HADNOUT E.13 - EXAMPLES ON TRANSFER FUNCTIONS, POLES AND ZEROS

Size: px

Start display at page:

Download "HADNOUT E.13 - EXAMPLES ON TRANSFER FUNCTIONS, POLES AND ZEROS"

Charlotte Francis
7 years ago
Views:

1 Paraura Leture 3 Augut HADNOUT E3 - EXAMPLES ON TRANSFER FUNCTIONS POLES AND ZEROS Exaple Deterine the tranfer funtion of the a-pring-daper yte The governing differential equation of a a-pring-daper yte i given by x x x F Taing the Laplae tranfor of the above equation auing zero initial ondition we have X X X F X F Equation repreent the tranfer funtion of the a-pring-daper yte Exaple Conider the yte given by the differential equation 4 t y y 3y r where rt i the input to the yte Aue zero initial ondition The Laplae tranfor yield Y 4Y 3Y R Y R 4 3 Equation repreent the tranfer funtion of the yte Exaple 3 Find the olution of the differential equation y t y t where y α and y β The Laplae tranfor of the above differential equation give Y α β Y α β Y

2 Paraura Leture 3 Augut After looing up in the tranfor table the two ter in the right ide of the above equation we get y t α o t β in t Exaple 4 Conider a RLC iruit The governing differential equation i given by di L Ri idt V dt C 3 But i dq dt Therefore equation 3 redue to d q dq L R q V dt dt C The Laplae tranfor of the above equation yield L Q RQ Q V C Q V L R C The above equation repreent the tranfer funtion of a RLC iruit Exaple 5 Deterine the pole and zero of the yte whoe tranfer funtion i given by 3 The zero of the yte an be obtained by equating the nuerator of the tranfer funtion to zero ie

3 Paraura Leture 3 Augut The pole of the yte an be obtained by equating the denoinator of the tranfer funtion to zero ie 3 Therefore - and - are the pole of the yte and -/ i the zero of the yte Exaple 6 Deterine the pole and zero of the yte whoe tranfer funtion i given by 3 6 H 4 3 The zero of the yte are given by Therefore 6 i the zero of the yte The pole of the yte are given by ± i6 4 i6 i3 i3 Therefore the pole of the yte are - i3 and - i3 3

4 Paraura Leture 3 Augut Exaple 7 Conider the a-pring-daper yte The governing differential equation of otion for the yte i given by x x x F 4 Let the tate of the yte be defined a x x x x 5 Fro the above relation it an be onluded that x x 6 Subtituting the relation given by equation 5 in equation 4 we get x x x F x x x F x x x F 7 Repreenting equation 6 and 7 in atrix forat we have x x x x F 8 If the output of the yte i the veloity of the a then writing the output relation in atrix forat we get y x x y [ x ] x 9 Equation 8 and 9 repreent the tate-pae repreentation of the a-pring-daper yte Obtaining the tranfer funtion fro the tate-pae repreentation iven the A B C and D atrie of the tate-pae equation the tranfer funtion of the yte i given by 4

5 Paraura Leture 3 Augut 5 D B A I C Fro equation 8 the tranfer funtion an be written a [ ] [ ] [ ] Equation repreent the tranfer funtion of the yte wherein the input to the yte i the fore applied to the yte and the output of the yte i the veloity of the a Exaple 8 Find the tranfer funtion for the blo diagra hown below Sol: Note that there are two negative feedba loop and one poitive feed-forward loop Reduing the negative feedba loop we get

6 Paraura Leture 3 Augut The above diagra an be further redued to Feed-forward loop when redued yield 6

7 Paraura Leture 3 Augut Aignent Deterine the tranfer funtion for the following yte whoe differential equation are given by J θ Bθ dia L dt Ri a K v a T i a K θ e The input to the yte i the voltage v a wherea the output i the angle θ Deterine the pole and zero of the yte whoe tranfer funtion are given by a b H Obtain the tate-pae repreentation of the yte whoe differential equation i given by the equation given in exaple 7

Lecture 14: Transformers. Ideal Transformers

Lecture 14: Transformers. Ideal Transformers White, EE 3 Lecture 14 Page 1 of 9 Lecture 14: Tranforer. deal Tranforer n general, a tranforer i a ultiort ac device that convert voltage, current and iedance fro one value to another. Thi device only