1. Block Diagrams David W. Graham EE 327

2. 2 Block Diagrams Symbolic representation of complex signals –Easier to understand the relationships between subsystems using block digrams than by looking at the equations representing them, or even schematics –Often easier to obtain a transfer function for the overall system by first drawing block diagrams

3. 3 Basic Elements 1. Block H(s) X(s)Y(s) = H(s)X(s) 2. Adder X 1 (s) + X 2 (s) Y(s) = X 1 (s) + X 2 (s) 3. Node X(s)

4. 4 Basic Connections Series / Cascade Parallel Feedback

5. 5 Series / Cascade Connections H 2 (s) W(s) = H 1 (s)X(s) H 1 (s) X(s) Y(s) = H 2 (s)W(s) = H 1 (s) H 2 (s)X(s) Equivalent to a single block of H 1 (s)H 2 (s) X(s)Y(s)

6. 6 Parallel Connections + H 1 (s) H 2 (s) X(s) Y(s) Y 1 (s) Y 2 (s) Y 1 (s) = H 1 (s)X(s) Y 2 (s) = H 2 (s)X(s) Y(s)= Y 1 (s) + Y 2 (s) = [H 1 (s) + H 2 (s)]X(s) Equivalent to a single block of H 1 (s) + H 2 (s) X(s)Y(s)

7. 7 Feedback Connection (Negative Feedback) + H 2 (s) X(s)Y(s) E(s) - H 1 (s) Y(s)= H 1 (s)E(s) E(s)= X(s) – H 2 (s)Y(s) Y(s)= H 1 (s)[X(s) – H 2 (s)Y(s)] = H 1 (s)X(s) – H 1 (s)H 2 (s)Y(s) Y(s)[1 + H 1 (s)H 2 (s)] = H 1 (s)X(s) Y(s) = H 1 (s) X(s) 1 + H 1 (s)H 2 (s) (Equivalent Block)

8. 8 Complex System Modeling 1.Obtain a transfer function for each subsystem 2.Determine equations that show the interactions of each subsystem 3.Draw a block diagram (Hint. Start from input and work to output or vice versa) 4.Perform block-diagram reduction H 1 (s)H 2 (s) X(s)Y(s) H 2 (s)H 1 (s) X(s)Y(s) A. + H 1 (s) H 2 (s) X(s) Y(s) B. H 1 (s) + H 2 (s) X(s)Y(s) + H 2 (s) X(s)Y(s) - H 1 (s) C. H 1 (s) 1 + H 1 (s)H 2 (s) X(s)Y(s) + H(s) X 2 (s) Y(s) = H(s)[X 1 (s) + X 2 (s)] X 1 (s) + H(s) X 2 (s) Y(s) = H(s)[X 1 (s) + X 2 (s)] H(s) D.

9. 9 Block-Diagram Reduction Example 1 + - + -

10. 10 Block-Diagram Reduction Example 2 + - + - Divide by 1/s block + + + - + - + +

11. 11 + - + - + + + - + +

12. 12 + - + + + - + + Feedback Parallel

13. 13 Large System Example Satellite Tracking System Trying to track a satellite with an antenna Input to the system is a control angle (i.e. the angle we want to point the antenna) Output is the actual angle of the antenna The overall system requires several components to complete its task +-+- Tracking Receiver θmθm θcθc θLθL Tachometer Motor Amplifier V1V1 V2V2 VaVa Gear Box -

14. 14 Subsystem Equations 1. Tracking Receiver Input  θ c, θ L Output  V 1 +-+- Tracking Receiver θmθm θcθc θLθL Tachometer Motor Amplifier V1V1 V2V2 VaVa Gear Box - H 1 (s) is a complicated transfer function 2. Amplifier Input  V 1 – V 2 Output  V a 3. Motor Input  V a Output  θ m System Input θ c, System Output θ L System Signals θ c, θ m, θ L, V 1, V 2, V a System Constants J, K a, K b, K t, K T 4. Tachometer Input  θ m Output  V 2 5. Gear Box Input  θ m Output  θ L

15. 15 Create a System Block Diagram (Let us work from input to output) + - + - Subsystem Equations 1. Tracking Receiver Input  θ c, θ L Output  V 1 H 1 (s) is a complicated transfer function 2. Amplifier Input  V 1 – V 2 Output  V a 3. Amplifier Input  V a Output  θ m 4. Tachometer Input  θ m Output  V 2 5. Gear Box Input  θ m Output  θ L Connect like nodes

16. 16 Perform Block-Diagram Reduction + - + - Series Connection + - + - Feedback Connection

17. 17 + - + -