ECE 140 Fall 2012 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 current, not just equal currents. Components in Parallel Circuit components are said to be in parallel if they are connected between the same pair of nodes. Components in parallel have the same voltage drop (again, not just equal voltage drops, but the same voltage drop). Connected Sources Several voltage sources in series may be replaced by an equivalent voltage source having a voltage equal to the algebraic sum of the individual sources. Parallel current sources may be combined by algebraicly adding the individual currents. Ideal voltage sources in parallel are permissible only when each has the same terminal voltage at every instant. Ideal current sources in series are permissible only when each has the same current (including sign) at every instant. Resistors in Series and Parallel We combine any number of resistors in series by ▯nding the algebraic sum of their resis- tances and replacing them with an equivelent resistor such that R = R + R + ▯▯▯ + R eq 0 1 n For resistors in parallel, we use ▯ ▯1 ▯1 ▯1▯▯1 Req = R 0 + R 1 + ▯▯▯ + Rn and the notation R eq= R 0=R =1:::==R n R R Note that we have the special case R0==R1where R eq= R0+R1 6 Voltage and Current Division Voltage division is used with resistors in series. To solve for the voltage across a single resistor when we have the voltage across all of them and their individual resistances, we have Rk vk= v R0+ R 1 ▯▯▯ + R +k▯▯▯ + R n Current division is used with resistors in parallel. If we have the network current and the resistances of each resistor, we can solve for the current across any one resistor with R0==R 1=:::==R =k:::==R n ik= i R k Ground In electrical circuits, we usually select a node in the circuit to be a ground. A ground is a node having a potential of zero volts. This allows use to reference all other voltages in a circuit to the ground. Nodal Analysis nodal analysis is a methodical circuit analysis technique based on KCL. An N node circuit will have N-1 unknown voltages, and thus will need N-1 equations, where each equations is a simple KCL equation. To perform nodal analysis, select one node to be a reference node, which will be the negative terminal of the voltages in the circuit. This reference node will be connected to ground and considered to have zero voltage. Usually, the reference node is the node with the greatest number of branches. Then, solve the system of equations formed by the KCL equations of each remaining node, with respect to the reference node. Supernodes A supernode is two nodes separated by an independant voltage source. In nodal analysis, we can treat this as a single node. Mesh Analysis Mesh analysis is a methodical circuit analysis technique based on KVL. It can only be applied to planar circuits. 7 Note: a planar circuit is a circuit for which it is possible to draw the circuit diagram on a plane surface in such a way that no branch passes over or under anyother branch. A mesh is a loop which does not contain any other loops within it. To perform mesh analysis, assign a unique mesh current (current that ows only around the perimeter of a mesh) to each mesh, and write the KVL equation for each mesh, using the mesh currents. Solving this system of equations will solve the circuit. If we have M meshes, we will have M Mesh currents and will write M KVL equations. The convention is to make all mesh currents clock-wise, but you can select any arbitrary direction. Supermeshes A supermesh is created from two meshes that have an independant current source as a common element. In mesh analysis, we can ignore the branch containing the current source and calculate the KVL for the resulting mesh. Note that if the current source lies on the perimeter of the circuit, there is no need to write KVL in for this mesh. Linearity and Superposition The superposition principle: The response (a desired current or voltage) in a linear circuit having more than one independant source can be obtained by adding the responses caused by the separate independant sources acting alone. The superposition principle can be applied to linear circuits only. Linear Elements A linear element is one which has a linear voltage-current relationship. A linear voltage- current relationship means that if we multiply the current through an element by a constant k, this results in multiplying the voltage across the component by the same constant k. Linear Circuits A linear circuit is one composed entierly from independant sources, linear dependant sources, and linear elements. For any linear circuit, if all independant voltage and cur- rent sources are multiplied by a constant k, all the current and voltage responses in the circuit will be multiplied by the same factor k. 8 The Principle of Superposition If we have two inputs (independant sources), which are also referred to as "forcing functions", and two responses (or "response functions"), then when the inputs change, the responses will change. Theorem: In any linear resistive network, the voltage across or the current through any resistor or source can be calculated by algebraically adding all the individual voltages or currents caused by the "seperate" independat sources acting along, with all other independent sources deactivated. To deactivate an independant voltage source, you make it into a short circuit where v = 0 (replace it with a wire). To deactivate an independant current source, you make it into an open circuit where i = 0 (remove that branch from the circuit). Dependant sources are left active at all times. For any linear resistive circuit with N independant sources, we must solve the circuit N times, each time considering only one independant source and deactivating all others. Note that if we have three independant sources, we can either consider only one source at a time, three times, or consider two sources at a time, three times. This applies to any N independant source case. A good way to think of superposition is as follows: each source has a contribution to your result, and that contribution does not depend on the contribution of any other sources. Source Transformation Practical Voltage Sources An ideal voltage source was de▯ned as a device whose terminal voltage is independant of the current through it. An ideal voltage source can provide an unlimited amount of power, and does not practically exist. A better approximation of a practical voltage source is an ideal voltage source in series with a resistor. This causes the voltage to be reduced when large currents are drawn from it. This allows us to use a combination of ideal circuit elements to model a real device. Note that any eral device is characterized by a certain current-voltage relationship at its terminals, thus we can try to develop some combination of ideal elements that can furnish a smiliar current-voltage characteristic. For any practical voltage source, as the current increases, the voltage approaches zero, even- tually reaching zero at some maximum current. The voltage when the current is zero is equal to the voltage provided by the ideal voltage source. 9 Note that the resistor is not "really" present as a separate component, but is only a part of the model we use to represent a practical voltage source in which the terminal voltage decreases as the load current increases. Practical Current Source Ideal current sources can deliver a constant current regardless of the load resistance connected to it or the voltage across its terminals. Such a device does not exist in the real world. A practical current source is de▯ned as an ideal current source in parallel with a resistor. As voltage across these elements increase, the current it provides decreases. Equivalent Practical Sources Two sources are said to be equivalent if they produce identical voltage and curretns when they are connected to identical resistances. Identical sources provide the same open-circuit voltage and short-circuit current, though they provide di▯erent amounts of power. In essence, they are equivalent only with respect to what appears at the load terminals, but thay are not equivalent internally. The goal of source transformation tends to be to end up with either all current sources or all voltage sources. To perform source transformation on a voltage source in series with a resistor, replace that network with a current source in parallel with a resistor. The resistor will have the same resistance, and the current source will have a current equal to the voltage of the original source divided by the resistance of the resistor. To perform source transformation on a current source in parallel with a resistor, replace that network with a voltage source in series with a resistor. The resistor will have the same resistance, and the voltage source will have a voltage equal to the current of the original source times the resistance of the resistor. Thevenin’s Equivalent Circuits Thevenin’s Theroem: It is possible to replace any linear circuit with an inde- pendant voltage source in series with a resistor. The response measured at the load resistor will remain unchanged, thus the original circuit and the equivalent circuit will be equivalent, from an electrical point of view. To use Thevenin’s Theorem: 10 1. Given any linear circuit, rearrange it in the form of two networks, connected by two wires. The network on the left will be made into an equivalent circuit. Ensure that the two parts of any dependant source are in the same network. 2. Disconnect the networks at the wire connections. 3. Find the open circuit voltage th across the broken wire connections (nodes). 4. To ▯nd the equivalent resistance Rth (chose one method) (a) For networks with only independant sources ▯ Deactivate any independant sources. ▯ Find R thbetween the two nodes. (b) For networks with dependant and/or independant sources, or networks without independant sources. ▯ Deactivate any independant sources. ▯ Leave any dependant sources untouched. ▯ Put a test voltage between the nodes and ▯nd the current across the nodes, Vt or put a test current source and ▯nd the voltage across. Then Rth = t (c) For networks with dependant and/or independant sources ▯ Leave all sources untouched. ▯ Find the short circuit current, assuming the nodes are attached. Then Rth= Vth Isc Norton’s Theorem Norton’s Theorem: It is possible to replace any circuit with an independant current source in parallel with a resistor. Norton’s method follows in the same way as Thevenin’s. Note the two equivalent circuits are related through source transformation. Maximum Power Transfer Theorem: An independant voltage source with resistance R or an indepen- S dant current source with resistance R S delivers a maximum power to the load resistance R for which R = R L S L Note that R Ss equal to the Thevenin or Norton equivalent resistance. 11 Thus we have 2 P = v max 4R 2 Operational Ampli▯ers "Op amps" are very useful every day electrical devices built on integrated circuits (IC). One IC may contain several op amps, and each op amp is built using many transistors. where we have vCC is the positive power supply terminalEE▯is the negative power supply + ▯ terminal (or ground), v is the non-inverting input, v is the inverting input,Ovis the output, and the o▯set null terminals are used for external adjustment to balance the op amp. For analysis, the inner circuitry of the om amp makes no di▯erence, and we are concerned only with the relationship between input and output. Ideal Op Amps In practice, most op amps perform so well that we can assume we are dealing with an ideal om amp. Characteristics of an ideal op amp ▯ No current can enter either input terminal + ▯ ▯ vo= A (o ▯ v ) where A isothe open loop gain and A =o1 for ideal op amps. ▯ The output terminal acts as an ideal voltage source (ie. does not vary depending on connected load). ▯ ▯v EE ▯ vo▯ v CC 12 Note that if v > v▯ then vo= v CC and if v < v ▯ then o = ▯v EE. The only way vocan be a ▯nite number is if EE = v CC In this course, we’ll focus on negative feedback systems. Inverting Ampli▯ers Since we have negative feedback and v is connected to the ground, v becomes a virtual ground. 13 This allows us to solve this circuit by following the current, which follows a linear path. The general form of this circuit gives us R f vo= ▯ R vin 1 Note that the relationship between
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