REFERENCES PROBLEMS. 62 Chapter 2 Models of Physical Systems

Transcription

1 62 hapter 2 Models of Physical Systems nonlinear, the linear models that we use in analysis and design are always the result of some type of linearization. The topic of linearization is covered more completel9 in hapter 14. The next chapter presents a method of modeling linear time-invariant systems that is different from the transfer function. This method is the state-variable procedure and is useful in simulation and in modern control analysis and design. REFERENES 1. EEE Standard Dictionary of Electrical and Electronic Terms. New York: EEE, rwin. Basic Engineering ircuit Analysis, Upper Saddle River, NJ: Prentice Hall, V. P. Nelson et al. Digital Logic ircuit Design and Analysis. Upper Saddle River, NJ: Prentice Hall, J. L. Agnew and R.. Knapp. Linear Algebra with Applications, 3rd ed. Pacific Grove, A: Brooks/ole, S.1 Mason. Feedback Theory Some Properties of Flow Graphs, Proc. RE, 41 (September 1953): McPherson and R. 0. Laramore. An ntroduction to Electrical Machines and Transformers. New York: Wiley, S.! hapman. Electrical Machinery Fundamentals. New York: McGraw-Hill, W. desilva. ontrol Sensors and Actuators. Englewood liffs, NJ: Prentice Hal, M. F. Hordeski. Design of Microprocessor Sensor & ontrol Systems. Reston, VA: Reston Publishing ompany, Wolf and R. Smith. Student Reference Manual for Electronic nstrumentation. Englewood liffs, NJ: Prentice Hall, E. B. Herceg. Handbook of Measurement and ontrol Pennsauken, NJ: Schaevitz Engineering, Trimmer. Response of Physical Systems. New York: Wiley, W. A. Blackwell and L. L. Grigsby. ntroductory Network Theory. Boston: Prindle, Weber & Schmidt, M. E. Van Valkenburg. Network Analysis. Englewood liffs, NJ: Prentice Hall, K. S. Fu, R.. Gonzalez, and. S. 0. Lee. Robotics: ontrol, Sensing, Vision, and ntelligence. New York: McGraw-Hill, Graupe. dentification of Systems. Huntington, NY: R. B. Kreiger, L. Ljung and E. 1. Ljung. System dentification: Theory for the User. Upper Saddle River, NJ: Prentice Hall, PROBLEMS Section 2.1 Problems 2.1. n circuit analysis, ideal voltage and current sources are commonly used to model physical voltage or current supply circuits. (a) onsider ways that these ideal sources fail to accurately model the physical circuits. (b) Show how the models of the physical circuits can be improved by the addition of resistors or other circuit elements onsider the operational amplifier circuits shown in Figure P2.2. What are some of the limita tions that must be assumed about the circuits in order for a linear mathematical model to be

2 Problems (a) B (b) 1 FGURE P2.2 (c) (d) Section 2.2 Problems t2,3. onsider the circuit of Figure P2.3. FGURE P2.3 (a) Find the voltage transfer function T/2(s)/V1(s). (h) Suppose that an inductor L2 is connected across the output terminals in parallel with R3. Find the transfer function V2(s)/V1(s). (c) A constant input voltage of 10 V is applied to the circuit. Using the final-value theorem of the Laplace transform (see Appendix B, Section B.2), find the steady-state values of the output voltages for the circuits of (a) and (b) onsider the circuit of Figure P OQ V1 V2

3 64 hapter 2 Models of Physical Systems (a) Find the voltage transfer function V2(s)/Vi(s). (b) Suppose that a 0.5 farad capacitor is connected across the output terminals in parallel with R2. Find the transfer function V2(s)1V1(s). (c) A constant input voltage of 10 V is applied to the circuit. Using the final-value theorem of the Laplace transform (see Appendix B, Section B.2), find the steady-state values of the output voltages for the circuits of (a) and (b) (a) Find the voltage transfer function V0(s)/V1(s) for each of the op-amp circuits in Figure P2.2. (b) Express each output voltage v0q) as a function of the input voltage and the circuit parameters Figure P2.6 shows an op-amp circuit that is commonly used in sensor systems to amplify the output of a transducer. The transducer output voltage is labeled e5. The 12V input is used to offset a constant bias voltage in e5. Express the circuit output voltage, v0, as a function of e~. -12 V 5OkQ 10kc~ 2 k 2 FGURE P (a) Design an op-amp circuit that realizes a voltage gain of 10. Do not use a resistance value of less than 10 kq. *(b) Repeat (a) for an op-amp circuit with a voltage transfer function Ga(s) = 10/s. (e) Repeat (a) for an op-amp circuit with a transfer function of Ge(s) = los/(s + 10). (d) Repeat (a) for an op-amp circuit with a transfer function of Gd(s) 10/(s + 10). (e) Repeat (a) for an op-amp circuit with a transfer function of Ge(s) = (los + 1). (1) Repeat (a) for an op-amp circuit with a transfer function of Gf(s) = ( /s). Section 2.3 Problems 2.8. onsider the flow graph shown in Figure P A FGURE P2.8 2 (a) Write the equations upon which the flow graph is based. (b) Solve the equations of (a) by matrix inversion. (,~\ ~ +k.-, t;.-..,.~ -s f..~ k,, fl,-...,,.,.,.,.

4 Problems 65 1 f (d) Solve the equations of (a) using MATLAB. (e) Verify your solution by direct substitution in the equations in (a). *2.9. onsider the block diagrams of Figure P2.9. E (a) E + (b) E FGURE P2.9 (c) (a) Find the transfer functions a and Gb such that the block diagram of (b) is equivalent to that of (a). (,) Find the transfer functions G~ and Gd such that the block diagram of (c) is equivalent to that of (a) Given the block diagrams of Figure P2.10, (a) Find the transfer functions Ga and G~ such that the block diagram of (b) is equivalent to that of (a). (b) Find the transfer functions G~ and 0d such that the block diagram of (c) is equivalent to that of (a). Section 2.4 Problems (a) Draw a flow graph for the e9uations given. Generate the node for the variable A from the first equation, the node for B from the second, and the node for from the third. A + 2B + 3 = 4 3A B = 0 A 7) fl_

5 66 hapter 2 Models of Physical Systems E (a) (b) E (c) FGURE P2.10 (b) Use Mason s gain formula to solve these equations. (c) Verify your solution in (b) by solving the equations by matrix inversion. (d) Verify your solution in (b) by solving the equations by ramer s rule. (e) Verify your solution by direct substitution into the given equations (a) Using Mason s gain formula, find the transfer function DR for the flow graph of Figure P2.12. One of the forward paths is easily overlooked. -1

6 Problems 67 (b) The equation for node A, in terms of the labeled nodes, is A=R-Q1A--04 n a like manner, write the equations for nodes B,, and D and confirm that DR is that found in part (a) (a) For the flow graph Figure P2.13, use Mason s gain formula to find the transfer function (s)/r(s). B G2 S G3 1 :1 FGURE P2.13 (b) Write three equations in the variables A, B, and. Then verify the results in (a) using ramer s rule. (c) Solve the equations in (b) using MATLAB (a) The flow graph of Figure P2.14 is called a simulation diagram. These diagrams are very useful in the analysis and design of systems. For this simulation diagram, find the transfer function (s)r(s) using Mason s gain formula. 3 B(s) (s) 1 FGURE P2.14 (b) Write three equations in th~ variables A(s), B(s), and (s). Then verify the results in (a) using MATLAB Determine the transfer function (s)r(s) for the system represented by the simulation diagram shown in Figure P (nh Derive the. tr2ncfer fnnrtinn r(v~r(~c~ Mr thp v ctnm vl,raun,, P.-,,,,.~ 0 ) 1 j(

7 68 chapter 2 Models of Physical Systems 1 s i s i R(s) (s) FGURE P (s) FGURE P2.16 (b) Repeat (a), given: 6(s + 2) 5+8 G2(s) = 2 s(s + 1) 03(s)= Hi(s) = 0, H2(s) = 1 (c) Repeat (a), given: 6(s + 2) s+8 G2(s) = 2 s(s + 1) G3(s) Hi(s) = 0.5s, H2(s) = 1 Section 2.5 Problems *2,17. (a) Write the differential equations for the mechanical system shown in Figure P2.17(a). There are no applied forces; the system is excited only by initial conditioas. (b) Repeat (a) for the system of Figure P2.17(b). (c) Write the differential equation model for the system of Figure P2.17(c). (d) Write the differential equation model for the system of Figure P2.17(d). *2.18. (a) A force f(t) is applied downward to the mass M in Figure P2.17(a). Find the transfer function from the applied force to the displacement, x1(t), of the mass; that is, find X1(s)F(s). (b) A force fq) is applied downward to the mass M in Figure P2.17(b). Find the transfer function v f._ drf..~

8 Problems 69 K1 B x2 ml (a) x2(t) (b) 0 x1(t) f(t) K 0) ~w ~) m HZB / Y/////////////77///////,7///////, massless point c) 0 0 F No friction d) No friction FGURE P2.17 (c) alculate a transfer function for the system shown in Figure P2.17(c). output andf(t) to be the input. Let in =~O.5, B 2, and K.9. (d) alculate a transfer function for the system shown in Figure P2.17(d). output and ftt) to be the ifiput onsider the mechanical system of Figure P2.19. (a) Write the differential equations that describe this system. onsider x1(t) to be the onsider xiq) to be the (b) Find the transfer function from the applied force ftt) to the displacement, y(t), of the mass; that is, find Y(s)F(s).