INPUT sensor transducer Signal A MEASUREMENT SYSTEM Signal B OUTPUT Signal C How does the measurement system respond to a given Input Static signals the input measurand is unchanging in the time required to make a measurement Dynamic signals signals change over time characteristics of time dependent signals may be important parameters in the measurement The measurement system that we use must be fast enough to keep up with the input signal the dynamic frequency response is critical The primary function of a measurement system is to sense an input signal and transform this to a useable output signal m is the bulb mass C is the bulb specific heat h is the heat transfer coefficient A is the surface area How about a simple bulb thermometer the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings hA TB TW TB0 TW exp t cm t TW TB0 TW exp cm hA Zero order responds instantly to measurand no instrument is truly zero order but many can be approximated as zero order First order show capacitive type energy storage effects mechanical analogs to capacitors are springs and devices that store thermal energy thermometer is a first order system Second order have inertial effects of inductance or accelerated mass as well as capacitive effects common spring mass systems bathroom scale Second order systems include a characteristic called damping or energy loss Specifications for the dynamic response include the response to a Step input transient response and the response to a range of Sinusoidal inputs frequency response Zero order Second order input Step transient response Frequency response Sinusoidal input magnitude First order frequency We will model our measurement system as an nth order ordinary differential equation The general form is dny d n 1 y d y an n an 1 n 1 a1 ao y F t dt dt dt The coefficients are determined by the type of system we are trying to model F t is the Forcing Function and can also be generalized in the form dmx d m 1 x d x F t bm m bm 1 m 1 b1 ao x m n dt dt dt Lets look at some general systems First the zero order system dny d n 1 y d y an n an 1 n 1 a1 ao y F t dt dt dt For a zero order system we keep the first term a0 y F t Dividing through by ao y KF t K 1 a0 K is called the static sensitivity or gain of the system Responds instantaneously to input valid at static equilibrium Consider the voltage divider circuit E IR1 e IR2 E R2 R1 x e E L Or we can write this as e IR2 E R2 E x R1 L Note however that if x varies extremely rapidly the approximation of a zero order system will break down Strain gages are also often considered zero order systems First order measurement systems Contain only a single mode of energy storage Electrical analogy RC circuit Temperature systems which contain thermal capacitance and resistance to heat flow dny d n 1 y d y an n an 1 n 1 a1 ao y F t dt dt dt d y a1 ao y F t dt Dividing through by a0 we get a1 d y 1 y F t a0 dt a0 d y y KF t dt is the time constant K is the static sensitivity We can now solve this system for different input functions F t First we consider a step function input F t AH t Where 0 t 0 H t 1 t 0 1 t d y We have y KAH t dt With the initial condition y 0 y0 and for t 0 y t KA y0 KA e t Due to the sharp instantaneous rise the step is ideal to test the response time of the system derivative of the step is the impulse y t KA y0 KA e Steady state t transient Here we can see the response is delayed and steady state not reached until the exponential burns out Consider the case of y0 0 transient term becomes t 0 KA KA 0 367 KA 2 71 KA t 2 0 135 KA 7 38 t SS y reaches 63 of its final value SS y reaches 87 of its final value The time constant gives the time that is takes for the system to reach 63 of its final value Can also look at this in a different way y t KA y0 KA e y t KA e y0 KA t t A thermocouple has a spherical junction with a diameter of 0 3 mm It is used to measure gas temperature in a combustion chamber When the flame is ignited it produces an approximate step change in T to 900 C The gas temperature before ignition is 300C The heat transfer coefficient is 500W m2 C The properties of platinum are r 21 450 kg m3 and c 134 J kg C Given the system is first order and follows the equation hA TG TT cm After how much time will the measurement error be less than 1 d y y KAH t dt dTT dt m is the bulb mass c is the bulb specific heat h is the heat transfer coefficient A is the surface area cm dTT TT TG H t hA dt Great so we know this solution from the previous slides T t TG T0 TG e t y t KA y0 KA e t Time constant Initial temperature Now we can solve for the error fraction and determine the wait time T t TG e 01 T0 TG t 4 605 287 1 32 s many physical processes have sinusoidal inputs Any periodic waveform can be represented in terms of sines and cosines can find the frequency response of the system d y y KF t dt F t A sin t How do we solve this It s a bit complicated but we can look up the solution dy f t y r t dt y t e h e h rdt c h f t dt d y 1 KA y sin t We have dt y t e We get t t KA t sin t dt c h e After integrating and a bit of trig we get the nice form y t ce t KA 1 2 2 arctan sin t y t ce t KA 1 2 2 sin t arctan Long time response purely sinusoidal linear system but Amplitude modified by quantity 2 2 What Happens if 2 2 1 What Happens if 2 2 1 How about the Phase The output follows the input The output goes to zero