Figure 1: For a purely resistive load, power = voltage * current, with both vectors in phase |

Electric power is the rate at which energy is transferred to a circuit, calculated in units of Watts, with a Watt being an energy-transfer rate of 1 Joule/s. For a purely resistive load fed by a single current vector and single voltage vector,

P = I

^{2}R = V^{2}/R = instantaneous V*Iwith both vectors perfectly in phase (Figure 1).

Figure 2: Phase angle is the angular difference between the voltage and current vectors |

The cosine of the phase angle is known as the "power factor" (PF or λ). Remember that PF = Cos(φ) only for purely sinusoidal waveforms. Power factor is always present in AC systems. It's a unit-less quantity ranging from 0 to 1. For a purely resistive load with V and I in phase, PF = 1. For a purely capacitive or a purely inductive load with V and I 90° out of phase, PF = 0. Power factor is typically not signed: current either leads (capacitive load) or lags (inductive load) voltage (Figure 2).

Figure 3: Phase angle can be directly measured between a pure voltage sine wave and a pure current sine wave |

At the zero crossing, we can also see that there's a time delay between the waveforms that can be expressed in degrees relative to a full period of 360°. That's the phase angle, and again, if these current and voltage waveforms were feeding a purely resistive load, they would be perfectly in phase and would be directly on top of each other. In this case, current leads voltage, indicating a capacitive load. Thus, for a single-phase, non-resistive load (meaning that it's either capacitive or inductive), P ≠ V*I, and we must account for the phase angle.

#### Three Types of Power

To break things down further, there are three different types of power:- Apparent power (S), which is expressed in Volt-Amperes (VA) and calculated by multiplying the RMS voltage and current at any given point in time for a given power cycle. This assumes that we can measure "true RMS" values.
- Real power (P), which is expressed in Watts and derived from the apparent power by multiplying apparent power times the power factor. This assumes two sinusoidal waveforms; this derivation is never true if that's not the case.
- Reactive power (Q), which is expressed in Volt-Amperes reactive (VAr) and is the quadrature subtraction of the real from the apparent power:

Q = √(S

^{2}-P^{2})Reactive power is not consumed, which is to say it's not transferred to the load during a power cycle, but rather simply circulates in the circuit during operation.

So to conclude, AC line voltages are very different from the 5- or 3.3-V logic voltages familiar to many engineers. Three-phase systems are a complex mix of magnitude, phase, and rotation, and they introduce different measurement challenges, such as the lag/lead of current vectors relative to voltage vectors. That results in more than one type of power (real, apparent, and reactive), so line "power" isn't a simple P = V*I situation.

Next time, we'll look at power calculations for distorted waveforms.

Previous posts in this series:

Back to Basics: The Fundamentals of Power

Back to Basics: Fundamentals of AC Line Power (Part II)

Back to Basics: Three-Phase Sinusoidal Voltages

More Basics of Three-Phase Sinusoidal Voltages

Back to Basics: AC Sinusoidal Line Current

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