Form factor (electronics)
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electronics Electronics is a scientific and engineering discipline that studies and applies the principles of physics to design, create, and operate devices that manipulate electrons and other Electric charge, electrically charged particles. It is a subfield ...
and
electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems that use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
, the form factor of an
alternating current Alternating current (AC) is an electric current that periodically reverses direction and changes its magnitude continuously with time, in contrast to direct current (DC), which flows only in one direction. Alternating current is the form in w ...
waveform (signal) is the ratio of the RMS (
root mean square In mathematics, the root mean square (abbrev. RMS, or rms) of a set of values is the square root of the set's mean square. Given a set x_i, its RMS is denoted as either x_\mathrm or \mathrm_x. The RMS is also known as the quadratic mean (denote ...
) value to the average value (mathematical mean of
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
s of all points on the waveform). It identifies the ratio of the
direct current Direct current (DC) is one-directional electric current, flow of electric charge. An electrochemical cell is a prime example of DC power. Direct current may flow through a conductor (material), conductor such as a wire, but can also flow throug ...
of equal power relative to the given alternating current. The former can also be defined as the direct current that will produce equivalent heat.


Calculating the form factor

For an ideal, continuous wave function over time T, the RMS can be calculated in
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
form: X_\mathrm = \sqrt The rectified average is then the mean of the integral of the function's absolute value: X_\mathrm = The
quotient In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...
of these two values is the form factor, k_\mathrm, or in unambiguous situations, k. k_\mathrm = \frac \mathrm \mathrm = \frac = \frac X_\mathrm reflects the variation in the function's distance from the average, and is disproportionately impacted by large deviations from the unrectified average value. It will always be at least as large as X_\mathrm, which only measures the absolute distance from said average. The form factor thus cannot be smaller than 1 (a square wave where all momentary values are equally far above or below the average value; see below), and has no theoretical upper limit for functions with sufficient deviation. \mathrm_\mathrm = \sqrt can be used for combining signals of different frequencies (for example, for harmonics), while for the same frequency, \mathrm_\mathrm = \mathrm_1 + \mathrm_2 + ... + \mathrm_n. As ARV's on the same domain can be summed as \mathrm_\mathrm = \mathrm_1 + \mathrm_2 + ... + \mathrm_n, the form factor of a complex wave composed of multiple waves of the same frequency can sometimes be calculated as k_ = \frac = \frac.


Application

AC measuring instruments are often built with specific waveforms in mind. For example, many multimeters on their AC ranges are specifically scaled to display the RMS value of a sine wave. Since the RMS calculation can be difficult to achieve digitally, the absolute average is calculated instead and the result multiplied by the form factor of a sinusoid. This method will give less accurate readings for waveforms other than a sinewave, and the instruction plate on the rear of an
Avometer AVOmeter is a British trademark for a line of multimeters and electrical measuring instruments; the brand is now owned by the Megger Group Limited. The first Avometer was made by the Automatic Coil Winder and Electrical Equipment Co. in 1923, ...
states this explicitly. The squaring in RMS and the absolute value in ARV mean that both the values and the form factor are independent of the wave function's sign (and thus, the electrical signal's direction) at any point. For this reason, the form factor is the same for a direction-changing wave with a regular average of 0 and its fully rectified version. The form factor, k_\mathrm, is the smallest of the three wave factors, the other two being
crest factor Crest factor is a parameter of a waveform, such as alternating current or sound, showing the ratio of peak values to the effective value. In other words, crest factor indicates how extreme the peaks are in a waveform. Crest factor 1 indicates no pe ...
k_\mathrm = \frac and the lesser-known averaging factor k_\mathrm = \frac. k_\mathrm \ge k_\mathrm \ge k_\mathrm Due to their definitions (all relying on the
Root Mean Square In mathematics, the root mean square (abbrev. RMS, or rms) of a set of values is the square root of the set's mean square. Given a set x_i, its RMS is denoted as either x_\mathrm or \mathrm_x. The RMS is also known as the quadratic mean (denote ...
, Average rectified value and maximum
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
of the waveform), the three factors are related by k_\mathrm = k_\mathrm k_\mathrm, so the form factor can be calculated with k_\mathrm = \frac.


Specific form factors

a represents the amplitude of the function, and any other coefficients applied in the vertical dimension. For example, 8 \sin(t) can be analyzed as f(t) = a \sin(t),\ a = 8. As both RMS and ARV are directly proportional to it, it has no effect on the form factor, and can be replaced with a normalized 1 for calculating that value. D = is the
duty cycle A duty cycle or power cycle is the fraction of one period in which a signal or system is active. Duty cycle is commonly expressed as a percentage or a ratio. A period is the time it takes for a signal to complete an on-and-off cycle. As a for ...
, the ratio of the "pulse" time \tau (when the function's value is not zero) to the full wave period T. Most basic wave functions only achieve 0 for infinitely short instants, and can thus be considered as having \tau = T, D = 1. However, any of the non-pulsing functions below can be appended with = = \sqrt to allow pulsing. This is illustrated with the half-rectified sine wave, which can be considered a pulsed full-rectified sine wave with D = , and has k_\mathrm = k_\sqrt.


References

{{DEFAULTSORT:Form Factor (Electronics) Electrical parameters