It is difficult to imagine a discussion of electricity that does not include some reference to voltage, current or both. Indeed, these quantities represent two of the three legs of the foundation described by Ohm’s Law. Although references to voltage and current are common, the values in an AC system can be expressed in a number of different ways, each of which can have a special significance in electrical design.
Peak values represent the peak or maximum voltage or current as measured from the zero reference. Though this is not used much in everyday calculations, peak values can be important in some calculations. For example, peak voltage is an important consideration when considering insulation and peak currents create peak flux values. These values are important in the design of equipment.
Sometimes AC values are stated as peak-to-peak values indicating the voltage or current deviation between the peak positive and peak negative value. Typically, the peak-to-peak value is double the peak value, though in rare cases the positive and negative deviation is asymmetrical.
Although not used much for practical calculations, understanding instantaneous values is important when studying other aspects of circuits. The instantaneous value is simply the value at a particular instant of time and can fall anywhere from zero to the peak value.
RMS or Effective Values
RMS (Root Mean Square) is the most common way to express electrical values since it models the equivalent work from a DC circuit with the same values. The RMS value of a sine wave can be easily calculated by multiplying the peak value by 0.7071 (1/√2), however, the method requires taking numerous instantaneous measurements for other types of waveforms. The arrows in the diagram below indicate points of instantaneous values being measured.
For brevity, let’s analyze the points on the green arrows. That gives us 8 points of measurement (counting the 2 zero crossing points). The measured values for points 1 through 8 are:
Each value is squared. The squared values are then added together and averaged (mean):
49 + 100 + 49 + 0 + 49 + 100 + 49 + 0 = 396
396/8 = 49.5
√49.5 = 7.0356
This measured value is very close to the expected 7.071 that we would expect with a peak value of 10. Increasing the number of sample points would increase the accuracy. Though this is not necessary for a sine wave, this system makes it possible to make accurate RMS measurements on non-sinusoidal waveforms.
The average value of a sine wave is zero since the sum of the positive and negative alternations is zero. Therefore, the average is actually stated as twice the value of the average of a half cycle. For a sine wave, this is stated mathematically as:
(2 x Peak) / π = 0.637
Or, if you prefer, calculate the average (mean) of a half cycle and multiply by two:
Peak / π = 0.318
2 x 0.318 = 0.637
As you can see, the value is slightly less than the RMS value.
Form factor is frequently used in transformer calculations and in some texts is expressed as a “constant” of 1.11. In reality, form factor is the ratio of the RMS and average values
F = RMS/Average
Use of the “constant” assumes the form factor is for a sine wave (or for other waveforms that are arranged to simulate a sine wave).
F = 0.707/0.637 = 1.11
Form factor for other waveforms will yield other multipliers, for example, a square waves RMS and average values are the same which give us a form factor of 1.0 for a square wave.
References and related links:
- Lowdon, Eric (1989). Practical Transformer Design Handbook (2nd Ed). Tab Books.