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Basic Operations

Reverse polarity, absolute value, multiply by constant, subtraction, addition, residual 3I₀, harmonic extraction, and envelope signals.

In Detego

Create this signal via the Compute Tab. This page covers the mathematical theory behind the computation.

Overview

Basic operations create new signals by combining or transforming one or more source channels using simple arithmetic. Despite their simplicity, these derived signals are essential tools in protection engineering — residual current reveals ground faults, harmonic extraction isolates specific frequency components for CT saturation analysis, envelope signals track how RMS or peak magnitude evolves over time, and arithmetic operations like addition and scaling enable flexible signal manipulation for KCL verification and CT ratio normalization.

Related Theory Pages

Differential current and three-winding differential are covered in Differential Protection. Superimposed delta quantities are covered in Protection Quality.

Arithmetic

1. Reverse Polarity

Reverse polarity

y[n] = -x[n]

Where

$x[n]$ Source channel samples
$y[n]$ Output signal (negated)

Negates every sample of the source channel. This is used for CT polarity correction when a current transformer is wired in reverse, or for phase reversal when the recording convention does not match the protection relay convention. In protection schemes, correct CT polarity is critical -- reversed polarity causes directional elements to see forward faults as reverse and vice versa, potentially blocking tripping for legitimate faults.

2. Absolute Value

Absolute value

y[n] = |x[n]|

Where

$x[n]$ Source channel samples
$y[n]$ Output signal (full-wave rectified)

Takes the absolute value of every sample, equivalent to full-wave rectification. The output preserves the magnitude of the signal without polarity information. Useful for visualizing the magnitude envelope of an AC waveform, comparing peak values regardless of polarity, or as a preprocessing step before further analysis.

3. Multiply by Constant

Multiply by constant

y[n] = k \cdot x[n]

Where

$x[n]$ Source channel samples
$k$ Scaling constant (user-defined)
$y[n]$ Output signal (scaled)

Scales every sample by a constant factor. Common uses include CT ratio normalization (e.g., multiplying secondary-side current by the CT ratio to obtain primary values), unit conversion (e.g., kV to V by multiplying by 1000), or applying a known gain/attenuation factor. A factor of −1 is equivalent to reverse polarity.

4. Subtraction

Channel subtraction

y[n] = x_1[n] - x_2[n]

Where

$x_1[n]$ First source channel
$x_2[n]$ Second source channel

General-purpose sample-by-sample subtraction of two channels. Common uses include computing line-to-line voltages from phase-to-ground measurements (e.g., $V_{ab} = V_a - V_b$ ), deriving the voltage across a series element, or comparing two measurements of the same quantity from different sources to identify discrepancies.

5. Addition (N Channels)

N-channel addition

y[n] = \sum_{i=1}^{N} x_i[n]

Where

$x_i[n]$ Source channel samples (2 to 8 channels)
$y[n]$ Output signal (sample-by-sample sum)

Supports 2 to 8 input channels. For exactly 3 current inputs, consider using 3I₀ Residual instead.

Sample-by-sample sum of N channels. Common uses include summing feeder currents at a busbar for KCL verification (the sum should equal the incoming current), combining partial measurements into a total quantity, or aggregating multiple parallel circuit contributions. Unlike 3I₀ Residual which is specifically designed for three-phase current summing, addition works with any number of channels and any signal type.

Derived

6. Residual Current (3I₀)

Residual (zero-sequence) current

3I_0 = I_a + I_b + I_c

Where

$I_a, I_b, I_c$ Three-phase current samples
$3I_0$ Residual current (three times zero-sequence current)

For a balanced three-phase system, the sum is zero. Any non-zero residual indicates ground fault current flowing through the neutral/earth path.

The sum of three-phase currents. Per Kirchhoff's current law, in a balanced system with no ground path, the three phase currents sum to zero. A non-zero sum indicates that current is flowing through the ground (earth) path, which is the hallmark of a ground fault. The residual current equals three times the zero-sequence component ( $3I_0$ ), making it a direct indicator of ground fault severity. Earth fault relays (ANSI 50N/51N) operate on this quantity. The Zero Sequence computed signal ( $I_0 = 3I_0 \,/\, 3$ ) uses the same time-domain sum divided by three.

7. Harmonic Extraction

Harmonic extraction via sliding DFT

y[n] = |H_k[n]| \cdot \cos\!\bigl(2\pi k f_0 t[n] + \angle H_k[n]\bigr)

Where

$H_k[n]$ Sliding DFT coefficient at the kth harmonic
$k$ Harmonic number (user-defined, e.g., 2 for H2)
$f_0$ Fundamental system frequency

Default output is a reconstructed waveform. Switch to Envelope mode for the smooth magnitude trace.

Extracts the Nth harmonic component from a channel using a sliding one-cycle DFT. Two output modes are available:

Waveform (default) — Reconstructs the oscillation at the harmonic frequency: $y[n] = \mathrm{Re}\!\cos(\omega t) + \mathrm{Im}\!\sin(\omega t)$ . Useful for overlaying the isolated harmonic on the original signal.
Envelope — Outputs the smooth magnitude: $y[n] = \sqrt{\mathrm{Re}^2 + \mathrm{Im}^2}$ . Shows how strong the harmonic is over time, ideal for trending H2/H1 ratios and identifying CT saturation windows.

Key use cases:

H2 (2nd harmonic) — Indicates CT saturation. When a CT saturates, the clipped waveform generates strong 2nd-harmonic content; monitoring H2/H1 ratio helps identify unreliable current measurements.
H3 (3rd harmonic) — Associated with transformer magnetizing current and ferroresonance. Third-harmonic blocking is used in some differential relay schemes.
H5 (5th harmonic) — Generated by non-linear loads (e.g., variable-frequency drives, power electronic converters). Monitoring H5 helps assess power quality and filter sizing.

8. Envelope (RMS)

RMS envelope (sliding one-cycle window)

x_{\mathrm{rms}}[n] = \sqrt{\frac{1}{N}\sum_{i=0}^{N-1} x[n-i]^{2}}

Where

$x[n]$ Source channel samples
$N$ Number of samples in one power-frequency cycle
$x_{\mathrm{rms}}[n]$ Instantaneous RMS value at sample n

The window length N equals the sample rate divided by the system frequency (e.g., 64 samples at 3200 Hz / 50 Hz).

Computes the true RMS value over a sliding one-cycle window, producing a smooth envelope that tracks how the effective magnitude of a signal changes over time. This is the same quantity that RMS-measuring instruments report, updated every sample rather than once per cycle. It responds to all frequency content in the window (fundamental plus harmonics plus noise), making it a faithful measure of total signal energy. Common uses:

Tracking voltage sag/swell depth and duration during fault events.
Monitoring current magnitude through a fault to identify the instant of breaker clearing versus arc extinguishing.
Providing a smooth reference trace when the raw oscillating waveform is hard to read visually.

9. Envelope (Peak)

Peak envelope (sliding one-cycle window)

x_{\mathrm{peak}}[n] = \max_{0 \le i < N} |x[n-i]|

Where

$x[n]$ Source channel samples
$N$ Number of samples in one power-frequency cycle
$x_{\mathrm{peak}}[n]$ Peak absolute value over the trailing cycle

For a pure sinusoid the peak envelope equals the crest value and is √2 times the RMS envelope.

Reports the maximum absolute sample value within a sliding one-cycle window. Unlike the RMS envelope which averages energy, the peak envelope captures the worst-case instantaneous magnitude. This is particularly relevant for:

Insulation stress assessment — Peak voltage determines dielectric stress, so the peak envelope directly tracks the quantity that matters for equipment withstand.
CT saturation detection — A peak current envelope that drops below the expected crest value can reveal cycles where the CT is saturating and clipping the waveform.
Transient identification — Switching surges and travelling waves produce short-lived peaks that the RMS envelope smooths away; the peak envelope preserves them.

PreviousEnvelope Analysis NextRecording Merge & Alignment

Loading documentation...

Basic Operations

Reverse polarity, absolute value, multiply by constant, subtraction, addition, residual 3I₀, harmonic extraction, and envelope signals.

In Detego

Create this signal via the Compute Tab. This page covers the mathematical theory behind the computation.

Overview

Related Theory Pages

Differential current and three-winding differential are covered in Differential Protection. Superimposed delta quantities are covered in Protection Quality.

Arithmetic

1. Reverse Polarity

Reverse polarity

y[n] = -x[n]

Where

$x[n]$ Source channel samples
$y[n]$ Output signal (negated)

2. Absolute Value

Absolute value

y[n] = |x[n]|

Where

$x[n]$ Source channel samples
$y[n]$ Output signal (full-wave rectified)

3. Multiply by Constant

Multiply by constant

y[n] = k \cdot x[n]

Where

$x[n]$ Source channel samples
$k$ Scaling constant (user-defined)
$y[n]$ Output signal (scaled)

4. Subtraction

Channel subtraction

y[n] = x_1[n] - x_2[n]

Where

$x_1[n]$ First source channel
$x_2[n]$ Second source channel

5. Addition (N Channels)

N-channel addition

y[n] = \sum_{i=1}^{N} x_i[n]

Where

$x_i[n]$ Source channel samples (2 to 8 channels)
$y[n]$ Output signal (sample-by-sample sum)

Supports 2 to 8 input channels. For exactly 3 current inputs, consider using 3I₀ Residual instead.

Derived

6. Residual Current (3I₀)

Residual (zero-sequence) current

3I_0 = I_a + I_b + I_c

Where

$I_a, I_b, I_c$ Three-phase current samples
$3I_0$ Residual current (three times zero-sequence current)

For a balanced three-phase system, the sum is zero. Any non-zero residual indicates ground fault current flowing through the neutral/earth path.

7. Harmonic Extraction

Harmonic extraction via sliding DFT

y[n] = |H_k[n]| \cdot \cos\!\bigl(2\pi k f_0 t[n] + \angle H_k[n]\bigr)

Where

$H_k[n]$ Sliding DFT coefficient at the kth harmonic
$k$ Harmonic number (user-defined, e.g., 2 for H2)
$f_0$ Fundamental system frequency

Default output is a reconstructed waveform. Switch to Envelope mode for the smooth magnitude trace.

Extracts the Nth harmonic component from a channel using a sliding one-cycle DFT. Two output modes are available:

Waveform (default) — Reconstructs the oscillation at the harmonic frequency: $y[n] = \mathrm{Re}\!\cos(\omega t) + \mathrm{Im}\!\sin(\omega t)$ . Useful for overlaying the isolated harmonic on the original signal.
Envelope — Outputs the smooth magnitude: $y[n] = \sqrt{\mathrm{Re}^2 + \mathrm{Im}^2}$ . Shows how strong the harmonic is over time, ideal for trending H2/H1 ratios and identifying CT saturation windows.

Key use cases:

H2 (2nd harmonic) — Indicates CT saturation. When a CT saturates, the clipped waveform generates strong 2nd-harmonic content; monitoring H2/H1 ratio helps identify unreliable current measurements.
H3 (3rd harmonic) — Associated with transformer magnetizing current and ferroresonance. Third-harmonic blocking is used in some differential relay schemes.
H5 (5th harmonic) — Generated by non-linear loads (e.g., variable-frequency drives, power electronic converters). Monitoring H5 helps assess power quality and filter sizing.

8. Envelope (RMS)

RMS envelope (sliding one-cycle window)

x_{\mathrm{rms}}[n] = \sqrt{\frac{1}{N}\sum_{i=0}^{N-1} x[n-i]^{2}}

Where

$x[n]$ Source channel samples
$N$ Number of samples in one power-frequency cycle
$x_{\mathrm{rms}}[n]$ Instantaneous RMS value at sample n

The window length N equals the sample rate divided by the system frequency (e.g., 64 samples at 3200 Hz / 50 Hz).

Tracking voltage sag/swell depth and duration during fault events.
Monitoring current magnitude through a fault to identify the instant of breaker clearing versus arc extinguishing.
Providing a smooth reference trace when the raw oscillating waveform is hard to read visually.

9. Envelope (Peak)

Peak envelope (sliding one-cycle window)

x_{\mathrm{peak}}[n] = \max_{0 \le i < N} |x[n-i]|

Where

$x[n]$ Source channel samples
$N$ Number of samples in one power-frequency cycle
$x_{\mathrm{peak}}[n]$ Peak absolute value over the trailing cycle

For a pure sinusoid the peak envelope equals the crest value and is √2 times the RMS envelope.

Insulation stress assessment — Peak voltage determines dielectric stress, so the peak envelope directly tracks the quantity that matters for equipment withstand.
CT saturation detection — A peak current envelope that drops below the expected crest value can reveal cycles where the CT is saturating and clipping the waveform.
Transient identification — Switching surges and travelling waves produce short-lived peaks that the RMS envelope smooths away; the peak envelope preserves them.

PreviousEnvelope Analysis NextRecording Merge & Alignment