Sensors And Actuators Sensor Physics Sander Stuijk

Transcript

Sensors and Actuators Sensor Physics Sander Stuijk ([email protected]) Department of Electrical Engineering Electronic Systems 4 SENSOR CHARACTERISTICS (Chapter 2) 5 Resistance  resistance of a material is defined as R  V i l  resistance depends on geometrical factors R   a  length of wire (l)  cross-sectional area (a) l m l  resistance depends on temperature R    2 a ne  a  number of free electrons (n)  mean time between collisions (τ)  resistor as temperature sensor  some types have almost linear relation between temperature t (°C) and resistance R (Ω)  example: platinum (PT100) sensor   Rt  R0 1  39.08 104 t  6 Transfer function  sensors translates input signal to electrical signal  transfer function gives relation between input and output signal Vout  R5 (3.01kΩ) V1 (5V) Rt, R0 = 100 Ω  Vout  Rt  R0 1  39.08 104 t  Rt V1 Rt  R5 7 Signal processing R5 (3.01 kΩ) R2 (11.8 kΩ) R1 (11 kΩ) + - V1 (5V) Rt R3 (105 kΩ) Vout R4 (12.4 kΩ) R5 // Rt R2 // Rt V  Vout  V1 R5 // Rt  R2 R2 // Rt  R5 R2 // Rt V1 R2 // Rt  R5 V  V  Vout  R5 // Rt R4  R4  R3 R5 // Rt  R2 V  Vout R4 R3  R4 Signal processing Vout sensor output voltage (v) 8 temperature (°C)  sensitivity increased from 0.63mV/°C to 6.67mV/°C  non-linearity has also been decreased... Nonlinearity 9  assumption: resistance has a linear dependency on temperature error (V) voltage (V) “ideal” linear transfer function “real” transfer function temperature (°C) temperature (°C) 10 Nonlinearity error (V) error (°C)  assumption: resistance has a linear dependency on temperature  error can be expressed as deviation from actual temperature temperature (°C) temperature (°C) 11 Nonlinearity  nonlinearity is the maximal deviation from the linear transfer function  nonlinearity must be deduced from the actual transfer function or from a calibration curve error (°C) ideal transfer function nonlinearity real transfer function temperature (°C) 12 Nonlinearity  nonlinearity is the maximal deviation from the linear transfer function  nonlinearity must be deduced from the actual transfer function or from a calibration curve raw sensor output R5 (3.01kΩ) R2 (11.8kΩ) error (°C) R1 (11kΩ) + V1 (5V) PT100 (100Ω) R3 (105kΩ) Vout R4 (12.4kΩ) temperature (°C)  nonlinearity can be reduced with signal processing electronics 13 Nonlinearity  nonlinearity is the maximal deviation from the linear transfer function  nonlinearity must be deduced from the actual transfer function or from a calibration curve raw sensor output error (°C) error (°C) network output temperature (°C) temperature (°C)  ~10x reduction in nonlinearity due to signal processing electronics 14 Errors  errors are deviations from the “ideal” transfer function  sources  nonlinearity  materials used  construction tolerances  aging  operational errors  calibration errors  impedance matching errors  noise  .... 15 Errors  errors are deviations from the “ideal” transfer function  types of errors  static errors: not time dependent  dynamic errors: time dependent  systemic errors: errors are constant at all times and conditions  random errors: different errors in a parameter or at different operating times 16 Accuracy  accuracy is a bound on the maximal deviation of the true input for any output of the sensor  example: if the accuracy is ±3°C, the measured temperature is the true value ±3°C  accuracy may be represented  in terms of measured value (Δ)  in percent of full scale input (%)  in terms of output signal (δ) 17 Accuracy  accuracy may be represented  in terms of measured value (Δ)  in percent of full scale input (%)  in terms of output signal (δ)  LM135 - precision temperature sensor  sensitivity: +10mV/°C  range: -55°C to +150°C  span: 150°C - (-55°C) = 205°C  input full scale: 150°C  output full scale: 4.2V  uncalibrated temp error: ±1°C  what is the accuracy of this sensor? 18 Accuracy  accuracy may be represented  in terms of measured value (Δ)  in percent of full scale input (%)  in terms of output signal (δ)  accuracy of the LM135  measured value: ±1°C  percentage: ±1°C/(150°C)∙100 = ±0.7%  output: ±10mV 19 Errors  errors are deviations from the “ideal” transfer function  sources of errors (seen so far)  nonlinearity  other sources of errors  calibration errors  repeatability  hysteresis  saturation  dead band  ... 20 Calibration error  calibration data is usually supplied by the manufacturer  calibration error is the inaccuracy permitted by the manufacturer when calibrating a sensor in the factory slope error  b   s2  s1 offset error   a  a1  a  s2  s1 measurement with error accurate measurement 21 Calibration and nonlinearity  nonlinearity needs to be considered when calibrating sensor  several calibration methods are used  use range points (line 1)  limit span to useful range and use these range points (line 2)  use tangent of single calibration point (line 3)  use linear best fit (line 4) 4 22 Errors  errors are deviations from the “ideal” transfer function  sources of errors (seen so far)  nonlinearity  calibration errors  other sources of errors  repeatability  hysteresis  saturation  dead band  ... 23 Repeatability  repeatability is the failure of a sensor to represent the same value under identical conditions when measured at different times  source: thermal noise, buildup charge, material plasticity, ... r   100% FS 24 Hysteresis  hysteresis is the deviation of the sensor’s output at any given point when approached from two different directions  caused by electrical or mechanical properties  mechanical friction  magnetization  thermal properties  loose linkages h  h 25 Example – magnetoresistive sensor  sensor can be used to measure the position of magnetic objects  resistivity of magnetoresistive sensor has relation with strength and position of magnetic field  sensor moved along X axis  Hx provides auxiliary field  variation in Hy is a measure for the displacement  sensor output voltage V0 follows Hy curve Hy Hx 26 Example – magnetoresistive sensor  sensor can be used to measure the position of magnetic objects  resistivity of magnetoresistive sensor had relation with strength and position of magnetic field  hysteresis error  too strong magnet or sensor to close to magnet  Hx exceeds maximal Hx  dipoles flip  sensor has hysteresis loop: ABCD Hy Hx 36 Static and dynamic characteristics  static characteristics  values given for steady state measurement  dynamic characteristics  values of the response to input changes  many sensors have a time-dependent behavior  output signal needs time to adapt to change in input  example - LM135 temperature sensor  voltage step at input  output needs time to settle 37 Dynamic error  dynamic error is the difference between the indicated value and true value of measured quantity when static error is zero  difference in sensor response when input is constant or varies  two important aspects  magnitude of error  speed of response (delay)  different inputs considered when analyzing dynamic characteristics  step (e.g., sudden temperature change)  ramp (e.g., gradual temperature change)  sinusoid (e.g., sound waves)  any real signal can be described as superposition of these signals 38 Transfer function  input-output behavior of sensor captured with constant-coefficient linear differential equation (sensor is linear time-invariant system)  general form linear differential equation d n y(t ) d n1 y(t ) d 1 y(t ) d m x(t ) d m1 x(t ) an  an1  ...  a1  a0 y(t )  bm  bm1  ...  b0 x(t ) n n 1 m m 1 dt dt dt dt dt     y(t) – output quantity x(t) – input quantity t – time ai, bi – constant physical parameters of system  solution to equation can be computed using Laplace transform  transfer function of a system is defined as Y ( s) bm s m  bm1s m1  ...  b1s  b0  X ( s) an s n  an1s n1  ...  a1s  a0 39 Transfer function  transfer function does not capture the instantaneous ratio of timevarying quantities Y ( s) bm s m  bm1s m1  ...  b1s  b0  X ( s) an s n  an1s n1  ...  a1s  a0  inverse Laplace transform is needed to go back to time domain dx(t ) d 2 x(t ) y(t )  Ax (t )  B C  ... 2 dt dt  Laplace form is convenient for combining transfer functions  initial condition can be ignored in transformation when all initial conditions are zero  this is true for many practical systems 40 Transfer function  complex sensors combine several transducers and a direct sensor  combination of transfer functions of all transducers gives transfer function of complex sensor measured quantity transducer direct sensor amplifier kt s  1 kd s 2 2s  1 2 ka n voltage n sensor measured quantity kt k d k a  s 2 2s  s  1 2   1  n n  voltage 41 Zero-order system  general form linear differential equation d n y(t ) d n1 y(t ) d 1 y(t ) d m x(t ) d m1 x(t ) an  an1  ...  a1  a0 y(t )  bm  bm1  ...  b0 x(t ) n n 1 m m 1 dt dt dt dt dt  many systems are simpler ...  example – potentiometric displacement sensor d (1-α)RT t vo Vr αRT D vo d t  this system is “memory” less y(t )  k  x(t ) vo (t )  d (t ) Vr D 42 Zero-order system  general form linear differential equation d n y(t ) d n1 y(t ) d 1 y(t ) d m x(t ) d m1 x(t ) an  an1  ...  a1  a0 y(t )  bm  bm1  ...  b0 x(t ) n n 1 m m 1 dt dt dt dt dt  many systems are simpler ...  differential equation for zero-order systems a0 y(t )  b0 x(t )  y (t )  b0 x(t )  k  x(t ) a0  static sensitivity given by k  note: S was used before when discussing static characteristics  S=k  zero-order system represents ideal or perfect dynamic performance Zero-order system 43  zero-order system represents ideal or perfect dynamic performance  demonstrated with response to step at input x(t) Ao/Ai K ci ω t φ y(t) kci ω t step input frequency response  no dynamic error present in zero-order systems  none of the elements in the sensor stores energy 44 First-order system  many systems are not ideal...  (parasitic) capacitance or inductance are often present  example – liquid-in-glass thermometer  input – temperature Ti(t) of environment  output – displacement xo of the thermometer fluid  liquid column has inertia (i.e. transfer function is not ideal) xo xo=0 Tf Ti(t) 45 First-order system  first-order system contains one energy storing element  differential equation for first-order system dy (t ) a1  a0 y(t )  b0 x(t ) dt  engineering practice to only consider x(t) and not its derivatives  solve equation to obtain transfer function b0 a1 dy (t )  y (t )  x(t ) a0 dt a0 b a k  0 ,  1 a0 a0   Y ( s) k     s  1 Y ( s )  k  X ( s )    X ( s) s  1    k – static sensitivity  τ – time constant 46 First-order system Y ( s) k b a  , with k  0 ,   1 X ( s) s  1 a0 a0  static input implies all derivatives are zero  static sensitivity (k) is the amount of output per unit input when the input is static (constant)  time constant (τ) determines the lag of the output signal on a change in the input signal small τ x(t) large τ y(t) ci kci t t step at input response at output 47 Example – liquid-in-glass thermometer  conservation of energy provides relation between fluid temperature (Tf) and liquid temperature (Ti) Vb C dT f dt  UAbT f  UAbTi xo Vb – volume of bulb [m3] ρ – mass density of thermometer fluid [kg/m3] C – specific heat of thermometer fluid [J/(kg°C)] U – overall heat-transfer coefficient across bulb wall [W/(m2°C)]  Ab – heat transfer area of bulb wall [m2]     xo=0 Tf Ti(t) 48 Example – liquid-in-glass thermometer  conservation of energy provides relation between fluid temperature (Tf) and liquid temperature (Ti) Vb C dT f dt  UAbT f  UAbTi xo  relation between liquid level (xo) and liquid temperature (Ti) K exVb xo  Tf Ac  xo – displacement from reference mark [m]  Kex – differential expansion coefficient of fluid and bulb [m3/(m3°C)]  Vb – volume of bulb [m3]  Ac – cross sectional area of capillary tube [m2]  what are sensitivity (k) and time constant (τ)? xo=0 Tf Ti(t) 49 Example – liquid-in-glass thermometer  conservation of energy provides relation between fluid temperature (Tf) and liquid temperature (Ti) Vb C dT f dt  UAbT f  UAbTi xo  relation between liquid level (xo) and liquid temperature (Ti) K exVb xo  Tf Ac  what are sensitivity (k) and time constant (τ)?  combining equations gives differential equation for whole system Tf dT f  UAbT f  UAbTi   CAc dxo UAb Ac dt  xo  UAbTi  K ex dt K exVb K exVb Ac xo  xo  Tf  Tf   Ac K exVb Vb C xo=0 Ti(t) 50 Example – liquid-in-glass thermometer  what are sensitivity (k) and time constant (τ)?  combining equations gives differential equation for whole system xo CAc dxo K ex UAb Ac  xo  UAbTi dt K exVb  general first-order system b b a a1 dy(t )  y (t )  0 x(t )  k  0 ,   1 a0 a0 a0 dt a0 K exVb  sensitivity [m/°C] k  Ac  time constant [s]   CVb UAb xo=0 Tf Ti(t) 51 Example – liquid-in-glass thermometer  what are sensitivity (k) and time constant (τ)? K exVb Ac CVb  time constant [s]   UAb  sensitivity [m/°C] k  xo  sensitivity and time constant related to physical parameters  larger sensitivity (k) requires larger bulb volume (Vb)  larger bulb volume (Vb) increases time constant (τ)  effect partially offset by increased contact area (Ab)  careful selection of parameters is required xo=0 Tf Ti(t)