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MEASUREMENT THEORY FUNDAMENTALS. Contents
5. Sources of errors
5.1. Impedance matching
5.4.1.
5.4.2.
5.4.3.
5.4.4.
5.2.
Anenergetic matching
Energic matching
Non-reflective matching
To match or not to match?
Noise types
5.2.1. Thermal noise
5.2.2. Shot noise
5.2.3. 1/f noise
5.3.
Noise characteristics
5.3.1. Signal-to-noise ratio, SNR
5.3.2. Noise factor, F, and noise figure, NF
5.3.3. Calculating SNR and input noise voltage from NF
5.3.4. Vn-In noise model
5.4.
Noise matching
5.4.1. Optimum source resistance
5.4.2. Methods for the increasing of SNR
5.4.3. SNR of cascaded noisy amplifiers
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5. SOURCES OF ERRORS
3
5. SOURCES OF ERRORS
Measurement errors can occur due to the undesirable
interaction between the measurement system and:
the object under test,
the environment,
observer.
Environment
Measurement
Object
Influence
Measurement
System
Matching
x+D x
Matching
Disturbance
y +Dy1
Observer
Influence
5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.1. Anenergetic matching
5.1. Impedance matching
Measurement errors can occur due to the undesirable
interaction between the measurement system and:
the object under test.
Environment
Measurement
Object
Influence
Measurement
System
Matching
x+D x
Matching
Disturbance
y +Dy1
Observer
Influence
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5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.1. Anenergetic matching
5
There are three types of impedance matching: anenergetic,
energetic, and non-reflective.
5.1.2. Anenergetic matching
Anenergetic matching is used to minimize the transfer of
energy between the measurement object and
the measurement system.
After matching, measurement system will not supply any
appreciable energy to, or receive from the measurement object.
Anenergetic matching is usually used in active measurement
systems, which do possess internal power amplification.
Reference: [1]
5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.1. Anenergetic matching
Example: Anenergetic matching
Measurement object
Rin >> RS 
RS
vin  vS
vS
the power supplied by the
object is small
most part of it is dissipated in
vin
Rin
Rin
Rin << RS 
Measurement object
RS
iin  iS
iS
the power supplied by the
object is small
most part of it is dissipated in
Measurement system
Rin
Measurement system
iin
Rin
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5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.2. Energic matching
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5.1.2. Energic matching
The aim of energic matching is to extract the maximum
available power from the measurement object, so that the
required power gain in the measurements system can be as
small as possible.
Energetic matching is especially important for passive
measurement systems, which do not possess internal power
amplification.
Reference: [1]
5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.2. Energic matching
To optimize the energic matching, let us consider the following
equivalent circuits of the measurement object and the
measurement system.
Measurement object
ZS= RS + XS
vS
Measurement system
iin
Zin= Rin + Xin
vin
The average power delivered to the measurement system can
be found as:
VS2 Rin
Pin = Iin2 Rin =
.
(RS+Rin)2 + (XS+Xin)2
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5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.2. Energic matching
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For a fixed non-zero RS, this power is maximal if the following
optimal matching is obtained:
Rin = RS and Xin = - XS
or
Zin= ZS* .
Therefore, the maximum power that a measurement object with
a fixed non-zero RS can deliver to a measurement system is:
V S2
Pin =
=
4 Rin
V S2
.
4 RS
Reference: [1]
5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.2. Energic matching
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If RS = 0, then the optimal matching is obtained when
RS = 0 and Xin = - Xo .
In this case, the maximum power a measurement object can
deliver to a measurement system is :
V s2
Pin =
.
Rin
Reference: [1]
5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.2. Energic matching
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Available power is defined as the maximum power that can be
delivered to a load from a source having fixed nonzero
resistance
Pa  Pin
Vin 2
=
.
4 RS
RS0
NB: The maximum power matching usually causes greater
measurement errors, since the input and output impedances of
the chain affect the measurement.
For this reason, the measurement systems almost always are
active ones (with built-in power gain).
Reference: [1]
5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.3. Non-reflective matching
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5.1.3. Non-reflective matching
Non-reflective or characteristic matching is used for transporting
high-frequency measurement signals along transmission lines. If
a transmission line is not terminated characteristically, reflections
off the ends of the line will cause standing waves on the line; the
line output signal is no longer a good measure for the line input
signal.
The characteristic impedance, Z0, of a transmission line equals
its input impedance if the transmission line length were infinite.
For a lossless transmission line with the series inductance per
meter L and the parallel capacitance per meter C,
Z0 = R0 =

L
.
C
Reference: [1]
5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.3. Non-reflective matching
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Illustration: Non-reflective matching: ZS = R0 = Zin
Measurement object
ZS
vS
Measurement system
Z0
Z0
Zin
vin = 0.5vS
NB: When ZS = R0 = Zin holds, energic matching is also
achieved simultaneously, since ZS = RS and Zin = Rin.
R0 is an apparent resistance that does not dissipate energy;
half of the energy delivered by vs is dissipated in RS and the
other half in Rin.
Reference: [1]
5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.3. Non-reflective matching
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Example: The characteristic impedances of different connections
Type of connection
DEFINITION
Characteristic
impedance
Coaxial cable
50 -
75 W
Printed circuit board traces
50 - 150 W
Twisted wire pairs
100 - 120 W
Ribbon cable
200 - 300 W
Free space
376 W
Reference: [1]
5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.4. When to match and when not?
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5.1.5. To match or not to match?
Do match by adjusting impedances, by adding voltage
buffers or by adding matching transformers:
To transfer maximum power to the load.
The source must be capable.
To minimise reflections from the load.
Important in audio, fast (high frequency) systems,
to avoid ringing or multiple pulses (e.g. in counting
systems).
To transmit fast pulses.
Pulse properties can contain important information.
Note that the same physics is encountered in other areas, e.g.
optical coatings, gel in ultrasound scans, optical grease, etc.
Reference: www.hep.ph.ic.ac.uk/Instrumentation/
5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.4. When to match and when not?
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Do not match:
High impedance source with small current signals.
Typical for many photodiode sensors, or other
sensors that must drive high impedance load.
Short cables are required to avoid difficulties.
Weak voltage source.
Drawing power from source would affect the result,
e.g. bridge circuits.
If you need to change properties of a fast pulse,
e.g. pulse widening for ease of detection.
Electronics with limited drive capabilities,
e.g. logic circuits, many are designed to drive other
logic, not long lines, CMOS circuits, even with
follower, are an example.
Reference: www.hep.ph.ic.ac.uk/Instrumentation/
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