Lecture 4: Sensor interface circuits

Transcription

1 Lecture : Sensor interface circuits g eview of circuit theory n oltage, current and resistance n apacitance and inductance n omplex number representations g Measurement of resistance n oltage dividers n Wheatstone Bridge n Temperature compensation for strain gauges g A bridges n Measurement of capacitance n Measurement of impedance

2 oltage, current, resistance and power g oltage n The voltage between two points is the energy required to move a unit of positive charge from a lower to a higher potential. oltage is measured in olts () g urrent n urrent is the rate of electric charge through a point. The unit of measure is the Ampere or Amp (A) g esistance n Given a piece of conducting material connected to a voltage difference, which drives through it a current I, the resistance is defined as g Power g g As you will recall, this is known as Ohm s Law An element whose resistance is constant for all values of is called an ohmic resistor n Series and parallel resistors I n The power dissipated by a resistor is P I I

3 Kirchhoff s Laws g st Law (for nodes) n The algebraic sum of the currents into any node of a circuit is zero g Or, the sum of the currents entering equals the sum of the currents leaving g Thus, elements in series have the same current flowing through them I I I I +I 3 g nd Law (for loops) n The algebraic sum of voltages in a loop is zero g Thus, elements in parallel have the same voltage across them. I 3 A B AB + B + D + DA D 3

4 apacitors and inductors g A capacitor is an element capable of storing charge n The amount of charge is proportional to the voltage across the capacitor Q g is known as the capacitance (measured in Farads) n Taking derivatives dq dt ( ) d dt I d dt n Therefore, a capacitor is an element whose rate of voltage change is proportional to the current through it g Similarly, an inductor is an element whose rate of current change is proportional to the voltage applied across it L di dt g L is called the inductance and is measured in Henrys

5 Frequency analysis g onsider a capacitor driven by a sine wave voltage I(t) (t) sin(ωt) n The current through the capacitor is d dt d dt ( sin( W) ) cos( t) I g Therefore, the current phase-leads the voltage by 9 and the ratio of amplitudes is (t) I(t) g What happens when the voltage is a D source? 5

6 oltages as complex numbers g At this point it is convenient to switch to a complex-number representation of signals n ecall that e jϕ cosϕ+jsin ϕ ircuit voltage versus time: (t) cos(ωt+ϕ) cos(ωt+ϕ) e jϕ Multiply by e jϕ and take real part From [HH89] omplex number representation: e jϕ a+jb g Applying this to the capacitor (t)/i(t) relationship I sin cos( ( W) cos( W ) t) I d dt e j cos( j e t) j H j 6

7 7 Impedance g Impedance () is a generalization of resistance for circuits that have capacitors and inductors n apacitors and inductors have reactance, while resistors have resistance g Ohm s Law generalized g Impedance in series and parallel / j & j - & j L I N P N S L L

8 Example: High-pass filter g High pass filter n The current through cap and resistor is I in + j n The output voltage is equal to the voltage differential across the resistor out n If we focus on amplitude and ignore phase in & I + j in & in out out + j in & in + & in ( ) + g Asymptotic behavior out g orner frequency ONE log log -3. db + in 8

9 Measurement circuits g esistance measurements n oltage divider (half-bridge) n Wheatstone bridge g A.. bridges n Measurement of capacitance n Measurement of impedance 9

10 oltage divider g Assumptions n Interested in measuring the fractional change in resistance x of the sensor: S (+x) g is the sensor resistance in the absence of a stimuli n Load resistor expressed as L k for convenience g The output voltage of the circuit is L k out S (+x) out S + S L ( + x) + x ( + x) + k + x + k g Questions n What if we reverse S and L? n How can we recover S from out? out / k. k k x

11 oltage divider g What is the sensitivity of this circuit? S d dx out d dx ( + x + k) ( + x) ( + x + k) ( + x + k) g For which L do we achieve maximum sensitivity? ds dk k + x + x + k n This is, the sensitivity is maximum when L S ( + x + k) k( + x + k) ( + x + k) d k ( ) dk + x + k S k k x k k + x

12 Wheatstone bridge g A circuit that consists of two dividers n A reference voltage divider (left) n A sensor voltage divider g Wheatstone bridge operating modes n Null mode g adjusted until the balance condition is met: n out n Deflection mode Advantage: measurement is independent of fluctuations in g The unbalanced voltage out is used as the output of the circuit 3 out 3 (+x) out n Advantage: speed

13 Wheatstone bridge g Assumptions n Want to measure sensor fractional resistance changes S (+x) n Bridge is operating near the balance condition: k out / k k k. g The output voltage becomes x out k k ( + x) ( x) k ( + x) kx + ( + x) k + ( + k)( + k + x) 3

14 Wheatstone bridge g What is the sensitivity of the Wheatstone bridge? d S dx out k ( + k)( + k + x) ( + k)( + k + x) kx( + k) ( + k) ( + k + x) k d dx ( + k + x) kx n The sensitivity of the Wheatstone bridge is the same as that of a voltage divider g You can think of the Wheatstone bridge as a D offset removal circuit g So what are the advantages, if any, of the Wheatstone bridge?

15 oltage divider vs. Wheatstone for small x g The figures below show the output of both circuits for small fractional resistance changes n The voltage divider has a large D offset compared to the voltage swing, which makes the curves look flat (zero sensitivity) g Imagine measuring the height of a person standing on top of a tall building by running a large tape measure from the street n The sensitivity of both circuits is the same! g However, the Wheatstone bridge sensitivity can be boosted with a gain stage n Assuming that our DAQ hardware dynamic range is -5D, <x<. and k, estimate the maximum gain that could be applied to each circuit.5 x -3.8 k. Divider.6. k Wheatstone.5 k k. k.5 k x x 5

16 ompensation in a Wheatstone bridge g Strain gauges are quite sensitive to temperature n A Wheatstone bridge and a dummy strain gauge may be used to compensate for this effect g The active gauge A is subject to temperature (x) and strain (y) stimuli g The dummy gauge D, placed near the active gauge, is only subject to temperature n The gauges are arranged according to the figures below n The effect of (+y) on the right divider cancels out D (+y) out A (+x)(+y) From [am96] 6

17 A bridges g The structure of the Wheatstone bridge can be used to measure capacitive and inductive sensors n esistance replaced by generalized impedance n D bridge excitation replaced by an A source g The balance condition becomes 3 A out n which yields two equalities, for real and imaginary components 3 X 3 3 X X + X 3 3 X X + X g There is a large number of A bridge arrangements n These are named after their respective developer X 7

18 A bridges g apacitance measurement n Schering bridge n Wien bridge g Inductance measurement n Hay bridge n Owen bridge 3 A A x x x L x Schering Hay 3 A A x x L x x Wien Owen 8

19 eferences [HH89] [am96] [Maz87] [Die7] [PAW9] [Gar9] [Fdn97] P. Horowitz and W. Hill, 989, The Art of Electronics, nd Ed., ambridge University Press, ambridge, UK D.. amsay, 996, Principles of Engineering Instrumentation, Arnold, London, UK F. F. Mazda, 987, Electronic instruments and measurement techniques, ambridge Univ. Pr., New York A. J. Diefenderfer, 97, Principles of electronic instrumentation, W. B. Sanuders o., Philadelphia, PA.. Pallas-Areny and J. G. Webster, 99, Sensors and Signal onditioning, Wiley, New York J. W. Gardner, 99, Microsensors. Principles and Applications, Wiley, New York. J. Fraden, 997, Handbook of Modern Sensors. Physics, Designs and Applications, AIP, Woodbury, NY 9