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Technology / Technical guide
Feedback Delay: Phase Lag and Stability Boundaries
A delay can leave a signal’s sinusoidal amplitude unchanged while shifting its phase. Inside feedback, that shift changes the timing between a correction and the observed response. Its effect depends on frequency and the rest of the loop, so delay duration cannot be judged independently of dynamics.
The same time delay has different phase effects.
Follow a delayed waveform into its frequency-domain phase, then retain the conditions behind any feedback interpretation.
01 / Time shift
A pure delay retains the waveform shape.
An ideal delay maps x(t) to x(t−τ) for positive duration τ.
An ideal delay maps x(t) to x(t−τ) for positive duration τ. It postpones the represented signal without smoothing or changing its shape. An observation can be physically older even when its displayed amplitude equals the value originally observed.
This operation differs from a filter that combines several input instants. The filter-delay guide addresses that transformation; here the question is the feedback effect of a declared time shift.
02 / Frequency effect
The delay factor has unity magnitude.
The Laplace factor is exp(−sτ), as represented in MathWorks time-delay documentation.
The Laplace factor is exp(−sτ), as represented in MathWorks time-delay documentation. At s = jω, its magnitude is one and its phase is −ωτ radians. In ordinary frequency f measured in hertz, the phase is −360fτ degrees.
The negative sign follows the delayed convention x(t−τ). Angular frequency in radians per second and frequency in cycles per second need the factor 2π between them. A duration expressed as a sample count also needs the corresponding sampling interval.
03 / Phase example
A small duration can have a larger high-frequency effect.
Hypothetical pure delay: τ = 0.05 seconds produces −36° at 2 Hz and −72° at 4 Hz.
Hypothetical pure delay: τ = 0.05 seconds produces −36° at 2 Hz and −72° at 4 Hz. Neither result changes the ideal sinusoid’s amplitude. The duration and frequencies are invented model inputs.
A phase shift is therefore not a fixed angular offset for every component. The same delay acting on a waveform with several frequencies shifts each component differently according to frequency, although their common time-domain translation preserves the original shape.
04 / Loop consequence
Magnitude preservation does not preserve feedback behaviour.
Adding this factor to a return ratio leaves its magnitude curve unchanged but rotates its frequency-dependent phase.
Adding this factor to a return ratio leaves its magnitude curve unchanged but rotates its frequency-dependent phase. That can bring a unity-magnitude crossing closer to the critical −1 condition, depending on the nominal loop and other crossings.
For an already-assessed stable loop with one relevant crossover, dividing a phase budget in radians by crossover angular frequency gives a model-based delay comparison. This reasoning requires its stability, crossing and fixed-delay assumptions; it is not a universal allowable network latency.
05 / Changing latency
A variable delay is not one fixed factor.
Transmission or computation can create latency that changes with time.
Transmission or computation can create latency that changes with time. Replacing every such observation with one mean duration discards its variation. A linear time-invariant pure-delay model does not automatically describe the resulting time-varying information path.
Held commands and computation order can add further effects. The hold guide separates a piecewise-constant output from an explicit delayed sample. Combining them under one unexplained delay label hides distinct mechanisms.
06 / Model scope
A phase calculation is not a tuning instruction.
A delay model does not include every saturation, actuator constraint, missing observation or change of operating condition.
A delay model does not include every saturation, actuator constraint, missing observation or change of operating condition. Nor does it guarantee that all endpoints represent the same physical time. The complete loop remains the relevant boundary.
No MARSTON latency, crossover or stability allowance is specified. This explanation supports the margin interpretation without recommending gain changes, logging alterations or live tests.
Related technical guides: Control Gain & Phase Margins: Conditions and Meaning · Discrete Control & Zero-Order Hold: Between Samples · Cross-Correlation: Lag Sign, Similarity & Time Delay.
General background: MathWorks: Input, output and internal time-delay representations. These references explain industry concepts; they are separate from the applicable MARSTON product and project documents.
Key concepts and conditions.
Read each quantity with its reference, units and stated conditions. A general relation does not establish a model-specific value.
| Concept / quantity | Meaning / relation | Conditions / limits |
|---|---|---|
| τ | Positive delay duration, seconds | x(t−τ) fixes the sign. |
| ω | Angular frequency, rad/s | ω = 2πf for f in Hz. |
| exp(−jωτ) | Ideal delay response | Unity magnitude with frequency-dependent phase. |
| −360fτ | Phase in degrees | Uses seconds and cycles per second. |
| Delay comparison | Phase budget divided by crossover ω | Requires the stated nominal-loop assumptions. |
Engineering questions.
Does pure delay attenuate a sine?
Not in the ideal model; it changes phase.
Is its phase shift identical at every frequency?
No. It is proportional to frequency.
Does unchanged loop magnitude preserve stability?
Not necessarily; phase also enters the feedback condition.
Can a mean latency describe every variable delay?
No. Variation makes the information path a different problem.
Is a delay budget automatically a safe operating limit?
No. It belongs to a stated model and assumptions.
Are 0.05 seconds actual equipment latency?
No. The value is hypothetical.
Technical enquiry
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