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Phasor Extraction

Fundamental-frequency DFT with DC offset removal for magnitude and phase angle extraction.

In Detego

View phasor diagrams in the Phasors tab. This page covers the mathematical theory behind the extraction.

Overview

A phasor is a complex number that represents the magnitude and phase angle of a sinusoidal waveform at the fundamental frequency. Detego extracts phasors using a one-cycle Discrete Fourier Transform (DFT) with explicit DC offset removal, producing accurate magnitude and angle values even in the presence of decaying DC transients.

Phasors are the foundation for nearly all advanced analysis: impedance calculation, directional determination, sequence component decomposition, and power measurement. Every protection relay internally computes phasors from its sampled inputs before making trip decisions.

Three-Phase Sinusoidal Waveforms

The diagram below shows balanced three-phase voltages separated by 120 degrees. Each waveform is a time-domain representation of a rotating phasor at the system frequency.

Three-phase balanced sinusoidal voltages (120-degree separation). Each phasor rotates at the system frequency, producing time-domain waveforms with one period T = 1/f.

DC Offset Removal

DC component (mean of window)

X_{\text{DC}} = \frac{1}{N} \sum_{k=0}^{N-1} x[k]

Where

$x[k]$ Sample values within the one-cycle window
$N$ Number of samples per cycle

The DC mean is subtracted from each sample before the DFT to prevent the decaying DC transient from corrupting the fundamental-frequency phasor.

DC offset removal is critical during faults. When a fault occurs, the sudden change in current produces a decaying exponential DC component whose magnitude depends on the point-on-wave at fault inception and the system X/R ratio. Without removal, this DC component leaks into the fundamental-frequency DFT bin, distorting both the magnitude and angle of the extracted phasor.

One-Cycle DFT Extraction

Real Component

Cosine correlation (real part)

X_{\text{real}} = \frac{2}{N} \sum_{k=0}^{N-1} \bigl(x[k] - X_{\text{DC}}\bigr) \cos\!\left(2\pi f\, t[k]\right)

Where

$f$ Fundamental frequency (50 or 60 Hz)
$t[k]$ Time of sample k in seconds
$X_DC$ DC offset (mean) computed above

Imaginary Component

Sine correlation (imaginary part)

X_{\text{imag}} = \frac{2}{N} \sum_{k=0}^{N-1} \bigl(x[k] - X_{\text{DC}}\bigr) \sin\!\left(2\pi f\, t[k]\right)

Where

$f$ Fundamental frequency (50 or 60 Hz)
$t[k]$ Time of sample k in seconds

RMS Magnitude

Phasor magnitude (RMS)

|X| = \frac{\sqrt{X_{\text{real}}^2 + X_{\text{imag}}^2}}{\sqrt{2}}

The factor of 1/sqrt(2) converts from peak to RMS. The DFT correlation yields peak amplitude; dividing by sqrt(2) gives the standard RMS phasor magnitude used in protection engineering.

Phase Angle

Phasor angle

\theta = \text{atan2}\!\left(-X_{\text{imag}},\; X_{\text{real}}\right) \times \frac{180}{\pi}

Where

$theta$ Phase angle in degrees, normalized to the range (-180, 180]

Convention: a pure cosine at t=0 has angle 0 degrees. The negative sign on the imaginary component follows the standard electrical engineering convention where lagging quantities have negative angles.

Angle Convention

Detego uses the standard protection engineering convention:

A pure cosine waveform at $t = 0$ has angle = 0 degrees.
Angles are normalized to the half-open interval $(-180\degree,\; 180\degree]$ .
Positive angles indicate leading quantities; negative angles indicate lagging quantities.
Phase relationships (e.g., voltage leading current) are computed as the difference between their respective angles.

Tip

When comparing phasor angles between phases, the absolute angle values depend on the reference time. The relative angles between phases (e.g., Va leading Vb by 120 degrees) are independent of the reference and are what matter for protection analysis.

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Phasor Extraction

Fundamental-frequency DFT with DC offset removal for magnitude and phase angle extraction.

In Detego

View phasor diagrams in the Phasors tab. This page covers the mathematical theory behind the extraction.

Overview

Three-Phase Sinusoidal Waveforms

The diagram below shows balanced three-phase voltages separated by 120 degrees. Each waveform is a time-domain representation of a rotating phasor at the system frequency.

Three-phase balanced sinusoidal voltages (120-degree separation). Each phasor rotates at the system frequency, producing time-domain waveforms with one period T = 1/f.

DC Offset Removal

DC component (mean of window)

X_{\text{DC}} = \frac{1}{N} \sum_{k=0}^{N-1} x[k]

Where

$x[k]$ Sample values within the one-cycle window
$N$ Number of samples per cycle

The DC mean is subtracted from each sample before the DFT to prevent the decaying DC transient from corrupting the fundamental-frequency phasor.

One-Cycle DFT Extraction

Real Component

Cosine correlation (real part)

X_{\text{real}} = \frac{2}{N} \sum_{k=0}^{N-1} \bigl(x[k] - X_{\text{DC}}\bigr) \cos\!\left(2\pi f\, t[k]\right)

Where

$f$ Fundamental frequency (50 or 60 Hz)
$t[k]$ Time of sample k in seconds
$X_DC$ DC offset (mean) computed above

Imaginary Component

Sine correlation (imaginary part)

X_{\text{imag}} = \frac{2}{N} \sum_{k=0}^{N-1} \bigl(x[k] - X_{\text{DC}}\bigr) \sin\!\left(2\pi f\, t[k]\right)

Where

$f$ Fundamental frequency (50 or 60 Hz)
$t[k]$ Time of sample k in seconds

RMS Magnitude

Phasor magnitude (RMS)

|X| = \frac{\sqrt{X_{\text{real}}^2 + X_{\text{imag}}^2}}{\sqrt{2}}

The factor of 1/sqrt(2) converts from peak to RMS. The DFT correlation yields peak amplitude; dividing by sqrt(2) gives the standard RMS phasor magnitude used in protection engineering.

Phase Angle

Phasor angle

\theta = \text{atan2}\!\left(-X_{\text{imag}},\; X_{\text{real}}\right) \times \frac{180}{\pi}

Where

$theta$ Phase angle in degrees, normalized to the range (-180, 180]

Angle Convention

Detego uses the standard protection engineering convention:

A pure cosine waveform at $t = 0$ has angle = 0 degrees.
Angles are normalized to the half-open interval $(-180\degree,\; 180\degree]$ .
Positive angles indicate leading quantities; negative angles indicate lagging quantities.
Phase relationships (e.g., voltage leading current) are computed as the difference between their respective angles.

Tip

PreviousRMS Calculation NextFrequency Tracking