Harmonic Analysis Reference (OPT-POWER-ADV)¶

This Chapter describes the measurement of

Harmonics (0 to 1000th Order)
Interharmonics (0.5 to 49.5th Order)
Higher Frequencies (2 to 9 kHz)
Supraharmonics (8 to 150 kHz)

The further mentioned calculation methods are complying with IEC 61000-4-7 for 50 Hz and 60 Hz operation. Beside this fundamental frequencies, the calculation and grouping methods are similar.

Calculation Interval

The Harmonic Analysis is synchronized to N periods of the fundamental.

Fig. 37 Period Aggregation on an example with 50 Hz Fundamental (Number of Periods = 10)¶

In the following Table you will find the number of periods which are used at different fundamental Frequencies:

Table 7 Used number of periods for each calculation interval¶
Fundamental Frequency	Number of Periods N	Additional Information
1–4.99	1
5–14.99	2
15–24.99	4
25–34.99	6
35–44.99	8
45–54.99	10	50 Hz Grid
55–64.99	12	60 Hz Grid
65–74.99	14
…	…	…

The Harmonic Calculation interval will always be linked to N periods but equates to about 200ms of a time window.

This is true for the Harmonics update rate setting “Default”. In case the Harmonics update rate setting is set to “Fast” the time window will half from the “Default” setting to 100 ms. For example in a 50 Hz Grid the “Default” Harmonic output will be after every 10 Periods the “Fast” Harmonic output will be after every 5 Periods, meaning double the output rate of the “Default” Harmonic update rate setting. See more in Settings below.

Settings¶

Power Group’s Tab for Settings
Details Tab for Settings of the Harmonic Analysis
Enable/Disable Harmonic Analysis
Selector for Grouping Type
Input Field for maximum order of Harmonics to be calculated and displayed
Input Field for number of Harmonics in THD calculation (must be lower or equal than (5))
Harmonic update rate, options: “Default” or “Fast”. “Fast” meaning double the output rate of “Default”
Enable/Disable supraharmonic analysis, options: Resolution of 2kHz or 200Hz
Enable/Disable Line-Line harmonics
Enable/Disable relative harmonics channels, options: set relative harmonics channels to be based on fundamental or nominal value
Enable/Disable impedance channels

Harmonic Grouping¶

Table 8 Legend¶
Symbol	Description
$\mathrm{Y}_{c}$	RMS Magnitude of FFT-Bin
$\mathrm{Y}_{h}$	RMS Magnitude of Harmonic Order h (Grouping “None”)
$\mathrm{Y}_{sg,h}$	RMS Magnitude of Harmonic Order h (Grouping “Type 1”)
$\mathrm{Y}_{g,h}$	RMS Magnitude of Harmonic Order h (Grouping “Type 2”)
$\mathrm{NP}_{c}$	Number of fundamental Periods
$\mathrm{h}$	Harmonic Order

Mode: “None”¶

When this mode is selected, only the harmonic bins are taken for generation of the harmonic data.

$\begin{aligned} Y_h &= Y_c[NP \cdot h] \end{aligned}$

Grouping Schema Mode "None" — Fig. 39 Grouping Schema Mode “None”¶

This example shows the grouping for 50 Hz fundamental Frequency.

Mode: “Type 1”¶

When this mode is selected, the harmonics are grouped according to IEC61000-4-7 Section 5.6.

$\begin{aligned} Y_{sg,h} &= Y_c[NP \cdot h] & NP < 2 \\ Y_{sg,h} &= \sqrt{\sum_{k=-1}^{1} Y_c \left[ NP \cdot h + k \right]^2} & NP >= 2 \end{aligned}$

Grouping Schema Mode "Type 1" — Fig. 40 Grouping Schema Mode “Type 1”¶

Mode: “Type 2”¶

When this mode is selected, the harmonics are grouped according to IEC61000-4-7 Section 5.5.1.

$\begin{aligned} Y_{g,h} &= Y_c[NP \cdot h] & NP < 2 \\ Y_{g,h} &= \sqrt{\frac{1}{2} Y_c \left[NP \cdot h - \frac{N}{2} \right]^2 + \sum_{k=\frac{-N}{2}+1}^{\frac{N}{2}-1} Y_c \left[ NP \cdot h + k \right]^2 + \frac{1}{2} Y_c \left[NP \cdot h + \frac{N}{2} \right]^2} & NP >= 2 \end{aligned}$

Grouping Schema Mode "Type 2" — Fig. 41 Grouping Schema Mode “Type 2”¶

Interharmonic Grouping¶

Table 9 Legend¶
Symbol	Description
$\mathrm{Y}_{c}$	RMS Magnitude of FFT-Bin
$\mathrm{Y}_{ih}$	RMS Magnitude of Interharmonic Order h (Grouping “None”)
$\mathrm{Y}_{isg,h}$	RMS Magnitude of Interharmonic Order h (Grouping “Type 1”)
$\mathrm{Y}_{ig,h}$	RMS Magnitude of Interharmonic Order h (Grouping “Type 2”)
$\mathrm{NP}$	Number of fundamental Periods
$\mathrm{h}$	Interharmonic Order

Mode: “None”¶

When this mode is selected, all bins (except of harmonic bin) are taken for generation of the interharmonic data.

$\begin{aligned} Y_{ih} &= NaN & NP = 1 \\ Y_{ih} &= \sqrt{\sum_{k=1}^{N-1} Y_c \left[ NP \cdot h + k \right]^2 } & NP > 1 \end{aligned}$

This example shows the grouping for 50 Hz fundamental Frequency.

Mode: “Type 1”¶

When this mode is selected, the Interharmonics are grouped according to IEC61000-4-7 Section 5.6.

$\begin{aligned} Y_{isg,h} &= NaN & NP <= 2 \\ Y_{isg,h} &= \sqrt{\sum_{k=2}^{N-2} Y_c \left[ NP \cdot h + k \right]^2} & NP > 2 \end{aligned}$

Mode: “Type 2”¶

When this mode is selected, the Interharmonics are grouped according to IEC61000-4-7 Section 5.5.1.

$\begin{aligned} Y_{ig,h} &= NaN & NP = 1 \\ Y_{ig,h} &= \sqrt{\sum_{k=1}^{N-1} Y_c \left[ NP \cdot h + k \right]^2} & NP > 1 \end{aligned}$

Higher Frequency Grouping¶

The Higher Frequency Grouping is orientated on fixed frequency bands (see IEC/EN61000-4-7 Appendix B). It is independent from the fundamental frequency and working between 10 and 1000 Hz.

Table 10 Legend¶
Symbol	Description
$\mathrm{Y}_{Cf}$	RMS Magnitude of FFT-Bin
$\mathrm{Y}_{B,b}$	RMS Magnitude of Higher Frequency Order b

$\begin{aligned} Y_{B,b} &= \sqrt{\sum_{f=b-95Hz}^{b+100Hz} Y_{C,f}^2} \end{aligned}$

This example shows the grouping for every fundamental frequency.

Supraharmonics Grouping¶

Unlike the other grouping methods, the Supraharmonics are aggregated in time. The Supraharmonics calculation will output channels with the Harmonic components in the frequency range between 8kHz and 150kHz.

Since the Line Resolution of 2kHz (or optional 200Hz) is much finer than the Harmonic Calculation Interval meaning the update rate of Supraharmonics is much faster than the Harmonic Interval the output channels will undergo a statistics evaluation (AVG, MIN, MAX). Each Supraharmonic Evaluation will result in 3 Channels with indices $\_shMIN, \_shMAX, \_shAVG$ . As an example for the first Voltage: $U1\_shMIN, U1\_shMAX, U1\_shAVG$ .

Table 11 Legend¶
Symbol	Description
$\mathrm{Y}_{C,b}$	RMS Magnitude of FFT-Bin b
$\mathrm{Y}_{SH,i,min}$	Minimum RMS Magnitude of Supraharmonics order i
$\mathrm{Y}_{SH,i,avg}$	Average RMS Magnitude of Supraharmonics order i
$\mathrm{Y}_{SH,i,max}$	Maximum RMS Magnitude of Supraharmonics order i
$\mathrm{NP}$	Number of fundamental periods (10 @ 50 Hz, 12 @ 60 Hz)
$\mathrm{Y}_{FFT}$	Number of Short Time FFTs in time interval of NP
$\mathrm{SR}$	Samplerate

$\begin{aligned} N_{FFT} &= \frac{\frac{NP}{F_{fund}} \cdot SR}{500} \\ Y_{SH,b,min} &= min(Y_{C,b}) \\ Y_{SH,b,avg} &= avg(Y_{C,b}) \\ Y_{SH,b,max} &= max(Y_{C,b}) \end{aligned}$

This graphic shows the time domain aggregation of the short time FFT results. The number of FFTs is orientated on the number of fundamental periods.

In the graphics above the Supraharmonics Resolution is set to the default 2 kHz. In case the option in the Power Group settings is set to 200 Hz the FFT-Bin at 150 kHz will be 750, not 75, and the line resolution will be 200 Hz.

Harmonic Element Extraction¶

Specific elements from the Harmonic Vector can be extracted as separate channels.

Open Channel Settings of Harmonic Vector
Select elements starting from zero
Each extracted element will create a new scalar channel under the Harmonic Vector

Voltage Channels (U)¶

U[i]_h[k]RMS¶

See Harmonic Grouping.

$\begin{aligned} Y_c &= abs(\underline{U}[i]_{fft}) \cdot \frac{1}{\sqrt{2}} & \\ U[i]\_h[k]_{RMS} &= \begin{cases} Y_h & \quad \text{if } MODE = "None" \\ Y_{sg,h} & \quad \text{if } MODE = "Type 1"\\ Y_{g,h} & \quad \text{if } MODE = "Type 2"\\ \end{cases} & Unit: \textbf{V} \end{aligned}$

U[i]_h[k]REL¶

Voltage relative harmonic value, referenced to fundamental. Must be enabled separately.

$\begin{aligned} U[i]\_h[k]_{REL} &= \frac{U[i]\_h[k]_{RMS}}{U[i]\_h[1]_{RMS}} \cdot 100\% & Unit: \textbf{\%} \end{aligned}$

U[ij]_h[k]RMS¶

Line-to-Line Harmonic Magnitude. Only available in 3-Phase Configurations (3P3W, 3P4W, 2V2A).

$\begin{aligned} \underline{U}[i]_{h} = U[i]\_h[k]_{RMS} \cdot ( & cos (U[i]\_h[k]_{PHI} \cdot \frac{\pi}{180}) + & \\ & j \cdot sin ( U[i]\_h[k]_{PHI} \cdot \frac{\pi}{180} )) & \\ \underline{U}[12]_{h} &= \underline{U}[1]_{h} - \underline{U}[2]_{h} & \\ \underline{U}[23]_{h} &= \underline{U}[2]_{h} - \underline{U}[3]_{h} & \\ \underline{U}[31]_{h} &= \underline{U}[3]_{h} - \underline{U}[1]_{h} & \\ U[ij]\_h[k]_{RMS} &= abs(\underline{U}[ij]_{h}) & Unit: \textbf{V} \end{aligned}$

U[i]_h[k]PHI¶

$\begin{aligned} PHI_{ref} &= arg(\underline{U1}_{fft}[1 \cdot NP]) + 90^\circ & \\ U[i]\_h[k]_{PHI} &= arg(\underline{U}[i]_{fft}[k \cdot NP]) + 90^\circ - k \cdot PHI_{ref} & Unit: \mathbf{deg} (^\circ) \end{aligned}$

U[i]_ih[k]RMS¶

See Interharmonic Grouping.

$\begin{aligned} Y_c &= abs(\underline{U}[i]_{fft}) \cdot \frac{1}{\sqrt{2}} & \\ U[i]\_ih[k]_{RMS} &= \begin{cases} Y_{ih} & \quad \text{if } MODE = "None" \\ Y_{isg,h} & \quad \text{if } MODE = "Type 1"\\ Y_{ig,h} & \quad \text{if } MODE = "Type 2"\\ \end{cases} & Unit: \textbf{V} \end{aligned}$

U[i]_ih[k]REL¶

Voltage relative interharmonic value, referenced to fundamental. Must be enabled separately.

$\begin{aligned} U[i]\_ih[k]_{REL} &= \frac{U[i]\_ih[k]_{RMS}}{U[i]\_h[1]_{RMS}} \cdot 100\% & Unit: \textbf{\%} \end{aligned}$

U[i]_hf[k]RMS¶

See Higher Frequency Grouping.

$\begin{aligned} Y_{C} &= abs(\underline{U}[i]_{fft}) \cdot \frac{1}{\sqrt{2}} & \\ U[i]\_hf[k]_{RMS} &= Y_{B,b} & Unit: \textbf{V} \end{aligned}$

U[i]_sh[k]MIN¶

Minimum Value of Voltage Supraharmonics Grouping. See Supraharmonics Grouping.

$\begin{aligned} U[i]\_sh[k]_{MIN} &= Y_{SH,b,min} & Unit: \textbf{V} \end{aligned}$

U[i]_sh[k]AVG¶

Average Value of Voltage Supraharmonics Grouping. See Supraharmonics Grouping.

$\begin{aligned} U[i]\_sh[k]_{AVG} &= Y_{SH,b,avg} & Unit: \textbf{V} \end{aligned}$

U[i]_sh[k]MAX¶

Maximum Value of Voltage Supraharmonics Grouping. See See Supraharmonics Grouping.

$\begin{aligned} U[i]\_sh[k]_{MAX} &= Y_{SH,b,max} & Unit: \textbf{V} \end{aligned}$

U[i]_THD¶

Voltage Total Harmonic Distortion, highest used harmonic is $k_{max}$ (selectable in menu in the Harmonics tab).

$\begin{aligned} U[i]_{THD} &= 100 \cdot \sqrt{\sum_{k=2}^{k_{max}} \left( \frac{U[i]_{h[k]RMS}}{U[i]_{h[1]RMS}} \right)^2 } & Unit: \mathbf{\%} \end{aligned}$

U[ij]_THD¶

Line-Line Voltage Total Harmonic Distortion, highest used harmonic is $k_{max}$ (selectable in menu in the Harmonics tab).

$\begin{aligned} U[ij]_{THD} &= 100 \cdot \sqrt{\sum_{k=2}^{k_{max}} \left( \frac{U[ij]_{h[k]RMS}}{U[ij]_{h[1]RMS}} \right)^2 } & Unit: \mathbf{\%} \end{aligned}$

U[i]_THV¶

Total Harmonic Voltage, highest used harmonic is $k_{max}$ (selectable in menu in the Harmonics tab).

$\begin{aligned} U[i]_{THV} &= \sqrt{\sum_{k=2}^{k_{max}} \left(U[i]_{h[k]RMS} \right)^2 } & Unit: \mathbf{V} \end{aligned}$

Current Channels (I)¶

I[i]_h[k]RMS¶

See Harmonic Grouping

$\begin{aligned} Y_c &= abs(\underline{I}[i]_{fft}) \cdot \frac{1}{\sqrt{2}} & \\ I[i]\_h[k]_{RMS} &= \begin{cases} Y_h & \quad \text{if } MODE = "None" \\ Y_{sg,h} & \quad \text{if } MODE = "Type 1"\\ Y_{g,h} & \quad \text{if } MODE = "Type 2"\\ \end{cases} & Unit: \textbf{A} \end{aligned}$

I[i]_h[k]REL¶

Current relative harmonic value, referenced to fundamental. Must be enabled separately.

$\begin{aligned} I[i]\_h[k]_{REL} &= \frac{I[i]\_h[k]_{RMS}}{I[i]\_h[1]_{RMS}} \cdot 100\% & Unit: \textbf{\%} \end{aligned}$

I[i]_h[k]PHI¶

$\begin{aligned} PHI_{ref} &= arg(\underline{U1}_{fft}[1 \cdot NP]) + 90^\circ & \\ I[i]\_h[k]_{PHI} &= arg(\underline{I}[i]_{fft}[k \cdot NP]) + 90^\circ - k \cdot PHI_{ref} & Unit: \mathbf{deg} (^\circ) \end{aligned}$

I[i]_ih[k]RMS¶

See Interharmonic Grouping.

$\begin{aligned} Y_c &= abs(\underline{I}[i]_{fft}) \cdot \frac{1}{\sqrt{2}} & \\ I[i]\_ih[k]_{RMS} &= \begin{cases} Y_{ih} & \quad \text{if } MODE = "None" \\ Y_{isg,h} & \quad \text{if } MODE = "Type 1"\\ Y_{ig,h} & \quad \text{if } MODE = "Type 2"\\ \end{cases} & Unit: \textbf{A} \end{aligned}$

U[i]_ih[k]REL¶

Current relative interharmonic value, referenced to fundamental. Must be enabled separately.

$\begin{aligned} I[i]\_ih[k]_{REL} &= \frac{I[i]\_ih[k]_{RMS}}{I[i]\_h[1]_{RMS}} \cdot 100\% & Unit: \textbf{\%} \end{aligned}$

I[i]_hf[k]RMS¶

See Higher Frequency Grouping.

$\begin{aligned} Y_{C} &= abs(\underline{I}[i]_{fft}) \cdot \frac{1}{\sqrt{2}} & \\ I[i]\_hf[k]_{RMS} &= Y_{B,b} & Unit: \textbf{A} \end{aligned}$

I[i]_sh[k]MIN¶

Minimum Value of Current Supraharmonics Grouping. See Supraharmonics Grouping.

$\begin{aligned} I[i]\_sh[k]_{MIN} &= Y_{SH,b,min} & Unit: \textbf{A} \end{aligned}$

I[i]_sh[k]AVG¶

Average Value of Current Supraharmonics Grouping. See Supraharmonics Grouping.

$\begin{aligned} I[i]\_sh[k]_{AVG} &= Y_{SH,b,avg} & Unit: \textbf{A} \end{aligned}$

I[i]_sh[k]MAX¶

Maximum Value of Current Supraharmonics Grouping. See Supraharmonics Grouping.

$\begin{aligned} I[i]\_sh[k]_{MAX} &= Y_{SH,b,max} & Unit: \textbf{A} \end{aligned}$

I[i]_THD¶

Current Total Harmonic Distortion, highest used harmonic is $k_{max}$ (selectable in menu in the Harmonics tab).

$\begin{aligned} I[i]_{THD} &= 100 \cdot \sqrt{\sum_{k=2}^{k_{max}} \left( \frac{I[i]_{h[k]RMS}}{I[i]_{h[1]RMS}} \right)^2 } & Unit: \mathbf{\%} \end{aligned}$

I[i]_THC¶

Current Total Harmonic Current, highest used harmonic is $k_{max}$ (selectable in menu in the Harmonics tab).

$\begin{aligned} I[i]_{THC} &= \sqrt{\sum_{k=2}^{k_{max}} \left(I[i]_{h[k]RMS} \right)^2 } & Unit: \mathbf{A} \end{aligned}$

Active Power Channels (P)¶

Grouping Mode always “None”

P[i]_h[k]¶

Active Power Harmonics.

$\begin{aligned} P[i]\_h[k] &= U[i]\_h[k]_{RMS} \cdot I[i]\_h[k]_{RMS} \cdot cos \left( U[i]\_h[k]_{PHI} - I[i]\_h[k]_{PHI} \right)& Unit: \mathbf{W} \end{aligned}$

P[i]_h[k]REL¶

Relative active Power Harmonics, referenced to fundamental

$\begin{aligned} P[i]\_h[k]_{REL} &= \frac{P[i]\_h[k]_{RMS}}{P[i]\_h[1]_{RMS}} \cdot 100\% & Unit: \textbf{\%} \end{aligned}$

P[i]_THP¶

Total Harmonic Power, highest used harmonic is $k_{max}$ (selectable in menu in the Harmonics tab).

$\begin{aligned} P[i]_{THP} &= \sum_{k=2}^{k_{max}} P[i]_{h[k]} & Unit: \mathbf{W} \end{aligned}$

Impedance Channels (RXZ)¶

R[i]_h¶

Active Impedance harmonic value. Must be enabled separately.

$\begin{aligned} R[i]\_h[k] &= \frac{U[i]\_h[k]_{RMS}}{I[i]\_h[k]_{RMS}} \cdot cos \left( U[i]\_h[k]_{PHI} - I[i]\_h[k]_{PHI} \right) & Unit: \textbf{Ohm} \end{aligned}$

X[i]_h¶

Reactive Impedance harmonic value. Must be enabled separately.

$\begin{aligned} X[i]\_h[k] &= \frac{U[i]\_h[k]_{RMS}}{I[i]\_h[k]_{RMS}} \cdot sin \left( U[i]\_h[k]_{PHI} - I[i]\_h[k]_{PHI} \right) & Unit: \textbf{Ohm} \end{aligned}$

Z[i]_h¶

Impedance harmonic value. Must be enabled separately.

$\begin{aligned} Z[i]\_h[k] &= \frac{U[i]\_h[k]_{RMS}}{I[i]\_h[k]_{RMS}} & Unit: \textbf{Ohm} \end{aligned}$