ECE 140 Fall 2012 1/2 Course Notes

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Electrical and Computer Engineering
ECE 140
Michael Ibrahim

ECE 140 - Linear Circuits Kevin Carruthers Fall 2012 Basic Components and Electric Circuits Units and Scales The international standard is SI units. Under this system, electric current is measured in Amperes (A), work and energy is measured in Joules (J), and power, or the rate in which work is done, is measured in Watts (W). ▯18 In increasing order, from 10 and ascending in factors of three, the pre▯xes we use are atto, femto, pico, nano, micro, milli, kilo, Mega, Giga, and tera. Charge, Current, Voltage, and Power Charge The basic unit of positive charge is the proton. The unit of negative charge is the electron. We can neither create nor destroy charges. Electric current is de▯ned as charge in motion, and follows the direction of the ow of positive charge (opposite the direction of electron ow). The fundamental unit of charge is the Coulomb (C). ▯19 A proton or electron has a charge of ▯1:602 ▯ 10 Current Moving charges create an electrical current. In moving charges from one place to another, we may also transfer energy. By changing the current (with respect to time), we can transfer information. Current, then, is the rate at which the charges are moving past a given reference point in a speci▯c direction. 1 The charge transfered between times t an0 t can1be expressed as Z Z 1 t1 dq(▯) = i(▯) d▯ 0 t0 and the total charge transfered is given by Z t1 q(t1) = q(t0) + i(▯) d▯ t0 Voltage Let a general 2-terminal circuit element have two terminals (ie. resistors, inductors, batter- ies...). There are thus two paths by which the current may enter or leave the element. If you want to push charges through a circuit element, you have to expend some energy. The energy used to push charge through an element is de▯ned as the voltage between the two terminals, or the potential di▯erence. In other words, the voltage across a terminal pair is a measure of the work required to move charge through that element. Voltage is measured in Volts (V), where 1J 1V = 1C A voltage can exist between a pair of terminals whether or not a current is owing. According to the Conservation of Energy, the energy expended in forcing charge through an element must appear elsewhere (ie. transformed into heat energy). Energy could be supplied to an element or by an element. Power Power is de▯ned as the rate at which work is done or energy is expended. It is measured in Watts (W), where 1J 1W = 1s If one Joule of energy is expended in transfering one Coulomb of energy through a device in 1J 1J 1C one second, then we have 1W = 1s= 1C ▯ 1s As such, we have P = V I If we have current entering the positive terminal of an element (ie. the terminal with a larger voltage), we follow the passive sign convention. As such, the power absorbed by the element is vi and the power generated or supplied by the element is ▯vi. 2 Voltage and Current Sources The mathematical models used for circuit analysis are only approximations. They depend on the relation between voltage across their terminals and the current going through them. Relationship Between v and i Element v / di Resistor vZ/ dt Inductor v / i(t) dt Capacitor v 6/ i Independant Voltage Source i 6/ v Independant Current Source v or i / v or i Dependant Source 0 0 Independant Voltage Sources The voltage of an independant voltage source is completely independant of the current. It is ideal because it does not exactly represent any real physical device, but it is a reasonable approximation of some. For example, household electrical outlets can be approximated as independant voltage sources providing 115 2cos(120▯▯t) volts. This approximation is valid for currents less than 20 A. If the terminal voltage is constant, we have a DC voltage source. Otherwise (ie. it provides a sinusoidal output) our source is AC. It is denoted by a circular shape with plus and minus symbols denoting the di▯erence in voltage. An optional tilde denotes AC voltage. Independant Current Sources The current of an independant current source is completely independant of the voltage across the source. If it provides constant current, we have a DC source, otherwise it is an AC source. Voltage across a current source is not known, it depends on the circuit connected to it. It is denoted by a circular shape with an arrow in the direction of current ow. An optional tilde denotes AC current. Dependant Sources, aka Controlled Sources A dependant source is one where the source quantity of either voltage or current is de- termined by a voltage or current elsewhere in the circuit. They usually appear in equivalent electrical models for devices such as operational ampli▯ers or transistors. 3 It is denoted by a diamond shape. Networks and Circuits An electrical network is the interconnection of two or more simple circuit elements. An electrical circuit is a network which contains at least one closed path. As such, every circuit is a network, but not all networks are circuits. An active network is one which contains at least one active element (ie. independant source). Alternatively, we have passive networks. Resistance Ohm’s Law Ohm’s Law describes the relationship between th voltage across a resistor and the current passing through it as v = iR 1V where resistance is measured in Ohms ( ) and 1 = 1A Note that a linear resistor is an idealized circuit element. Actual resistors only act like linear resistors within certain ranges of current, voltage, or power, and also depend on temperature and other environmental factors. Power Absorbtion Assuming we are following the passive sign convention, we have v 2 P = vi = i R = R is the power absorbed by a resistor. Note that resistors dissipate energy in the form of heat and/or light as they cannot store or generate it. For a complete circuit, the absorbed power is equal to the generated power, as per conser- vation of energy. Resistance of Wires Each material has a property called resistivity (▯) which is the measure of how "easily" electrons can travel through that material. The units of resistivity are ▯ m, thus we have l R = ▯ A Usually, the resistance of wires can be approximated as 0 4 Conductance The conductance is the inverse of the resistance, or G = 1 R Short or Open Circuits Short circuits are ones where R = 0 thus V = 0V for any i. Open circuits are ones where R = 1 thus i = 0A for any v. We will assume wires to be a perfect short circuit. Voltage and Current Laws Nodes, Paths, Loops, and Branches A node is a point at which two or more elements in a circuit have a common connection. Since we assume all wires are perfectly conducting, any wire attached to a node is considered part of a node. Note that each element has a node at each end. A path is a route through a circuit where no node is encountered more than once. If we end at the same node as we started at, we have a path. A branch is a single path in a network composed of one simple element and the nodes at each end of that element. Kircho▯’s Current Law (KCL) Theorem: The algebraic sum of all currents entering any node is zero. KCL is based on the principle of conservation of charge. Basically, charge can not accumulate at a node, so any charge entering a node must leave it. Consider an intersection: the number of cars entering an intersection is exactly equal to the number of cars leaving that intersection. KCL can also be written as "the sum of the currents entering a node is equal to the sum of currents leaving a node". Kircho▯’s Voltage Law (KVL) Theorem: The algebraic sum of the voltage around any closed path is zero. This is based on the fact that the enrgy required to move a charge from point A to point B must have the same value independant of the path between points. 5 Series and Parallel Connections Components in Series Circuit components that carry the same current are said to be in series. Note that they must have the same cur
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