# Last lecture. Today s menu. Capacitive sensing elements. Capacitive sensing elements (cont d...) Examples. General principle (A) (B) (C)

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1 Last lecture esistive sensing elements: Displacement sensors (potentiometers). Temperature sensors. Strain gauges. Deflection briges. Toay s menu Capacitive sensing elements. Inuctive sensing elements. eactive Deflection briges. Electromagnetic sensing elements. Thermoelectric sensing elements. Elastic sensing elements. Piezoelectric sensing elements. 1 2 Capacitive sensing elements General principle Capacitive sensing elements (cont...) Examples Consier two metal plates with areas A, separate by a istance by some ielectric meium: (A) (B) (C) x x x E 1 E 2 E A (D) (E) l The capacitance is then given by C = ε 0εA. 3 4

2 Capacitive sensing elements (cont...) Examples (cont..) (A) Variable isplacement sensor: C = ε 0εA + x. (B) Variable area isplacement sensor: C = ε 0ε (A wx). Capacitive sensing elements (cont...) Examples (cont..) (D) Capacitive pressure sensor: ΔC C = (1 ν2 )a 2 16Et 3 P. (E) Differential capacitive isplacement sensor: C 1 = εε 0A +x, C 2 = εε 0A x (C) Variable area isplacement sensor: C = ε 0w [ε 2l (ε 2 ε 1 )x]. 5 6 Capacitive sensing elements (cont...) Capacitive sensing elements are incorporate in a.c. eflection brige circuits or oscillator circuits. The sensor is not purely capacitive, but also has a resistance in parallel to represent losses in the ielectric. The quality of the ielectric is often expresse in terms of the loss tangent, tan δ = 1 ωc Inuctive sensing elements Variable inuctance/reluctance sensors First, some comments on magnetic circuits: In an electrical circuit, an electromotive force (e.m.f.) rives the current through the circuit e.m.f. = current resistance In a magnetic circuit, the magnetomotive force (m.m.f.) which rives a flux φ through a magnetic circuit is: m.m.f. = flux reluctance = φ. The reluctance limits the flux through the circuit, just as resistance limits current flow through an electric circuit. 7 8

3 Inuctive sensing elements (cont...) Inuctive sensing elements (cont...) The flux in one turn is given by φ = ni weber The total flux is given by N = nφ = n2 i i n turns The reluctance of a magnetic circuit is given by: = l μμ 0 A, where l is the total length of the flux path, μ is the relative permeability of the circuit material, μ 0 =4π 10 7 H/m is the permeability of free space, The self-inuctance is efine as A is the cross-sectional area of the flux path. L = N i = n Inuctive sensing elements (cont...) The inuctive isplacement sensor Inuctive sensing elements (cont...) The inuctive isplacement sensor (cont... Air gap L core permeability c i n turns raius r air gap armature permeability A 11 12

4 Inuctive sensing elements (cont...) The inuctive isplacement sensor (cont... The total reluctance is TOT = COE + GAP + AMATUE eactive eflection briges Typical capacitive brige 2 3 where COE = GAP = AMATUE = μ 0 μ C r 2 2 μ 0 πr 2 μ 0 μ A rt C 0 E Th V S ~ C h eactive eflection briges Typical inuctive brige E Th Electromagnetic sensing elements These elements are use for measure linear an angular velocity an are base on Faraay s law. This means that if a flux N linke to a conuctor is changing with time, then the back electromotive force inuce in the conuctor is E = N t, i.e. proportional to the rate of change of the flux N. L 1 L 2 V S ~ 15 16

5 Thermoelectric sensing elements Thermoelectric or thermocouple sensing elements are commonly use for measuring temperature. The reason for this is that, if two metals A an B are joine together, there will be a ifference in electrical potential across the junction. The potential epens on the types of metal an the temperature. A Thermoelectric sensing elements (cont...) The junction potential can be escribe by a power series of the form E T = a 1 T + a 2 T 2 + a 3 T For the temperature ifference between two junctions, the potential ifference is then E T 1 E T 2 = a 1 (T 1 T 2 )+a 2 (T 2 1 T 2 2 )+a 3 (T 3 1 T 2 2 )+... E T1 E T2 T 1 T 2 B ET 1 - E T Elastic sensing elements The general principle of elastic sensing elements is to convert a force to an output isplacement, which is then escribe by a change in impeance. Elastic elements are often use to measure: torque = force istance pressure = force / area acceleration = force / mass. The ynamic behavior of these can often be escribe by secon-orer systems. See the text book for etails. Piezoelectric sensing elements If a force is applie to a crystal, the atoms of the crystal are isplace from their normal positions, as x = 1 k F, where k is the stiffness of the crystal. The ynamics can be escribe by a secon-orer system. In a piezoelectric crystal, the isplacement results in a charge q = Kx = K k F

6 Piezoelectric sensing elements (cont...) The piezoelectric effect is reversible, which means that if we apply a charge across a crystal, its imensions will change accoringly. In orer to measure the charge q the faces of the crystals are coate with metal electroes, resulting in a capacitor, with capacitance C N = ε 0εA, t where A is the area, t is the thickness ε the permittivity of the crystal, an ε 0 the permittivity of free space. Piezoelectric sensing elements (cont...) The crystal can therefore be represente as a charge generator q in parallel with a capacitance C N, or as an equivalent Norton circuit with a current source i N in parallel with C N, where i N = q t = K x t. In the Laplace omain this is (in transfer function form) Δī N (s) =Ks. Δ x Piezoelectric sensing elements (cont...) If the crystal is connecte to a resistive loa using a capacitive cable, we have the following system: Piezoelectric sensing elements (cont...) The transfer function of the system, incluing the piezoelectric crystal, a capacitive cable, an a recorer is: G (s) = ΔV L Δi N (s) = L s (C N + C c )+1, i N C N C C L an relating it to the input force we get ΔV L ΔF (s) =ΔV L Δi N Δi N Δx Δx ΔF Piezoelectric crystal Capacitive cable ecorer See the text book for the etails of this erivation. The piezoelectric crystal generates a current, proportional to the velocity of a force acting on its surface

7 Summary Capacitive sensing elements. Inuctive sensing elements. eactive Deflection briges. Electromagnetic sensing elements. Thermoelectric sensing elements. Elastic sensing elements. Piezoelectric sensing elements. Next lecture The remainer of chapter 9. Have covere only eflection briges so far. Ieal amplifiers are covere in other courses. ea up on this, an we can focus on limitations an errors. There are more etails an some other examples of sensing elements in the book. ea this on your own. The elements presente at the lecture are only examples. Make sure you get the big picture ecommene exercises Questions? , 8.4,

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