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 1 R 2 R S v   v0v0 I0I0

BASIC STRATEGY USED IN ANALYSIS

MATHEMATICAL ANALYSIS DEVELOP A SET OF MATHEMATICAL EQUATIONS 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 TECHNIQUES TO DEVELOP MATHEMATICAL MODELS FOR ELECTRIC CIRCUITS THE MATHEMATICS CLASSES - LINEAR ALGEBRA, DIFFERENTIAL EQUATIONS- PROVIDE THE TOOLS TO SOLVE THE MATHEMATICAL MODELS FOR THE FIRST PART WE WILL BE EXPECTED TO SOLVE SYSTEMS OF ALGEBRAIC EQUATIONS LATER THE MODELS WILL BE DIFFERENTIAL EQUATIONS OF THE FORM THE MODELS THAT WILL BE DEVELOPED HAVE NICE MATHEMATICAL PROPERTIES. IN PARTICULAR THEY WILL BE LINEAR WHICH MEANS THAT THEY SATISFY THE PRINCIPLE OF SUPERPOSITION Model Principleof Superposition yTu TuuTuTu   ()()() 

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

UNITS Standard SI Prefixes – pico (p) –10 -9 nano (n) –10 -6 micro (  ) –10 -3 milli (m) –10 3 kilo (k) –10 6 mega (M) –10 9 giga (G) –10 12 tera (T)

BASIC 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)

CURRENT Time rate of change of charge Units: 1 A = 1 C/s Constant currentTime varying current i(t) 1 Question: i = 10 mA, t = 5  S, Chang of Charge? A B

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) tt DC AC

Positive versus negative current 2 A -2 A

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 + – v(t) A B + _ i(t) Given a component, is it supplying or absorbing electrical energy ? i(t)

Positive versus negative voltage + – – + 2 V -2 V

POWER (P) Rate of change of energy P = W/t since P(t) = v(t) i(t) We obtain

PASSIVE 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 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) + v(t) i(t) independent voltage sourceindependent current source

Circuit Elements (cont’d) + _ _ + Voltage controlled (dependent) voltage source + _ Current controlled (dependent) voltage source Q: What are the units for and r?

Circuit Elements (cont’d) _ + Voltage controlled (dependent) current source Current controlled (dependent) current source Q: What are the units for and g?

CONSERVATION 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!