Chapter 31c-32a - Power in (harmonic) AC circuits The power...

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Power in (harmonic) AC circuits The instantaneous power is still, a) it determines the energetic (and therefore monetary) cost of running the circuit, b) it determines the efficiency with which the circuit uses that energy (a simple change might improve this), and, c) it determines the heat that will be generated by the circuit (which might require cooling lest the circuit melt). p = iv (lower case time varying quantities) The power is important because:
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md sin( t) ξ=ξ ω d iI s i n (t ) φ p = i ξ So with & The instantaneous power is, dm d p [Isin( t )][ sin( t)] φ ξ ω The average of a time varying function f(t) over interval t 1 to t 2 is, 2 1 t t Ave 21 f(t)dt f tt = Since the instantaneous power is also periodic it is sufficient to take the average over one period, T , so TT Ave d m d 00 11 P pdt t )][ sin( t)]dt == ω φ ξ ω ∫∫
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T m Ave d d 0 I P sin( t )sin( t)dt T ξ φ ω The first sine function is expanded using the trig identity, sin(A B) sinAcosB cosAsin B −= T m Ave d d d 0 I P [sin( t)cos cos( t)sin ]sin( T ξ φ ω φ ω T m Ave d d d d 0 I P [sin( t)sin( cos( t)sin ]dt T ξ ω φ ω ω φ The second sine function is distributed, TT 2 mm Ave d d d 00 Ic o s Is i n P s i n (t ) d t c o s ) s i n ) d t ξφ ξ φ ω ω ∫∫ 0 T/2 m Ave I Pc o s 2 ξ So,
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md sin( t) ξ=ξ ω d iI s i n (t ) = ω− φ So given & The average power is, m Ave I Pc o s 2 ξ = φ Recall that we can express I and ξ m in terms of root mean square values (what AC ammeters and voltmeters measure) where, RMS I I 2 = m RMS 2 ξ ξ= & m Ave I o s 22 ξ = φ Then, Ave RMS RMS PI c o s = ξφ Where the phase factor is (still) the offset in phase between the drive voltage and the current. Its cosine is called the power factor .
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Ave RMS RMS PI c o s = ξφ What this says is that average power available to do work (per unit time) in the load depends on the phase between the power supply voltage and the current in the circuit. When the two are in phase ( φ = 0, cos φ = 1 ) this is maximized but otherwise the power is reduced (energy gets reflected back to the power supply). To better appreciate this we’ll consider a specific case for which we need the average power in a different form.
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Recall the phasor diagram for the RLC circuit. φ L V R V m ξ LC VV C V This gave that, m I Z ξ = 22 2 mR L C V( ) ξ= + R VI R = CC X = LL X = , , using, this gave that, where ZR ( X X ) =+ From the plot (& these expressions) we can also write that, R m R R cos IZ Z φ= = = ξ m IZ ξ =
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Ave RMS RMS RMS RMS R PI c o s I Z φ Then, & since m I Z ξ = RMS RMS I Z ξ = 2 2 2 RMS RMS Ave RMS RMS 22 RR R P ZZ R ξξ = ξ = So 2 2 RMS Ave LC R P R R( X X ) ξ = +− This expression, equivalent to is what we want.
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This note was uploaded on 03/12/2010 for the course PHY PHY taught by Professor Mueller during the Spring '09 term at University of Florida.

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Chapter 31c-32a - Power in (harmonic) AC circuits The power...

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