# Equivalence

In electrical engineering , especially in the field of electrical measurement technology and theoretical electrical engineering , the term equivalence stands for arithmetic mean or linear time mean . It is an application of the arithmetic mean to variable quantities of a stationary process that are continuously present over time . It indicates the constant component if there is a superposition of alternating and constant quantities.

## approach

Is the mathematical definition of the arithmetic mean value is applied to a continuously available size, the result is with single values which in the same time intervals during an observation period has been obtained, ${\ displaystyle {\ overline {x}} = {\ frac {1} {n}} \ sum _ {i = 1} ^ {n} x_ {i}}$${\ displaystyle \ Delta t}$${\ displaystyle \ tau = n \ cdot \ Delta t}$

{\ displaystyle {\ begin {aligned} {\ overline {x}} & = \ lim _ {n \ to \ infty} {\ frac {1} {n}} \ sum _ {i = 1} ^ {n} x_ {i} \\ & = \ lim _ {\ tau \ to \ infty} {\ frac {1} {\ tau}} \ sum _ {i = 1} ^ {\ tau / \ Delta t} x_ {i } \ cdot \ Delta t \ quad {\ text {or}} \\ & = \ lim _ {\ Delta t \ to 0} {\ frac {1} {\ tau}} \ sum _ {i = 1} ^ {\ tau / \ Delta t} x_ {i} \ cdot \ Delta t \. \ end {aligned}}}

The last line leads to an integral if the size can be represented by an integrable function.

In practice, a representative finite period is sufficient as the observation period.

## Equivalent for periodic processes

Sinusoidal alternating voltage, rectified, squared; the equivalents in each case

Using the example of an electrical voltage with the instantaneous value, its equivalent value ${\ displaystyle u}$

• the mean height of all stress-time areas   or${\ displaystyle u \, \ cdot \, \ Delta t}$
• the sum of all stress-time areas during an observation period divided by the observation period.${\ displaystyle \ tau}$

In the case of periodic processes with the period duration , the observation period can be limited to a number of periods ( , integer) and the equivalent value can be calculated using the sum ${\ displaystyle T}$${\ displaystyle m}$${\ displaystyle m \ geq 1}$

${\ displaystyle {\ overline {u}} = {\ frac {1} {mT}} \ cdot \ sum u \ cdot \ Delta t \.}$

It is necessary to record as precisely as possible with many individual values. One chooses or . (Also has to be.) If the function is known , the sum is replaced by the integral over a period ( ) with a point in time that can be selected as desired${\ displaystyle \ Delta t \ ll \ tau = m \ cdot T}$${\ displaystyle \ Delta t \ ll T}$${\ displaystyle m \ gg 1}$${\ displaystyle \ Delta t ${\ displaystyle u}$${\ displaystyle m = 1}$${\ displaystyle t_ {1}}$

${\ displaystyle {\ overline {u}} = {\ frac {1} {T}} \ int \ limits _ {t_ {1}} ^ {t_ {1} + T} u \ \ mathrm {d} t \ .}$

When AC voltage is called a voltage whose polarity changes in regular repetition, but whose time average is zero. The curve shape of the voltage is irrelevant and in no way tied to the sine curve. The area of ​​the voltage above the zero line is just as large in magnitude as the area below the zero line; the sum of the positive area (above the zero line) and the negative area (below the zero line) is then equal to zero.

In the case of mixed voltage , the constant component is obtained from the height of a horizontal straight line, in which the areas above and below complement each other to zero.

As further symbols except be used: ${\ displaystyle {\ overline {u}}}$

${\ displaystyle {\ overline {u}} = {\ overline {U}} = U _ {-} = U _ {\ text {av}} = U _ {\ text {DC}}}$; av stands for average, DC for direct current.

## Measurement of equivalency

### Analog measuring method

With this measuring method , the moving coil of the moving-coil measuring mechanism is deflected by a force that is proportional to the current . An alternating voltage generates an alternating positive and negative current and a corresponding force. Since the mechanical measuring mechanism cannot follow the rhythm of technical alternating voltages, only the mean force is recorded and thus the direct voltage component of the mixed voltage is displayed. ${\ displaystyle i}$${\ displaystyle U _ {-}}$

### Digital measuring method

Digital multimeters often use an analog-to-digital converter using the two-ramp method . Here, too, the integration is carried out in the input stage using analog technology.

When measuring in the DC measuring range , a capacitor is charged for a fixed period of time; it integrates current. A DC voltage charges the capacitor linearly over time. With AC voltage, the capacitor is charged and discharged again to the same extent; after a whole number of periods, e.g. B. after 300 ms at 50 Hz or 60 Hz, the capacitor is just as much or less charged by the AC voltage as by the DC voltage alone. The level of the capacitor charge is decisive for the display. This means that only the direct voltage component of the mixed voltage is measured in the DC range.

### Procedure for changing sizes

Since, by definition, an alternating quantity has the equivalent value zero, it is meaningless to measure it for this quantity. The simplest method of characterizing an alternating quantity by measurement is to determine its rectified value . With regard to energy transfer, the measured rms value is more meaningful.

## Individual evidence

1. a b DIN 40110-1: 1994 alternating current quantities
2. DIN 5483-1: 1983 Time-dependent quantities