates that this independent voltage source deliver the power p = vsi to the circuit. On the other hand, the independent voltage source in Fig. 1.9c absorb the power p = vsi from the circuit.

In the case that vs varies sinusodially with time the independent voltage source is termed "an independent ac voltage source" represented by the symbols shown in Fig. 1.10c.


vsVs

Figure 1.10 (a) and (b) DC independent voltage source symbols. (c) AC independent voltage source symbol.

(a) (b) (c)

V

1.3.2 Independent Current Sources

Similarly, an ideal independent current source provides a specified current completely independent of the voltage across the source. That is, the current source delivers to the circuit whatever voltage is necessary to maintain the designated current.


(a) (b)

Figure 1.11 (a) A time-varying independent current source. (b) A constant (DC) independent current source.

i I

Fig. 1.11 illustrates the symbols of independent current source where the arrow indicates the direction of current i or I. In Fig. 1.11a, lower-case "i" indicates a time-varying independent current source which is normally a sinusoidal function (ac current). In Fig. 1.11b, on the other hand, upper-case "I" indicates an independent dc current source.


1.3.3 Dependent Sources

The dependent, or controlled, source is a source in which the source voltage or current is determined by another voltage or current elsewhere in the circuit being analyzed. It is usually represented by diamond-shaped symbol, as shown in Fig. 1.12.

(a) (b)

Figure 1.12 (a) A dependent voltage source. (b) A dependent current source.

v i


Since the dependent source can be controlled by either voltage or current of some other element in the circuit, we can categorize the dependent source into four types, as shown in Fig. 1.13, which are:

1. A voltage-controlled voltage source (VCVS),

2. A current-controlled voltage source (CCVS),

3. A voltage-controlled current source (VCCS),

4. A current-controlled current source (CCCS).

(a)

i

v = vx

VCVS:

v = vx

vx is somewhere (not shown) and is a constant.

i = whatever


(b)

i

v = ix

CCVS:

v = ix

i = whateverix is somewhere (not shown) and is a constant.

(c)

i = vx

VCCS:

i = vx

vx is somewhere (not shown) and is a constant.

v = whatever

+

_v


(d)

i = ix

CCCS:

i = ix

v = whateverix is somewhere (not shown) and is a constant.

+

_v

Figure 1.13 Four possible types of dependent sources: (a) a voltage-controlled voltage source (VCVS), (b) a current-controlled voltage source (CCVS), (c) a voltage-controlled current source (VCCS), (d) a current- controlled current source (CCCS).

Controlled sources are used primarily to model electronic devices. For example, the junction field-effect transistor (JFET) and the bipolar junction transistor (BJT) are modeled as shown in Fig 1.14. These devices are used to construct electronic circuits


such as amplifiers and digital computers. Without dependent sources we would not be able to model these important electrical components.

JFET

BJT

G

D

S

GD

S

ro

+

vGS

_

gmvGS

B

C

E

BD

S

ro

iB

ro

Figure 1.14 The electronic devices, JFET and BJT, can be modeled by controlled sources.

iB

BYST Circuit -F2003: Introduction 7 CPE220 Electric Circuit Analysis Chapter 1: Introduction.

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