**5.3.3 Non-linearity effect**

When two sinusoidal signals are simultaneously input, such as the 1st and 5th orders, and when the input circuit of the meter (or the output circuit of the source) exhibits non-linearity of frequencies, two new sinusoidal signals, the 4th and 6th orders, appear in the results. This effect occurs too in the single tone experiment.

We design and implement an experiment, in which a signal with the fundamental and 5th order is measured. Another signal with only the 5th order is measured. The difference between the two above-mentioned DFT results at the 4th and 6th orders reflects this nonlinearity. According to the experimental results, the contribution of this effect is 2.2 μV/V (*k* = 1), including other possible influences from the signal source.

#### **5.4 Conclusion**

158 Modern Metrology Concerns

The harmonic components are smaller in practical non-sinusoidal signals. The uncertainty estimation at full range in Table 5.2.1 cannot be directly applied to smaller amplitudes. A general method for non-linear amplitudes can be employed, where the ratio of (*Vk* /*V*1 ) is replaced by [0.8(*Vk*/*V*1)+0.2], in which a minimum value of 0.2 (or 20%) is maintained even if *Vk* = 0. The ratio 0.8:0.2 (or 80%:20%) is taken from the experimental results. Eq. (4.1.4)

 *u*(*Vk*/*V*1) = [0.8(*Vk*/*V*1))+0.2] × *u*(*Vk*/*Vk*) . (5.3.1)

When a frequency signal (for example, a sinusoidal signal corresponding to the 10th order harmonics) is tested in the single tone experiment, other components (from the DC to the 60th order) aside from the input signal itself appear in the DFT computational results. These components should be zero but are non-zero in practice possibly because the output of the source is not a pure single tone but with side bands; this result may also be attributed to

To overcome the side bands of the source, 3 sets of data (7th, 8th, 9th) ahead of the input signal (10th) and 3 sets of data behind (11th, 12th, 13th) that same signal in the DFT

The rest of the data reflect mainly leakage and noise. An "enlargement test" is designed and implemented to distinguish between both parts. The effect of the non-integer-period is enlarged 10 times (Δ is enlarged from 0.04 to 0.4). The difference between the two groups of data is regarded as the influence from leakage only, and the remainder is

The error of the *k*-th harmonic includes all leakages from the DC to the 60th order aside from itself. Therefore, all the contributions from the other harmonics are accumulated into the

The contribution of the leakage from the experimental results is 2.5 μV/V, and the noise

When two sinusoidal signals are simultaneously input, such as the 1st and 5th orders, and when the input circuit of the meter (or the output circuit of the source) exhibits non-linearity of frequencies, two new sinusoidal signals, the 4th and 6th orders, appear in the results. This

We design and implement an experiment, in which a signal with the fundamental and 5th order is measured. Another signal with only the 5th order is measured. The difference between the two above-mentioned DFT results at the 4th and 6th orders reflects this nonlinearity. According to the experimental results, the contribution of this effect is 2.2 μV/V

**5.3.1 Effect of the small component** 

then becomes

**5.3.2 Leakage and noise** 

regarded as noise.

uncertainty of the *k*-th harmonic.

**5.3.3 Non-linearity effect** 

dependent on frequency is 6.0 μV/(*k* = 1).

effect occurs too in the single tone experiment.

(*k* = 1), including other possible influences from the signal source.

both leakage due to faulty compensation and noise.

computational results are excluded in the analysis below.

The small component effect of Eq. (5.3.1) is a part of the uncertainty. This component is called the *traceability component* because of its dependence on the voltage unit.

The leakage, noise, and non-linearity are other parts, but they are independent of the voltage unit. They can be regarded as the *resolution component*, expressed as SQR (2.52 +6.02 +2.22) = 6.85 μV/V (*k* = 1) on the basis of the results in Section 5.3.

Considering the two components (traceability and resolution), the fiducial uncertainty of Eq. (5.3.1) becomes

$$
\mu\_c \text{\textdegree (\Delta V\_k/V\_1)} = \left[ 0.8 (V\_k/V\_1) + 0.2 \right] \text{\textdegree 2} \left( \Delta V\_k / V\_k \right) + 6.85^2 \text{\textdegree \textdegree (5.4.1)}.\tag{5.4.1}
$$

where the parameters of *u*(*Vk*/*Vk*) can be introduced from Table 5.2.1, and the ratio of (*Vk*/*V*1) can be adopted from the characterizing signal in Table 4.2.1. The final uncertainties are calculated and shown in Table 5.4.1.


Table 5.4.1. Fiducial uncertainty under the characterizing signal condition, *u*c (*Vk*/*V*1)/(V/V) (*k* = 1)

An estimation of 30 V/V (*k* = 2) for all the voltage ranges can be obtained. At 100 V, however, this estimation amounts to 20 V/V.

#### **6. Traceability of harmonic current and its uncertainty**

#### **6.1 Determination of harmonic current**

When Eq. (3.1.1) expresses a non-sinusoidal current signal, and Eq. (3.1.3) denotes the sampling data under the sampling model of Eq. (3.1.2), after computation according to Eq. (3.3.7), the DC current, fundamental current, and harmonic current components can be obtained thus:

0 0 *I a* , (6.1.1)

$$I\_1 = \sqrt{a\_1^2 + b\_1^2} \tag{6.1.2}$$

$$\varphi\_{l1} = \arctan(b\_1 \,/\, a\_1) \,. \tag{6.1.3}$$

$$I\_k = \sqrt{a\_k^2 + b\_k^2} \tag{6.1.4}$$

$$
\varphi\_{lk} = \arctan(b\_k \,/\, a\_k) \,. \tag{6.1.5}
$$

where φ*I*1, φ*Ik* are the phase angles (against a reference time point) of the fundamental and harmonics, respectively, and *k* = 2,3,…*w*.

Similar experiments and computations are carried out for the harmonic current specifications of the NIM harmonic power standard equipment.

#### **6.2 Frequency characteristic**

The shunts in the equipment are tested. The relative differences/errors are not considered as corrections. The current traceability uncertainty in the full range of every shunt is estimated and shown in Table 6.2.1, including the contribution from the Primary Standards (DC resistance, AC/DC transfer, and DC voltage).


Table 6.2.1. Traceability uncertainty in the full range of shunts without correction, *u*(*Ik*/*Ik*)/(A/A) (*k =* 1)

An experiment is designed and performed to check the aforementioned estimation (Fig. 6.2.1). An inductive shunt is developed [Zhang J.T. et al., 2007] to produce two equal currents with very high accuracy. In a general series connection of two resistors, a potential at the middle point causes error. The defect is addressed in this study. The results obtained satisfy the estimation in Table 6.2.1.

Fig. 6.2.1. Circuit of current traceability.

#### **6.3 Harmonic characteristic**

Similar to the voltage case, the *fiducial uncertainty of harmonic current u*(*Ik*/*I*1) is obtained as

$$
\mu\_c \mathbf{\hat{\epsilon}} (\mathbf{A} \mathbf{l}\_k / \mathbf{l}\_1) = \mathbf{u}^2 \left(\mathbf{A} \mathbf{l}\_k / \mathbf{l}\_1\right) + 7.28^2 \text{ \AA} \tag{6.3.1}
$$

where

160 Modern Metrology Concerns

where φ*I*1, φ*Ik* are the phase angles (against a reference time point) of the fundamental and

Similar experiments and computations are carried out for the harmonic current

The shunts in the equipment are tested. The relative differences/errors are not considered as corrections. The current traceability uncertainty in the full range of every shunt is estimated and shown in Table 6.2.1, including the contribution from the Primary Standards (DC

> Order 1 10 20 30 40 50 60 ≤5A 10.6 11.8 11.8 11.7 10.3 11.3 17.0

20 A 14.6 15.5 19.1 19.0 22.5 27.4 34.5

An experiment is designed and performed to check the aforementioned estimation (Fig. 6.2.1). An inductive shunt is developed [Zhang J.T. et al., 2007] to produce two equal currents with very high accuracy. In a general series connection of two resistors, a potential at the middle point causes error. The defect is addressed in this study. The results obtained

Table 6.2.1. Traceability uncertainty in the full range of shunts without correction,

harmonics, respectively, and *k* = 2,3,…*w*.

resistance, AC/DC transfer, and DC voltage).

10 A,

**6.2 Frequency characteristic** 

*Ik*/*Ik*)/(A/A) (*k =* 1)

satisfy the estimation in Table 6.2.1.

Fig. 6.2.1. Circuit of current traceability.

*u*(

specifications of the NIM harmonic power standard equipment.

$$
\mu\left(\varDelta l\_{k}/I\_{1}\right) = \left[0.7(I\_{k}/I\_{1}) + 0.3\right]\mu\left(\varDelta l\_{k}/I\_{k}\right) \tag{6.3.2}
$$

is the traceability component in which non-linearity of amplitude is considered. For the resolution component, 2.5, 6.0, and 3.3 A/A are experimentally determined for leakage, noise, and non-linearity of frequencies, respectively.

The digital computational results for the characteristic signal condition are expressed in Table 6.3.1.


Table 6.3.1. Combined fiducial uncertainty in the characteristic signal, *u*c (*Ik*/*I*1)/(A/A) (*k* = 1)

The table shows that the experimental results for 50 A are derived from the current transformer. It also shows that the maximum result is 36 A/A (*k* = 2), which can be used as an estimation for the harmonic measurement of the standard equipment.

#### **7. Determination of phase shift and its uncertainty**

The phase difference between voltage and current is an important quantity in power measurement. In harmonic power measurement, every phase difference between the harmonic voltage and harmonic current at the same order from the 2nd to the 60th must be determined. Two problems are discussed: how the phase difference is measured and how its uncertainty is evaluated.

#### **7.1 Measurement of the phase difference of two signals**

#### **7.1.1 Phase difference of two voltage signals**

Phase difference can be measured using the sampling approach. The phase difference between two voltages can be calculated using the DFT sampling data results [Lu et al., 2006]:

$$\varphi\_k = \arctan(b\_{2k} \mid a\_{2k}) - \arctan(b\_{1k} \mid a\_{1k}) \,\,\,\,\,\tag{7.1.1}$$

where φ*k* is the phase difference of the *k*-th harmonics between voltages *y*2 and *y*1:

$$\begin{aligned} y\_1 &= a\_{10} + \sum\_{k=1}^{w} (a\_{1k} \sin kot + b\_{1k} \cos kot) \\\\ y\_2 &= a\_{20} + \sum^{w} (a\_{2k} \sin kot + b\_{2k} \cos kot) \end{aligned}$$

*k k*

 2 20 2 2 1

*k*

 $a\_{1k}$   $b\_{1k}$  ( $a\_{2k}$   $b\_{2k}$ ) are the Fourier coefficients of  $y\_1$  ( $y\_2$ ).

The method involves sampling the two voltage signals simultaneously using two ADCs (i.e., DVMs in the standard equipment), and applying DFT to the sampling data and then computing phase difference according to Eq. (7.1.1).

The algorithm introduced in Section 3 can be applied here for precise Fourier coefficients to result in precise phase difference.

Because two different ADCs are used, an intrinsic phase difference occurs, thereby affecting the results. A voltage can be connected parallel to the two channels and the above-mentioned method can be implemented to determine this intrinsic error, which can then be corrected in succeeding measurements [Svensson,1998]. This process can be called **producing a 0° standard**.

#### **7.1.2 Uncertainty of phase difference**

The phase angle definition is included in Eq. (7.1.1); that is,

$$
\mathfrak{op} = \arctan(b/a). \tag{7.1.2}
$$

Therefore, its uncertainty can be expressed by the amplitude

$$
\mu^2(\wp) = \mu^2(\text{arc } \tan(b/a)) = [b \ u(a)/(a^2 + b^2)]^2 + [a \ u(b)/(a^2 + b^2)]^2 \dots
$$

If *u*(*a*) ≥ *u*(*b*),

$$
\mu(\varphi) \le \mu(a) / \operatorname{sqrt}(a^2 + b^2) = \mu(a) / \operatorname{c},\tag{7.1.3}
$$

where *u*(*a*) is an absolute uncertainty, whose concept is similar to absolute uncertainty *u*(*Vk*) in Section 4.

In general, *a* = (10.71) *c c*; thus,

$$\mu(\mathfrak{q}\_{Vk}) = (\mathbf{1}\text{-}0.71)\ \mu(\Delta V\_k)/V\_k \le \mu(\Delta V\_k)/V\_k = \mu(\Delta V\_k/V\_k)\ .$$

The uncertainty of phase difference *u* (*Vk*) between two voltages can be expressed as

$$
\mu^2(\Lambda \mathfrak{q}\_{Vk}) = \mu^2(\Lambda V\_{1k}/V\_{1k}) + \mu^2(\Lambda V\_{2k}/V\_{2k}).\tag{7.1.4}
$$

The relative uncertainty of *u*(*Vk*/*Vk*) is discussed in Section 5 for voltage (and Section 6 for current), and determined in experiments. However, some uncertainty factors can be discussed further:


#### **7.1.3 Phase difference between a voltage signal and a current signal**

Only the shunt in the equipment is discussed in this section. The input signal is a current and the output signal is a voltage, shown in its equivalent circuit in Fig. 7.1.1, wherein points A and B denote the current input terminal, and C and D represent the voltage output terminal. *R* denotes resistance, *L* represents inductance, and *C* denotes capacitance. When the input current is taken as reference, the output voltage has a phase shift φ:

$$
\rho \approx o\mathcal{L} \;/\; R-o\mathcal{R}\mathcal{C} \;. \tag{7.1.5}
$$

Fig. 7.1.1. Equivalent circuit of shunts.

162 Modern Metrology Concerns

*y a a kt b kt* 

*y a a kt b kt* 

The method involves sampling the two voltage signals simultaneously using two ADCs (i.e., DVMs in the standard equipment), and applying DFT to the sampling data and then

The algorithm introduced in Section 3 can be applied here for precise Fourier coefficients to

Because two different ADCs are used, an intrinsic phase difference occurs, thereby affecting the results. A voltage can be connected parallel to the two channels and the above-mentioned method can be implemented to determine this intrinsic error, which can then be corrected in succeeding measurements [Svensson,1998]. This process can be called **producing a 0° standard**.

*u*2( ) = *u*2(arc tan(*b*/*a*) ) = [*b u*(*a*)/(*a*2 + *b*2)]2 + [*a u*(*b*)/(*a*2 + *b*2)]2 .

where *u*(*a*) is an absolute uncertainty, whose concept is similar to absolute uncertainty

*u*(*Vk* ) = (10.71) *u*(*Vk*)/*Vk u*(*Vk*)/*Vk* = *u*(*Vk*/*Vk*) .

 *u*2(*Vk*) = *u*2(*V*1*<sup>k</sup>*/*V*1*<sup>k</sup>*) + *u*2(*V*2*<sup>k</sup>*/*V*2*<sup>k</sup>*). (7.1.4) The relative uncertainty of *u*(*Vk*/*Vk*) is discussed in Section 5 for voltage (and Section 6 for current), and determined in experiments. However, some uncertainty factors can be

1. Given that the phase angle is dependent on the ratio of Fourier coefficients, the

uncertainty of the voltage standard from traceability has no function here.

2. The standard deviation of measurement is an important factor.

The uncertainty of phase difference *u* (*Vk*) between two voltages can be expressed as

( sin cos )

( sin cos )

= arctan(*b*/*a*). (7.1.2)

) *u*(*a*)/sqrt(*a*2 + *b*2) = *u*(*a*)/*c*, (7.1.3)

*k k*

*k k*

1 10 1 1 1

2 20 2 2 1

*w*

*k*

*k*

*a*1*<sup>k</sup>*, *b*1*<sup>k</sup>* (*a*2*<sup>k</sup>*, *b*2*<sup>k</sup>*) are the Fourier coefficients of *y*1 (*y*2).

computing phase difference according to Eq. (7.1.1).

The phase angle definition is included in Eq. (7.1.1); that is,

Therefore, its uncertainty can be expressed by the amplitude

result in precise phase difference.

**7.1.2 Uncertainty of phase difference** 

 *u*(

In general, *a* = (10.71) *c c*; thus,

3. Leakage and noise are also factors.

If *u*(*a*) ≥ *u*(*b*),

*u*(*Vk*) in Section 4.

discussed further:

 

 

*w*

When an ADC is used to measure the voltage, its input capacitor *CD* may generate a new measurement result:

$$
\sigma \approx o\text{L} \;/\; R-o\text{R}\text{C}-o\text{R}\text{C}\_{D'} \tag{7.1.6}
$$

where the function of *CD* cannot be disregarded. To overcome such influencing factors, a substitution method-based sampling measurement is developed [Wang et al., 2008].

Fig. 7.1.2A. Phase shift measurement step 1.

Fig. 7.1.2B. Phase shift measurement step 2.

In Fig.7.1.2A, the shunts, one's phase shift known and another's unknown, are connected in series. Two ADCs simultaneously measure the voltages: one for a shunt near the earth point, and another for the full voltage. The relevant phase difference of φ1-φ2 as the measurement result can be obtained according to Eq. (7.1.2). Fig.7.1.2B shows that the positions of the two shunts are carefully exchanged, and the new phase difference of φ3-φ4 can be obtained.

The influence function of *CD*1 and *CD*2 can be eliminated in the difference of both results of Δφ = (φ1-φ2)-( φ3-φ4). Suitable wiring may enable *CS*1, *CS*2 to be disregarded, and yields

$$
\Delta\phi \approx o(L\_2 \;/\; R\_2 - R\_2 C\_2) - o(L\_1 \;/\; R\_1 - R\_1 C\_1) \,. \tag{7.1.7}
$$

On the basis of the phase shift definition of the shunt in Eq. (7.1.5), we infer that the result from Eq. (7.1.7) requires measurement, and then the phase shift of the unknown shunt is determined using the known shunt.

#### **7.2 Phase shift of voltage dividers**

#### **7.2.1 Phase shift measurement**

The phase shift of the voltage divider between its output and input, and relative uncertainty is determined in an experiment. Every one of the sets of dividers at the ranges 8, 15, 30, 60, 120, 240, and 500 V is measured. The experiment is based on a step-up procedure, discussed below.

**Step-up Procedure** Fig. 7.2.1 depicts the step-up procedure for measuring the phase shift of the dividers.

**Preparation**: A voltage of 0.8 V is connected in parallel to both DVMs to eliminate their intrinsic phase difference; that is, the 0º standard is formulated.


In Fig.7.1.2A, the shunts, one's phase shift known and another's unknown, are connected in series. Two ADCs simultaneously measure the voltages: one for a shunt near the earth point, and another for the full voltage. The relevant phase difference of φ1-φ2 as the measurement result can be obtained according to Eq. (7.1.2). Fig.7.1.2B shows that the positions of the two shunts are carefully exchanged, and the new phase difference of φ3-φ4 can be obtained.

The influence function of *CD*1 and *CD*2 can be eliminated in the difference of both results of Δφ = (φ1-φ2)-( φ3-φ4). Suitable wiring may enable *CS*1, *CS*2 to be disregarded, and yields

2 2 22 1 1 11

On the basis of the phase shift definition of the shunt in Eq. (7.1.5), we infer that the result from Eq. (7.1.7) requires measurement, and then the phase shift of the unknown shunt is

The phase shift of the voltage divider between its output and input, and relative uncertainty is determined in an experiment. Every one of the sets of dividers at the ranges 8, 15, 30, 60, 120, 240, and 500 V is measured. The experiment is based on a step-up procedure, discussed

**Step-up Procedure** Fig. 7.2.1 depicts the step-up procedure for measuring the phase shift of

**Preparation**: A voltage of 0.8 V is connected in parallel to both DVMs to eliminate their

**Step 1.** A voltage of 0.8 V is applied parallel to both DVMs (DVM2 at 10 V and DVM1 at 1

**Step 2.** A voltage of 8 V is applied parallel to the 8 V divider (a divider with a range of 8 V)

V) without any divider. The phase difference between the 10 and 1 V range of

and DVM2 in its 10 V range. The output of the divider (i.e., 0.8 V) is connected to

(/ ) (/ ) *L R RC L R RC* 

. (7.1.7)

Fig. 7.1.2B. Phase shift measurement step 2.

 

intrinsic phase difference; that is, the 0º standard is formulated.

determined using the known shunt.

**7.2 Phase shift of voltage dividers 7.2.1 Phase shift measurement** 

DVM2 is measured.

below.

the dividers.

DVM1 in its 1 V range. The phase difference between the output and input of the 8 V divider is measured.

**Step 3.** The 15 V divider is measured, and the 8 V divider is taken as a standard. A voltage of 8 V is applied parallel to both dividers, and the outputs of both dividers are connected to two DVMs. The phase shift of the 15 V divider is measured.

Similar procedures are repeated up to the 480 V divider. The measurement results are shown in Table 7.2.1 and their standard deviations are listed in Table 7.2. 2.

Fig. 7.2.1. Step-up procedure for measuring the phase shift of the voltage dividers.


Table 7.2.1. Test results of phase shift φ*Vk*/rad


Table 7.2.2. Standard deviation /rad

### **7.2.2 Uncertainty of phase shift measurement**

The uncertainty of the phase shift is evaluated according to Eq. (7.1.4).

The 0º standard uncertainty, determined in an experiment, is shown in Table 7.2.3.


Table 7.2.3. Uncertainty of the 0° standard/(rad) (*k* = 1)

Another factor, the half range effect, is considered. A divider obtains the value in its half range but provides the value in its full range to the next divider.

This effect can be measured by comparing two cases: the full and 50% ranges. The results are shown in Table 7.2.4.


Table 7.2.4. Variation of phase difference between half and full range/(rad)

Thus, Table 7.2.5 shows the uncertainty of the phase shift of every divider, including the standard deviation in Table 7.2.2, uncertainty of the 0° standard in Table 7.2.3, half range effect in Table 7.2.4, and uncertainty of its foregoing divider.


Table 7.2.5. Uncertainty of phase shift of dividers after correction, *u*(*Vk*)/(rad) (*k* = 1)

The 120 V divider under a 100 V fundamental has a phase shift of 14.6 rad, and its uncertainty is 13 rad (*k* = 2). However, these results can be applied only to the sinusoidal signal and in full range because the experiments are implemented under these conditions.

#### **7.3 Phase shift of shunts**

A similar step-up procedure (Fig. 7.3.1) is conducted for the shunts from 0.1 to 20 A, and 50 A. An original standard is necessary; that is, a standard resistor of a known time constant developed early at the NIM. With the parameters *R* = 10 Ω, *τ* = 0.03 × 10-9, phase shift is less than 6 × 10-7 within 3 kHz.

Order 1 10 20 30 40 50 60

Another factor, the half range effect, is considered. A divider obtains the value in its half

This effect can be measured by comparing two cases: the full and 50% ranges. The results

Order 1 10 20 30 40 50 60

Thus, Table 7.2.5 shows the uncertainty of the phase shift of every divider, including the standard deviation in Table 7.2.2, uncertainty of the 0° standard in Table 7.2.3, half range

> Order 1 10 20 30 40 50 60 60 V 6.03 5.41 7.64 9.22 11.4 13.9 15.2 120 V 6.38 6.11 8.56 10.6 12.9 15.6 17.2 240 V 6.71 6.47 9.16 11.3 14.0 16.9 18.5 480 V 7.04 6.91 10.3 13.5 17.2 22.8 20.1

The 120 V divider under a 100 V fundamental has a phase shift of 14.6 rad, and its uncertainty is 13 rad (*k* = 2). However, these results can be applied only to the sinusoidal signal and in full range because the experiments are implemented under these conditions.

A similar step-up procedure (Fig. 7.3.1) is conducted for the shunts from 0.1 to 20 A, and 50 A. An original standard is necessary; that is, a standard resistor of a known time constant developed early at the NIM. With the parameters *R* = 10 Ω, *τ* = 0.03 × 10-9, phase shift is less

*Vk*)/(rad) (*k* = 1)

Table 7.2.4. Variation of phase difference between half and full range/(rad)

2.0 2.0 3.0 4.0 5.0 6.0 6.5

) 3.01 2.12 1.84 2.06 1.70 3.14 3.66

**7.2.2 Uncertainty of phase shift measurement** 

Table 7.2.3. Uncertainty of the 0° standard/(rad) (*k* = 1)

range but provides the value in its full range to the next divider.

effect in Table 7.2.4, and uncertainty of its foregoing divider.

Table 7.2.5. Uncertainty of phase shift of dividers after correction, *u*(

*u*(

are shown in Table 7.2.4.

**7.3 Phase shift of shunts** 

than 6 × 10-7 within 3 kHz.

The uncertainty of the phase shift is evaluated according to Eq. (7.1.4).

The 0º standard uncertainty, determined in an experiment, is shown in Table 7.2.3.

Fig. 7.3.1. Step-up procedure for measuring the phase shift of shunts.

Fig. 7.3.2. Measurement the half range effect of the shunts.

All the shunts and 50 A transformers are measured using the method described in Section 7.1.3. The measurement results for all the shunts are obtained. The results for the shunt of 5 A, including its phase shift and uncertainty, are expressed in Table 7.3.1.

In the uncertainty, all the factors are considered and computed. The half range effect is determined in an experiment, as shown in Fig. 7.3.2. R2 is measured initially at the half current of 0.5*I*, and then at the full current of *I*. The half range effect can be determined when R4 is used as reference.


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.
