Lecture 20 - AC voltage sources As we have shown, the use...

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AC voltage sources • As we have shown, the use of rotating elements in generators and the ability to use transformers led to the widescale adoption of AC voltage sources. • For these, the voltage varies with time as (t)=V os(2 )= V os( . V(t) V 0 cos(2 π ft) V 0 cos( ω t). •V 0 is the amplitude. In common US household applications V 0 170 V. Huh? What about 120 V? In Europe, the analogous V 0 is 311 V. • f is the frequency. In the USA, f=60 Hz. Europe uses 50 z. This was an unlucky choice as voltages at f 60 Hz Hz. This was an unlucky choice as voltages at f 60 Hz present very high relative shock hazards. It’s too late to change.
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resistor connected to AC coltage A resistor connected to AC coltage • Notice we complete the circuit with ground connections. • We have V cos(2 π ft)-i(t)R=0, so i(t)=(V /R)cos(2 π ft). 0 ( ) () , () ( 0 )( ) • The current is in phase with the voltage. That is, if we expressed i(t) as i(t)=i 0 cos(2 π ft- φ ), with φ the phase onstant we have 0 for a resistor constant , we have φ =0 for a resistor. • The power delivered is P(t)=V(t)i(t)=V 0 2 cos 2 (2 π ft)/R. The power is not constant, but the fast oscillation is often not portant important. • Of more interest is the average power.
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verage power and RMS voltage Average power and RMS voltage • The average power is <P>=<V 0 cos(2 π ft)i 0 cos(2 π ft- φ )>, where <x> means the “time average of x”, and to be general we let the current has a phase constant. • This can be written P>=V cos 2 )cos cos(2 )sin(2 )sin <P> V 0 i 0 <cos (2 π ft)cos φ +cos(2 π ft)sin(2 π ft)sin φ > =(V 0 i 0 /2)cos φ , • So <P>=V 0 2 /2R for a resistive load. It is conventional to rite this as <P>=V where V V ] /2 V 2 write this as <P>=V rms 2 /R, where V rms [<V 2 >] 1/2 =V 0 / 2. • It is V rms that equals 120V in USA household electricity applications, and 220V in European household electricity. • One can similarly define i rms =[<i 2 >] 1/2 , so in general <P> = V s i s cos φ .
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This note was uploaded on 05/11/2010 for the course PHY 214 taught by Professor Timothybolton during the Spring '10 term at Kansas State University.

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Lecture 20 - AC voltage sources As we have shown, the use...

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