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Envelope Analysis

RMS and peak envelope signals for tracking magnitude changes, fault inception, and DC offset effects through a COMTRADE recording.

In Detego

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

Overview

Envelope analysis extracts the slowly-varying magnitude profile of an AC waveform. While the instantaneous samples oscillate at the power frequency (50 or 60 Hz), the envelope traces the upper and lower bounds of the waveform over time. This reveals magnitude changes -- such as the step increase in current at fault inception or the exponential decay of DC offset -- that are difficult to see in the raw waveform.

Detego provides two envelope types: the RMS envelope, which gives a true power-equivalent magnitude, and the peak envelope, which tracks instantaneous peaks and is more sensitive to asymmetrical waveform features.

Instantaneous waveform with RMS envelope overlay. The RMS value is computed using a sliding one-cycle window. At fault inception, the RMS steps up reflecting the increased fault current magnitude.

Computed Signals

1. RMS Envelope

The RMS envelope computes the root-mean-square value over a sliding one-cycle window. This is the same RMS calculation used by the standalone RMS signal, but visualized as a continuous envelope overlaid on the waveform. The RMS value at each time point represents the equivalent DC value that would deliver the same power to a resistive load.

Sliding one-cycle RMS

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

Where

$x_rms[n]$ RMS value at sample n
$N$ Number of samples per cycle (= sample rate / frequency)
$x[n-i]$ Instantaneous sample value, i samples before n

The window slides one sample at a time, producing an RMS value at every sample. The first N-1 samples have incomplete windows and may show transient values.

For a pure sinusoidal waveform with peak amplitude $A$ , the RMS value is $A / \sqrt{2} \approx 0.707 A$ . The RMS envelope therefore sits at 70.7% of the peak value for undistorted waveforms. During a fault, the RMS envelope shows a step change corresponding to the increase (or decrease, for voltage) in magnitude.

Settling Time

After a step change in magnitude (e.g., at fault inception), the RMS envelope takes one full cycle to settle to its new value. During this transition period, the RMS value ramps between the pre-fault and post-fault levels. This one-cycle settling time is inherent to any sliding-window RMS calculation and matches the behavior of numerical relay measurement algorithms.

2. Peak Envelope (Adjusted)

The peak envelope tracks the absolute maximum instantaneous value within a sliding one-cycle window. Unlike the RMS envelope, which averages over the full cycle, the peak envelope captures the single highest (or lowest) sample in each window. This makes it more sensitive to asymmetrical peaks caused by DC offset, harmonic distortion, or CT saturation.

Peak envelope (one-cycle sliding window)

x_{\text{peak}}[n] = \max_{i=0}^{N-1} |x[n - i]|

Where

$x_peak[n]$ Peak envelope value at sample n
$N$ Number of samples per cycle
$|x[n-i]|$ Absolute value of the sample i positions before n

For a pure sinusoidal waveform, the peak envelope equals the crest value $A$ , which is $\sqrt{2}$ times the RMS value. The ratio between the peak and RMS envelopes is the crest factor:

Crest factor for sinusoidal waveforms

\text{Crest Factor} = \frac{x_{\text{peak}}}{x_{\text{rms}}} = \sqrt{2} \approx 1.414

A crest factor significantly greater than √2 indicates waveform asymmetry from DC offset, and a crest factor less than √2 may indicate clipping from CT saturation.

DC Offset Detection

The gap between the peak and RMS envelopes is a powerful indicator of DC offset. In a symmetrical waveform, the peak envelope is exactly

\sqrt{2}

times the RMS envelope. When DC offset is present, the positive and negative peaks become asymmetrical -- one side is larger, the other smaller. The peak envelope captures the larger side, making it visibly higher than the

\sqrt{2}

ratio would predict, while the RMS envelope remains relatively stable because the squared averaging treats positive and negative values equally.

Engineering Context

Identifying Fault Inception

Both envelope types provide a clear indication of fault inception. Before the fault, the envelope is at the steady-state load level. At fault inception, the current envelope steps up (fault current is higher than load current) and the voltage envelope steps down (voltage sag at the fault location). The sharpness of this step depends on the fault severity and the electrical distance from the measurement point to the fault.

The RMS envelope is preferred for identifying fault inception time because it provides a single, unambiguous step. The peak envelope may show a more gradual transition if the DC offset builds up over the first cycle after fault inception.

DC Offset and Asymmetry

When a fault occurs, the current waveform typically contains a decaying DC component whose initial magnitude depends on the point-on-wave at the instant of fault inception. The DC offset makes the waveform asymmetrical: one half-cycle has larger peaks than the other. This asymmetry is visible as a divergence between the positive and negative peak envelopes.

The peak envelope is particularly useful for assessing the severity of DC offset because it tracks the absolute maximum in each cycle. A high peak-to-RMS ratio (crest factor significantly above $\sqrt{2}$ ) indicates substantial DC offset, which has implications for CT performance, circuit breaker duty, and relay measurement accuracy.

CT Saturation Effects

Current transformers (CTs) can saturate when the primary current contains a large DC component or exceeds the CT's rated burden-times-accuracy limit. Saturation causes the secondary waveform to flatten on one or both half-cycles, reducing the measured magnitude below the true primary current.

RMS envelope during CT saturation: drops below the true value because the clipped waveform has less energy per cycle. The RMS will recover partially during the unsaturated portion of each cycle.
Peak envelope during CT saturation: may appear relatively normal during the unsaturated half-cycle but drops dramatically during the saturated half-cycle. This creates an oscillating peak envelope that is a distinctive signature of CT saturation.

CT Saturation and Protection

CT saturation can cause relay underreach (measuring less current than actually flows) and slow operation. When the envelope analysis shows signs of CT saturation (oscillating peak envelope, depressed RMS), all downstream calculations -- phasors, impedance, harmonics -- should be interpreted with caution. The Protection Quality category in the Compute tab includes a dedicated CT saturation detector.

Envelope Comparison Summary

Property	RMS Envelope	Peak Envelope
Calculation	$\sqrt{(1/N)\sum x_i^2}$	$\max \|x_i\|$ over one cycle
Sinusoidal ratio	$A / \sqrt{2}$	$A$ (peak)
DC offset sensitivity	Low (averaged out)	High (tracks peaks)
CT saturation detection	Shows magnitude depression	Shows oscillating pattern
Best for	Overall magnitude tracking, fault inception timing	Instantaneous peak assessment, asymmetry detection

PreviousFrequency Tracking NextSignal Operations

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Envelope Analysis

RMS and peak envelope signals for tracking magnitude changes, fault inception, and DC offset effects through a COMTRADE recording.

In Detego

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

Overview

Instantaneous waveform with RMS envelope overlay. The RMS value is computed using a sliding one-cycle window. At fault inception, the RMS steps up reflecting the increased fault current magnitude.

Computed Signals

1. RMS Envelope

Sliding one-cycle RMS

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

Where

$x_rms[n]$ RMS value at sample n
$N$ Number of samples per cycle (= sample rate / frequency)
$x[n-i]$ Instantaneous sample value, i samples before n

The window slides one sample at a time, producing an RMS value at every sample. The first N-1 samples have incomplete windows and may show transient values.

Settling Time

2. Peak Envelope (Adjusted)

Peak envelope (one-cycle sliding window)

x_{\text{peak}}[n] = \max_{i=0}^{N-1} |x[n - i]|

Where

$x_peak[n]$ Peak envelope value at sample n
$N$ Number of samples per cycle
$|x[n-i]|$ Absolute value of the sample i positions before n

For a pure sinusoidal waveform, the peak envelope equals the crest value $A$ , which is $\sqrt{2}$ times the RMS value. The ratio between the peak and RMS envelopes is the crest factor:

Crest factor for sinusoidal waveforms

\text{Crest Factor} = \frac{x_{\text{peak}}}{x_{\text{rms}}} = \sqrt{2} \approx 1.414

A crest factor significantly greater than √2 indicates waveform asymmetry from DC offset, and a crest factor less than √2 may indicate clipping from CT saturation.

DC Offset Detection

The gap between the peak and RMS envelopes is a powerful indicator of DC offset. In a symmetrical waveform, the peak envelope is exactly

\sqrt{2}

\sqrt{2}

ratio would predict, while the RMS envelope remains relatively stable because the squared averaging treats positive and negative values equally.

Engineering Context

Identifying Fault Inception

DC Offset and Asymmetry

CT Saturation Effects

RMS envelope during CT saturation: drops below the true value because the clipped waveform has less energy per cycle. The RMS will recover partially during the unsaturated portion of each cycle.
Peak envelope during CT saturation: may appear relatively normal during the unsaturated half-cycle but drops dramatically during the saturated half-cycle. This creates an oscillating peak envelope that is a distinctive signature of CT saturation.

CT Saturation and Protection

Envelope Comparison Summary

Property	RMS Envelope	Peak Envelope
Calculation	$\sqrt{(1/N)\sum x_i^2}$	$\max \|x_i\|$ over one cycle
Sinusoidal ratio	$A / \sqrt{2}$	$A$ (peak)
DC offset sensitivity	Low (averaged out)	High (tracks peaks)
CT saturation detection	Shows magnitude depression	Shows oscillating pattern
Best for	Overall magnitude tracking, fault inception timing	Instantaneous peak assessment, asymmetry detection

PreviousFrequency Tracking NextSignal Operations