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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 sin ) ( = This EMF is time dependent, has an amplitude , and varies with angular frequency . Other time dependent EMFs include square wave and sawtooth. 31.1: Phasors and Alternating currents 3 f 2 = angular frequency in rads/sec frequency in cycles/sec or Hz The current in a resistor is still given by Ohms law: t I t R R t t I sin sin ) ( ) ( = = = The current has an amplitude of I = /R. 4 What are the averages of (t) and I(t) over one cycle? cos 1 sin sin = = = T t dt dt t t T T T The problem here is that the average value of sin t over one complete cycle is zero! This is not a useful way to characterize the quantities (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 = = I I 2 1 4 2 sin 2 sin sin 2 2 = = = T t t dt dt t t T T T ( ) t 2 rms = The average value of cos 2 t is 6 Phasors and phasor diagrams ( ) ( ) + = 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) ( t+ ) 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 cos cos = = = = 9 Consider a single capacitor connected to the EMF. ( ) ( ) ( ) ( ) ( )  = = = = = = = 2 cos sin sin cos 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 = = The current amplitude is Where the capacitive reactance is C X C 1 = This now looks like Ohms law: C C IX V = 11 Consider a single inductor connected to the EMF....
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 Fall '07
 Fuchs
 Current, Power

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