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)
3.4.1 Mathematical representation of linear systems 5. Frequency preservability f →)y(t and for x(t=xejot then the output j(at+)
5. Frequency preservability If and for then the output 3.4.1 Mathematical representation of linear systems x(t)→ y(t) ( ) j t x t x e = 0 ( ) ( + ) = j t y t y e0
3.4.1 Mathematical representation of linear systems Proof: According to the proportionality property 2 x(t)→>O (38) According to the differentiation property (3.9) dt d x(t oxt+ oy (3.10) Since x(t J d2x(o) Jo)xoe
Proof: According to the proportionality property According to the differentiation property Since 3.4.1 Mathematical representation of linear systems x(t) y(t) 2 2 → (3.8) ( ) ( ) 2 2 2 2 dt dy t dt d x t → (3.9) ( ) ( ) ( ) ( ) → + + 2 2 2 2 2 2 dt dy t y t dt d x t x t (3.10) ( ) j t x t x e = 0 ( ) ( ) x(t) x e j x e dt d x t j t j t 2 0 2 0 2 2 2 = − = − =
3.4.1 Mathematical representation of linear systems Letting the left-hand side of eq. 3.10) be zero dolt 2x(t)+ then the right-hand side of eq ( 3.10) must also be zero 0 y(t)+ t Solving the equation yields yoe J(at+o where is the phase shift
Letting the left-hand side of Eq. (3.10) be zero, then the right-hand side of Eq. (3.10) must also be zero, Solving the equation yields: where φ is the phase shift. 3.4.1 Mathematical representation of linear systems ( ) ( ) 0 2 2 2 + = dt d x t x t ( ) ( ) 0 2 2 2 + = dt d y t y t ( ) ( + ) = j t y t y e0
3.4.2 Representation of system's characteristics in terms of transfer function or frequency response 1。 Transfer function 日 Definition: For tso, y(t=0, the Laplace transform Y(s) (3.1 of y(t) is defined as y(o e where s is the laplace operator: s=a+jb for a>0
1. Transfer function ❑ Definition: For t0, y(t)=0, the Laplace transform Y(s) of y(t) is defined as where s is the Laplace operator: s=a+jb for a>0. 3.4.2 Representation of system’s characteristics in terms of transfer function or frequency response ( ) ( ) − = 0 Y s y t e dt st (3.11)
3.4.2 Representation of system 's characteristics in terms of transfer function or frequency response If all the systems initial conditions are zero, making Laplace transform of Eg (3.3) gives then the expression y(slans"+amS+.+a,s+ao X((bns"+bnsm+…+b1s+b。) The transfer function H(s) y(s)bm5+bm-ISm-++b,5+b H 1 (3.12) )as"+as 十a1S+a 0 The transfer function H(s) represents the transfer characteristics of a system
If all the system’s initial conditions are zero, making Laplace transform of Eq. (3.3) gives then the expression The transfer function H(s): ❖The transfer function H(s) represents the transfer characteristics of a system. 3.4.2 Representation of system’s characteristics in terms of transfer function or frequency response ( )( ) ( )( ) 1 0 1 1 1 0 1 1 X s b s b s b s b Y s a s a s a s a m m m m n n n n = + + + + + + + + − − − − ( ) ( ) ( ) 1 0 1 1 1 0 1 1 a s a s a s a b s b s b s b X s Y s H s n n n n m m m m + + + + + + + + = = − − − − (3.12)
