Fundamentals of Measurement Technology (6) Prof Wang Boxiong
Fundamentals of Measurement Technology (6) Prof. Wang Boxiong
3.4 Dynamic characteristics of measuring systems FOr dynamic measurement, the measuring system must be a linear one We can only process linear systems mathematically It is rather difficult to perform nonlinear corrections in situations of dynamic measurement a Practical systems may be considered as linear systems within a certain range of operation and permissible error limits lt is of general significance to study linear systems
❑For dynamic measurement, the measuring system must be a linear one. ▪ We can only process linear systems mathematically. ▪ It is rather difficult to perform nonlinear corrections in situations of dynamic measurement. ▪ Practical systems may be considered as linear systems within a certain range of operation and permissible error limits. ❖It is of general significance to study linear systems. 3.4 Dynamic characteristics of measuring systems
3.4.1 Mathematical representation of linear systems uThe input-output relationship of a linear system d n-1 ∴+a d dx(t (3.3) m-1 +…+b +box() where x(t=input of the system y(t=output of the system an, al, ao, and bm, b1, bo are systems parameters Ua linear constant-coefficient system or linear time-invariant(LTD) system: the parameters are constants
❑The input-output relationship of a linear system: where x(t)= input of the system y(t)= output of the system an , a1 , a0 , and bm, b1 , b0 are system’s parameters. ❑A linear constant-coefficient system or linear time-invariant (LTI) system: the parameters are constants. 3.4.1 Mathematical representation of linear systems ( ) ( ) ( ) ( ) ( ) ( ) ( ) b x(t) dt dx t b dt d x t b dt d x t b a y t dt dy t a dt d y t a dt d y t a m m m m m m n n n n n n 1 0 1 1 1 1 0 1 1 1 = + + + + + + + + − − − − − − (3.3)
3.4.1 Mathematical representation of linear systems 日 Properties 1. Superposition property(superposability) If for x1()->y ()→>y2( then x,((+x2()->y,()+y2(t) (3.4) 2. Proportional x(t)→ en ax(t)→>ay(t) (3.5) Where a is a constant
❑ Properties: 1. Superposition property (superposability): If for then 2. Proportionality If then Where a is a constant. 3.4.1 Mathematical representation of linear systems x (t) y (t) 1 → 1 x (t) y (t) 2 → 2 x (t) x (t) y (t) y (t) 1 + 2 → 1 + 2 (3.4) x(t)→ y(t) ax(t)→ ay(t) (3.5)
3.4.1 Mathematical representation of linear systems 3. Differentiation x(t)→y(t) dx(t)、d(t then 4. Integration fx()->y(t) and for a zero initial condition of the system then x()→y(h
3. Differentiation If then 4. Integration If and for a zero initial condition of the system, then 3.4.1 Mathematical representation of linear systems x(t)→ y(t) ( ) ( ) dt dy t dt dx t → (3.6) x(t)→ y(t) ( ) ( ) → t t x t dt y t dt 0 0 (3.7)