**3. Algorithm**

144 Modern Metrology Concerns

The digital sampling technique is widely adopted for commercial harmonic analyzers. For standard equipment, this technique must be developed to a higher level to satisfy calibration requirements. The standard equipment of the NIM is also based on digital sampling techniques. Unlike that proposed in general sampling theory, however, the synchronization between sampling rate and signal frequency is not a stringent requirement for the NIM equipment. The leakage effect that results from such an asynchronous case is compensated for by the introduction of a novel algorithm. The engineering practicality of this algorithm is demonstrated and its calculation is limited to a couple of seconds, without

The uncertainty evaluation for the standard equipment is based primarily on experiments. The standard equipment is tested using the national AC voltage and AC current standards at variable frequency points to determine frequency characteristics. Some special factors for harmonic conditions are then considered; these include the small harmonic components, leakage between harmonics, noise, and nonlinear effect of frequency. Finally, some

The principles and methods of the algorithm are designed to achieve higher accuracy. The uncertainty evaluation is carried out from the frequency to the harmonic feature. These attributes would be of significant reference for researchers, engineers, and students in developing higher quality commercial instruments or in general study. Based on these principles and methods the

Further investigation to extend the capability of the algorithm for impedance measurement,

The equipment system is introduced in Section 2. The algorithm is described in Section 3. The uncertainty of harmonic measurement is conceptually shown in Section 4. The harmonic voltage and harmonic current, their phase shifts, and harmonic power are discussed in Sections 5, 6, 7, and 8, respectively. The experimental validation is presented in Section 9.

The core materials on which this chapter is based are taken from [Lu et al., 2010], and some concepts in [Lu et al., 2008a, 2008b, 2008c] are extended. A primary form of the algorithm is discussed in [Lu, 1988], and its detailed analysis and application can be found in [Lu, 1991].

A block diagram of the NIM harmonic power standard equipment, including some general hardware, is shown in Fig. 2.1. Two commercial high-accuracy digital sampling voltmeters (DVMs) with type of HP3458A are employed as A/D converters (ADCs) to measure the instantaneous values of voltage and current signals. Non-sinusoidal signals are provided by two commercial programmable signal generators, one with the type of Fluke 6100A and another NST3500 from Chinese manufacturer. A set of resistive dividers have been developed to extend voltage ranges to 8, 15, 30, 60, 120, 240, and 500 V using selected resistors with a low time constant and low temperature coefficient. The resistance values are designed in such a way that the operating current does not introduce a significant heating effect. Another set of resistive dividers provide protection to reduce the effect of stray capacitance. A set of shunts (provided by SP of Sweden) are used to extend current ranges

to 0.1, 0.2, 0.5, 1, 2, 5, 10, and 20 A, while a current transformer is adopted for 50 A.

higher frequency measurement, or other AC measurements would also be valuable.

experiments are designed and implemented to validate the uncertainty.

readers can improve system and reach a higher accuracy and a better function.

the need for an especially large computing space.

**2. System** 

Generally, power sources and digital sampling meters have their own internal time clocks. Ensuring synchronous sampling capability in both instruments is difficult and often impossible. Such an asynchronous condition generates leakage errors in harmonic analysis because of the noncorrespondence between the accumulated values of the sampling data and the integral values in a precise period.

To resolve this issue, the authors in [Ihlenfeld et al., 2003] developed a method in which one time clock is adopted in the standard equipment, not only for the source (source of standard equipment, SSE), but also for the meter (meter of standard equipment, MSE). When a meter under test (meter under test, MUT) is calibrated, the SSE provides output signals for the MSE and MUT, resulting in synchronous sampling and appropriate calibration results. However, problems occur when calibrating another source (source under test, SUT) that has its own time clock. In such a case, due to two different time clocks are used, MSE calibration under synchronous sampling is difficult to accomplish, consequently resulting in poor calibration.

Non-integer-period sampling (NIPS) resolves this problem [Lu,1991]. It generates good results for both the MUT and SUT, without needing any special time clock and other equipment. The leakage effect persists, but can be overcome by the algorithm developed in this study for more general-purpose applications. The following section describes the algorithm.

#### **3.1 Non-integer-period sampling**

A period signal can be written as

$$y(\mathbf{x}) = a\_0 + \sum\_{k=1}^{w} a\_k \sin k\mathbf{x} + b\_k \cos k\mathbf{x} \tag{3.1.1}$$

where *x t* 2 / , is the signal period or fundamental period, and *k* denotes the harmonic order.

All cases can be expressed as

$$(n + \Delta)h = 2\text{rm} \,\,,\,\tag{3.1.2}$$

and the sampling data become

$$y\_j = a\_0 + \sum\_{k=1}^{n} a\_k \sin jkh + b\_k \cos jkh$$

$$j = 0, 1, 2, \dots, n. \dots \tag{3.1.3}$$

where *h* is the sampling interval, *n* denotes the number of samples in *m* periods, and Δ represents a small quantity with the same means as *n*, noted as a deviation degree of the non-integer-period.

A sampling will be entitled as integer-period sampling (IPS) when Δ = 0, and quasi-integerperiod sampling (QIPS) when |Δ|<1. Generally, it will be expressed as |Δ*h*|<*π*; that is, NIPS.

Fig. 3.1 explains this concept, in which three periods of a signal is divided by 13 samples at the same interval. For case (a), an IPS sample is obtained when 13 samples are computed; for case (b), an NIPS sample is derived when 7 samples are computed; for case (c), a QIPS sample is obtained when 4 samples are calculated.

The key point lies in how many samples are obtained or treated. At any specified uncertainty, a certain IPS sample can always be found for any NIPS sample when the sample number is not limited, so that the NIPS sample is a part of the IPS sample [Lu, 1991].

Similar QIPS samples also exist, as shown in Fig. 3.1. These are the groups of five, eight, and nine samples.

#### **3.2 Orthogonal check of the trigonometric function in NIPS**

An accumulation operation is used in measuring harmonic quantities from samples of periodic signals. It is denoted as a linear operator/calculator *A***.** 

A few detailed forms of *A* were introduced in [Lu, 1988]. These are the trapezoid formula

$$Ay\_j = \frac{1}{n} \left( 0.5y\_0 + \sum\_{j=1}^{n-1} y\_j + 0.5y\_n \right) \tag{3.2.1}$$

rectangular (Stenbakken's) compensation

$$A y\_j = \frac{1}{n + \Delta} \left( \sum\_{j=0}^{n-1} y\_j + \Delta y\_n \right) \tag{3.2.2}$$

trapezoid compensation

146 Modern Metrology Concerns

0

0

1 sin cos *j kk*

*k*

 

1 ( ) sin cos *k k k*

*y x a a kx b kx* 

(*n*+Δ)*h* = 2π*m* , (3.1.2)

*y a a jkh b jkh*

 *j* = 0,1,2,…,*n*. , (3.1.3) where *h* is the sampling interval, *n* denotes the number of samples in *m* periods, and Δ represents a small quantity with the same means as *n*, noted as a deviation degree of the

A sampling will be entitled as integer-period sampling (IPS) when Δ = 0, and quasi-integerperiod sampling (QIPS) when |Δ|<1. Generally, it will be expressed as |Δ*h*|<*π*; that is,

Fig. 3.1 explains this concept, in which three periods of a signal is divided by 13 samples at the same interval. For case (a), an IPS sample is obtained when 13 samples are computed; for case (b), an NIPS sample is derived when 7 samples are computed; for case (c), a QIPS

The key point lies in how many samples are obtained or treated. At any specified uncertainty, a certain IPS sample can always be found for any NIPS sample when the sample number is not limited, so that the NIPS sample is a part of the IPS sample [Lu, 1991]. Similar QIPS samples also exist, as shown in Fig. 3.1. These are the groups of five, eight, and

An accumulation operation is used in measuring harmonic quantities from samples of

1

 

, (3.2.1)

1 <sup>1</sup> 0.5 0.5 *n j j n j Ay y y y*

A few detailed forms of *A* were introduced in [Lu, 1988]. These are the trapezoid formula

0

*n*

, (3.1.1)

is the signal period or fundamental period, and *k* denotes the harmonic

**3.1 Non-integer-period sampling**  A period signal can be written as

All cases can be expressed as

and the sampling data become

sample is obtained when 4 samples are calculated.

**3.2 Orthogonal check of the trigonometric function in NIPS** 

periodic signals. It is denoted as a linear operator/calculator *A***.** 

non-integer-period.

NIPS.

nine samples.

where *x t* 2 / , 

order.

$$Ay\_j = \frac{1}{n+\Delta} \left[ \sum\_{j=1}^{n-1} y\_j + 0.5(1+\Delta)(y\_0+y\_n) \right]. \tag{3.2.3}$$

Fig. 3.1. (a) is an IPS sample, (b) is a NIPS sample, and (c) is a QIPS sample

In NIPS, the orthogonality of the trigonometric function appears as a deviation. If

$$A\sin jkh = \alpha\_{k\_{\text{max}}} \tag{3.2.4}$$

cos *A jkh k* . (3.2.5)

Using the characteristic of the trigonometric function proves that

$$2A(\sin jklt \cos jlh) = \alpha\_{k+l} + \alpha\_{k-l} \tag{3.2.6}$$

$$2A \langle \sin jkl \sin jlh \rangle = -\beta\_{k+l} + \beta\_{k-l} \tag{3.2.7}$$

$$2A \text{(cos } jkl \text{cos } jlh) = \beta\_{k+l} + \beta\_{k-l} \text{ .} \tag{3.2.8}$$

In particular, when *k* = 0,

$$
\alpha\_0 = A \sin j0\\h = A \cdot 0 = 0 \quad , \tag{3.2.9}
$$

$$
\beta\_0 = A \cos j0\\h = A \cdot 1 = 1\ .\tag{3.2.10}
$$

#### **3.3 Fourier coefficient of the signal in NIPS**

If the Fourier coefficients of the signal described in Eq. (3.1.1) are 0 , , *k k aab* , the following hold:

$$
\widehat{a}\_0 = a\_0 + \sum\_{k=1}^{\infty} a\_k a\_k + \beta\_k b\_k \, \tag{3.3.1}
$$

$$
\widehat{a}\_k = 2a\_k a\_0 + \sum\_{l \neq k}^{1, \infty} (-\beta\_{k+l} + \beta\_{k-l}) a\_l + (1 - \beta\_{2k}) a\_k \, \, \, \, \, \,
$$

$$
+ \sum\_{l \neq k}^{1, \infty} (a\_{k+l} + a\_{k-l}) b\_l + a\_{2k} b\_k \, \, \, \, \, \, \tag{3.3.2}
$$

$$
\widehat{b}\_k = 2\beta\_k a\_0 + \sum\_{l \neq k}^{1, \infty} (a\_{k+l} - a\_{k-l}) a\_l + a\_{2k} a\_k \, \, \, \, \, \, \} \, \, \, \, \, \, \,
$$

$$
\widehat{b}\_{1, \infty}
$$

$$+\sum\_{l\neq k} (\mathcal{J}\_{k+l} + \mathcal{J}\_{k-l}) b\_l + (1 + \mathcal{J}\_{2k}) b\_k \ . \tag{3.3.3}$$
 
$$\dots \quad \dots \quad \dots \quad \dots \quad \dots \quad \dots \quad \dots \quad \dots \quad \dots \quad \dots \quad \dots \quad \dots \quad \dots \quad (2.3.4)$$

If *w* is the maximum limit of the harmonic order of the signal shown in Eq. (3.1.1), the Fourier coefficients of the signal can be denoted by a vector **a** with order 2*w*+1:

$$\mathbf{a} = (a\_0, a\_1, a\_2, \cdots; \; a\_{w\_2}, b\_1, b\_2, \cdots; \; b\_w)^\mathrm{T}. \tag{3.3.4}$$

Then, its DFT result can be expressed as another vector . **a**ˆ :

$$\hat{\mathbf{a}} = \left(\hat{a}\_0, \hat{a}\_1, \hat{a}\_2, \dots, \hat{a}\_w, \hat{b}\_1, \hat{b}\_2, \dots, \hat{b}\_w\right)^T. \tag{3.3.5}$$

According to Eqs. (3.3.1), (3.3.2), and (3.3.3), we have

$$
\hat{\mathbf{a}} = \mathbf{F}\_{\mathrm{R}} \mathbf{a} \,. \tag{3.3.6}
$$

Thus, the results of **a** can be calculated as follows:

$$\mathbf{a} = \mathbf{F}\_{\mathbb{R}}^{-1} \hat{\mathbf{a}}\tag{3.3.7}$$

where **FR** is a matrix with order 2*w*+1:

148 Modern Metrology Concerns

*A jkh* sin

cos *A jkh*

2 (sin cos ) *A jkh jlh*

2 (sin sin ) *A jkh jlh*

2 (cos cos ) *A jkh jlh*

If the Fourier coefficients of the signal described in Eq. (3.1.1) are 0 , , *k k aab* , the following

1

0 2 2 ( ) (1 ) *k k kl kl l k k*

<sup>2</sup> ( ) *kl kl l k k*

0 2 2( ) *k k kl kl l kk*

,

<sup>2</sup> ( ) (1 ) *kl kl l k k*

If *w* is the maximum limit of the harmonic order of the signal shown in Eq. (3.1.1), the

,

*k aa a b* 

> 

1,

*l k b a aa*

Fourier coefficients of the signal can be denoted by a vector **a** with order 2*w*+1:

 

*kk k k*

 *b b*

 *b b*

 

*kl kl* 

*kl kl* 

*kl kl* 

*A jh A* sin 0 0 0 , (3.2.9)

*A jh A* cos 0 1 1 . (3.2.10)

, (3.3.1)

 *a a*

, (3.3.2)

 

. (3.3.3)

*k* , (3.2.4)

*k* . (3.2.5)

, (3.2.6)

, (3.2.7)

. (3.2.8)

In NIPS, the orthogonality of the trigonometric function appears as a deviation.

Using the characteristic of the trigonometric function proves that

0 

0 

0 0

1,

*l k*

1,

*l k*

1,

*l k*

**3.3 Fourier coefficient of the signal in NIPS** 

*a a* 

If

In particular, when *k* = 0,

hold:

$$\mathbf{F}\_{\rm R} = \begin{bmatrix} \mathbf{F}\_{11} & \mathbf{F}\_{12} & \mathbf{F}\_{13} \\ \mathbf{F}\_{21} & \mathbf{F}\_{22} & \mathbf{F}\_{23} \\ \mathbf{F}\_{31} & \mathbf{F}\_{32} & \mathbf{F}\_{33} \end{bmatrix}. \tag{3.3.8}$$

Its detailed form will be dependent on the sampling rate and signal period, and will be independent of the amplitude and phase angle of the signal. That is, matrix **FR** is available for all signals when both the sampling rate and signal period are fixed.

Some interesting and useful characteristics of **FR** are as follows:

$$
\mathbf{F}\_{11} = 1
$$

$$
\mathbf{F}\_{12} = \frac{1}{2} (\mathbf{F}\_{21})^T = (a\_{1'}, a\_{2'}, \cdots, a\_w)
$$

$$
\mathbf{F}\_{13} = \frac{1}{2} (\mathbf{F}\_{31})^T = (\beta\_{1'} \beta\_{2'}, \cdots, \beta\_w)
$$

$$
\{\mathbf{F}\_{22}\}\_{kl} = \{-\beta\_{k+l} + \beta\_{k-l}\}
$$

$$
\{\mathbf{F}\_{23}\}\_{kl} = \{a\_{k+l} + a\_{k-l}\}
$$

$$
\{\mathbf{F}\_{32}\}\_{kl} = \{a\_{k+l} - a\_{k-l}\}
$$

$$
\{\mathbf{F}\_{33}\}\_{kl} = \{\beta\_{k+l} + \beta\_{k-l}\}
$$

where *k, l* = 1,2, to *w*.

When *w* = 3, the matrix of **FR** can be constructed as


#### **3.4 Determination of the period**

In the algorithm, the period must first be determined. For the standard equipment, the signal period is cursorily known. A more accurate result can be calculated using Δ.

The period end point can be assumed located between points *y*n and *y*n+1. Its value is exactly *y*0. The period exists in the relationship of (*y*n, *y*0, *y*n+1). The quantity is calculated [Lu, 1986] as

$$
\Delta = (y\_{0^\bullet} y\_n) / \left( \text{\textquotedbl{}y\_n + y\_{n^\bullet 1}\text{\textquotedbl{}y\_{n^\bullet 1}}} \right). \tag{3.4.1}
$$

The other similar relationships of (*y*0, *y*n+1, *y*1) can also be used. A more precise formula is [Zhang J.Q., 1996]

$$
\Delta = (y\_0 + y\_1 \text{-} y\_{\text{n}} \text{-} y\_{\text{n}+1}) / \left(\text{-} y\_0 \text{+} y\_1 \text{-} y\_{\text{n}} + y\_{\text{n}+1}\right) \,. \tag{3.4.2}
$$

The selection of the starting point of *y*0 is important; it must be located in a region with a large slope. Under sinusoidal conditions, for example, it should be a zero-crossing point, or a point with a minimum value among sampling data.

Other approaches to determining the period are also available. For example, the relationships of the calculated values of phase angles as (5.1.5) about the first period and the second period [Dai, 1989] can be used.

#### **3.5 Practical implementation of DFT and compensation**

For input sampling data, the practical implementing process of DFT is as follows:


Where an important point is that the base functions of sin *jkh* and cos *jkh* needed by DFT come from the practical case including the signal period and the sampling rate, but not come only from the sampling rate of ADC.

When application of Eq. (3.2.3) in DFT, the term of *<sup>j</sup> y* shall be replaced by sin *<sup>j</sup> y jkh* or cos *<sup>j</sup> y jkh* , *j n* 0,1,2, .

After this DFT a further compensation will be carried out for higher accuracy. For this aim, in the fact, the eqn. (3.3.7) is recognized by many researchers, but its calculation is very difficult. A large-capacity RAM and long computing time are required to calculate **F**R-1 because of the huge matrix. Its 2*w*+1 order, for example, is 121 when *w* = 60 in the NIM standard equipment.

For practical implementation, **F**R must be simplified.

150 Modern Metrology Concerns

1 2 1

**3.4 Determination of the period** 

as

[Zhang J.Q., 1996]

2 1

a point with a minimum value among sampling data.

**3.5 Practical implementation of DFT and compensation** 

1. to determine Δ according to Eq. (3.4.1) or (3.4.2);

second period [Dai, 1989] can be used.

only from the sampling rate of ADC.

cos *<sup>j</sup> y jkh* , *j n* 0,1,2, .

 

2 1

 

2 1

 

 

2 1

 

signal period is cursorily known. A more accurate result can be calculated using Δ.

2 1

In the algorithm, the period must first be determined. For the standard equipment, the

The period end point can be assumed located between points *y*n and *y*n+1. Its value is exactly *y*0. The period exists in the relationship of (*y*n, *y*0, *y*n+1). The quantity is calculated [Lu, 1986]

The other similar relationships of (*y*0, *y*n+1, *y*1) can also be used. A more precise formula is

The selection of the starting point of *y*0 is important; it must be located in a region with a large slope. Under sinusoidal conditions, for example, it should be a zero-crossing point, or

Other approaches to determining the period are also available. For example, the relationships of the calculated values of phase angles as (5.1.5) about the first period and the

For input sampling data, the practical implementing process of DFT is as follows:

3. to calculate the base functions of sin *jkh* and cos *jkh* which are needed by DFT; 4. to implement DFT with the trapezoid compensation calculator *A* in Eq.(3.2.3).

Where an important point is that the base functions of sin *jkh* and cos *jkh* needed by DFT come from the practical case including the signal period and the sampling rate, but not come

When application of Eq. (3.2.3) in DFT, the term of *<sup>j</sup> y* shall be replaced by sin *<sup>j</sup> y jkh* or

2. to calculate the sampling/discrete interval of *h* according to Eq. (3.1.2)

 

 

1231 23

 

 

> 

 

 

> 

Δ = (*y*0-*y*n)/(-*y*n+*y*n+1) . (3.4.1)

Δ = (*y*0+*y*1-*y*n-*y*n+1)/(-*y*0+*y*1-*y*n+ *y*n+1) . (3.4.2)

 

 

 

> 

 

 

1 2 1 3 2 4 2 3 1 42 213 4 15 31 4 51 32415 6 4251 6 1 2 3142 2 1324 231 4 5113 4 15 34251 6 2415 6

 

> 

 

  As mentioned in Section 3.3, only two types of quantities, *αk* and *βk ,* are necessary for all the elements of the matrix. They should be expressed in analyzable forms. According to the expression of *αk* and *βk* in Eqs. (3.2.4) and (3.2.5), as well as the expression of trapezoid compensation operator *A* in Eq. (3.2.3), the quantities can be analyzed and described as

$$\alpha\_k = \frac{1}{n+\Delta} \left( \sin^2 \frac{\Delta k \hbar}{2} ctg \frac{k \hbar}{2} - \frac{\Delta}{2} \sin \Delta k \hbar \right),\tag{3.5.1}$$

$$\beta\_k = \frac{1}{n+\Delta} \left( -\frac{1}{2} \sin \Delta k h \, \text{ctg} \, \frac{k\hbar}{2} + \Delta \cos^2 \frac{\Delta k \hbar}{2} \right). \tag{3.5.2}$$

However, *βk* =1, *αk* = 0 when *k* = 0. Alternatively, *α-k* = -*αk* , *β-k* = *βk*.

Some special controls are adopted in the NIM standard equipment to let **X** 1, where **FR = I + X** and **I** is a unit matrix. Thus, **FR** -1 ≈ **I** – **X**, where **X** can be directly provided according to Eqs. (3.3.6), (3.5.1), and (3.5.2). The calculation of **FR** -1 becomes a very easy process. The author notes here that it will be not necessary for the hardware control in principle.

The precision result, **a**, is directly calculated without any intermediate process. A complete calculation that includes DFT and compensation can be completed within a couple of seconds, without the necessity for an especially large computing space.

The compensation results are dependent on the ratio of the computational error/uncertainty of Δ to *n* [Lu, 2008d].

## **3.6 Simulation and the effect of noise**

A simulation test is designed to verify the effect of the algorithm and results, as evaluated against those derived from the general DFT method with no compensation. In the general method the sampling interval is regarded as 2π*m*/*n*, but in the compensation method used in this study, it is 2π*m*/(*n*+Δ), as shown in Eq. (3.1.2). However in this DFT this Δ value is asked to calculated from the sampling data.

The sampling data from a sinusoidal signal with an amplitude of 0.8 V (to check for potential computational errors) are simulated on a computer with same interval of 2π*m*/(*n*+Δ) (it becomes a NIPS or QIPS precisenly) . The sampling rate is set at 60 sampling data in one period with different Δ values. The general DFT and proposed algorithm are then used to handle the same sampling data. Their computing values for signal amplitude are provided and then compared with the set value (0.8 V) to reveal the relative errors (Table 3.6.1).


Table 3.6.1. Simulation results for amplitude, *y* 0.8sin*t* (V), *n*=60

The results show that when *n* = 60 and is controlled within 0.05, the magnitude of the compensation effect can be as much as 1000 times. The relative error reaches the 10-7 level.

A similar simulation is conducted for the phase difference between two signals with the same sampling rate, but the phase difference is set at 60°. The results are shown in Table 3.6.2.


Table 3.6.2. Simulation results for phase difference, *n*=60

On the basis of this analysis, we design a practical sampling method in which 1680 samples over about 4 periods and a value controlled within 0.04 is applied in the equipment.

For a fixed , the phase-angle values in units of rad are different in view of varied harmonic components. At the fundamental, if the value is 2*m* , but at the *k*-th harmonics, it is 2*km* . Therefore, the same for higher harmonics produces different errors.

In simulating this case, the same sampling point construction is used for all the harmonics of the non-sinusoidal signal. This approach, which we call "harmonic discrete division," is the foundation of the investigation on harmonic sampling measurement.

This method is adopted to investigate the performance of the algorithm for harmonics. A practical condition is considered; i.e., 1680 samplers (with Δ) over 4 fundamental periods. When the 10th order harmonics is simulated, the 1680 points are constructed over 40 sinusoidal waveforms. The results are shown in Table 3.6.3.

0.5 3905 10.0 0.2 1626 1.9 0.1 822 0.8 0.05 413 0.4 0.02 166 0.1 0.01 83 0.1

The results show that when *n* = 60 and is controlled within 0.05, the magnitude of the compensation effect can be as much as 1000 times. The relative error reaches the 10-7 level. A similar simulation is conducted for the phase difference between two signals with the same sampling rate, but the phase difference is set at 60°. The results are shown in Table

<sup>Δ</sup> Error/rad

0.5 –3714 –13 0.2 –1615 –2.7 0.1 –828 –1.1 0.05 –420 –0.5 0.02 –169 –0.2 0.01 –85 –0.1

On the basis of this analysis, we design a practical sampling method in which 1680 samples over about 4 periods and a value controlled within 0.04 is applied in the equipment.

For a fixed , the phase-angle values in units of rad are different in view of varied harmonic

In simulating this case, the same sampling point construction is used for all the harmonics of the non-sinusoidal signal. This approach, which we call "harmonic discrete division," is the

This method is adopted to investigate the performance of the algorithm for harmonics. A practical condition is considered; i.e., 1680 samplers (with Δ) over 4 fundamental periods. When the 10th order harmonics is simulated, the 1680 points are constructed over 40

*km* . Therefore, the same for higher harmonics produces different errors.

foundation of the investigation on harmonic sampling measurement.

sinusoidal waveforms. The results are shown in Table 3.6.3.

*m* , but at the *k*-th harmonics, it is

Rel. error/(V/V) General DFT Proposed

General DFT Proposed algorithm

algorithm

*t* (V), *n*=60

Δ

Table 3.6.1. Simulation results for amplitude, *y* 0.8sin

Table 3.6.2. Simulation results for phase difference, *n*=60

components. At the fundamental, if the value is 2

3.6.2.

2


Table 3.6.3. Amplitude error after compensation for harmonic components, *y* 0.8sin*t* (V) for every harmonic, =0.04, *n*=1680

The errors that occur after compensation reaches the 10-10 level for the fundamental, and are less than 3 10-7 up to the 60th order harmonics. This result indicates that the leakage caused by the non-integer-period has been resolved.

Many other tests, including some complex waveforms, are conducted. A characterizing waveform signal discussed below is tested, in which all harmonic components from DC to 60th order harmonic exist with a relative high percent of amplitude, the largest error is 0.2 × 10-6. This result can be disregarded for the final estimated uncertainty.

The practical specifications are determined in experiments, such as the calibration traceable to the primary standard of the AC voltage and AC current of the NIM. In such cases, the noise becomes the primary determining factor for uncertainties.

The experimental results show that the noise effect for sinusoidal signal is about (0.5 to 2.0) × 10-6, as validated by Monte Carlo test results [Xue, 2011]. This means that the error from the algorithm (less than 1.0 × 10-6) is smaller than the effect of noise, and additional efforts to derive higher level algorithm results are not necessary.
