Zero-OrderSystemsThe simplest model of a measurement systems and one usedwith static signals is the zero-order system model. This isrepresented by the zero-order differential equation:aoy = F(t)y(t) = KF(t),K =1/aoK is called the static sensitivity or steady gain of the system
Zero-Order Systems The simplest model of a measurement systems and one used with static signals is the zero-order system model. This is represented by the zero-order differential equation: a0y = F(t) y(t) = KF(t), K =1/a0 K is called the static sensitivity or steady gain of the system
First-OrderSystemsMeasurement systems that contain storage elements donot respond instantaneously to changes in inputIn general, systems with a storage or dissipative capability butnegligible inertial forces may be modelled using a first-orderdifferential equation of the formdy1aj+αoy= F(t);dxDividing through by ao givesi+ y= KF(t);The parameter t is called the time constant of the system
First-Order Systems Measurement systems that contain storage elements do not respond instantaneously to changes in input. In general, systems with a storage or dissipative capability but negligible inertial forces may be modelled using a first-order differential equation of the form dx dy a1 y a0 y F(t); y Dividing through by a0 gives y y KF(t); The parameter is called the time constant of the system
StepFunctionInputThe stepfunction, AU(t), is defined asUt)1AU(t) = 0, t ≤ 0-AU(t) = 0, t ≥ O+Os20Timewhere A is the amplitude of the step function andU(t) is defined as the unit step function
Step Function Input • The step function, AU(t), is defined as AU(t) = 0, t ≤ 0- AU(t) = 0, t ≥ 0+ where A is the amplitude of the step function and U(t) is defined as the unit step function
Step FunctionTo illustrate this, let us apply a step function as an input tothe general first-order system. Setting F(t) = AU(t)ti+ y = KAU(t):with an arbitrary initial condition denoted by, y(O) = yo. Solving fort ≥ 0+ yieldsKAKA)e2+stady responsetimeresponsetransient response
Step Function To illustrate this, let us apply a step function as an input to the general first-order system. Setting F(t) = AU(t) y y KAU(t); with an arbitrary initial condition denoted by, y(0) = y0 . Solving for t ≥ 0+ yields transient response t stady response time response y t KA y KA e / ( ) ( )
Step Function ResponsetheerrorfractionKAT= y()- yα= e-/- 0.632 (KA-yo)y-y.1.0223145Ct/T0.80.60.40.3680.210.0012345t/r
/ ( ) t e y y y t y the error fraction Step Function Response