etrology
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ETH
How Loading Errors Arise in Metrology
Lesson
Karl H. Ruhm
Institute of Machine Tools and Manufacturing (IWF), ETH Zurich, Switzerland
ruhm@ethz.ch
08. 03. 2010
Version 04; 24.08.2011
www.mmm.ethz.ch/dok01/d0000933.pdf
There exists a German version of this document: d0000765
Aim
Sense and understand
loading structures of mutual relations
and their consequences
on measurement quality.
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2
Administrative Matters
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3
• What is Covered?
Overview on Different Fundamental Topics
Overview on Interrelations of these Topics (Structures)
Theory and Practice
Examples and Applications
Block Diagrams (Signal Effect Diagrams) for Visualisation
• What is not Covered?
Recipes for Today and for Everybody
"Measurement Uncertainty" (→ GUM)
"How to Measure …"
Instrumentation Design
Product Information
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Content
Administrative Matters
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Problems and Concepts
2
Measurement Setting –
Process und Measurement Process
3
Principle "Loading" –
Qualitative Examples
4
Principle "Loading" –
Quantitative Examples
5
Generalisation –
Back-Loading Structures
Summary
5
1
Problems and Concepts
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Statements
• Measurement procedures are never ideal:
Nonideal Measurement.
• However, the concept of
Ideal Measurement
is immensely convenient to define and to fight
measurement errors and uncertainties.
• So, nonideal processes are viewed and judged
from the perspective of virtual, ideal processes.
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Supplement → Module "Ideal Measurement Process"
Supplement → Module "Nonideal Measurement Process"
7
Statements
There are three, and only three, causes of measurement errors:
•
nonideal transfer response behaviour of the measurement path
(transfer errors)
•
disturbing quantities and nonideal transfer response behaviour of the
disturbance path (disturbance errors)
•
nonideal transfer response behaviour of the loading path and resulting
loading quantities (loading errors)
Here, we will treat the last topic only:
Loading Errors
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Supplement → Module "Nonideal Measurement Process"
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Statements
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Properties and behaviour of processes of interest
and their quantities may be influenced and disturbed
by attached measurement processes.
•
Within the fields of natural and technological sciences the causes
of these disturbances are always of pure physical nature.
•
Withdrawal of information goes with withdrawal of energy;
so called
"loading effects", "back-loading effects", "burden effects"
will occur.
•
Unwanted "Loading" can be understood as a generalised
"impedance mismatch".
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Statements
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•
Loading (burden) effects provoke
deviations of the quantities to be measured.
•
Deviations of the quantities to be measured
are not visible in the immediate measurement results.
•
Therefore,
we detect loading (burden) errors only seldom.
•
Whether loading (burden) errors are disturbing or
acceptable, is judged individually by means of the
independent Quality Assurance Process Q.
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Prerequisites and Methods – Qualitative
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•
The physical relations within the process without instrumentation
must be known.
•
The physical relations within the measurement process must be known.
•
The physical interrelations between process and measurement process
in the closed loop, due to loading, must be known.
•
Modelling of the processes is done using concepts of
Signal- and System Theory.
•
The definition of the loading (burden) error is given by
Measurement Theory.
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2
Measurement Setting –
Process und Measurement Process
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Methods – Qualitative
Starting point: "Ideal Circumstances"
• Model of the ideal process P
• Model of the ideal measurement process M
• Model of the ideal series connection of process and
measurement process
• No disturbing feedback effects at all
quantities
actually
measured y(t)
u(t)
!
process P
yˆ (t) − y(t) = o
B1124
process domain
instrumental process domain
measurement
process M
process under measurement PUM
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y(t)
ˆ
resulting
quantities
Supplement → Module "Ideal Measurement Process"
13
The ideal structure of measurement
is described by an ideal series connection:
only feed-forward acting quantities
B1356
u(t) = u 2(t)
y 1(t) = y(t)
y 2 (t) = u 1(t)
2
1
→ no feed-back quantities, no loading quantities ←
This concept
• is useful as a working hypothesis
• is often far from reality and inapplicable
• has to be improved
ETH
Supplement → Module "Group-Notation of Signals"
Supplement → Module "Elements of the Signal Effect Diagram"
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The ideal structure of measurement
is described by an ideal series connection:
only feed-forward acting quantities
Example, approximately ideal:
Principle of
Optical Distance, Velocity, Vibration Measurement
Figure: Polytec
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→ no feed-back quantities, no loading quantities ←
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The ideal structure of measurement
is described by an ideal series connection:
only feed-forward acting quantities
Example,
series connection of process P and measurement process M
approximately ideal:
Principle of
Optical Stress and Strain Measurement
Figure: Polytec
→ no feed-back quantities, no loading quantities ←
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Supplement → External Document "Strain Measurement"
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Methods – Qualitative
Reality: "Nonideal Circumstances"
Example
series connection of three "nonideal" processes:
Principle of
Piezoelectric Sensor, Cable and Charge-to-Voltage Converter
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Figure: Kistler
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Methods – Qualitative
Reality: "Nonideal Circumstances"
• Model of the nonideal process P
• Model of the nonideal measurement process M
• Model of the series connection with feedback-effects
→ closed loop connections involved ←
↓
We have to investigate those structures
in order to understand the relevant relations!
↓
ETH
Supplement → Module "Nonideal Measurement Process"
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3
Principle "Loading" –
Qualitative Examples
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General Principle "Loading"
anywhere in
• Natural Sciences and Technology
• Business Administration, Economics, Sociology, Psychology
• Community
not only in Metrology
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The ideal structure of measurement
is described by an ideal series connection:
only feed-forward effecting quantities
B1356
u(t) = u 2(t)
y 1(t) = y(t)
y 2 (t) = u 1(t)
2
1
→ no feed-back quantities, no loading quantities ←
Nondynamic Transfer Response Equation
y(t) = g1 g2 u(t) [{y}]
with
gi [{yi ui−1}] static transfer response values
Supplement → Module "Group-Notation of Signals"
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Supplement → Module "Elements of the Signal Effect Diagram"
Supplement → Module "Concept and Term «Transfer» Concerning Systems"
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nonideal series connection:
feed-back effecting quantities
• Extended series connection
• Series of interlaced loop connections
Example 1
Electric current i(t) through a RC-Process
B1354
source
ETH
Δu 1
Δu 2
Δu 3
Δu 4
R1
R2
R3
R4
u0 (t)
C1
C2
C3
C4
u(t)
sink
i(t)
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nonideal series connection:
feed-back effecting quantities
• Extended series connection
• Series of interlaced loop connections
Example 2
* through the layers of a wall
Heat current Q(t)
R1
ϑ0 (t)
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R3
R4
*
Q(t)
B1355
source
R2
C1
C2
Δu 1
Δu2
C3
C4
ϑ(t) sink
Δu 3 Δu 4
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Example 3
Extended series connection of multiple sub-processes
Process Diagram
tank
T
power
electronics
E
uE(t)
i (t)
M
M
=
M
motor
M
* T(t)
m
p
(t)
A(t) cons
gearbox
GB
ω
(t)
GB
* V(t)
m
pC(t)
compressor
C
B0630
u(t)
ω (t)
valve
V
How does the
Signal Effect Diagram
look like?
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Example 3
Extended series connection of multiple sub-processes
Process Diagram
tank
T
power
electronics
E
M
=
M
motor
M
i (t)
* V(t)
m
pC(t)
* T(t)
m
p
(t)
A(t) cons
gearbox
GB
ω
M
compressor
C
(t)
GB
B0630
uE(t)
u(t)
ω (t)
valve
V
Signal Effect Diagram
iE(t)
ML
* (t)
m
T
(t)
GB
ω M(t)
u(t)
pcons(t)
p (t)
C
* V(t)
m
B0629
A(t)
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uE(t)
power
electronics E
iM(t)
motor M
ω (t)
gearbox GB GB compressor C
ML (t)
C
tank T
pT (t)
valve V
* (t)
m
V
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Generalisation
Extended series connection → Series of interlaced loop connections
B0003
or
or
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Extended series connection → Series of interlaced loop connections
basic physical structure
currents (matter, energy, pulse, momentum)
flow through extended series connections of
resistors and storages
B1358
special basic system structure
quadrupole
u 1 (t)
y1(t)
u1(t)
y2 (t)
or u (t)
u 2 (t)
2
y1(t)
y2(t) or
u(t)
y(t)
general basic mathematical structure
band matrices
B1357
0
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0
Supplement → Module "Generalised Kirchhoff-Relations"
Supplement → Module "Principle Resistance"
27
Example
simple electric quadrupole
Voltage Divider
Process Diagram
Δu R(t)
iR(t) R
1
i(t)
u1(t) i1(t)
R1
Signal Effect Diagrams
detail
u(t)
u1(t)
i(t)
i1(t)
1
R1
R
process
iR (t)
i(t)
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Σ
– Δu (t)
R
B0648
u(t)
+
u1(t)
B0668
global
iR (t)
B0667
u(t)
1
Σ
+
i1(t)
+
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Generalised "Quadrupole"
for example
Bidirectional Closed Information Loop in Automatic Control
matter
matter
energy
momentum
energy
momentum
B1227
process P
actuation
process A
process domain
instrumental process domain
measurement
process M
control
process C
information
information
controlled process CP
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Remember: All lines represent potential multiple signals, modelled as vectors.
29
Importance of extended series connection
•
•
•
•
•
forward and backward acting quantities
exchange of matter, energy, pulse, moment etc.
energy and power relevant
practice oriented
demanding concerning modelling
source extensive quantities
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B0005
intensive quantities
drain
Supplement → Module "Extensive, Intensive Signals"
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Examples of Power Demand
electrical power source
thermal power source
∗
P(t) = c m(t) Δϑ(t) [W]
P(t) = i(t) Δu(t) [W]
Δϑ(t)
P(t)
Π
B1352
B1352
u(t)
Π
* = P(t)
Q(t)
*
c m(t)
i(t)
and so on.
Note: The quantity "Power" is always composed of several individual quantities.
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Supplement → Examples "Physical Power"
31
4
Principle Loading –
Quantitative Examples
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Attempt
Nonideal and error related items will be shown in red,
ideal items will be shown in green.
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Phenomenon Loading
Quantitative Example 1
Loading of a nonideal voltage source
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Application
Sensors as Voltage Sources
Always as First Question
How would the ideal loaded voltage source look like?
Equivalent Circuit Diagram
source
u0
iL
B1091
Q
uL
resistance
uK
RL (load)
L
i
Fundamental Relations
(non-dynamic transfer response functions)
current
output voltage
i=
1
u0
RL
[A]
uK = f(u0 ) = 1⋅ u0
[V]
No other resistances (impedances)!
ETH
Supplement → Module "Loading of a Voltage Source"
35
Equivalent Circuit Diagrams
ideal loaded voltage source
iL
B1091
Q
u0
uL
resistance
uK
RL (load)
L
uQ iQ
source
Q
RQ
uL
u0
iL
B1090
source
nonideal loaded voltage source
i
i
Fundamental Relations
(non-dynamic transfer response functions)
current
output voltage
resistance
uK
RL (load)
L
1
i=
u
RL 0
i=
[A]
uK = f(u0 ) = 1⋅ u0
[V]
1
u
RL + RQ 0
uK = f(RQ ;RL ;u0 ) =
[A]
1
u
RQ 0
1+
RL
[V]
≠1
ETH
Supplement → Module "Loading of a Voltage Source"
36
Consequences
Deviation in the output voltage uK by the voltage drop uQ
static transfer response characteristic
output voltage uK = f(i)
uK /V
uQ iQ
RQ
u0
uL
iL
B1090
Q
u0
uK(i)
resistance
uK
RL (load)
L
B1081
source
uQ = u0 u K = eu (i)
i
0
0
i [A]
Remark
A controlled voltage source would enforce a source resistance (impedance) of RQ = 0 Ω
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Supplement → Module "Loading of a Voltage Source"
37
uQ iQ
RQ
source
uL
u0
iL
B1090
Q
resistance
uK
RL (load)
L
i
Next Step for a System Analysis
According to System Theory we look for the model of the nonideal process
given by a
• signal effect diagram of the expected closed loop
u
+
B1444
= u0 [V]
–
Σ
gv
gr
y
= uK [V]
g
• overall transfer response function y = f(u) = g u
• overall transfer response value g =
ETH
gv
1 + gv gr
Supplement → Module "Concept and Term «Transfer» Concerning Systems"
38
uQ iQ
RQ
source
iL
B1090
Q
u0
uL
resistance
uK
RL (load)
L
i
Model of the Nonideal Process
signal effect diagram
u0
+_
Σ
B1082
uQ
RQ
i Q = iL
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1
u = guL ,u0 u0
RQ 0
1+
RL
1
RL
uL
load L
source Q
transfer response function uK = f(u0) = g u0
uK = uL =
u K = uL
u 0 uQ
[V]
g uL,u0
transfer response value g
guk ,u0 =
gv
1
!
=
= 1 [−]
R
1 + gv gr
1+ Q
RL
Supplement → Module "Concept and Term «Transfer» Concerning Systems"
39
u0
+_
Σ
uQ
B1082
u K = uL
u 0 uQ
RQ
source Q
i Q = iL
1
RL
uL
load L
g uL,u0
Goal
No Voltage Drops
No Loading
No Loading Error
Possible, if
Source Impedance RQ = 0
or
Load Impedance RL = ∞
What is to be done, if we cannot reach the goal?
→ Correction within the following Reconstruction Process R ←
(which we need anyway)
ETH
Supplement → Module "Loading of a Voltage Source"
40
Phenomenon Loading
Quantitative Example 2
Back-loading error at a voltage divider
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41
Application
All measurement principles with voltage dividers
e.g.: position, distance or angle measurement
Measurement Principle
100%
R1
B0437
u0
R2
x
u
0%
R +R =R
1
2
Draw-Wire Sensor
(Figures: MICRO-EPSILON)
ETH
Supplement → Interactive Animation "Potentiometer"
42
ideal
source
u1
R1 x
R2 i2
i
u0
static transfer response characteristic
u
u
real
drain
RL
1.0
B0435
B0436
equivalent circuit diagram
0
R=R +R
1
0.8
iL
u
RL
=
R
supply
potentiometer
2
measurement
device
oo
0.6
2
1
0.5
e NL
0.4
We may call the error
loading error eload(t)
with regard to its real cause
or
nonlinearity error eNL(t)
with regard to its appearance.
0.2
0
0
0.2
0.4
0.6
0.8
1
x=
R2
R
ETH
Supplement → Interactive Animation "Potentiometer"
43
Phenomenon Loading
Quantitative Example 3
Back-loading error in a thermocouple TC
caused by an amplifier V
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44
Loading as an error process E attached to a sensor process S
ϑ
2
Δϑ = ϑ1 ϑ2 [K]
bei ϑ1 > ϑ2
RQ
u
ˆV
u
ˆ Δϑ RL
amplifier V
ϑ
1
thermocouple TC
RQ
uΔϑ
B1020
B1019
Δu B
uˆΔϑ
ûV
RL
iL
g uˆ ,uˆ
V Δϑ
Again a voltage divider!
Figure: Analog Devices
What can be done?
Can we reduce the systematic error eload(t)?
→ Yes ←
ETH
Supplement → Example "Electrical Loading of a Thermocouple"
45
Loading as an error process E attached to a sensor process S
Δu B
RQ
B1020
uΔϑ
uˆΔϑ
ûV
RL
iL
g uˆ ,uˆ
V Δϑ
What would be ideal?
→ The Nominal Sensor Process SN ←
Δϑ
Δϑ
process domain
instrumental process domain
uV
B1059
nominal sensor process SN
–
Σ
+
error
at the output of the
sensor process
e uv
ˆV
u
sensor process S
ETH
Supplement → Module "Ideal Measurement Process"
Supplement → Example "Electrical Loading of a Thermocouple"
46
Development of the nonideal sensor process S
Δϑ
Δϑ
[°C]
[°C]
process domain
instrumental process domain
u Δϑ
g u,Δϑ
[V]
nominal
thermocouple TCN
gu
uV
V
,u
[V]
nominal
amplifier VN
Note: this is just the model of the
process and not physical reality!
nominal sensor process SN
_
+
g u,Δϑ
u Δϑ
+_
RQ
B1315
ETH
[V]
+
u
ˆ Δϑ
[V]
ΔuB = e uΔϑ
(source Q)
_
Δϑ
Σ
Σ
thermocouple TC
eu
iL
[V]
Σ
eu
V
[V]
u
ˆV
gu
V
,u
1
RL
amplifier V
(load L)
We identify the error process E,
positioned in series between the
thermocouple TC and the amplifier
V (red loop).
[V]
Cause of loading errors: Voltage
divider with poorly matched
parameters (parameter mismatch).
Ideal would be:
RQ = 0 and / or
RL = ∞,
then the loading path would be
tight (closed).
[A]
nonideal sensor process S
Supplement → Module "Nonideal Measurement Process"
Supplement → Example "Electrical Loading of a Thermocouple"
47
Details
concerning the Error Process E
uΔϑ
+
Σ
_
Δu L = e uΔϑ
[V]
[V]
[V]
RQ
iL
[A]
1
RL
eu
↓
B1447
Δϑ
[V]
error process E
The transfer response function
of the
error process E
is
1
ûΔϑ = gE uΔϑ =
u
RQ Δϑ
1+
RL
ETH
numerical
example
RQ = 1.506 Ω
RL = 1046 Ω
u
ˆ Δϑ
[V]
gE = 0.9985 [V / V]
The error concerning the
output voltage of the
thermocouple TC
is
euΔQ
RQ
uΔϑ
=
RQ + RL
Supplement → Module "Nonideal Measurement Process"
Supplement → Example "Electrical Loading of a Thermocouple"
[V]
48
Δϑ
Δϑ
[°C]
[°C]
process domain
instrumental process domain
u Δϑ
g u,Δϑ
gu
[V]
nominal
thermocouple TCN
We know already,
ideal would be:
uV
V
,u
[V]
• RQ = 0 and / or
• RL = ∞
nominal
amplifier VN
nominal sensor process SN
_
+
g u,Δϑ
u Δϑ
+_
RQ
B1315
[V]
+
gu
[V]
iL
V
,u
1
RL
[V]
amplifier V
[A]
nonideal sensor process S
ETH
Σ
eu
then the loading path
would be closed.
V
[V]
But, that's impossible:
u
ˆV
u
ˆ Δϑ
ΔuB = e uΔϑ
(source Q)
_
Δϑ
Σ
Σ
thermocouple TC
eu
(load L)
[V]
We have to compensate
error sources
of the
sensor process S
within the following
reconstruction process R.
Supplement → Module "Nonideal Measurement Process"
Supplement → Example "Electrical Loading of a Thermocouple"
49
Any Sensor Process S needs a Reconstruction Process R
(Fundamental Axiom of Metrology).
B0398
ideal
• Sensor Process S (mapping)
pressure
sensor
• Reconstruction Process R (inversion)
7.52 bar
ˆ
p/bar
2.71Volt
u p /V
Example
p/bar
actually measured quantity y(t)
u(t)
process P
Σ
B1406
process domain
instrumental process domain
+
!
yˆ (t) − y(t) = o
sensor quantity
y(t) = u S(t)
OpS {...}
yS(t) = uR(t)
1
OpS {...}
yR (t)
sensor
reconstruction
process S
process R
measurement process M
ETH
error quantity
e y(t)
=
y(t)
ˆ
erroneous
resulting quantity
Supplement → Module "Sensor Process"
Supplement → Module "Reconstruction Process – A Survey"
50
Any Sensor Process S needs a Reconstruction Process R.
nonideal
• Sensor Process S (mapping)
• Error Process E (inference) within the sensor process S
in a series connection
• Reconstruction Process R (inversion of mapping and inference)
y(t)
B1454
OpS {...}
ETH
y S (t)
sensor
process S
OpE {...}
yˆ S (t)
1
OpE {...}
yScorr(t)
1
OpS {...}
ˆ
y(t)
reconstruction
process R
measurement process M
Supplement → Module "Sensor Process"
Supplement → Module "Reconstruction Process – A Survey"
Supplement → Module "Inversion of Interconnected Systems"
51
The usual incomplete reconstruction process R at a loaded sensor process S
Δϑ
Δϑ
[°C]
process domain
instrumentation domain
[°C]
u Δϑ
g u,Δϑ
[V]
nominal
thermocouple TCN
gu
uV
V
,u
[V]
nominal
amplifier VN
nominal sensor process SN
_
+
g u,Δϑ
u Δϑ
RQ
+_
(source Q)
_
[V]
+
Δϑ
Σ
u
ˆ Δϑ
Σ
[V]
ΔuB = e uΔϑ
thermocouple TC
eu
[V]
Σ
eu
_
V
[V]
u
ˆV
gu
V
,u
[V]
(load L)
[A]
B1018
1
gu
,u
V
[°C]
u
ˆ Δϑ
1
ˆ
Δϑ
[V]
g u,Δϑ
[°C]
1
RL
amplifier V
iL
+
Σ
eΔϑ
nonideal
sensor process S
incomplete
invers model of
amplifier V
incomplete
invers model of
thermocouple TC
incomplete
reconstruction process R
nonideal measurement process M
ETH
Supplement → Module "Reconstruction Process – A Survey"
Supplement → Example "Electrical Loading of a Thermocouple"
52
Possibilities shown by the
Transfer Response Function
Ideal Measurement Process
Δϑˆ =
1
1
gu,Δϑ guV ,u
guV ,u gu,Δϑ = 1⋅ Δϑ [°C]
Incomplete Reconstruction Process R
Δϑˆ =
1
1
gu,Δϑ guV ,u
guV ,ugEgu,ΔϑΔϑ = gEΔϑ [°C]
=
Complete Reconstruction Process P
Δϑˆ =
1
gu,Δϑ
1 1
guV ,ugEgu,ΔϑΔϑ = 1⋅ Δϑ [°C]
gE guV ,u
The usual incomplete reconstruction process R at a loaded sensor process S leads to
loading errors eDJ.
ETH
Supplement → Module "Reconstruction Process – A Survey"
Supplement → Example "Electrical Loading of a Thermocouple"
53
The complete reconstruction process R at a loaded sensor process S
(Assumption: Ideal Reconstruction Process R; no numerical errors)
Δϑ
Δϑ
[°C]
[°C]
process domain
instrumental process domain
u Δϑ
g u,Δϑ
gu
[V]
nominal
thermocouple TCN
uV
V
,u
[V]
nominal
amplifier VN
nominal sensor process SN
_
+
g
u Δϑ
u,Δϑ
RQ
+_
B1017
_
[V]
+
Δϑ
Σ
u
ˆ Δϑ
Σ
gu
[V]
Δu L = e uΔϑ
thermocouple TC
(source Q)
eu
iL
V
,u
amplifier V
(load L)
_
V
[V]
u
ˆV
1
RL
[V]
Σ
eu
[V]
+
u
ˆ Δϑ
1
g u V ,u
[V]
1
RL
+
ûΔϑcor
1
[V]
gu,Δϑ
Σ
+
Δu L = e uΔϑ
invers model of
amplifier V
[A]
ˆi L
[V]
Σ
eΔϑ= 0
[°C]
Δϑ
[°C]
RQ
invers model of
thermocouple TC
[A]
nonideal sensor process S
reconstruction process R
measurement process M
ETH
Supplement → Module "Inversion of Interconnected Systems"
Supplement → Module "Reconstruction Process – A Survey"
54
The complete reconstruction process R at a loaded sensor process S
Example
Parameter from a Calibration Process CP
transfer response value thermocouple TC:
gu,Δϑ = 52.8 μV °C−1
transfer response value amplifier V:
gu ,u = 0.992mV μV −1
V
RQ = 1.506 Ω
source resistance thermocouple TC:
RL = 1004 Ω
load resistance of amplifier V:
Transfer Response Value gS of the Sensor Process S:
gS = guV ,u
1
1 + RQ
1
RL
gu,Δϑ = 52.299
[mV °C−1]
Transfer Response Value gR of the Reconstruction Process R:
gR =
ETH
1
gu,Δϑ
(1 + RQ
1
1
= 0.01912 [°C mV −1]
)
RL guV ,u
Supplement → Module "Inversion of Interconnected Systems"
Supplement → Module "Reconstruction Process – Survey"
55
Intermediate Results
• Unmatched parameters within measurement processes M
lead to systematic loading errors eload(t).
• Loading effects within a measurement process can be identified by a
calibration procedure of the measurement process (modelling).
• Loading effects within a measurement process M can be
systematically compensated by a complete reconstruction R process.
• For the example at hand the back-loading error eload(t) is available on
demand, since all information is known. Unfortunately, this is not
always the case.
ETH
56
Prerequisites
for these results:
knowledge (model) about process P and sensor process S:
→ Model Based Measurement ←
Assumptions and Constraints
for these results:
•
•
•
•
linear behaviour of all elements and signal interactions
nondynamic behaviour of all elements
no deterministic or random disturbance signals
ideal behaviour of the reconstruction process R (no numerical errors)
→ More or Less Unrealisable in Practice ←
But doing nothing is worse!
ETH
57
Further Things to Do
• Dynamic Sensor Process S
• Dynamic Back-Loading
• Dynamic Reconstruction
• Impedance Matching
• Dynamic Process P
• Extended Reconstruction Process
Some Examples and Results
ETH
58
Phenomenon Loading
Quantitative Example 4
Dynamic thermal loading
by a temperature sensor
ETH
59
Again, loading acts as a disturbance quantity at the process
approximate
signal effect diagram
process diagram
R(t) [Ω]
*
Q23(t)
3
ϑ (t)
1
3
ϑ 1 (t)
environment
3
B0747
ϑ1(t)
3
ϑ 2 (t) m 3 = oo ; ϑ3 ; QSp
* (t)
Q
13
S
ϑ2(t)
IC
u(t)
2
Pel(t)
* (t)
Q
21
2
m 1 ; ϑ 1 ; Q Sp
1
R(t)
1
sensor process
process
B0745
P(t) [W]
m2 ; ϑ2 ; QSp
ϑ 2 (t)
2
Fundamental Relations
storage equations
QSp (t) = mc Δϑ(t)
ETH
current equations
[J]
Q(t) = α A Δϑ(t) [Js −1]
*
Supplement → Exercise "Thermal Back-Loading"
60
Again, loading acts as a disturbance quantity at the process
detailed signal effect diagram
α1A13
–
Σ
* (t)
Q
13
Σ
+
αA
1 13
d
Q (t)
dt Sp1
–
+
QSp (t)
1
sensor
process S
1
1
m 1 c1
ϑ 1(t)
QSp (0)
1
αA
–
* (t)
Q
21
2 21
Σ
+
α2A21
ϑ3
environment
ETH
+
αA
3 23
* (t)
Q
23
–
Σ
Σ
–
2
QSp (t)
2
2
–
+
2
αA
1
m 2 c2
ϑ 2 (t)
B0748
Pel (t)
QSp (0)
process P
d
dt QSp (t)
3 23
Supplement → Exercise "Thermal Back-Loading"
61
Phenomenon Loading
Quantitative Example 5
Dynamic mechanical loading
by an oscillatory structure
ETH
62
Loading acts as a disturbing quantity at the process
hs (t)
.
h s(t)
..
h s(t)
h(t)
.
h(t)
..
h(t)
m
κ
c
cs
F(t)
ms
B0911
Fs (t)
schwingende
Struktur
κs
Sensor
single differential equations
process P :
ma(t) + c(v(t) − v S (t)) + κ(h(t) − hS (t)) = f(t) [N]
sensor process S : mS aS (t) + c S (v S (t) − v(t)) + κS (hS (t) − h( t)) = 0
ETH
[N]
Supplement → Example "Mechanical loading by an oscillatory structure"
63
Loading acts as a disturbing quantity at process P
κ
m
c
m
v(0)
h(0)
– –
f(t)
1
m
1
m
+
Σ
a(t)
v(t)
h(t)
h(t)
v(t)
+
a(t)
process domain
instrumentation domain
process P
κS
mS
cS +
Σ
mS
–
vS (0)
+ +
Σ
+
Σ
–
hS(0)
a S(t)
vS(t)
fa (t)
S
g u ,f
u f (t)
a
f a
B1122
mS
h S(t)
ETH
back-load path
sensor process S
Supplement → Example "Mechanical loading by an oscillatory structure"
64
5
Generalisation –
Back-Loading Structures
ETH
65
Three Causes of Nonideal Measurement
1.
2.
3.
nonideal transfer response (structure, parameter)
disturbing quantities and disturbance paths
loading quantities and load paths
u M(t)
B1360
= disturbed
measurement
quantities ydist(t)
vM(t)
measurement
path
Σ
measurement /
load path
yM(t)
estimating
quantities y(t)
ˆ
disturbance
path
disturbance /
load path
Σ
zM(t)
loading
quantities
disturbing
quantities
measurement process M
ETH
Supplement → Module "Nonideal Measurement Process"
66
Generalised Fundamental Axiom of Measurement:
The transfer response function matrix
of the measurement process M is ideal,
if
firstly,
the transfer response function matrix
of the measurement path equals the unit matrix I:
yˆ (t) = yM (t) = I uM (t) = I y(t)
u M(t)
= disturbed
measurement
quantities ydist(t)
!
=I
!
=0
B1164
!
=0
vM(t)
!
=0
measurement
path
measurement /
load path
estimating
quantities y(t)
ˆ
disturbance
path
disturbance /
load path
disturbing
quantities
ETH
Σ
yM(t)
zM(t)
Σ
loading
quantities
measurement process M
Supplement → Module "Nonideal Measurement Process"
67
Generalised Fundamental Axiom of Measurement:
The transfer response function matrix
of the measurement process M is ideal,
if
secondly,
the transfer response function matrix
of the disturbance path equals the zero matrix 0:
yˆ (t) = yM (t) = 0 vM (t)
u M(t)
= disturbed
measurement
quantities ydist(t)
!
=I
!
=0
B1164
!
=0
vM(t)
!
=0
measurement
path
Σ
measurement /
load path
yM(t)
estimating
quantities y(t)
ˆ
disturbance
path
disturbance /
load path
disturbing
quantities
zM(t)
Σ
loading
quantities
measurement process M
ETH
Supplement → Module "Nonideal Measurement Process"
68
Generalised Fundamental Axiom of Measurement:
The transfer response function matrix
of the measurement process M is ideal,
if
thirdly,
the transfer response function matrixes
of the loading paths equal each the zero matrix 0:
zM (t) = 0 uM (t) und
u M(t)
= disturbed
measurement
quantities ydist(t)
!
=I
!
=0
B1164
!
=0
vM(t)
!
=0
zM (t) = 0 vM (t)
measurement
path
Σ
measurement /
load path
yM(t)
estimating
quantities y(t)
ˆ
disturbance
path
disturbance /
load path
disturbing
quantities
zM(t)
Σ
loading
quantities
measurement process M
ETH
Supplement → Module "Nonideal Measurement Process"
69
y nom(t)
not measurable
nominal process PN
(open loop observer, OLO)
(unequipped process)
measurement
quantities of interest
–
+
Σ
ey
(t)
nom
error quantities
due to nonideal
interrelations
nominal domain
process domain
Nonideal
Measurement Process M
• concerning nonideal (dynamic)
transfer response behaviour
• concerning disturbance quantities
• concerning loading quantities
u(t)
y dist(t)
v(t)
unknown
w(t)
measurement
quantities,
disturbed
process P
process domain
instrumental
process domain
–
B1128
+
uM(t)
yM(t)
v M(t)
zM(t)
Σ
nonideal
measurement process M
ETH
e y (t)
error quantities
due to nonideal
measurement
=
ŷ(t)
Two types of
measurement errors:
1. error quantities due to
nonideal measurement
2. error quantities due to
back-loading
resulting quantities,
erroneous,
estimating
Supplement → Module "Nonideal Measurement Process"
70
Generalised Model
described by a set of equations
of the classical System Theory
(State Space Description)
non-dynamic subsystem
D
x(0)
dynamic subsystem
u(t)
(M)
B
Σ
.
x(t)
x(t)
(N)
(N)
C
y(t)
Σ
(P)
x (t) = A x(t) + Bu(t)
x(0) = x 0
y(t) = C x(t) + Du(t)
B0080
A
dynamic system
u(t)
Reduction:
(M)
A B
C D
u(t)
(M)
ETH
y(t)
(P)
y(t)
P
(P)
⎡ x (t)⎤ ⎡ A B ⎤ ⎡ x(t)⎤
⎢ y(t) ⎥ = ⎢ C D ⎥ ⎢ u(t) ⎥
⎥⎦ ⎣
⎣
⎦ ⎢⎣
⎦
⎡ x (t)⎤
⎡ x(t)⎤
⎢ y(t) ⎥ = P ⎢ u(t) ⎥
⎣
⎦
⎣
⎦
Supplement → Module "Basic Structures of State Space Representation"
71
Detailed model in state space representation (time domain)
with
disturbances, back-loading and nonideal transfer response
D
u(t)
B
input
quantities
C
x(0)
x(t)
.
B1230
x(t)
y(t)
Σ
output
quantities
F
⎡ x (t)⎤ ⎡ A B E⎤ ⎡ x(t)⎤
⎡ x(t)⎤
⎢ y(t) ⎥ = ⎢ C D F ⎥ ⎢ u(t) ⎥ = P ⎢ u(t) ⎥
⎢
⎥ ⎢
⎥⎢
⎥
⎢
⎥
⎢⎣ z(t) ⎥⎦ ⎢⎣ G H J ⎥⎦ ⎢⎣ v(t) ⎥⎦
⎢⎣ v(t) ⎥⎦
A
H
disturbing
quantities
ETH
E
G
J
y(t) = C x(t) + Du(t) + F v(t)
z(t) = G x(t) + Hu(t) + Jv(t)
Σ
v(t)
x (t) = A x(t) + Bu(t) + E v(t)
Σ
z(t)
back-loading
quantities
non-ideal, dynamic, disturbed and back-loading system
Supplement → Module "Basic Structures of State Space Representation"
72
Generalisation
1.
u P (t)
y P(t)
B1161
vP(t)
PP
2.
u P (t)
z P(t)
vP(t)
PP
y P(t)
process P
process P
process domain
instrumentation domain
process domain
instrumentation domain
uM(t)
vM(t)
zM(t)
yM(t)
PM
zM(t)
u1(t)
y1(t)
PP
uM(t)
PM
vM(t)
nonideal
measurement process M
3.
z P(t)
yM(t)
nonideal
measurement process M
4.
u1(t)
y1(t)
u (t) S {PP , PM } y (t)
2
2
u2(t)
ETH
PM
y2(t)
"P" = parameter matrix
observed process OP
described by
Redheffer-Star-Product Matrix S
73
Special Series Connection in State Space Description:
u1(t)
Multiplication of two
parameter matrices PM und PP
in the so-called
y1(t)
Redheffer-Star-Product Matrix S
B1162
u2(t) S {PP , PM } y2(t)
observed process OP
described by
Redheffer-Star-Product-Matrix S
Reconstruction of Process und Measurement Process:
Inversion of the Redheffer-Star-Product Matrix S
Matlab-Function
"starp"
ETH
74
Summary
ETH
75
ETH
•
Measurement processes load processes –
loading errors arise
•
Loading errors depend on properties of the measurement process as
well as on properties of the process –
impedance matching task
•
Loading errors differ structurally from all other measurement errors –
loading errors appear in the surroundings of the process
•
Loading errors are only seldom recognised –
loading errors are not discernable by conventional calibration
processes
•
Loading errors have to be considered and errors have to be analysed
and discussed concerning specific demands –
task for quality assurance
•
Loading errors can be avoided by load-free series connections –
demanding task for theory and practice
76