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Block Diagrams

Block Diagrams. x. y. G. A line is a signal A block is a gain A circle is a sum Due to h.f. noise, use proper blocks : num deg ≤ den deg Try to use just horizontal or vertical lines Use additional “ ” to help e.g. y = Gx. +. x. s. Σ. +. -. s = x + z - y. y. z. Σ. +.

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Block Diagrams

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  1. Block Diagrams x y G • A line is a signal • A block is a gain • A circle is a sum • Due to h.f. noise,use proper blocks: num deg ≤ den deg • Try to use just horizontal or vertical lines • Use additional “ ” to help e.g. y = Gx + x s Σ + - s = x + z - y y z Σ + x s + + z - y

  2. Block Diagram Algebra x y • Series: • Parallel: G1 G2 x y G1 G2 G1 + x y x y  G1 + G2 + G2

  3. Feedback: • Proof: + e x x y G1 y  - b G2

  4. + G1 + G2 + -

  5. >> s=tf('s') Transfer function: s >> G1=(s+1)/(s+2) Transfer function: s + 1 ----- s + 2 >> G2=5/(s+5) Transfer function: 5 ----- s + 5 >> G=G1*G2 Transfer function: 5 s + 5 -------------- s^2 + 7 s + 10 >> H=G1+G2 Transfer function: s^2 + 11 s + 15 --------------- s^2 + 7 s + 10 >> HF=feedback(G1, G2) Transfer function: s^2 + 6 s + 5 --------------- s^2 + 12 s + 15

  6. >> delay1=tf(1,1,'inputdelay',0.05) Transfer function: exp(-0.05*s) * 1 >> H2=HF*delay1 Transfer function: s^2 + 6 s + 5 exp(-0.05*s) * --------------- s^2 + 12 s + 15 >> stepresp=H2*1/s Transfer function: s^2 + 6 s + 5 exp(-0.05*s) * ------------------- s^3 + 12 s^2 + 15 s >> step(H2)

  7. Quarter car suspension Series R(s) + y - R(s) + y Feedback - R(s) y

  8. >> b=sym('b'); >> m=sym('m'); >> k=sym('k'); >> s=sym('s'); >> G1=b*s+k G1 = b*s+k >> G2=1/m*1/s*1/s G2 = 1/m/s^2 >> G=G1*G2 G = (b*s+k)/m/s^2 >> Gcl=G/(1+G) Gcl = (b*s+k)/m/s^2/(1+(b*s+k)/m/s^2) >> simplify(Gcl) ans = (b*s+k)/(m*s^2+b*s+k)

  9. pick-up point summation • Move a block (G1) across a into all touching lines: • If arrow direction changes, invert • If arrow direction remains, no change in block • e.g. along arrow no change along arrow x y x y G1 G2 G1 G2 no change z G3 G1 along arrow along arrow z G3

  10. x G1 G2 x G1 G2 y y  z G3 z G3 1/G2 x G1 G3 1/G3 G2 x G1 G2 y y  z z G3

  11. I2 I1 - Vc U + y + - I2 - Vc U + y + - - U + y + -

  12. - U + y - U + y U y

  13. Find TF from U to Y: • No pure series/parallel/feedback • Needs to move a block, but which one? Key: move one block to create pure series or parallel or feedback! So move either left or right. + U + + Y + - -

  14. + U + + Y + - - + U + + Y + - - + U + Y + -

  15. fig_03_16 fig_03_16 is wrong Should be H1(s)G2(s)

  16. fig_03_17 fig_03_17 is wrong

  17. fig_03_18b

  18. fig_03_19 Can use superposition: First set D=0, find Y due to R Then set R=0, find Y due to D Finally, add the two component to get the overall Y

  19. fig_03_20 First set D=0, find Y due to R

  20. Then set R=0, find Y due to D fig_03_21 G2

  21. Finally, add the two component to get the overall Y fig_03_19

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