ME_446_-_Lec_08-Block_Diagrams

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

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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 ...
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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.

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