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Unformatted text preview: 1 Chapter 31: Alternating current 2 A power supply can be set to give an EMF of the form: t t ω ε ε sin ) ( = This EMF is time dependent, has an amplitude ε , and varies with angular frequency ω . Other time dependent EMFs include square wave and sawtooth. §31.1: Phasors and Alternating currents 3 f π ω 2 = angular frequency in rads/sec frequency in cycles/sec or Hz The current in a resistor is still given by Ohm’s law: t I t R R t t I ω ω ε ε sin sin ) ( ) ( = = = The current has an amplitude of I = ε /R. 4 What are the averages of ε (t) and I(t) over one cycle? cos 1 sin sin = = = ∫ ∫ T t dt dt t t T T T ω ω ω ω The “problem” here is that the average value of sin ω t over one complete cycle is zero! This is not a useful way to characterize the quantities ε (t) and I(t). 5 To fix this problem we use the root mean square (rms) as the characteristic value over one cycle. 2 and 2 rms rms ε ε = = I I 2 1 4 2 sin 2 sin sin 2 2 = = = ∫ ∫ T t t dt dt t t T T T ω ω ω ω ( ) t 2 rms ε ε = The average value of cos 2 ω t is 6 Phasors and phasor diagrams ( ) ( ) φ ω + = t A t A sin max Consider the timedependent quantity A(t) The value of A(t) [in blue] can be determined from the projection of A max [in red] onto the yaxis A max A(t) ( ω t+ φ ) 7 Consider a single resistor connected to the EMF. §31.2: Resistance and reactance 8 The current and the EMF have the same time dependence so they are in phase. The voltage amplitude is V R = IR. ( ) ( ) ( ) t R t I t IR t IR t ω ε ω ε ε ε cos cos = = = = 9 Consider a single capacitor connected to the EMF. ( ) ( ) ( ) ( ) ( )  = = = = = = = ∫ 2 cos sin sin cos π ω ω ω ω ω ω ω ε t C I t C I C t Q V t I dt t I t Q t I t I C Q t C Note: The time dependence of the voltage and current are different. The current reaches a maximum before the voltage. It is said that “the current leads the voltage”. 10 C C C X V CV I = = ω The current amplitude is Where the capacitive reactance is C X C ω 1 = This now looks like Ohm’s law: C C IX V = 11 Consider a single inductor connected to the EMF....
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 Fall '07
 Fuchs
 Current, Power, Alternating Current, RLC, Inductor

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