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Harmonic Analysis

One-cycle DFT harmonic decomposition with windowing strategies, THD calculation, and true RMS computation for power quality and protection assessment.

In Detego

View harmonic spectra in the Harmonics tab. This page covers the mathematical theory behind the analysis.

Overview

Harmonic analysis decomposes a periodic waveform into its constituent frequency components. Detego uses a one-cycle Discrete Fourier Transform (DFT) to compute the magnitude and phase angle of each harmonic order up to the 25th (by default). DC offset is removed before computation, and an order-dependent windowing strategy is applied to minimize spectral leakage.

Each harmonic result includes: harmonic order, frequency, magnitude (RMS), peak magnitude, percentage of the fundamental, and phase angle. These quantities are essential for CT saturation detection, transformer inrush identification, and power quality assessment.

Harmonic spectrum of a typical 6-pulse rectifier load showing characteristic 5th and 7th harmonics. THD (Total Harmonic Distortion) quantifies the total non-fundamental content relative to the fundamental.

DFT Computation

One-Cycle DFT

The one-cycle DFT evaluates the Fourier coefficient at a specific harmonic order $k$ by correlating the signal with sine and cosine basis functions of the corresponding frequency. The window length equals exactly one cycle of the fundamental frequency, which ensures that the fundamental and all its integer harmonics are orthogonal within the window.

DFT coefficient for harmonic order k

X_k = \frac{2}{N} \sum_{n=0}^{N-1} w[n] \cdot x[n] \cdot e^{-j\,2\pi\,k\,n\,/\,N}

Where

$X_k$ Complex Fourier coefficient for harmonic order k
$N$ Number of samples in one fundamental-frequency cycle
$w[n]$ Window function applied to sample n
$x[n]$ Input signal sample after DC offset removal
$k$ Harmonic order (0 = DC, 1 = fundamental, 2 = 2nd harmonic, ...)

DC offset is removed before the DFT by subtracting the mean of the one-cycle window from each sample. This prevents the DC component from leaking into the fundamental and harmonic magnitudes, which is particularly important for fault current waveforms that contain significant decaying DC offset.

Windowing Strategy

Spectral leakage occurs when the signal frequency does not perfectly match the DFT bin frequency. In power systems, the actual system frequency may deviate slightly from nominal (49.95 Hz instead of 50.0 Hz), causing the DFT window to not contain an exact integer number of cycles for higher harmonics. Detego applies a different window function for each harmonic order to optimize the accuracy-resolution tradeoff:

Order	Window	Rationale
0 (DC)	None (simple mean)	DC is computed as the arithmetic mean of the window. No windowing is needed because DC has no frequency to leak.
1 (Fundamental)	Rectangular	The one-cycle window is exactly matched to the fundamental frequency, so the rectangular window provides maximum frequency resolution with zero leakage for the fundamental component.
2-25	Hanning	Higher harmonics are more sensitive to frequency deviation because the window-to-period ratio increases. The Hanning window suppresses side-lobe leakage at the cost of slightly wider main lobes.

Hanning window function

w[n] = 0.5 \left(1 - \cos\!\left(\frac{2\pi n}{N-1}\right)\right)

Where

$w[n]$ Window coefficient for sample n
$N$ Total number of samples in the window

Hanning Amplitude Correction

The Hanning window has a coherent gain of 0.5, meaning it attenuates the measured amplitude by a factor of 2. Detego applies an amplitude correction factor of 2 to all Hanning-windowed results to compensate. This ensures that the reported harmonic magnitudes are correct regardless of which window was used.

DC Component (Order 0)

The DC component (harmonic order 0) is computed as the arithmetic mean of all samples in the one-cycle window, before any windowing is applied. It represents the average offset of the waveform from zero and is reported with zero phase angle.

In fault recordings, a significant DC component typically indicates a decaying DC offset from an asymmetrical fault inception angle. The DC component is included in the harmonic detail table and contributes to the true RMS calculation, but is excluded from THD computation (THD only considers orders 2 and above relative to the fundamental).

Computation Limits

The harmonic analysis engine enforces several limits to ensure valid results:

Nyquist frequency clipping: The maximum harmonic order is clamped so that the harmonic frequency does not exceed the Nyquist frequency ( $f_s / 2$ ). For example, at a 1600 Hz sample rate with 50 Hz fundamental, the maximum computable harmonic is order 16 (800 Hz). Harmonics above this limit are silently excluded.
Minimum window size: The analysis window must meet minimum sample requirements. If the one-cycle window has too few samples (extremely low sample rates), the DFT returns a zero-magnitude result.
Window duration threshold: The cursor time must be past a minimum window duration threshold from the recording start for a valid analysis window. If the cursor is placed too early in the recording (before enough data for a full cycle), no results are returned.

Derived Quantities

Total Harmonic Distortion (THD)

THD quantifies the total harmonic content of a waveform relative to the fundamental. It is expressed as a percentage and is the standard metric used in power quality standards (IEEE 519, IEC 61000-3-6) to assess waveform distortion.

Total Harmonic Distortion

\text{THD\%} = \frac{\sqrt{\sum_{n=2}^{n_{\max}} H_n^2}}{H_1} \times 100

Where

$H_n$ RMS magnitude of the nth harmonic
$H_1$ RMS magnitude of the fundamental (1st harmonic)
$n_max$ Maximum harmonic order included (default: 25)

THD is undefined when H₁ = 0 (no fundamental component). In this case, Detego reports THD as 0%.

True RMS from Harmonics

The true RMS of a signal can be reconstructed from its harmonic spectrum using Parseval's theorem. This provides an independent verification of the sliding-window RMS calculation and accounts for all frequency components including DC and harmonics.

True RMS from harmonic components

x_{\text{rms}} = \sqrt{\sum_{n=0}^{n_{\max}} H_n^2}

Where

$H_0$ DC component magnitude
$H_1$ Fundamental RMS magnitude
$H_n$ nth harmonic RMS magnitude (n ≥ 2)

For a pure sinusoid with no harmonics and no DC offset, true RMS equals H₁. Any harmonic content increases the true RMS above the fundamental.

Per-Harmonic Results

For each harmonic order, Detego computes and reports:

Field	Unit	Description
Order	--	Integer harmonic order (0 = DC, 1 = fundamental, 2 = 2nd, ...)
Frequency	Hz	$f_n = n \times f_{\text{fundamental}}$
Magnitude	A or V (RMS)	RMS magnitude of this harmonic component
Peak Magnitude	A or V (peak)	$\sqrt{2} \times$ RMS magnitude
Percentage	%	$(H_n / H_1) \times 100$
% True RMS	%	$(H_n / x_{\text{rms}}) \times 100$ — percentage of the true RMS rather than the fundamental, useful when the fundamental is small or absent
Angle	degrees	Phase angle relative to the start of the DFT window

Harmonic Extraction (Computed Signal)

In addition to the snapshot spectrum at the cursor position, Detego provides a Harmonic Extraction computed signal (created via the Compute Tab) that isolates a single harmonic order from a channel as a continuous time-domain waveform. It uses a sliding one-cycle DFT to track the selected harmonic throughout the entire recording.

Two output modes are available:

Waveform (default) — Reconstructs the oscillating harmonic component at the selected frequency: $y[n] = \mathrm{Re}\!\cos(\omega_k t) + \mathrm{Im}\!\sin(\omega_k t)$ . Useful for overlaying the isolated harmonic on the original signal to see where distortion occurs.
Envelope — Outputs the smooth magnitude: $y[n] = \sqrt{\mathrm{Re}^2 + \mathrm{Im}^2}$ . Shows how the harmonic strength varies over time, producing a non-oscillating trace that is ideal for trending and comparison.

See Basic Operations — Harmonic Extraction for the full mathematical treatment.

Engineering Applications

CT Saturation Detection

Current transformer saturation produces distinctive harmonic signatures that can be detected through harmonic analysis:

Elevated even harmonics (2nd, 4th): CT saturation typically affects one half-cycle more than the other (due to residual flux or DC offset), creating waveform asymmetry. This asymmetry generates even harmonics that are normally absent in power system waveforms.
Elevated odd harmonics (3rd, 5th): The clipping effect of saturation flattens the waveform peaks, which generates odd harmonics similar to a square wave.
Waveform asymmetry: Unilateral saturation (only one half-cycle saturated) produces a strong 2nd harmonic component, typically 20-40% of the fundamental.

CT Saturation Indicator

A 2nd harmonic ratio above 10-15% in a fault current waveform is a strong indicator of CT saturation. However, this must be distinguished from transformer inrush, which also produces elevated 2nd harmonics. The key differentiator is that CT saturation occurs in current channels while inrush is the actual primary current.

Transformer Inrush Identification

Transformer magnetizing inrush current has a characteristic harmonic profile that differential relays use to distinguish inrush from internal faults:

High 2nd harmonic: Inrush current has a large 2nd harmonic component, typically 30-70% of the fundamental. This is the primary restraint signal used by transformer differential relays (2nd harmonic blocking/restraint).
Moderate 3rd harmonic: The 3rd harmonic is also elevated during inrush, typically 10-30% of the fundamental.
Decreasing higher orders: The harmonic content decreases with increasing order, with the 4th and higher harmonics typically below 10%.

2nd harmonic restraint criterion

\frac{H_2}{H_1} > k_{\text{restraint}} \quad \Rightarrow \quad \text{Block trip (inrush detected)}

Where

$H_2$ 2nd harmonic magnitude
$H_1$ Fundamental magnitude
$k_restraint$ Restraint threshold, typically 15-20%

Transformer Overexcitation

Transformer overexcitation (overvoltage or under-frequency operation that drives the core into saturation) produces a different harmonic profile than inrush:

High odd harmonics (3rd, 5th): Overexcitation produces symmetrical saturation on both half-cycles, generating predominantly odd harmonics. The 3rd harmonic is typically 30-60% of the fundamental, and the 5th harmonic is 10-30%.
Low 2nd harmonic: Unlike inrush, overexcitation has a relatively low 2nd harmonic because the saturation is symmetrical. This means 2nd harmonic restraint alone cannot block overexcitation-driven false differential trips.
5th harmonic blocking: Many modern transformer differential relays include 5th harmonic blocking specifically to prevent tripping during overexcitation conditions.

5th harmonic restraint criterion

\frac{H_5}{H_1} > k_{5} \quad \Rightarrow \quad \text{Block trip (overexcitation detected)}

Where

$H_5$ 5th harmonic magnitude
$H_1$ Fundamental magnitude
$k₅$ 5th harmonic restraint threshold, typically 25-35%

Power Quality Assessment

Harmonic analysis is the foundation of power quality assessment per IEEE 519 and IEC 61000-3-6. These standards define limits for individual harmonic magnitudes and total harmonic distortion based on the voltage level and the ratio of short-circuit current to load current.

Standard	Scope	Key Limits
IEEE 519	Current and voltage harmonics at PCC	Voltage THD < 5% (below 69 kV); individual harmonic < 3%
IEC 61000-3-6	Emission limits for MV/HV systems	Planning levels vary by voltage level and harmonic order; uses compatibility levels for assessment

Harmonic Sources in COMTRADE Recordings

COMTRADE recordings from fault events may contain harmonic content that differs significantly from steady-state power quality measurements. During faults, harmonic content is dominated by CT saturation and transformer effects rather than load-generated harmonics. The harmonic analysis in Detego is most useful during fault analysis for CT saturation and inrush detection, but can also be applied to steady-state recordings for power quality assessment.

Harmonic Signature Summary

The following table summarizes the characteristic harmonic profiles for common power system phenomena:

Phenomenon	Dominant Harmonics	H2/H1 Ratio	Key Indicator
CT saturation	2nd, 4th (even)	15-40%	Waveform asymmetry + elevated even harmonics
Transformer inrush	2nd, 3rd	30-70%	Very high 2nd harmonic with decaying magnitude
Overexcitation	3rd, 5th (odd)	< 10%	High odd harmonics, low 2nd harmonic
6-pulse rectifier	5th, 7th, 11th, 13th	< 5%	Characteristic harmonics at $6k \pm 1$
Internal fault	Low harmonics	< 15%	Clean fundamental with minimal distortion (absent CT saturation)

Extended Metrics

K-Factor

The K-Factor quantifies the additional heating effect of harmonics on transformers. It is used per IEEE C57.110 to determine transformer derating when supplying non-linear loads. A K-Factor of 1 indicates a pure sinusoidal load; higher values indicate progressively more harmonic heating.

K-Factor

K = \frac{\sum_{n=1}^{n_{\max}} n^2 \cdot I_n^2}{\sum_{n=1}^{n_{\max}} I_n^2}

Where

$n$ Harmonic order
$I_n$ RMS magnitude of the nth harmonic

K = 1 for a pure sinusoid. K-rated transformers are available for K = 4, 9, 13, 20, 30, 40, and 50.

K-Factor Severity

K < 4: normal (standard transformer). K 4-10: moderate harmonic heating (K-4 rated transformer recommended). K > 10: heavy harmonic loading (requires K-13 or higher rated transformer).

Crest Factor

Crest factor is the ratio of peak value to RMS value. For a pure sinusoid, crest factor is exactly $\sqrt{2} \approx 1.414$ . Harmonic distortion and waveform clipping change the crest factor, providing insight into waveform shape without requiring full spectral analysis.

Crest Factor

CF = \frac{|x|_{\text{peak}}}{x_{\text{rms}}}

Where

$|x|_{\text{peak}}$ Maximum absolute value in the analysis window
$x_{\text{rms}}$ RMS value of the signal in the analysis window

Harmonic Ratios (H2, H3, H5)

Detego extracts the 2nd, 3rd, and 5th harmonic ratios as named metrics because these specific orders are the primary indicators used in protection and power quality diagnostics:

H2/H1 (h2Ratio): 2nd harmonic ratio — primary indicator for transformer inrush detection and CT saturation. Threshold typically 15% for inrush blocking.
H3/H1 (h3Ratio): 3rd harmonic ratio — elevated during both inrush and overexcitation. Also common in systems with single-phase non-linear loads (triplen harmonics).
H5/H1 (h5Ratio): 5th harmonic ratio — primary indicator for overexcitation detection. Used in 5th harmonic blocking schemes for transformer differential relays.

Visualizing Harmonic Ratios Over Time

The Harmonic Extraction computed signal in Envelope mode provides an alternative way to visualize how a specific harmonic magnitude changes throughout the recording. For example, extracting the 2nd harmonic in Envelope mode produces a smooth trace of

\sqrt{\mathrm{Re}^2 + \mathrm{Im}^2}

that directly shows H2 magnitude evolution — complementing the H2/H1 ratio shown in the Trend view.

Even Harmonic Ratio

The even harmonic ratio measures the proportion of total harmonic energy carried by even-order harmonics. In normal power systems, even harmonics are negligible because symmetrical waveforms produce only odd harmonics. An elevated even harmonic ratio indicates waveform asymmetry, which is characteristic of CT saturation.

Even Harmonic Ratio

\text{Even Ratio \%} = \frac{\sum_{n \in \text{even}} H_n^2}{\sum_{n=1}^{n_{\max}} H_n^2} \times 100

Where

$n \in \text{even}$ Even harmonic orders: 2, 4, 6, ...
$H_n$ RMS magnitude of the nth harmonic

Used in combination with H2/H1 > 10% to distinguish CT saturation (even ratio > 30%) from transformer inrush (predominantly 2nd harmonic only).

IEEE 519 Compliance

IEEE 519-2014 defines voltage distortion limits at the point of common coupling (PCC). Detego checks individual harmonic magnitudes and total THD against these limits based on the selected voltage class. Non-compliant harmonics are highlighted in the spectrum chart and detail table.

Voltage Class	Individual Harmonic	THD Limit
≤ 69 kV	3.0%	5.0%
69 kV – 161 kV	1.5%	2.5%
> 161 kV	1.0%	1.5%

Voltage Class

The default voltage class is ≤ 69 kV. For transmission-level recordings, the stricter limits at higher voltage classes apply. Currently Detego uses the ≤ 69 kV limits by default.

Harmonic Trend Analysis

The Trend view computes harmonic metrics at each cycle throughout the recording, displaying THD%, H2/H1%, and H5/H1% as time-series traces on the chart. Behind the scenes, K-Factor, Crest Factor, and fundamental magnitude are also computed at each cycle point and available in the underlying trend data. This reveals how harmonic content evolves during fault events, motor starting, transformer energization, and other transient conditions.

Rising H2/H1 with decaying envelope: Characteristic of transformer inrush — the 2nd harmonic ratio starts high and decreases as the inrush current decays over several cycles.
Sudden THD spike at fault inception: CT saturation causes an abrupt increase in THD when fault current exceeds the CT rating. The THD may remain elevated throughout the fault duration.
Steady elevated H5/H1: Consistent 5th harmonic content suggests overexcitation or 6-pulse rectifier loading.

Performance

The trend computation steps one cycle at a time and is capped at 500 data points to keep the UI responsive. For long recordings, the trend is automatically downsampled.

PreviousImpedance Analysis NextDistance Protection

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Harmonic Analysis

One-cycle DFT harmonic decomposition with windowing strategies, THD calculation, and true RMS computation for power quality and protection assessment.

In Detego

View harmonic spectra in the Harmonics tab. This page covers the mathematical theory behind the analysis.

Overview

DFT Computation

One-Cycle DFT

DFT coefficient for harmonic order k

X_k = \frac{2}{N} \sum_{n=0}^{N-1} w[n] \cdot x[n] \cdot e^{-j\,2\pi\,k\,n\,/\,N}

Where

$X_k$ Complex Fourier coefficient for harmonic order k
$N$ Number of samples in one fundamental-frequency cycle
$w[n]$ Window function applied to sample n
$x[n]$ Input signal sample after DC offset removal
$k$ Harmonic order (0 = DC, 1 = fundamental, 2 = 2nd harmonic, ...)

Windowing Strategy

Order	Window	Rationale
0 (DC)	None (simple mean)	DC is computed as the arithmetic mean of the window. No windowing is needed because DC has no frequency to leak.
1 (Fundamental)	Rectangular	The one-cycle window is exactly matched to the fundamental frequency, so the rectangular window provides maximum frequency resolution with zero leakage for the fundamental component.
2-25	Hanning	Higher harmonics are more sensitive to frequency deviation because the window-to-period ratio increases. The Hanning window suppresses side-lobe leakage at the cost of slightly wider main lobes.

Hanning window function

w[n] = 0.5 \left(1 - \cos\!\left(\frac{2\pi n}{N-1}\right)\right)

Where

$w[n]$ Window coefficient for sample n
$N$ Total number of samples in the window

Hanning Amplitude Correction

DC Component (Order 0)

Computation Limits

The harmonic analysis engine enforces several limits to ensure valid results:

Nyquist frequency clipping: The maximum harmonic order is clamped so that the harmonic frequency does not exceed the Nyquist frequency ( $f_s / 2$ ). For example, at a 1600 Hz sample rate with 50 Hz fundamental, the maximum computable harmonic is order 16 (800 Hz). Harmonics above this limit are silently excluded.
Minimum window size: The analysis window must meet minimum sample requirements. If the one-cycle window has too few samples (extremely low sample rates), the DFT returns a zero-magnitude result.
Window duration threshold: The cursor time must be past a minimum window duration threshold from the recording start for a valid analysis window. If the cursor is placed too early in the recording (before enough data for a full cycle), no results are returned.

Derived Quantities

Total Harmonic Distortion (THD)

Total Harmonic Distortion

\text{THD\%} = \frac{\sqrt{\sum_{n=2}^{n_{\max}} H_n^2}}{H_1} \times 100

Where

$H_n$ RMS magnitude of the nth harmonic
$H_1$ RMS magnitude of the fundamental (1st harmonic)
$n_max$ Maximum harmonic order included (default: 25)

THD is undefined when H₁ = 0 (no fundamental component). In this case, Detego reports THD as 0%.

True RMS from Harmonics

True RMS from harmonic components

x_{\text{rms}} = \sqrt{\sum_{n=0}^{n_{\max}} H_n^2}

Where

$H_0$ DC component magnitude
$H_1$ Fundamental RMS magnitude
$H_n$ nth harmonic RMS magnitude (n ≥ 2)

For a pure sinusoid with no harmonics and no DC offset, true RMS equals H₁. Any harmonic content increases the true RMS above the fundamental.

Per-Harmonic Results

For each harmonic order, Detego computes and reports:

Field	Unit	Description
Order	--	Integer harmonic order (0 = DC, 1 = fundamental, 2 = 2nd, ...)
Frequency	Hz	$f_n = n \times f_{\text{fundamental}}$
Magnitude	A or V (RMS)	RMS magnitude of this harmonic component
Peak Magnitude	A or V (peak)	$\sqrt{2} \times$ RMS magnitude
Percentage	%	$(H_n / H_1) \times 100$
% True RMS	%	$(H_n / x_{\text{rms}}) \times 100$ — percentage of the true RMS rather than the fundamental, useful when the fundamental is small or absent
Angle	degrees	Phase angle relative to the start of the DFT window

Harmonic Extraction (Computed Signal)

Two output modes are available:

Waveform (default) — Reconstructs the oscillating harmonic component at the selected frequency: $y[n] = \mathrm{Re}\!\cos(\omega_k t) + \mathrm{Im}\!\sin(\omega_k t)$ . Useful for overlaying the isolated harmonic on the original signal to see where distortion occurs.
Envelope — Outputs the smooth magnitude: $y[n] = \sqrt{\mathrm{Re}^2 + \mathrm{Im}^2}$ . Shows how the harmonic strength varies over time, producing a non-oscillating trace that is ideal for trending and comparison.

See Basic Operations — Harmonic Extraction for the full mathematical treatment.

Engineering Applications

CT Saturation Detection

Current transformer saturation produces distinctive harmonic signatures that can be detected through harmonic analysis:

Elevated even harmonics (2nd, 4th): CT saturation typically affects one half-cycle more than the other (due to residual flux or DC offset), creating waveform asymmetry. This asymmetry generates even harmonics that are normally absent in power system waveforms.
Elevated odd harmonics (3rd, 5th): The clipping effect of saturation flattens the waveform peaks, which generates odd harmonics similar to a square wave.
Waveform asymmetry: Unilateral saturation (only one half-cycle saturated) produces a strong 2nd harmonic component, typically 20-40% of the fundamental.

CT Saturation Indicator

Transformer Inrush Identification

Transformer magnetizing inrush current has a characteristic harmonic profile that differential relays use to distinguish inrush from internal faults:

High 2nd harmonic: Inrush current has a large 2nd harmonic component, typically 30-70% of the fundamental. This is the primary restraint signal used by transformer differential relays (2nd harmonic blocking/restraint).
Moderate 3rd harmonic: The 3rd harmonic is also elevated during inrush, typically 10-30% of the fundamental.
Decreasing higher orders: The harmonic content decreases with increasing order, with the 4th and higher harmonics typically below 10%.

2nd harmonic restraint criterion

\frac{H_2}{H_1} > k_{\text{restraint}} \quad \Rightarrow \quad \text{Block trip (inrush detected)}

Where

$H_2$ 2nd harmonic magnitude
$H_1$ Fundamental magnitude
$k_restraint$ Restraint threshold, typically 15-20%

Transformer Overexcitation

Transformer overexcitation (overvoltage or under-frequency operation that drives the core into saturation) produces a different harmonic profile than inrush:

High odd harmonics (3rd, 5th): Overexcitation produces symmetrical saturation on both half-cycles, generating predominantly odd harmonics. The 3rd harmonic is typically 30-60% of the fundamental, and the 5th harmonic is 10-30%.
Low 2nd harmonic: Unlike inrush, overexcitation has a relatively low 2nd harmonic because the saturation is symmetrical. This means 2nd harmonic restraint alone cannot block overexcitation-driven false differential trips.
5th harmonic blocking: Many modern transformer differential relays include 5th harmonic blocking specifically to prevent tripping during overexcitation conditions.

5th harmonic restraint criterion

\frac{H_5}{H_1} > k_{5} \quad \Rightarrow \quad \text{Block trip (overexcitation detected)}

Where

$H_5$ 5th harmonic magnitude
$H_1$ Fundamental magnitude
$k₅$ 5th harmonic restraint threshold, typically 25-35%

Power Quality Assessment

Standard	Scope	Key Limits
IEEE 519	Current and voltage harmonics at PCC	Voltage THD < 5% (below 69 kV); individual harmonic < 3%
IEC 61000-3-6	Emission limits for MV/HV systems	Planning levels vary by voltage level and harmonic order; uses compatibility levels for assessment

Harmonic Sources in COMTRADE Recordings

Harmonic Signature Summary

The following table summarizes the characteristic harmonic profiles for common power system phenomena:

Phenomenon	Dominant Harmonics	H2/H1 Ratio	Key Indicator
CT saturation	2nd, 4th (even)	15-40%	Waveform asymmetry + elevated even harmonics
Transformer inrush	2nd, 3rd	30-70%	Very high 2nd harmonic with decaying magnitude
Overexcitation	3rd, 5th (odd)	< 10%	High odd harmonics, low 2nd harmonic
6-pulse rectifier	5th, 7th, 11th, 13th	< 5%	Characteristic harmonics at $6k \pm 1$
Internal fault	Low harmonics	< 15%	Clean fundamental with minimal distortion (absent CT saturation)

Extended Metrics

K-Factor

K = \frac{\sum_{n=1}^{n_{\max}} n^2 \cdot I_n^2}{\sum_{n=1}^{n_{\max}} I_n^2}

Where

$n$ Harmonic order
$I_n$ RMS magnitude of the nth harmonic

K = 1 for a pure sinusoid. K-rated transformers are available for K = 4, 9, 13, 20, 30, 40, and 50.

K-Factor Severity

K < 4: normal (standard transformer). K 4-10: moderate harmonic heating (K-4 rated transformer recommended). K > 10: heavy harmonic loading (requires K-13 or higher rated transformer).

Crest Factor

CF = \frac{|x|_{\text{peak}}}{x_{\text{rms}}}

Where

$|x|_{\text{peak}}$ Maximum absolute value in the analysis window
$x_{\text{rms}}$ RMS value of the signal in the analysis window

Harmonic Ratios (H2, H3, H5)

Detego extracts the 2nd, 3rd, and 5th harmonic ratios as named metrics because these specific orders are the primary indicators used in protection and power quality diagnostics:

H2/H1 (h2Ratio): 2nd harmonic ratio — primary indicator for transformer inrush detection and CT saturation. Threshold typically 15% for inrush blocking.
H3/H1 (h3Ratio): 3rd harmonic ratio — elevated during both inrush and overexcitation. Also common in systems with single-phase non-linear loads (triplen harmonics).
H5/H1 (h5Ratio): 5th harmonic ratio — primary indicator for overexcitation detection. Used in 5th harmonic blocking schemes for transformer differential relays.

Visualizing Harmonic Ratios Over Time

\sqrt{\mathrm{Re}^2 + \mathrm{Im}^2}

that directly shows H2 magnitude evolution — complementing the H2/H1 ratio shown in the Trend view.

Even Harmonic Ratio

\text{Even Ratio \%} = \frac{\sum_{n \in \text{even}} H_n^2}{\sum_{n=1}^{n_{\max}} H_n^2} \times 100

Where

$n \in \text{even}$ Even harmonic orders: 2, 4, 6, ...
$H_n$ RMS magnitude of the nth harmonic

Used in combination with H2/H1 > 10% to distinguish CT saturation (even ratio > 30%) from transformer inrush (predominantly 2nd harmonic only).

IEEE 519 Compliance

Voltage Class	Individual Harmonic	THD Limit
≤ 69 kV	3.0%	5.0%
69 kV – 161 kV	1.5%	2.5%
> 161 kV	1.0%	1.5%

Voltage Class

The default voltage class is ≤ 69 kV. For transmission-level recordings, the stricter limits at higher voltage classes apply. Currently Detego uses the ≤ 69 kV limits by default.

Harmonic Trend Analysis

Rising H2/H1 with decaying envelope: Characteristic of transformer inrush — the 2nd harmonic ratio starts high and decreases as the inrush current decays over several cycles.
Sudden THD spike at fault inception: CT saturation causes an abrupt increase in THD when fault current exceeds the CT rating. The THD may remain elevated throughout the fault duration.
Steady elevated H5/H1: Consistent 5th harmonic content suggests overexcitation or 6-pulse rectifier loading.

Performance

The trend computation steps one cycle at a time and is capped at 500 data points to keep the UI responsive. For long recordings, the trend is automatically downsampled.

PreviousImpedance Analysis NextDistance Protection