Figure 1: The sum of many sine waves, of varying amplitudes and frequencies, comprises the rough- looking square wave shown in red |

Among the best-known examples of a distorted waveform is a pulse-width-modulated (PWM) waveform from a motor-drive or inverter output. So that's what we'll use for examples as we look over these power calculations for distorted waveforms.

Figure 2: The output of a PMSM drive, though ostensibly sinusoidal, betrays sawtooth characteristics upon closer inspection (see zoom inset) |

Figure 3: A distorted waveform can be acquired as an analog signal and digitally sampled for its displayed representation |

The way to handle power calculations for these sorts of distorted waveforms is with digital sampling techniques (Figure 3). An oscilloscope acquires the analog signal through sampling at a given rate and represents the signal as a collection of sampled points that are joined together by an algorithm. The result appears on the oscilloscope display as the waveform, and from there we can apply any number of math processes to the waveform to obtain power information.

Figure 4: An algorithm determines the signal's measurement period, which is denoted by a highlighted overlay |

Figure 5: Shown is a sine wave with two measurement periods indicated; from them algorithms will calculate per-cycle values |

Table 1: Formulas used for per-cycle calculations of voltage, current, and real, apparent, and reactive power |

Table 2: Formulas used for per-cycle calculation of power factor and phase angle |

Likewise, Table 2 shows the formulas for power factor (the ratio of real to apparent power) and phase angle (the inverse cosine of the power factor).

So to sum up, the textbook descriptions of power calculations typically assume sinusoidal waveforms for single-phase systems (one voltage and one current). But the output of a power electronics converter/inverter is a distorted waveform that requires calculation methodologies that may be unfamiliar to most engineers. There's no practical way to measure phase angle between distorted voltage and current waveforms, so the only alternative is digital sampling techniques as described here (which also work for pure sine waves).

Next time, we'll discuss three-phase power calculations.

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

Power Calculations for Pure Sine Waves

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