ENGR 12
Ch 10 Notes on Power
From the beginning, we have used p = iv for power, where positive means absorbing and
negative means delivering. With sinusoidal signals, this formula still applies, but there are
many variations/ways of applying it to AC circuits
1) Instantaneous Power p(t) = i(t) v(t) (Watts) Defined at every instant, t, p(t) fluctuates in
sinusoidal circuits, so is not very useful for analysis. But it underlies all other approaches to power.
For example, if
v(t) = Vm cos( wt + v)
and i(t) = Im cos( wt + i)
Various trig identities can be used to show that
p(t) = Vm Im cos(v - i) + Vm Im cos(v - i) cos (2t) - Vm Im sin(v - i) sin (2t)
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2
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(1)
2) Average and Reactive Power Eq 1 can be expressed as
p(t) = P + P cos(2t) - Q sin(2t) , where
P = Vm Im cos(v - i)
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AVERAGE or REAL Power
Q = Vm Im sin(v - i)
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REACTIVE power
P, the Average or Real power is the power that is transformed from electric to non-electric energy.
3) Purely Resistive Circuits ( v - i = 0o) , p = P + P cos (2t) (unit: Watts)
Which is always positive (or 0), always dissipating energy.
4) Purely Inductive Circuits p = - Q sin(2t) (unit: VAR = volt-amp reactive)
Due to the phase offset of inductive circuits ( v - i = 90o), P goes to zero, and Q reduces to (Vm Im)/2
Average power is zero, instantaneous power is sometimes positive, sometimes negative
Since Q is positive, the minus sine above means p is an inverted sinewave
5) Capacitive Circuits p = - Q sin(2t) (VAR) Again, due to the phase offset of capacitive circuits (this
time v - i = -90o), P goes to zero, and Q now reduces to -(Vm Im)/2
Average power is zero, instantaneous power is sometimes positive, sometimes negative
Since Q is negative, the minus sine above means p is a non-inverted sinewave
Inductors demand (or absorb) magnetizing VARs, Capacitors furnish (or deliver) VARs.
6) Power Factor pf = cos(v - i) and is considered a lagging power factor (inductive load) if current
lags voltage, that is, (v > i) and a leading power factor (capacitive load) if (v < i)
This quantity determines how resistive the load is and what type the reactive portion is.
7) RMS Value and Power
For a purely resistive circuit, P = <v(t)2/R> which for a sinusoid is Vm2/2R = V2rms/R
(also P = Vrms Irms = Irms2R)
RMS value is also referred to as effective value because an AC source of a given rms voltage delivers
the same energy in one period that a DC source of the same voltage. For example, a 100 Vrms AC
source delivers the same power to resistor R that a 100 V DC source does.
For reactive circuits, average power can be written as P = V rms I rms cos(v - i),
reactive as Q = V rms I rms sin(v - i).
8) Complex Power S = P + j Q (unit: Volt-Amps or VA) the above power relationships can be
encapsulated better in a complex number, which can be computed directly from the Voltage and
Current Phasors for a circuit.
This also incorporates the power factor angle = v - i as the arctan of Q/P
9) Apparent Power
The magnitude of S, |S| = sqrt(P2 + Q2) is measured also in VA, which indicates the volt-amp capacity
needed to supply the average power. (What you pay for from the electric company). Average Power P
is what you actually get in terms of converting to non-electric energy uses (heat, mechanical).
Therefore it’s best to balance inductive loads (like refrigerators, motors, fluorescent lights) with
capacitive ones to reduce the power factor angle near to zero, where, |S| = P and you “get what you
pay for” in terms of electrical energy.
Instead of calculating P and Q the laborious way (plug in Vm, (v - i), etc) we can get directly to
complex power by further trig analysis.
S
= Vm Im cos(v - i) + j Vm Im sin(v - i)
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2
S
= Vm Im (cos(v - i) + j sin(v - i) )
2
S
= Vm Im e j(v - i) = VmIm < (v - i) = Vrms Irms< (v - i) = Vrms I*rms
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2
In other words, S = the rms phasor for V multiplied by complex conjugate of rms Phasor for I
OR, using regular non-rms phasors: S = VmIm*
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