**8.1 Determination of Harmonic Power**

In the Section 5, DC component *V*0, fundamental component *V*1 and its phase angle φ*V*1, and *k*-th harmonic components *Vk* and their phase angle φ*V*k, (*k* = 2,3,…,*w*) are obtained for a non-sinusoidal voltage signal.

Section 6 details the derivation of DC component *I*0, fundamental component *I*1 and its phase angle φ*I*1, *k*-th harmonic components *Ik* and their phase angle φ*I*k, and (*k* = 2,3,…,*w*) for a non-sinusoidal current signal.

From these, we can calculate the harmonic power.

DC power is

$$P\_0 = V\_0 l\_0.\tag{8.1.1}$$

Fundamental active power, reactive power, and apparent power are

$$P\_1 = V\_1 I\_1 \text{cost} \mathbf{p}\_{\text{l}\nu} \tag{8.1.2}$$

$$Q\_1 = V\_1 I\_1 \text{sign} \mathbf{p}\_{1\prime} \tag{8.1.3}$$

$$S\_1 = V\_1 I\_1.\tag{8.1.4}$$

The *k*-th harmonic active power, reactive power, and apparent power are as follows:

$$P\_k = V\_k I\_k \text{COSap}\_k\,\,\,\,\,\tag{8.1.5}$$

 *Qk* = *VkIk*sinφ*k* , (8.1.6)

$$S\_k = V\_k I\_k \tag{8.1.7}$$

where φ1 = φ*V*1–φ*I*1, φ*k* = φ*Vk*–φ*Ik*, and *k* = 2,3,…,*w*.

#### **8.2 Uncertainty analysis**

According to the definition in Section 4.1, the *fiducial error of harmonic active power* can be written as

$$
\Delta P\_k = \Delta P\_k / \left(V\_1 \, I\_1\right) \tag{8.2.1}
$$

where *k* = 1,2,…,*w*.

The cases of reactive power and apparent power can be similarly analyzed.

The uncertainty of *Pk*, *fiducial uncertainty of harmonic active power*, *u*(*Pk*/(*V*1*I*1)) in reference to fundamental apparent power has four parts:

$$
\mu^2(\Delta \mathcal{P}\_k / \left(V\_1 \, I\_1\right)) = \mu\_1 \mathcal{I} + \mu\_2 \mathcal{I} + \mu\_3 \mathcal{I} + \mu\_4 \mathcal{I} \, \tag{8.2.2}
$$

where

168 Modern Metrology Concerns

current of 0.5*I*, and then at the full current of *I*. The half range effect can be determined

Order 1 10 20 30 40 50 60

Table 7.3.1. Phase shift and its uncertainty (after correction) of the 5A shunt, /rad, (*k* = 1)

The 5 A shunt under a 5 A fundamental has a phase shift of –1.3 rad, and its uncertainty is 12 rad (*k* = 2). The results can be applied only to the sinusoidal signal and in full range.

The uncertainty evaluation is discussed to measure phase difference by applying DFT to sampling data in NIPS. The step-up procedures are described for the voltage dividers and shunts. The phase shifts and their uncertainties are given. The uncertainty of the phase shift in the fundamental is 13 rad (*k* = 2) for the 120 V divider, and 12 rad (*k* = 2) for the 5 A shunt. These specifications fall under the special frequency characteristic. They are

In the Section 5, DC component *V*0, fundamental component *V*1 and its phase angle φ*V*1, and *k*-th harmonic components *Vk* and their phase angle φ*V*k, (*k* = 2,3,…,*w*) are obtained for a

Section 6 details the derivation of DC component *I*0, fundamental component *I*1 and its phase angle φ*I*1, *k*-th harmonic components *Ik* and their phase angle φ*I*k, and (*k* = 2,3,…,*w*) for

 *P*0 = *V*0*I*0. (8.1.1)

 *P*1 = *V*1*I*1cosφ1, (8.1.2)

 *Q*1 = *V*1*I*1sinφ1, (8.1.3)

 *S*1 = *V*1*I*1. (8.1.4)

 *Pk* = *VkIk*cosφ*k* , (8.1.5)

The *k*-th harmonic active power, reactive power, and apparent power are as follows:

discussed further when the results are applied to harmonic power measurement.

*Ik* –1.3 –3.0 –0.7 –.3 –7.0 –12.4 –0.5

*Ik*) 5.99 5.38 7.64 10.1 12.4 15.2 16.9

when R4 is used as reference.

**8. Harmonic power measurement 8.1 Determination of Harmonic Power** 

non-sinusoidal voltage signal.

a non-sinusoidal current signal.

DC power is

From these, we can calculate the harmonic power.

Fundamental active power, reactive power, and apparent power are

*u*(

**7.4 Conclusion** 

$$u\_1 = \begin{pmatrix} I\_k/I\_1 \end{pmatrix} \cos \mathfrak{p}\_k \text{ } u \text{ (}\Delta \ V\_k/V\_1\text{) },\tag{8.2.3}$$

$$u\_2 = \begin{pmatrix} V\_k/V\_1 \end{pmatrix} \cos \mathfrak{p}\_k \text{ } \mathfrak{u} \text{ (}\Delta \operatorname{I}\_k/I\_1\text{) },\tag{8.2.4}$$

$$\mu\_{\mathbb{S}} = \left[ \text{sirqọ} \; I\_k \; V\_k / \left( V\_1 \; I\_1 \right) \right] \mu \left( \Lambda \text{q} \mu\_{\mathbb{R}} \right) \;,\tag{8.2.5}$$

$$\mu\_4 = \begin{bmatrix} \sin \mathfrak{p}\_k \ I\_k \ V\_k \end{bmatrix} \begin{Bmatrix} V\_1 \ I\_1 \end{Bmatrix} \begin{Bmatrix} \mu(\Delta \mathfrak{p}\_{\mathbb{R}}) \end{Bmatrix} . \tag{8.2.6}$$

Eqs. (8.2.3) and (8.2.4) are dependent on the fiducial uncertainty of the harmonic voltage and current, or *u* ( *Vk*/*V*1) and *u* (*Ik*/*I*1), respectively. They result from the ratio errors of the dividers and shunts.

Eqs. (8.2.5) and (8.2.6) are dependent on the relative uncertainty of the phase difference of the harmonic voltage and current, or *u*(*Vk*) and *u*(*Ik*), respectively. These are analyzed in Section 7. They result from phase angle errors.

When *k* = 0, Eqs. (8.2.5) and (8.2.6) equal zero; when *k* = 90, Eqs. (8.2.3) and (8.2.4) equal zero.

Eq. (8.2.5), *u*3, can be re-written as

$$\mu\_3 = \left[ \sin \mathfrak{q} \mathbb{k} \times (l\_k / l\_1) \right] \left[ \mu (\varDelta \mathfrak{q} \,\_{Vk}) \times (V\_k / \,\_{V1}) \right].$$

where *u*(*Vk*) is a relative uncertainty of the phase difference of harmonic voltage, which is related to *Vk*, and its values in the standard equipment are determined by sampling measurement. The product of *u*(*Vk*)×(*Vk*/*V*1) indicates that the reference of *u*(*Vk*) should be converted from *Vk* to *V*1. As a concept of fiducial uncertainty, this product can be called *fiducial uncertainty of the phase difference of voltage*, denoted as *u*(*Vk*/*V*1). Similar to the case of voltage (current), we consider the effects of the small component, leakage, noise, and non-linearity.

$$
\mu(\mathsf{Aq}\_{Vk}/V\_1) = (V\_k/V\_1)\,\,\mu(\mathsf{Aq}\_{Vk}).
$$

Considering the non-linearity of amplitude under the harmonic conditions in Section 5.2, the ratio of (*Vk*/*V*1) can be converted to [0.8(*Vk*/*V*1) + 0.2]. With the addition of the resolution component, a new expression is derived:

$$\mu^2(\mathcal{A}\mathfrak{p}\_{Vk}/V\_1) = [0.8(V\_k/V\_1) + 0.2]^2 \mu^2(\mathcal{A}\mathfrak{p}\_{Vk}) + 6.852\dots$$

Thus, Eq. (8.2.5) becomes

$$\mu\_{\mathcal{I}} = \left[ \text{sinc} \mathfrak{p}\_k \left( \text{l}\_k / \text{l}\_{\mathcal{I}} \right) \right] \text{u} \big( \Delta \mathfrak{p}\_{\mathcal{I} \mathbb{R}} / V\_{\mathcal{I}} \big). \tag{8.2.7}$$

Eq. (8.2.6), *u*4, can be similarly treated.

$$\mu\_4 = \left[ \text{sinc}\, \upmu\_k \left( V\_k / V\_1 \right) \right] \left( \mu \left( \Delta \upmu\_l d\_1 \right) \right) \tag{8.2.8}$$

*u*2(I*k*/*I*1) = [0.7(*Ik*/*I*1) + 0.3]2 *u*2(*Ik*)+ 7.28 2.

The digital estimation of *u*3 and *u*4 can be provided.

#### **8.3 Uncertainty evaluation**

The uncertainty of harmonic active power can be calculated according to Eq. (8.1.5), where parameters *Vk*/*V*1, *Ik*/*I*1 are taken from the characterizing signal in Table 4.2.1, and uncertainties *u*(*Vk*/*V*1), *u* (*Ik*/*I*1), *u*(*Vk*), and *u*(*Ik*) are taken from the experimental results in Tables 5.4.1, 6.3.1, 7.2.5, and 7.3.1, respectively. For the cases *k* = 0, 30, 60, and 90, the calculated results are shown in Table 8.3.1.


Table 8.3.1. Fiducial uncertainty of active power at 500 V, 20 A under the characterizing signal condition, *U* (*Pk*/(*V*1*I*1))/(W/VA) (*k* = 2)

In Table 8.3.1 the maximum estimated uncertainty (*k* = 2) is 42 W/VA, which is the uncertainty index of the harmonic active power. At 100 V and 5 A, however, the maximum estimated uncertainty (*k* = 2) is 36 W/VA.

In practical measurement, in order to obtain the best uncertianty of measurement, for example in the important comparison, the parameters of *Vk*/*V*1, *Ik*/*I*1 may use of the measuring values instead of the values of the characteristic signal. When the measuring signal consists of only a few harmonics, the resolution component in eqn.(5.4.1), (6.3.1), (8.2.7), (8.2.8) may take smaller values according the practical case. In the NIM's standard equipment for the non-sinusoidal signal including 1 to 2 harmonics only, under the best experimental conditions, the value may take 1.00 but not 6.85 or 7.28.

Some factors of interest, such as noise and jitter effects, are already included in the experimental results. For jitter, it exists practically as an uncertainty factor in the time point of sampling. However the author takes the concept that the sampling was at the precise time points still according to the Section 3.1, but, as equivalence, the signal instantaneous amplitudes (the sampling values) were attached additional errors. As far as the aliasing effect is concerned, no analysis can be carried out beyond the 40th harmonic component; the analysis is restricted because the power source of the standard equipment does not provide harmonic component signals with orders higher than the 40th in satisfying the specified maximum harmonic components of the 40th order, as per the IEC standard [IEC, 2002, 2003].
