RMS value

Due to the periodic voltage curves, each time step has a different voltage value. In this case, only the peak value can be specified directly, but this cannot always be used for calculations. To ensure useful comparability of voltage values, the effective value is used. The RMS value describes, for example, the identical power that would be generated at a resistor if it were connected to a direct current source. For a further explanation of the RMS value and the necessary fundamentals, this chapter discusses the following topics:

1 Fundamentals: Mean square and addition theorem

In addition to the arithmetic mean, which is used in alternating current calculations to determine the mean value, the quadratic mean is used to determine the RMS value. The quadratic mean (RMS – root mean square) is used to enable the comparability of alternating voltage values. First, a function f(t) is squared. Then, the mean value of the squared function is calculated. The final step is to take the square root of the calculated mean value. The mathematical relationship of the RMS is shown in equation 1.

\begin {equation} RMS=\sqrt {\frac {1}{T}\int _{0}^T f(t)^2 dt} \label {GleichungEff1} \end {equation}

When calculating the quadratic mean, the sine function is also squared. To assist with this, an addition theorem is presented to simplify the calculation. Addition theorems refer to the relationships in trigonometry used to transform a function. The transformation of the quadratic sine function using the addition theorem is shown in equation 2. Here, the quadratic sine function becomes a function that depends only on a simple cosine function.

\begin {equation} \sin ^2(x)=\frac {1}{2}(1-\cos (2x)) \label {GleichungEff2} \end {equation}

2 Amplitude

The amplitude of a function describes the maximum deflection during one period. For an alternating voltage, the amplitude value indicates the highest instantaneous voltage value. In Figure 1, voltage signals with a frequency of \(50\,Hz\) and thus with an identical period duration of \(T=20\,ms\) are shown. They differ in their amplitude values. The peak values of the amplitudes of the voltages shown are \(\hat {U}_1=1\,V\), \(\hat {U}_2=2\,V\) and \(\hat {U}_3=4\,V\). The amplitude is a kind of factor for calculating the RMS value. Thus, the height of the RMS value depends directly on the amplitude value.

3 Waveform

Neben der Amplitude eines Signals ist die Kurvenform des Signals ausschlaggebend für den Effektivwert. Bei der Berechung des Effektivwerts wird der Scheitelfaktor \(C\) (engl. für crest factor) bestimmt, welcher mit dem Ampltudenwert verrechnet wird. In der Gleichung 3 wird die Beziehung zwischen dem Amplitudenwert und den Effektivwert über den Scheitelfaktor angegeben.

\begin {equation} \hat {U} = C \cdot U_\mathrm {RMS} \label {GleichungKur1} \end {equation}

This crest factor is \(C_S=\sqrt {2}\) for sine waves, \(C_D=\sqrt {3}\) for triangular waves, and \(C_R=1\) for square waves. Thus, the RMS value of a sine wave voltage is approximately 70.7 per cent of the peak value of the amplitude. In addition to the crest factor as the relationship between the peak value and the effective value, the form factor \(F\) is also used. The form factor is defined as the quotient of the RMS value and the rectified value.

4 RMS value

The RMS value, for example of a sinusoidal voltage, should be used for the power consumption as an instantaneous value that causes the same power consumption that would correspond to the power value of a DC voltage under comparable conditions.

The RMS value is also applicable to Ohm’s law and Kirchhoff’s laws and is therefore suitable for all fundamental laws. The calculation of the RMS value will be explained using an alternating current signal \(i(t)\). If the definition of the current signal is inserted into equation 4, the result is:

\begin {equation} f(t) = i(t) = \hat {I} \cdot \sin (\omega t) \rightarrow I_\mathrm {Eff}=\sqrt {\frac {1}{T}\int _{0}^T {\hat {I}}^2 \cdot \sin ^2(\omega t) dt} \label {GleichungEff3} \end {equation}

The square root of the peak value is not time-dependent and can be moved outside the integral sign. In addition, the factor from the addition theorem can be removed from the integral, resulting in the following equation 5:

\begin {equation} I_\mathrm {RMS}=\sqrt {\frac {\hat {I}^2}{2T}\int _{0}^T {1-\cos (2\omega t) dt}} \label {GleichungEff4} \end {equation}

According to the difference rule, differences in an integrand can be determined separately. In this way in the following equation 6, the integrals for the two operands of the difference are determined separately.

\begin {equation} \text {with} \int _0^T 1 dt = T \text {and} \int _0^T \cos (2 \omega t) dt = 0 \label {GleichungEff5} \end {equation}

Solving the integral yields:

\begin {equation} I_\mathrm {RMS} = \sqrt {\frac {\hat {I}^2}{2T}\cdot (T - 0)} \label {GleichungEff6} \end {equation}

Here, the period duration \(T\) can be reduced from the numerator and denominator, resulting in the following relationship:

\begin {equation} I_\mathrm {RMS} = \sqrt {\frac {\hat {I}^2}{2}} \label {GleichungEff7} \end {equation}

Now the root is taken separately from the numerator and denominator. The same relationship can also be defined for the voltage:

\begin {equation} I_\mathrm {RMS} = \frac {\hat {I}}{\sqrt {2}} \qquad \text {and} \qquad U_\mathrm {RMS} = \frac {\hat {U}}{\sqrt {2}} \label {GleichungEff8} \end {equation}

Figure 2 shows a sinusoidal voltage with an amplitude value of \(4\,V\). The RMS value associated with the sinusoidal voltage is plotted at \(4\,V/\sqrt {2}\). The situation is similar with voltages in the home. For example, a voltage measurement in the household would show an RMS value of \(230\,V\). However, the amplitude value of the actual alternating voltage is \(230\,V\cdot \sqrt {2}\), i.e. \(325\,V\).

RMS valueRMS-value:

If the voltage is measured at a domestic socket, for example, the RMS value of 230V is specified.

The following tasks should be completed to gain a better understanding of the RMS value:

• Calculation of the amplitude value of the voltage at a domestic power socket.
• Calculation of the RMS value assuming that it is a rectangular signal.

a)

The amplitude value at a socket with an RMS value of 230V is: \begin {align} \hat {U} &= I_\mathrm {RMS} \cdot \sqrt {2} = 230V \cdot \sqrt {2} \nonumber \\ \hat {U} &= 325,269V \nonumber \end {align}

b)

The RMS value of a square wave voltage with a peak value of 325V is: \begin {align} U_\mathrm {Rectangle} &= \frac {\hat {U}}{\sqrt {3}} = \frac {325V}{\sqrt {3}} \nonumber \\ U_\mathrm {Rectangle} &= 187,639V \nonumber \end {align}