In order to remove the ambiguity with respect to phase variation when using a component as a phase calibration element, proper alignment is required. This 4-slide-teaser puts things black and white. References are provided for those who are interested to go more in depth.

1. “triple A”:
alignment addresses
ambiguity

2. Ambiguous Statement
“Based on the graph below ...
the phases appear to be stable after at least two hours.”
phase variation comb generator
phase (relative to situation @ 2 hours)
(deg)
4f0
30
3f0
25
2f0
f0
20
15
10
2 hours
5
time (s)
1000 2000 3000 4000 5000 6000 7000
Possible naive conclusion: “One has to wait two hours before
one can use the component as a phase calibration element.”
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3. Back-to-Basics Statements
 “A phase calibration element needs to have a stable phase
relationship between its fundamental f0 and harmonics k.f0.”
 “An arbitrary delay has no impact on the phase relationship
between the fundamental f0 and the harmonics k.f0.”
 “In order to study the phase variation of a component or
signal, one has to remove this arbitrary delay.”
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4. Possible Alignments
original phase variation(*) phase variation(*) after alignment
phase
(deg)
phase using fundamental f0 @ 2 hours
(deg)
30
1
25 4f0 0.75
20 3f0 0.5
2f0 0.25
15
f0 time (s)
10 1000 2000 3000 4000 5000 6000 7000
0.25
5 0.5
0.75
time (s) [A]
1000 2000 3000 4000 5000 6000 7000 1
phase variation(*) after alignment
phase using f0 and harmonics @ 2 hours
(deg)
1
0.75
Observation #1: phase variation is proportional
to harmonic index  almost pure delay ! 0.5
0.25
Observation #2: from 31 degrees max. variation time (s)
1000 2000 3000 4000 5000 6000 7000
0.25
to [A] 1.3 degrees to [B] 0.6 degrees !
0.5
0.75
[B]
1
(*)
relative to signal @ 2 hours
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5. Back-to-Basics Conclusions
 “Alignment reduces the phase variation significantly
whenever a delay is involved.”
 “The remaining phase variation depends on the alignment
method.”
 “In the above example, alignment based on the fundamental
reduces the maximum phase variation by 95%.”
 “Taking all spectral components into account reduces the
overall phase variation even further.”
 “One has to wait much less than 2 hours before one can use
the component as a phase calibration element.”
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6. References
Frans Verbeyst, Rik Pintelon, Yves Rolain, Johan Schoukens and Tracy S.
Clement, “System Identification Approach Applied to Drift Estimation”,
Instrumentation and Measurement Technology Conference, IMTC 2007,
Warsaw, Poland, May 1-3, 2007.
http://www.nmdg.be/library/contributions/IM-7123.pdf
Frans Verbeyst, “Contribution to Large-Signal Network Analysis”,
PhD dissertation, September 2006.
http://www.nmdg.be/index_jump.html?goto=library_center
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