Electronics – 96032
Alessandro SpinelliPhone: (02 2399) [email protected] home.deib.polimi.it/spinelli
Wheatstone Bridge and Sensors
Alessandro Spinelli – Electronics 96032
Slides are supplementary material and are NOT a
replacement for textbooks and/or lecture notes
Disclaimer 2
Alessandro Spinelli – Electronics 96032
Acquisition chain 3
Sensor Filter ADC
small signal
noise
amplifiedsignal
amplifiednoise
amplifiedsignal
reducednoise
Amp
next lessons
Alessandro Spinelli – Electronics 96032
• At this point, we know how to analyze and design simpleamplifiers
• Effective amplifier design depend upon the input signalcharacteristics (impedance, bandwidth,…)
• In this part of the class we discuss a few sensor arrangement: Wheatstone bridge (this lesson) Deformation and temperature sensors (next lesson) Sensor technologies (optional overview)
Purpose of the lesson 4
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• Wheatstone bridge and sensitivity• Effect of wire resistance• Temperature compensation• Sensors: general definitions
Outline 5
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• Resistors whose value changes with variation in a physicalquantity S (light, heat, stress,…)
• Among the most common in instrumentation• For small changes in S, a linear approximation holds:
𝑅𝑅 = 𝑅𝑅0 1 + 𝛼𝛼𝛼𝛼 = 𝑅𝑅0 1 + 𝑥𝑥
where
Resistive sensors 6
𝛼𝛼 =1𝑅𝑅0
�𝑑𝑑𝑅𝑅𝑑𝑑𝛼𝛼 𝑅𝑅0
Alessandro Spinelli – Electronics 96032
• Noise and fluctuations in ground potential and 𝑅𝑅0degrade performance
• Can be used for high-levelsignals, low noise, short distance environments
Single-ended measurements 7
≈
𝑉𝑉𝑠𝑠𝐼𝐼 𝑅𝑅0(1 + 𝑥𝑥)
Δ𝑉𝑉𝐺𝐺
𝑉𝑉𝑠𝑠 = 𝐼𝐼𝑅𝑅0 + 𝐼𝐼𝑅𝑅0𝑥𝑥 + Δ𝑉𝑉𝐺𝐺 + 𝐼𝐼Δ𝑅𝑅0 + 𝐼𝐼Δ𝑅𝑅0𝑥𝑥Offset + Signal + Disturbs
Alessandro Spinelli – Electronics 96032
Insensitive to ground potential and resistance fluctuations
Wheatstone bridge 8
𝑉𝑉𝑠𝑠 = 𝑉𝑉𝑐𝑐𝑐𝑐𝑅𝑅4
𝑅𝑅3 + 𝑅𝑅4−
𝑅𝑅2𝑅𝑅1 + 𝑅𝑅2
𝑉𝑉𝑐𝑐𝑐𝑐 𝑉𝑉𝑠𝑠
𝑅𝑅1 𝑅𝑅3
𝑅𝑅2 𝑅𝑅4
Alessandro Spinelli – Electronics 96032
• We set 𝑉𝑉𝑠𝑠 = 0 for 𝑥𝑥 = 0:
11 + 𝑅𝑅3/𝑅𝑅4
−1
1 + 𝑅𝑅1/𝑅𝑅2= 0 ⇒
𝑅𝑅1𝑅𝑅2
=𝑅𝑅3𝑅𝑅4
= 𝑘𝑘
• We pick 𝑘𝑘 by requiring maximum 𝑉𝑉𝑠𝑠 sensitivity to resistance variation:
𝑑𝑑𝑉𝑉𝑠𝑠𝑑𝑑𝑅𝑅1
=𝑉𝑉𝑐𝑐𝑐𝑐
1 + 𝑅𝑅1𝑅𝑅2
21𝑅𝑅2
=𝑉𝑉𝑐𝑐𝑐𝑐𝑅𝑅1
𝑘𝑘(1 + 𝑘𝑘)2
Bridge balancing 9
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Let’s assume 𝑅𝑅1 = 𝑅𝑅2 = 𝑅𝑅3 = 𝑅𝑅; 𝑅𝑅4 = 𝑅𝑅(1 + 𝑥𝑥). We have:
𝑉𝑉𝑠𝑠 = 𝑉𝑉𝑐𝑐𝑐𝑐𝑅𝑅(1 + 𝑥𝑥)𝑅𝑅(2 + 𝑥𝑥)
−12
= 𝑉𝑉𝑐𝑐𝑐𝑐𝑥𝑥
2(2 + 𝑥𝑥)≈ 𝑉𝑉𝑐𝑐𝑐𝑐
𝑥𝑥4
• The non-linearity relative error is
Unbalanced bridge 11
𝜀𝜀 =𝑥𝑥4
2(2 + 𝑥𝑥)𝑥𝑥
− 1 =|𝑥𝑥|2
Alessandro Spinelli – Electronics 96032
Double sensitivity 12
𝑉𝑉𝑠𝑠 = 𝑉𝑉𝑐𝑐𝑐𝑐𝑅𝑅(1 + 𝑥𝑥)𝑅𝑅(2 + 𝑥𝑥)
−𝑅𝑅
𝑅𝑅(2 + 𝑥𝑥)= 𝑉𝑉𝑐𝑐𝑐𝑐
𝑥𝑥2 + 𝑥𝑥
≈ 𝑉𝑉𝑐𝑐𝑐𝑐𝑥𝑥2
𝑉𝑉𝑐𝑐𝑐𝑐
𝑅𝑅(1 + 𝑥𝑥) 𝑅𝑅
𝑅𝑅(1 + 𝑥𝑥)𝑅𝑅
𝑉𝑉𝑠𝑠
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Maximum sensitivity 13
𝑉𝑉𝑠𝑠 = 𝑉𝑉𝑐𝑐𝑐𝑐𝑅𝑅(1 + 𝑥𝑥)
2𝑅𝑅−𝑅𝑅(1 − 𝑥𝑥)
2𝑅𝑅= 𝑉𝑉𝑐𝑐𝑐𝑐𝑥𝑥
Needs both positive and negative equaldependences
𝑅𝑅(1 + 𝑥𝑥)
𝑅𝑅(1 + 𝑥𝑥)
𝑅𝑅(1 − 𝑥𝑥)
𝑅𝑅(1 − 𝑥𝑥)
𝑉𝑉𝑠𝑠𝑉𝑉𝑐𝑐𝑐𝑐
Alessandro Spinelli – Electronics 96032
• Sensitivity: voltage output when 𝑉𝑉𝑐𝑐𝑐𝑐 = 1 V and 𝑥𝑥 = 𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚. Usuallyexpressed in mV/V
• Accuracy: difference with respect to the linear characteristics. Expressed in %
• Resistance: resistance of the bridge measured between the output terminals
Bridge parameters 14
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Bridge amplifiers 15
A
A
𝑅𝑅(1 + 𝑥𝑥)𝑅𝑅
𝑅𝑅 𝑅𝑅
𝑉𝑉𝑐𝑐𝑐𝑐
𝑅𝑅/2
𝑅𝑅/2𝑉𝑉𝑐𝑐𝑐𝑐/2
𝑉𝑉𝑐𝑐𝑐𝑐 𝑥𝑥/4
𝑉𝑉𝑜𝑜
𝑉𝑉𝑜𝑜
Alessandro Spinelli – Electronics 96032
• High gain 𝑉𝑉𝑐𝑐𝑐𝑐 = 10 V, 𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚 = 1 %, i.e., 𝑉𝑉𝑠𝑠,𝑚𝑚𝑚𝑚𝑚𝑚 = 25 mV. If 𝑉𝑉𝑜𝑜,𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑉𝑉𝑐𝑐𝑐𝑐 = 10 V ⇒𝐺𝐺 = 400
• High input resistance 𝑅𝑅 = 100 Ω and an error smaller than 1‰ is required ⇒𝑅𝑅𝑖𝑖 ≥ 1000𝑅𝑅 = 100 kΩ
• High CMRR with 8-bit resolution 𝑉𝑉𝑠𝑠,𝐿𝐿𝐿𝐿𝐿𝐿 = 25 mV/28 ≈ 100 μV and 𝑉𝑉𝐶𝐶𝐶𝐶 = 5 V ⇒
CMR𝑅𝑅 ≥ 𝑉𝑉𝐶𝐶𝐶𝐶/𝑉𝑉𝑠𝑠,𝐿𝐿𝐿𝐿𝐿𝐿 = 94 dB
Amplifier requirements (example) 16
Alessandro Spinelli – Electronics 96032
• Wheatstone bridge and sensitivity• Effect of wire resistance• Temperature compensation• Sensors: general definitions
Outline 17
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• In remotely located bridges, cable resistances and noise pickupare the biggest problems
• Cable resistances give an offest error (which can be compensated), but…
• Changes in cable resistances during operation (e.g., with temperature) produce an error signal (gain error) at the bridge output
Wiring resistance in remote sensor 18
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2-wire connection 19
𝑉𝑉𝑠𝑠 = 𝑉𝑉𝑐𝑐𝑐𝑐 −12
+𝑅𝑅(1 + 𝑥𝑥) + 2𝑅𝑅𝐿𝐿𝑅𝑅(2 + 𝑥𝑥) + 2𝑅𝑅𝐿𝐿
≈ 𝑉𝑉𝑐𝑐𝑐𝑐𝑥𝑥4
+𝑅𝑅𝐿𝐿2𝑅𝑅
𝑅𝑅(1 + 𝑥𝑥)𝑅𝑅
𝑅𝑅 𝑅𝑅𝑅𝑅𝐿𝐿
𝑅𝑅𝐿𝐿𝑉𝑉𝑐𝑐𝑐𝑐
𝑉𝑉𝑠𝑠
Alessandro Spinelli – Electronics 96032
• We could add 2𝑅𝑅𝐿𝐿 in series to the lower-left bridge resistor• The problem now is the temperature dependence of 𝑅𝑅𝐿𝐿: 𝑉𝑉𝑐𝑐𝑐𝑐 = 10 V, 𝑅𝑅 = 350 Ω, 𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚 = 1%, 𝑅𝑅𝐿𝐿 = 10 Ω
𝑉𝑉𝑜𝑜 𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚 = 10370 + 3.5720 + 3.5
−370720
= 23.52 mV
When temperature is included (𝑇𝑇𝑇𝑇𝑅𝑅 = 0.385% /°C, Δ𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚 = 10°C)
𝑉𝑉𝑜𝑜 0 = 10370 + 0.77720 + 0.77
−370720
= 5.19 mV
𝑉𝑉𝑜𝑜 𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚 = 10370 + 3.5 + 0.77720 + 3.5 + 0.77
−370720
= 28.66 mV
What if we compensate? 20
Alessandro Spinelli – Electronics 96032
When temperature is accounted for, we have 𝑉𝑉𝑜𝑜 0 = 0 and
𝑉𝑉𝑜𝑜 𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚 = 10360 + 3.5 + 0.385720 + 3.5 + 0.77
−12
= 24.16 mV
3-wire connection 21
𝑉𝑉𝑠𝑠 = 𝑉𝑉𝑐𝑐𝑐𝑐 −12
+𝑅𝑅(1 + 𝑥𝑥) + 𝑅𝑅𝐿𝐿𝑅𝑅(2 + 𝑥𝑥) + 2𝑅𝑅𝐿𝐿
≈ 𝑉𝑉𝑐𝑐𝑐𝑐𝑥𝑥4
1 −𝑥𝑥2−𝑅𝑅𝐿𝐿𝑅𝑅
𝑅𝑅(1 + 𝑥𝑥)𝑅𝑅
𝑅𝑅 𝑅𝑅𝑅𝑅𝐿𝐿
𝑅𝑅𝐿𝐿
𝑉𝑉𝑠𝑠 𝑅𝑅𝐿𝐿𝑉𝑉𝑐𝑐𝑐𝑐
(it is 24.19 mVwhen ΔT = 0)
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Kelvin (4-wire) connection 22
𝑉𝑉𝑐𝑐𝑐𝑐
𝑉𝑉𝑐𝑐𝑐𝑐′
𝑅𝑅𝑅𝑅
𝑅𝑅 𝑅𝑅
𝑅𝑅𝐿𝐿
𝑅𝑅𝐿𝐿
𝑅𝑅𝐿𝐿
𝑅𝑅𝐿𝐿
𝑅𝑅𝐿𝐿
𝑅𝑅𝐿𝐿𝑉𝑉𝑠𝑠
Alessandro Spinelli – Electronics 96032
• The 3-wire method works well for remote elements several tensof meters away
• Connecting wires must have the same characteristics• Stability of 𝑉𝑉𝑐𝑐𝑐𝑐 remains a concern• The 4-wire connection is required for remote bridges, e.g. with 4
active elements• The Kelvin connection is actually a six-lead assembly. Constant-
current excitation can reduce it to 4
Comparison 23
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• Wheatstone bridge and sensitivity• Effect of wire resistance• Temperature compensation• Sensors: general definitions
Outline 24
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• The bridge output is𝑉𝑉𝑠𝑠 = 𝑉𝑉𝑐𝑐𝑐𝑐𝑥𝑥 = 𝑉𝑉𝑐𝑐𝑐𝑐𝛼𝛼𝛼𝛼
𝛼𝛼 =1𝑅𝑅0
�𝑑𝑑𝑅𝑅𝑑𝑑𝛼𝛼 𝑅𝑅0
• In reality, 𝛼𝛼 = 𝛼𝛼(𝑇𝑇), which introduces inaccuracies in the output (unless we are measuring the temperature)
Temperature dependence 25
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𝑑𝑑𝑉𝑉𝑠𝑠𝑑𝑑𝑇𝑇
= 𝛼𝛼 𝛼𝛼𝑑𝑑𝑉𝑉𝑐𝑐𝑐𝑐𝑑𝑑𝑇𝑇
+ 𝑉𝑉𝑐𝑐𝑐𝑐𝑑𝑑𝛼𝛼𝑑𝑑𝑇𝑇
= 0
⇒1𝑉𝑉𝑐𝑐𝑐𝑐
𝑑𝑑𝑉𝑉𝑐𝑐𝑐𝑐𝑑𝑑𝑇𝑇
= −1𝛼𝛼𝑑𝑑𝛼𝛼𝑑𝑑𝑇𝑇
= −𝛽𝛽
The bridge excitation voltage must be temperature-dependent and have an opposite rate of variation with respect to 𝛼𝛼
Temperature compensation 26
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T-independent resistor in series
𝑉𝑉𝑐𝑐𝑐𝑐 = 𝑉𝑉𝑐𝑐𝑐𝑐′𝑅𝑅𝐿𝐿
𝑅𝑅𝐿𝐿 + 𝑅𝑅𝑇𝑇𝑑𝑑𝑉𝑉𝑐𝑐𝑐𝑐𝑑𝑑𝑇𝑇
= 𝑉𝑉𝑐𝑐𝑐𝑐′𝑅𝑅𝑇𝑇
𝑅𝑅𝐿𝐿 + 𝑅𝑅𝑇𝑇 2𝑑𝑑𝑅𝑅𝐿𝐿𝑑𝑑𝑇𝑇
1𝑉𝑉𝑐𝑐𝑐𝑐
𝑑𝑑𝑉𝑉𝑐𝑐𝑐𝑐𝑑𝑑𝑇𝑇
=𝑅𝑅𝑇𝑇
𝑅𝑅𝐿𝐿 + 𝑅𝑅𝑇𝑇1𝑅𝑅𝐿𝐿
𝑑𝑑𝑅𝑅𝐿𝐿𝑑𝑑𝑇𝑇
−𝛽𝛽 = 𝑇𝑇𝑇𝑇𝑅𝑅𝑅𝑅𝑇𝑇
𝑅𝑅𝐿𝐿 + 𝑅𝑅𝑇𝑇
27
= 1𝑅𝑅𝑑𝑑𝑅𝑅𝑑𝑑𝑇𝑇
= 𝑇𝑇𝑇𝑇𝑅𝑅(for a bridge with
equal resistances)= −𝛽𝛽
𝑅𝑅𝑅𝑅
𝑅𝑅 𝑅𝑅𝑉𝑉𝑐𝑐𝑐𝑐
𝑉𝑉𝑐𝑐𝑐𝑐′ 𝑅𝑅𝑇𝑇
Alessandro Spinelli – Electronics 96032
𝑅𝑅𝑇𝑇 = −𝛽𝛽
𝛽𝛽 + 𝑇𝑇𝑇𝑇𝑅𝑅𝑅𝑅
• Very simple and popular solution, but with a few disadvantages: Only possible if 𝛽𝛽 < 0 and TCR > |𝛽𝛽| 𝛽𝛽 and 𝑇𝑇𝑇𝑇𝑅𝑅 must be precisely known Output signal is reduced
• Usually adopted in the range 25 ± 15°C
Result 28
Alessandro Spinelli – Electronics 96032
• Wheatstone bridge and sensitivity• Effect of wire resistance• Temperature compensation• Sensors: general definitions
Outline 29
Alessandro Spinelli – Electronics 96032
• Convert an input physical property (the stimulus) to a differentone (usually an electrical signal). Sensors are «energy converters»
• You can find many disquisitions on the difference betweensensors and transducers, which I gladly leave to your rainy dayreading
Sensors 30
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• Measurand Temperature, pressure, velocity, current,…
• Detection mean Biological, chemical, electrical, mechanical,…
• Sensor material Semiconductor, organic, liquid,…
• Field of application Scientific, industrial, medical,…
Sensor classification 31
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• Static parameters Transfer function, accuracy, resolution,…
• Dynamic parameters Frequency response, settling time,…
• Other parameters Operating and storage conditions, reliability,…
Sensor characteristics 32
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I-O characteristic 33
From [1] From [2] From [3]
When used as detectors, the inverse function is needed
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• Defined as the ratio between output and input variations
𝛼𝛼 =𝑑𝑑𝛼𝛼𝑜𝑜𝑑𝑑𝛼𝛼𝑖𝑖
• Linear sensors have constant sensitivity• Linear approximations can be used in other cases, over a limited
input range. Otherwise, data processing is required
Sensitivity 35
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Maximum difference between the real transfer function and its linear approximation
(Non)Linearity error 36
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Different straight lines can be defined (end points, least squares,…), giving different NL errors
Which linear characteristic? 37
From [5]
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Adopts the straight line thatminimizes the maximum (absolute) NL error
Independent nonlinearity 38
From [4]
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• Is the smalllest increment in stimulus that can be sensed, specified in absolute quantity or percentage of FS input
• Resolution is ultimately determined by the noise of the sensoritself
• Other factors (noise of electronics front-end, digitization,…) can further degrade it
Resolution 39
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• Is the ability of the sensor to reproduce the same result afterrepetitive experiments
• Precision is not resolution A digital clock may have ms resolution but worse precision The terms are often (mis)used interchangeably
Precision 40
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Accuracy
• Accuracy is the maximum deviation from the ideal value
• The average value should be considered for each sensor in presence of a strong random component
41
From [4]
Deviationsfrom the ideal (true) value
Alessandro Spinelli – Electronics 96032
Accuracy vs. precision 42
PrecisionPrecision
Precision PrecisionFrom [6], modified
Alessandro Spinelli – Electronics 96032
• Frequency response• Response/settling time• Bandwidth• …
Dynamic parameters 43
Alessandro Spinelli – Electronics 96032
1. http://www.analog.com/media/en/training-seminars/design-handbooks/Practical-Design-Techniques-Sensor-Signal/Section2.PDF
2. J. Fraden, «Handbook of modern sensors», Springer (2004)3. www.scienceprog.com/characteristics-of-sensors-and-transducers/4. sales.hamamatsu.com/assets/applications/SSD/nmos_kmpd9001e04.pdf5. iopscience.iop.org/00223727/45/22/225305/article6. J. Fraden, «Handbook of modern sensors», Springer (2010)7. J. Webster, ed., «Measurement, instrumentation and sensor handbook», CRC
Press (1999)8. www.sensortips.com/pressure/accuracy-vs-resolution/
References 44