ME_446_-_Lec_08-Block_Diagrams

ME_446_-_Lec_08-Block_Diagrams - Block Diagrams 446-8 446-...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Block Diagrams 446-8 446- Prof. Neil A. Duffie University of Wisconsin-Madison Wisconsin- 8 ® Neil A. Duffie, 1996 All rights reserved 1 Utility of Block Diagrams • “Visualization” of control systems - shows transfer functions of components - reflects physical structure of system - nature of control inputs and outputs • Convenient analysis of control systems - based on transfer functions - straightforward manipulation of blocks - simplification of block diagrams 8 - obtaining system transfer functions 2 Blocks Input X(s) x(t) G(s) = Y (s ) X(s) G(s) Y(s) y(t) Output • Blocks are unidirectional (input  output). output). • G(s) is the transfer function of the block (the ratio of the Laplace transform of the output variable y(t) to the Laplace transform of the input variable x(t)). 8 3 Proportional Block Input K Output potentiometer x(t) [mm] V [volts] e(t) [volts] Model e(t) = K pot x(t) Block Diagram X(s) E(s) Kpot 4 8 Time Constant Block Input Rotating System I b K τs + 1 Output Transfer Function 1 Ω(s) G(s) = =b T(s) τs + 1 Block Diagram X(s) 1 E(s) b τs + 1 5 T(t) ω(t) dω( t) I + bω(t) = T(t) dt dω(t) 1 τ + ω (t) = T(t) dt b 8 I τ= b Integrating Block Input K s Output Hydraulic Cylinder q(t) y(t) A dy (t ) 1 = q(t) dt A 8 Transfer Function Y (s ) 1 G(s) = = Q(s) As Block Diagram Q(s) 1 As Y(s) 6 Second-Order Block SecondMotor-Amplifier Transfer Function MotorKa Ω ( s) = 2 ⎞ ⎛ V(s) LJs RJs ⎜ + + 1⎟ ⎟ ⎜ (K + K K )K (K + K K )K ⎠ ⎝e av t e av t ωn = (K e + K aK v )K t LJ ζ= ω nRJ 2(K e + K aK v )K t Block V(s) Diagram 8 Ka s 2 2ζs + +1 2 ωn ωn Ω(s) 7 Velocity Command Integral Control R(s) + E(s) Kc s System Block Diagram Motor-Amplifier MotorV(s) Ka s 2 2 ωn + 2ζs +1 ωn Ω(s) Velocity Feedback Velocity Error e(t) = r(t) − Ω (t) E(s) = R(s) − Ω (s) Integral of Error t v(t) = K c ∫0 e(t)dt 8 V(s) = Control Kc E(s) Gain 8 s Closed-Loop Transfer Function ClosedControl Process Output Command R(s) + E(s) M(s) C(s) Gc(s) Gp(s) - B(s) H(s) Feedback C( s) = Gp (s )M( s) C(s) = G c (s)Gp (s)E(s) C(s) = G c (s)Gp (s)(R(s) − B(s)) 8 C(s) = G c (s)Gp (s)(R(s) − H(s)C(s)) 9 Closed-Loop Transfer Function ClosedR(s) + E(s) - Gc(s) M(s) Gp(s) C(s) B(s) H(s) C( s) = G c (s )Gp (s )R(s ) − G c (s )Gp (s )H( s)C (s) C(s) + Gc (s)Gp (s)H(s)C(s) = G c (s)Gp (s)R(s) (1 + Gc (s)Gp (s)H(s))C(s) = Gc (s)Gp (s)R(s) 8 Gc (s)Gp (s) C(s) = 1 + G c (s)Gp (s)H(s) R(s) General Equation 10 Block Diagram Reduction R(s) + E(s) Gc(s) M(s) Gp(s) C(s) - B(s) H(s) R(s) Gc (s)Gp (s) 1 + G c (s)Gp (s)H(s) C(s) 8 11 Combining Cascaded Blocks E(s) Gc(s) M(s) Gp(s) C(s) C( s) M( s) C( s) = = G c (s)Gp (s) E(s) E(s) M(s) E(s) 8 Gc(s) Gp(s) C(s) 12 Exchanging Summing Points C(s) A(s) + - - D(s) + B(s) D(s) = A(s) - B(s) - C(s) = A(s) - C(s) - B(s) C(s) A(s) - + 8 + D(s) B(s) 13 Exchanging Pickoff Points A(s) B(s) C(s) A(s) B(s) C(s) 8 B(s) = A(s) C(s) = A(s) A(s) C(s) B(s) 14 Moving Pickup Point Ahead of Block A(s) B(s) B(s) = G(s) A(s) A(s) B(s) 8 G (s) B(s) G (s) B(s) G (s) 15 Moving Pickup Point Behind Block A(s) A(s) A(s) = G(s) 1 A(s) G(s) G (s) B(s) A(s) G (s) 1 G (s) B(s) A(s) 8 16 Moving Summing Point Behind Block A(s) + - G (s) B(s) C(s) C(s) = G(s)[A(s) - B(s)] = G(s)A(s) - G(s)B(s) A(s) G (s) + - C(s) B(s) 17 G (s) 8 A(s) Moving Summing Point Ahead of Block C(s) + G (s) G (s) - B(s) ⎛ ⎞ 1 C(s) = G(s)A(s) − B(s) = ⎜ A(s) − B(s)⎟ G(s) G(s) ⎝ ⎠ A(s) + 8 G (s) 1 G (s) C(s) B(s) 18 Pickup and Summing Points (They cannot be exchanged!) B(s) A(s) + D(s) C(s) B(s) A(s) 8 D(s) = C(s) = A(s) - B(s) D’(s) = A(s) ≠ D(s) C(s) 19 + D’(s) Examples of Block Diagram Reduction (worked in class) 8 20 ...
View Full Document

This note was uploaded on 04/27/2010 for the course ME 446 taught by Professor Nd during the Spring '10 term at Universität für Bodenkultur Wien.

Ask a homework question - tutors are online