**4.1 Fiducial error and fiducial uncertainty**

Relative error and relative uncertainty are two concepts often used in calibrations for general quantities. These concepts have also been used in many expressions of harmonic measurement. However, the example given in this study illustrates the need to revisit these concepts.

A harmonic analyzer is used to measure a non-sinusoidal voltage signal with a fundamental 100 V and a 5th order harmonic 10 V. It measures the 5th order harmonics as 10.0101 V, of which 10 mV is a leakage error from the fundamental (100 V), whereas 0.1 mV is an error only from the 5th order harmonic (10 V). In this case, the relative measurement error of the 5th order harmonic of the analyzer is 10.1 mV/10 V = 0.1%.

In another measurement, the analyzer measures a signal with 100 V of the fundamental plus 0.1 V of the 5th order harmonic. In this case, the leakage error from the fundamental (10 mV) persists because of the fact that the 100 V fundamental is unchanged, but the error that comes only from the 5th order harmonic may be 1 μV. Hence, the relative measurement error is 10 mV/0.1V = 10%, and not 0.1% as in the first case.

The example illustrates that the specifications of an analyzer are dependent on the magnitude of the signal to be measured for the relative error concept. Some error components may come from the fundamental. In such a case, the harmonic relative error of the analyzer is variable when the harmonic amplitude varies. This attribute does not correspond with that indicated in the typical error concept.

The fundamental is the main component in a general non-sinusoidal signal, whereas the harmonics are secondary components. Therefore, the selection of the measurement range in characterizing harmonic analyzers is dependent/determined primarily by the value of the fundamental component.

However, the concept of total harmonic distortion (THD) is related only to the fundamental component or the total RMS value of the harmonics including the fundamental, and not to the individual harmonic component itself.

If the fundamental component is used as reference, then the error is the same in both cases, as shown in the earlier example. That is, 10 mV/100V = 0.01%. However, we introduce a new error concept called the *fiducial harmonic error,* developed by the International Vocabulary of Metrology (VIM, Second edition) [BIPM et al., 1993a] (regrettably, the term is not embodied in the Third edition, JCGM 200:2008). This concept is defined as follows:

$$
\delta\mathcal{S}\,V\_k = \Delta V\_k / V\_1\,\tag{4.1.1}
$$

where *Vk* is the absolute error of the *k*-th harmonic, and *V*1 is value of the fundamental.

The harmonic uncertainty concept is expanded with a relative uncertainty concept, *u* (*Vk*/*Vk*), when the quantity itself, *Vk*, is taken as the reference and the absolute uncertainty is defined as *u* (*Vk*):

$$
\mu \left( \Delta \, V\_k / V\_k \right) = \left[ \mu \left( \Delta \, V\_k \right) \right] / V\_k. \tag{4.1.2}
$$

Changing the reference from *Vk* into *V*1, we introduce a new harmonic uncertainty concept and denote it as *u* ( *Vk*/*V*1):

$$
\mu \left( \Delta V\_k / V\_1 \right) = \left[ \mu \left( \Delta V\_k \right) \right] / V\_1. \tag{4.1.3}
$$

This harmonic uncertainty is defined as the *fiducial harmonic uncertainty*.

The relationship between the fiducial and relative harmonic uncertainty is

$$
\mathfrak{u}(\Delta V\_k/V\_1) = \mathfrak{u}(\Delta V\_k/V\_k) \times (V\_k/V\_1). \tag{4.1.4}
$$

The fiducial harmonic uncertainty is a reasonable choice in expressing harmonic measurement uncertainties, as explained previously.

#### **4.2 Characterizing waveform signal**

In Eq. (4.1.4), the ratio of *Vk*/*V*1 is an indeterminate variable. To estimate the uncertainty of the equipment, this ratio should be a fixed value to cover all the possible calibration cases. The IEC documents, international recommendations for electricity meters, and EMC provide some typical waveforms [IEC, 2001, 2002, 2003] that can be used to estimate these fixed ratio values. The National Research Council of Canada (NRC) proposed eight different reference waveforms based on actual field-recorded distorted waveforms that can be used for same aim [Arseneau et al., 1995a].

In conclusion, after considering all the different potential test waveforms that can be used, some rules can be assumed as follows:

The fundamental is the main component in a general non-sinusoidal signal, whereas the harmonics are secondary components. Therefore, the selection of the measurement range in characterizing harmonic analyzers is dependent/determined primarily by the value of the

However, the concept of total harmonic distortion (THD) is related only to the fundamental component or the total RMS value of the harmonics including the fundamental, and not to

If the fundamental component is used as reference, then the error is the same in both cases, as shown in the earlier example. That is, 10 mV/100V = 0.01%. However, we introduce a new error concept called the *fiducial harmonic error,* developed by the International Vocabulary of Metrology (VIM, Second edition) [BIPM et al., 1993a] (regrettably, the term is not embodied in the Third edition, JCGM 200:2008). This concept is defined as follows:

 *Vk* = *Vk*/*V*1 , (4.1.1)

This harmonic uncertainty is defined as the *fiducial harmonic uncertainty*. The relationship between the fiducial and relative harmonic uncertainty is

measurement uncertainties, as explained previously.

**4.2 Characterizing waveform signal** 

some rules can be assumed as follows:

aim [Arseneau et al., 1995a].

where *Vk* is the absolute error of the *k*-th harmonic, and *V*1 is value of the fundamental.

The harmonic uncertainty concept is expanded with a relative uncertainty concept, *u* (*Vk*/*Vk*), when the quantity itself, *Vk*, is taken as the reference and the absolute

 *u* ( *Vk*/*Vk*) = [ *u* ( *Vk*)]/*Vk* . (4.1.2) Changing the reference from *Vk* into *V*1, we introduce a new harmonic uncertainty concept

 *u* (*Vk*/*V*1)= [ *u* (*Vk*)]/*V*<sup>1</sup> . (4.1.3)

 *u*(*Vk*/*V*1) = *u*(*Vk*/*Vk*) × (*Vk*/*V*1). (4.1.4) The fiducial harmonic uncertainty is a reasonable choice in expressing harmonic

In Eq. (4.1.4), the ratio of *Vk*/*V*1 is an indeterminate variable. To estimate the uncertainty of the equipment, this ratio should be a fixed value to cover all the possible calibration cases. The IEC documents, international recommendations for electricity meters, and EMC provide some typical waveforms [IEC, 2001, 2002, 2003] that can be used to estimate these fixed ratio values. The National Research Council of Canada (NRC) proposed eight different reference waveforms based on actual field-recorded distorted waveforms that can be used for same

In conclusion, after considering all the different potential test waveforms that can be used,

fundamental component.

the individual harmonic component itself.

uncertainty is defined as *u* (*Vk*):

and denote it as *u* ( *Vk*/*V*1):

$$V\_0 / V\_1 \le 0.5 \,\, \_ {\prime}$$

$$V\_k / V\_1 \le 1 \,\, \_ {\prime} \text{ when } 2 \le k \le 10 \,\, \_ {\prime}$$

$$V\_k / V\_1 \le \,\, \_ {\prime} / k \,\, \_ {\prime} \text{ when } 10 < k \le 60 \,\, \_ {\prime}$$

where *V k* is the amplitude of the *k*-th harmonic component, *V* 0 denotes the DC component, *V* 1 represents the fundamental, and *γ* is a ratio factor.

A characterizing waveform signal, determined according to the aforementioned limit values, is proposed (Table 4.2.1). Its corresponding waveform is shown in Fig 4.2.1. This is only a theoretical signal (*THD* = 280%), conjured to estimate the uncertainty of the harmonic power standard equipment. The signal covers all the possible cases under calibration, so that the harmonic uncertainty claim has universality. It is frequently used for the general analysis and digital evaluation of uncertainty for the harmonic power standard of the NIM [Lu, 2010].


Table 4.2.1. Amplitude ratio of the characterizing signal, *Vk*/*V*1

Fig. 4.2.1. Waveform of the characterizing signal.
