Transcript Lecture 4. Dynamic Response to Sinusoidal Input

Announcements
• Will return Temp labs when they are all done.
• 9-noon March 6, 2015
field trip to NIST for pressure laboratory
World’s best manometer.
• Friday we will do the RH lab. Think about how to
calibrate the Davis RH probe; the Vaisala T & RH
probes are on the Cessna for now.
• You should be working or the precip lab – esp the
part with variability with height and distance.
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AOSC 634
Air Sampling and Analysis
Lecture 4
Measurement Theory
Dynamic Performance of Sensor Systems
Response of a first order instrument to
Sinusoidal input
See Brock et al. Chapter 2.
Copyright Brock et al. 1984; Dickerson 2015
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Goal
Geophysical fluid phenomena often occur in waves.
• Weather depends on the flux of heat, momentum, and
moisture from or into the surface.
• The soil flux of trace gases such as CO2, CH4, NO, N2O, is
critical to understanding air quality and climate.
• What is required to measure those fluxes?
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Objective
We need to analyze geophysical wave processes. In a useful
instrument, the steady state response to sinusoidal input should
be predictable. The amplitude will be diminished and the phase
shifted, but we want all spectral components to have the same
relative amp. and phase angle as the input. We are looking for:
• A flat amplitude response (independent of frequency).
• A linear phase response (change in phase proportional to
input freq. and lag independent of frequency).
• A Fourier transform of the output should return the input freq
distribution.
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Dynamic Response
Sensor output in response to changing input.
Sinusoidal input
X I (t) = X 0 + Asin(w I t)
Where A = amplitude
w I = Input angular frequency (s-1 )
The controlling first-order diff equation is
dX
+ wb X = wb XI
dt
where w b = 1 / t , the angular freq instrument resp.
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Dynamic Response
The input, in normalized coordinates (t' = t , and A(amp) = 1), is:
X ' I (t ') =1+ sin(a t ') for all t > 0.
Where a = w I / w b the dimensionless frequency number.
The response function is:
X '(t ') = A(a ){sin [a t '- f (a )] + eat ' sin [f (a )]}
-1/2
A(a ) = éë1+ a 2 ùû
the normalized output amp function
and f = tan -1 (a ) the phase angle function.
The steady state portion is underlined in red; the rest is transient.
Further details can be found in Chapter 2 of Brock et al. and
Doebelin (1990).
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Response to input with wI ≈ wb
Normalized input and output
Phase shift
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Dynamic Response
After a few time constants, the transient part goes to zero
and the response is characterized by:
Output angular frequency A(a)
Phase shift f(a) in radians.
• When the response (output) angular freq >> input ang
freq the amplitude is nearly the same and there is no
discernable phase lag.
• When the response (output) angular freq << input ang
freq the amplitude is zero.
• When wb = wI then a =  
• Output lags input by 0 to p/2 radians, and this
increases with increasing a.
• When a = 1 (or wb = wI then f = p4.
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Dynamic Response
When a = 1 (or wb = wI then f = p4.
Remember the argument of a trig fnx is always in degrees
or radians.
And that power is proportional to amplitude squared.
Therefore the output power is ½ input power when a = 1.
The next diagrams show that the output amplitude is
always less than the input amplitude.
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A(a) = Output Amplitude (normalized
a = ratio input/output freq = wI/wb)
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a = wI/wb)
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Phase lag (radian)
Applications
• Anderson et al. CH4 flux from melting Arctic
Tundra. http://www.arp.harvard.edu/publication/
• NO and N2O emissions form fertilized corn
fields.
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Real world example
Civerolo et al., 1999.
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Summary
• Because all real measurement systems have a finite response
time, response to sinusoidal input will be, at best, sinusoidal
output with reduced amplitude and a phase shift.
• If the input frequency in the same as the response freq of the
system the amplitude will be reduced to 1/√2 (~70%) of the
input.
• The phase shift will be in the desirable range only for input
frequencies of a few times the instrument response freq.
• For input frequencies > about 10 times the response frequency
the output will be barely discernable as a sine wave. Peak power
in vertical flux of water, momentum, heat, CO2, O3 etc. is about
5Hz. Good instruments can measure these fluxes!
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