Download - Wheatstone bridge and sensors - Politecnico di Milanohome.deib.polimi.it/spinelli/corsi/ele/L6.pdf•Wheatstone bridge and sensitivity •Effect of wire resistance •Temperature compensation

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


Acquisition chain 3

Sensor Filter ADC

small signal

noise

amplifiedsignal

amplifiednoise

amplifiedsignal

reducednoise

Amp

next lessons


• 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


• Wheatstone bridge and sensitivity• Effect of wire resistance• Temperature compensation• Sensors: general definitions

Outline 5


• 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


• 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


Insensitive to ground potential and resistance fluctuations

Wheatstone bridge 8

𝑉𝑉𝑠𝑠 = 𝑉𝑉𝑐𝑐𝑐𝑐𝑅𝑅4

𝑅𝑅3 + 𝑅𝑅4−

𝑅𝑅2𝑅𝑅1 + 𝑅𝑅2

𝑉𝑉𝑐𝑐𝑐𝑐 𝑉𝑉𝑠𝑠

𝑅𝑅1 𝑅𝑅3

𝑅𝑅2 𝑅𝑅4


• 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


Bridge sensitivity to R 10


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


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


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


• 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


Bridge amplifiers 15

A

A


𝑅𝑅 𝑅𝑅


𝑅𝑅/2

𝑅𝑅/2𝑉𝑉𝑐𝑐𝑐𝑐/2

𝑉𝑉𝑐𝑐𝑐𝑐 𝑥𝑥/4

𝑉𝑉𝑜𝑜

𝑉𝑉𝑜𝑜


• 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



Outline 17


• 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


2-wire connection 19

𝑉𝑉𝑠𝑠 = 𝑉𝑉𝑐𝑐𝑐𝑐 −12

+𝑅𝑅(1 + 𝑥𝑥) + 2𝑅𝑅𝐿𝐿𝑅𝑅(2 + 𝑥𝑥) + 2𝑅𝑅𝐿𝐿


+𝑅𝑅𝐿𝐿2𝑅𝑅


𝑅𝑅 𝑅𝑅𝑅𝑅𝐿𝐿

𝑅𝑅𝐿𝐿𝑉𝑉𝑐𝑐𝑐𝑐

𝑉𝑉𝑠𝑠


• 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


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


1 −𝑥𝑥2−𝑅𝑅𝐿𝐿𝑅𝑅


𝑅𝑅 𝑅𝑅𝑅𝑅𝐿𝐿

𝑅𝑅𝐿𝐿

𝑉𝑉𝑠𝑠 𝑅𝑅𝐿𝐿𝑉𝑉𝑐𝑐𝑐𝑐

(it is 24.19 mVwhen ΔT = 0)


Kelvin (4-wire) connection 22


𝑉𝑉𝑐𝑐𝑐𝑐′

𝑅𝑅𝑅𝑅

𝑅𝑅 𝑅𝑅

𝑅𝑅𝐿𝐿

𝑅𝑅𝐿𝐿

𝑅𝑅𝐿𝐿

𝑅𝑅𝐿𝐿

𝑅𝑅𝐿𝐿

𝑅𝑅𝐿𝐿𝑉𝑉𝑠𝑠


• 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



Outline 24


• 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


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


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


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



Outline 29


• 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


• 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


• Static parameters Transfer function, accuracy, resolution,…

• Dynamic parameters Frequency response, settling time,…

• Other parameters Operating and storage conditions, reliability,…

Sensor characteristics 32


I-O characteristic 33

From [1] From [2] From [3]

When used as detectors, the inverse function is needed


I-O ranges 34

From [4]

Full-scale input

Full-scale output


• 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


Maximum difference between the real transfer function and its linear approximation

(Non)Linearity error 36


Different straight lines can be defined (end points, least squares,…), giving different NL errors

Which linear characteristic? 37

From [5]


Adopts the straight line thatminimizes the maximum (absolute) NL error

Independent nonlinearity 38

From [4]


• 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


• 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


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


Accuracy vs. precision 42

PrecisionPrecision

Precision PrecisionFrom [6], modified


• Frequency response• Response/settling time• Bandwidth• …

Dynamic parameters 43


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