Example Sheet 2 With Solutions
1. LongQuestionExample
(a) Explain the Seebeck effect and how it is used in thermocouple design. [8 marks]
An EMF is generated across a material with a temperature gradient. The size of effect depends on the conductor and the Seebeck parameter for that material. The Seebeck series relates the size of the effect for different materials and includes: Bi,Ni,Co,Pt,Cu,Mn,Hg,Pb,Sn,Au,Ag,Zn,Cd,Fe,As,Sb,Te. Making a thermocouple from more than one material is required and the size fo the effect can be enhanced by chosing the right combinations – the bigger the distance between materials on the Seebeck series the bigger the effect.
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You can’t measure the effect directly as when you add connections it changes things so you combine two materials to make a junction with a stable effect.
Thermocouples combine two materials, The EMF can be measured between the junctions. This EMF is determined by the difference in the Seebeck coefficients for the materials and the difference in temperature between the hot and cold junctions.
The Seebeck parameter changes with temperature so this simple approach is only good for small range of temperature changes.
(b) Sketch a graph for how the EMF of a thermocouple changes with temperature, you should add labels and explain the neutral temperature, the inversion temperature and explain how the sensitivity can be obtained from this graph.
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The neutral temperature is when the device has no sensitivity i.e. when 𝑑𝑉 = 0,
which is when T=Tn = -a/2b
The Inversion Temperature is when T = Ti= 0 = T0-a/b here the EMF continues to change but now has opposite sign.
Tn is property of the thermocouple, due to materials used and is independent of cold junction temperature.
Tc is the cold junction temperature, (zero in example graph above)
Ti is dependant on Tc as Tn is (Ti+Tc)/2
S is the local gradient of this curve so for a particular device and operating range it can be calculated.
If the temperature range over which the device will be used is small then a linear approximation can be used – in the example graph, the region around 0-50 degrees is almost linear.
(c) Using table X work out the sensitivity of a thermocouple which is made from copper and constantan.
S = 6.5-(-35) = 41.5 μV/K
(d) What is the measured voltage if the hot junction is at 50 degrees C and the cold
junction is at room temperature?
Room temperature is 298K so:
41.5 μV/K x (323K-298K) = 1.0375mV
(e) Suggest a circuit and the required gain to match this to a digitisation circuit operating over +/-1 Volts with a depth of12 bits if the temperature is to be measured to 0.25K accuracy
Full range is 1- -1 = 2V
2V/2^12 bits = LSB = 4.8828e-4 V/bit
LSB / S*Gain = 4.8828e-4 / (41.5e-6*Gain) = 11.6/Gain k-1 We need 0.25 k resolution so
0.25=11.6/Gain
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Voltage Gain is therefore = 46.4 Gain in dB = 20log10(46.4) = 33dB
Circuit suggestion is the non inverting op AMP configuration, here gain is 1+Rf/R2, BW usually low for temperature measurements so it is easy to get high gain with stable operation.
We could use inverting amplifier arrangement as well as well as ADC is +/- 1v range.
(f) What is the max temp that can be measured assuming the cold junction is at 0 degrees C
Delta T = 1V range / (41.5e-6*46.4) = +519 deg C
2. What is the frequency of a 550nm light source? How much energy does a single photon contain?
f = 2.998×108 / λ
E = hf = 6.626×10-34 x f
λ (nm) f (THz) E (J)
550 545.1 3.612 x 10-19
3. Whatisthephotonflux(photonspersecond)foralaserwiththefollowing specifications? λ = 800 nm, P = 100 mW
E = hf = hc / λ = 2.483 x 10-19 J
P = 100 x 10-3 Js-1
N = P / E = 4.03 x 1017 photons.s-1
4. A photodiode has a dark leakage current of 10 nA, a shunt resistance of 750 MΩ and a responsivity of 0.6 AW-1 at the wavelength we are using, if the bandwidth B is 1 Hz: standard room temperature (25 degrees Celsius):
calculate: shot noise current, Johnson noise current, total noise current, NEP.
shot noise current = 5.66 x 10-14 A
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=√2×1.6×10−19 ×10×10−9 ×1
Johnson noise current = 4.60 x 10-15 A
= √4×1.33×10−23 ×298×1 750 × 106
total noise current = 5.68 x 10-14 A
Nep = total noise current / Resposnivity NEP = 5.68 x 10-14 /0.6 = 9.47x 10-14 W
5. A signal has the form: 𝑠(𝑡) = 𝑎 + 𝑏 sin(2𝜋𝑓𝑡) volts Where a = 2.5V, b = 1mV and f=1MHz.
If the ADC being used to digitise this signal has an input range of 0-5V, and we require at least 10 levels to capture the full scale of the signal. How many bits must the ADC have and what would be a suitable sampling frequency.
Full signal swing is +/-1mV = 2mV, so LSB should be full swing / 10 = 0.2mV
The range on the ADC is 0-5V so the number of levels in this range is 5V/0.2mW = 25000
The nearest power of two is ceil(log(25000)/log(2)) = ceil(14.6) so need 15 bits, in reality a 16 bit ADC will be used (as 15 bit devices not common)
Sampling frequency needs to be at least 2f but better to use higher than this, typically 4-10 times ok, so a 10MHz 16 bit ADC seems a sensible choice.
6. Explain how quantisation noise arises during analogue to digital conversion and how the size of the noise is related to the number of bits used.
In digitising the data we have errors (e) for each point that is pulled down or rounded up to the next LSB.
The number of bits (N) and the input range control the size of the LSB, this is the minimum change in voltage (∆) that can take place. This ∆ controls the size of the error.
∆= 2𝑁 𝑖𝑛𝑝𝑢𝑡 𝑟𝑎𝑛𝑔𝑒
The max error is +/- Δ/2 and this Quantisation error is uniformly distributed across that range and integrates to 1. Expression for the noise can be determined in terms of delta as:
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2 1 2 1𝑒3 2 1 ቀ2ቁ ቀ−2ቁ 𝜎𝑄=න∆𝑒𝑑𝑒=ቈ∆3∆=∆൦3− 3 ൪
2 1∆3 −∆3 ∆2 ∆
𝜎𝑄 = ∆ ቈ24 − 24 = 12 𝜎𝑄 = ξ12 = 𝑉𝑄_𝑟𝑚𝑠
7. Describe how signals and signal processing are used in biomedical applications. Explain the impact this has to the wider world giving examples of specific applications. Discuss the types of techniques and approaches used.
Medical ultrasound uses an array of piezoelectric transducers to produce images of soft tissues according to mechanical contrast, particularly at discontinuities in acoustic properties. Although various modes of operation are available, typically B- scans are used to produce images along the direction of propogation of ultrasound energy – where back-reflections are presented as varying contrast along each penetration vector. Applications include scanning the abdomen of pregnant women to track fetal development and morphology, or to detect injuries or malformations of soft tissues. Impacts include the diagnosis of disease, abnormalities, ultrasound- guided surgery and injection.
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