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Berkeley ELENG 42 - Lecture Notes

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EECS 42, Spring 2005 Week 2a 1Electric CurrentDefinition: rate of positive charge flowSymbol: iUnits: Coulombs per second ≡ Amperes (A)i = dq/dtwhere q = charge (in Coulombs), t = time (in seconds)Note: Current has polarity.EECS 42, Spring 2005 Week 2a 2Electric Potential (Voltage)• Definition: energy per unit charge• Symbol: v• Units: Joules/Coulomb ≡ Volts (V)v = dw/dqwhere w = energy (in Joules), q = charge (in Coulombs)Note: Potential is always referenced to some point.Subscript convention:vabmeans the potential at aminus the potential at b.abvab≡ va-vbEECS 42, Spring 2005 Week 2a 3Electric Power• Definition: transfer of energy per unit time• Symbol: p• Units: Joules per second ≡ Watts (W)p = dw/dt = (dw/dq)(dq/dt) = vi• Concept:As a positive charge q moves through a drop in voltage v, it loses energy energy change = qv rate is proportional to # charges/secEECS 42, Spring 2005 Week 2a 4The Ideal Basic Circuit ElementAttributes:• Two terminals (points of connection)• Mathematically described in terms of current and/or voltage• Cannot be subdivided into other elements+v_i• Polarity reference for voltage can beindicated by plus and minus signs• Reference direction for the currentis indicated by an arrowEECS 42, Spring 2005 Week 2a 5- v +A problem like “Find the current” or “Find the voltage”is always accompanied by a definition of the direction: In this case, if the current turns out to be 1 mA flowing to the left, we would say i = -1 mA.In order to perform circuit analysis to determine the voltages and currents in an electric circuit, you need to specify reference directions. There is no need to guess the reference direction so that the answers come out positive, however.A Note about Reference DirectionsiEECS 42, Spring 2005 Week 2a 6Suppose you have an unlabelled battery and you measure its voltage with a digital voltmeter (DVM). It will tell you the magnitude and sign of the voltage.With this circuit, you are measuring vab. The DVM indicates −1.401, so vais lower than vbby 1.401 V. Which is the positive battery terminal?−1.401DVM+abNote that we have used the “ground” symbol ( ) for the reference node on the DVM. Often it is labeled “C” for “common.”Sign Convention ExampleEECS 42, Spring 2005 Week 2a 7Sign Convention for Power• If p > 0, power is being delivered to the box. • If p < 0, power is being extracted from the box.+v_iPassive sign convention_v+ip = vi+v_i_v+ip = -viEECS 42, Spring 2005 Week 2a 8Find the power absorbed by each element:Power Calculation Examplevi (W)918- 810-12- 400- 2241116p (W)Conservation of energyÎ total power deliveredequals total power absorbedAside: For electronics these are un-realistically large currents – mA is more typical than AEECS 42, Spring 2005 Week 2a 9Circuit Elements• 5 ideal basic circuit elements:– voltage source– current source–resistor– inductor–capacitor• Many practical systems can be modeled with just sources and resistors• The basic analytical techniques for solving circuits with inductors and capacitors are the same as those for resistive circuitsactive elements, capable ofgenerating electric energypassive elements, incapable ofgenerating electric energyEECS 42, Spring 2005 Week 2a 10Electrical Sources•An electrical source is a device that is capable of converting non-electric energy to electric energy and vice versa.Examples:– battery: chemical electric– dynamo (generator/motor): mechanical electricÆElectrical sources can either deliver or absorb powerEECS 42, Spring 2005 Week 2a 11Ideal Independent and Dependent Voltage Sources• Circuit element that maintains a prescribed voltage across its terminals, regardless of the current flowing in those terminals.– Voltage is known, but current is determined by the circuit to which the source is connected.• The voltage can be either independent or dependenton a voltage or current elsewhere in the circuit, and can be constant or time-varying.Circuit symbols:+_vs+_vs=µ vx+_vs=ρ ixindependentvoltage-controlledcurrent-controlledEECS 42, Spring 2005 Week 2a 12EECS 42, Spring 2005 Week 2a 13Other Independent Voltage Source Symbolsv(t) = Vpeaksin(ωt)Veffective= Vpeak\/2(In US, 120 V, soVpeak= 170 V)Sinusoidal AC sourceBattery (realistic source)+VSEECS 42, Spring 2005 Week 2a 14EE 42 Lecture 314Spring 2005Realistic Voltage Source A real-life voltage source, like a battery or the function generator in lab, cannot sustain a very high current. Either a fuse blows to shut off the device, or something melts… Additionally, the voltage output of a realistic source is not constant. The voltage decreases slightly as the current increases. We usually model realistic sources considering the second of these two phenomena. A realistic source is modeled by an ideal voltage source in series with an “internal resistance”, RS.+−VsRSEECS 42, Spring 2005 Week 2a 15I-V Plot for a Real BatteryEECS 42, Spring 2005 Week 2a 16Ideal Independent and Dependent Current Sources• Circuit element that maintains a prescribed current through its terminals, regardless of the voltage across those terminals.– Current is known, but voltage is determined by the circuit to which the source is connected.• The current can be either independent or dependenton a voltage or current elsewhere in the circuit, and can be constant or time-varying.Circuit symbols:isis=α vxis=β ixindependentvoltage-controlledcurrent-controlledEECS 42, Spring 2005 Week 2a 17Electrical Resistance• Resistance: Electric field is proportional to current density, within a resistive material. Thus, voltage is proportional to current. The circuit element used to model this behavior is the resistor.Circuit symbol:Units: Volts per Ampere ≡ ohms (Ω) • The current flowing in the resistor is proportional to the voltage across the resistor:v = i Rwhere v = voltage (V), i = current (A), and R = resistance (Ω)R(Ohm’s Law)EECS 42, Spring 2005 Week 2a 18Resistance of an actual resistorWLTMaterial resistivity= ρ (Ω-cm)Resistance = resistivity x length/(cross-sectional area)R = ρ (L/WT)EECS 42, Spring 2005 Week 2a 19Electrical Conductance• Conductance is the reciprocal of resistance.Symbol: GUnits: siemens (S) or mhos ( )Example:Consider an 8 Ω resistor. What is its conductance?ΩEECS 42, Spring 2005 Week 2a 20Short Circuit and Open CircuitWire (“short circuit”):• R = 0 Æ no voltage


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Berkeley ELENG 42 - Lecture Notes

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