An Electrical Network IS AN INTERCONNECTION OF ELECTRICAL COMPONENTS.

TYPICAL LINEAR CIRCUIT

• To analyze, design and measure a number of quantities (e.g. current, voltage) of linear analog electrical network systems, across engineering disciplines and within sub-disciplines of Electrical Engineering.

OBJECTIVES

+-

L

C

1R

2R

Sv

v0I0

BASIC STRATEGY USED IN ANALYSIS

MATHEMATICAL ANALYSISDEVELOP A SET OF MATHEMATICALEQUATIONS THAT REPRESENT THE CIRCUIT - A MATHEMATICAL MODEL -

LEARN HOW TO SOLVE THE MODEL TO DETERMINE HOW THE CIRCUIT WILL BEHAVE IN A GIVEN SITUATION

THIS COURSE TEACHES THE BASIC TECHNIQUESTO DEVELOP MATHEMATICAL MODELS FORELECTRIC CIRCUITS

THE MATHEMATICS CLASSES - LINEAR ALGEBRA,DIFFERENTIAL EQUATIONS- PROVIDE THE TOOLSTO SOLVE THE MATHEMATICAL MODELS

FOR THE FIRST PART WE WILL BE EXPECTEDTO SOLVE SYSTEMS OF ALGEBRAIC EQUATIONS

20642

0164

84912

321

321

321

VVV

VVV

VVV

LATER THE MODELS WILL BE DIFFERENTIALEQUATIONS OF THE FORM

fdt

dfy

dt

dy

dt

yd

fydt

dy

4384

3

2

2

THE MODELS THAT WILL BE DEVELOPED HAVENICE MATHEMATICAL PROPERTIES.IN PARTICULAR THEY WILL BE LINEAR WHICHMEANS THAT THEY SATISFY THE PRINCIPLE OFSUPERPOSITION

Model

Principle of Superposition

y Tu

T u u T u T u

( ) ( ) ( ) 1 1 2 2 1 1 2 2

BASIC CONCEPTS

LEARNING GOALS

•System of Units: The SI standard system; prefixes

•Basic Quantities: Charge, current, voltage, power and energy

•Circuit Elements: Active and Passive

UNITSUNITS

• Standard SI Prefixes– 10-12 pico (p)– 10-9 nano (n)– 10-6 micro ()– 10-3 milli (m)– 103 kilo (k)– 106 mega (M)– 109 giga (G)– 1012 tera (T)

BASIC QUANTITIESBASIC QUANTITIES

• Electric charge (q)– in Coulombs (C)

• Current (I)– in Amperes (A)

• Voltage (V)– in Volts (V)

• Energy (W)– in Joules (J)

• Power (P)– in Watts (W)

CURRENTCURRENT

• Time rate of change of charge

• Units: 1 A = 1 C/s

t

qI dttdqti /)()(

Constant current Time varying current

i(t)

tIq

t

dxxitq )()(

1 Question: i = 10 mA, t = 5 S, Chang of Charge?

A B

CURRENT (cont’d)CURRENT (cont’d)• Notation: Current flow represents the flow of

positive charge– Q: How does the positive charge move in an

electrical field?• Alternating versus direct current (AC vs DC)

– Any examples?

i(t) i(t)

t t

DCAC

Positive versus negative currentPositive versus negative current

2 A -2 A

VOLTAGE (POTENTIAL)VOLTAGE (POTENTIAL)

• Voltage between any two points– the difference in the energy level of 1 C charge

located at each of the two points

• Units: 1 V = 1 J/C

qWV / dqdwv /

+

–

v(t)

A

B

+ _v(t)

i(t)

Given a component, is it supplying or absorbing electrical energy ?

i(t)

Positive versus negative voltagePositive versus negative voltage

+

–

–

+

2 V -2 V

POWER (P)POWER (P)

• Rate of change of energy

P = W/t dttdwtp /)()(

since

dt

dw

dt

dq

dq

dw

P(t) = v(t) i(t)We obtain

PASSIVE SIGN CONVENTIONPASSIVE SIGN CONVENTION

• Used to determine the electrical power is being absorbed or supplied– if P is positive (+), power is absorbed– if P is negative(–), power is supplied

+

–

v(t)

i(t)p(t) = v(t) i(t)

v(t) is defined as the voltage with positive reference at the same terminal that the current i(t) is entering.

CIRCUIT ELEMENTSCIRCUIT ELEMENTS

• Passive elements (cannot generate energy)– e.g., resistors, capacitors, inductors, etc.

• Active elements (capable of generating energy)– batteries, generators, etc.

• Important active elements– Independent voltage source– Independent current source– Dependent voltage source

• voltage dependent and current dependent

– Dependent current source• voltage dependent and current dependent

Circuit Elements (cont’d)Circuit Elements (cont’d)

+v(t) i(t)

independent voltage source independent current source

Circuit Elements (cont’d)Circuit Elements (cont’d)

+_

_

+

Sv Svv

Voltage controlled (dependent) voltage source

+_ Sriv

Current controlled (dependent) voltage source

Si

Q: What are the units for and r?

Circuit Elements (cont’d)Circuit Elements (cont’d)

_

+

SvSgvi

Voltage controlled (dependent) current source

Current controlled (dependent) current source

Si

Q: What are the units for and g?

Sii

CONSERVATION OF ENERGYCONSERVATION OF ENERGY

• Important Principle: The electric circuits (under consideration) satisfy the conservation of energy!– The energy supplied by the active elements is

equivalent to the energy absorbed by the passive elements!