t t t Numerically, this is an extension of the basic definition of the average for a discrete

Transcription

1 Average and alues of a Periodic Waveform: (Nofziger, 8) Begin by defining he average value of any ime-varying funcion over a ime inerval as he inegral of he funcion over his ime inerval, divided by : f f f (.) Numerically, his is an exension of he basic definiion of he average for a discree variable, N xi i x, alied o a coninuously-varying funcion. N Grahically, his means ha he area under he funcion (beween imes and ) is equivalen o he area of a recangle of heigh f mulilied by wih ) : The average of a sine wave over one half-cycle: Consider a sine wave of eak amliude and frequency f: ( f ( ) f (.) sin ( πf ) sin( ω) (.) The eriod of he wave,, is given as: π f ω (.) The average value of one half-cycle in ime, is hen calculaed as: sin( ω) sin( ω) (.5) ( ω) cos ω cos ( )cos() ω ω cos( π ) ω + ω π π (.6) (.67) calculaed over one half-cycle of he sine wave (.7) π

2 The average of a sine wave over one full-cycle: sin( ω) (.8) ( ω) cos ω ω ω [ cos ( ) ] π + [ + ] ω ω [ cos cos()] calculaed over one full-cycle of he sine wave (.9) While his resul is mahemaically correc, i doesn rovide any hysical insigh ino wha his sine wave can accomlish over one comlee cycle. Consider he AC sinusoidal volage delivered by Tucson Elecric Power a a wall oule. s average value may be, ye we know from exerience ha his sine wave will ligh u fluorescen bulbs, hea u ungsen filamens in ligh bulbs, hea wires in oasers, ec. This is because hese devices absorb energy (ower) from he sine wave, wheher he volage is osiive-going or negaive-going. Useful work is done during boh half-cycles of he sinusoidal waveform. The roblem, mahemaically, is ha we have added u negaive as well as osiive values. To accoun for his and make sure ha all ars of he waveform conribue o is calculaed effecive value, firs square he funcion f () so ha all values are osiive. Then add u (inegrae) all values of f () over one comlee cycle, and finally ake he square roo. This is known as he Roo Mean Square or value of any imevarying (or saially-varying) waveform, and is defined as: { f ) } AG Y ( f (.) The hysical meaning of he value is his i is he consan, or DC value ha would cause he same hysical effec as he acual ime-varying waveform does, during one comlee eriod. This migh be o deliver he same ower in a circui, o cause he same heaing effec in a oaser, o ligh u a bulb wih he same brighness, ec. Noe ha, in general, a eriodic waveform may no have symmery like sinusoidal or riangular waveforms have. n his case, you sill calculae he value according o equaion (.), by inegraing over one comlee cycle. Modern-day insrumenaion (he DMMs and oscilloscoes in our lab) digially samle a waveform, and numerically inegrae he values o calculae he value, according o equaion (.).

3 alue of a Sinusoidal Waveform: Consider a sinusoidal volage of eak amliude and frequency f: sin ( πf ) sin( ω) (.) The value is calculaed as: sin ( ω) (.) ( cosω ) cos(ω) (.) sin(ω) ( ) (.) ω sin(ω ) sin( π ) ω ω (.5) [ ] + (.6) (.77) for a sinusoidal waveform (.7) (Noe ha you ge he same resul wheher you inegrae from ( ), ( ), or even from ( ), because of he symmery of his waveform. For non-symmerical waveforms, you have o inegrae over he comlee cycle.) The quoed value of AC for he volage a he wall socke, as delivered by Tucson Elecric Power is, in fac, he value of he (6Hz) sinusoidal volage. The eak volage is herefore 69, and he eak-o-eak value is 9!!!

4 alue of a Triangular Waveform: Consider a riangular volage waveform ha is bi-olar, has a 5% duy-cycle (symmerical abou ), and a frequency f. Le he maximum volage be, and he minimum volage be. Because of he symmery, inegrae from ) ( o calculae he value: ) ( (valid beween and ) (.8) ) ( AG (.9) (.) 6 6 (.).577 for a riangular waveform (.)

5 alue of a Recangular Waveform: Consider a recangular volage waveform ha is bi-olar, has a 5% duy-cycle (symmerical abou ), and a frequency f. Le he maximum volage be, and he minimum volage be. Because of he symmery, inegrae from ( ) o calculae he value: () (valid beween and ) (.) AG () (.) ( ) (.5) for a recangular waveform (.6)