King fahd university of petroleum & minerals college of computer sciences & engineering
Review of Solutions of Linear Ordinary Differential Equations
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- EXPERIMENT 2: Basic Operations of the Analog Computer
Review of Solutions of Linear Ordinary Differential EquationsSince the analog computer deals mainly with linear or nonlinear differential equations describing the behaviors of some physical systems, it is very important that the student remembers some fundamental results in the solution of differential equations that he learned in his freshman. Therefore, we will devote this small section of this manual to review some of these results. However, the student is also expected to review these from some good books (reference [1] is a good source). First order Equations A linear first order equation with constant coefficient can be written as  (1)
where y(t) is the output of the system with initial value  . f(t) is the input, which is a given time function, and  , are constant coefficient. The solution to the above system (1) is comprised of two parts. A complementary solution (CS), and a particular solution (PS). The complementary solution is the solution of the system when the input function f(t) = 0. It is also referred to as the zero input solution. Furthermore, it governs the transient response of the system. To get the complementary solution of the above system, we first obtain the auxiliary equation of the system which is given by
 (2)
in terms of the auxiliary variable. The solution (root) of this ordinary equation gives the system mode (or time constants), and in this case is given by  (3)
Finally, we can now write the complementary solution as  (4)
where C is an arbitrary constant to be determined from the initial conditions. For the case of the first order system,  . Therefore, the solution becomes
 (5)
Now regarding the particular solution of the system, this depends on the input function f(t). The particular solution also determines the steady state behavior of the solution. Also, there are various methods of obtaining it. Some of these methods are : the method of integrating functions; the method of variation of parameters; the method of undetermined coefficients and the Laplace transform method (see reference j1 for more details on this). The Laplace transform method is probably the easiest for the student. It also has the advantage of giving both the complementary solution (CS) and the particular solution (PS). The sum of these two solutions gives the whole solution of the system. Second order Equations A second order equation can similarly be represented as  (6)
Without any lost of generality, we can divide the above equation throughout by a and write the system as  (7)
where  , , g(t) are the corresponding coefficients and input respectively.
To find the complementary solution to the above system, we find the auxiliary equation of the system as in the first order case. This is given by  (8)
Let the roots of this quadratic equation be OC ,a Then the complementary solution of the above system is given by  (9)
where  , are constants, to be determined from the initial conditions. The equation (9) is the general form of the complementary solution. If however, the roots are complex conjugates, then equation (9) can further be simplified in the following way:
Let the roots be in this case  (10)
Then the complementary solution is given by  (11)
where  , are another set of arbitrary constants to be determined from the initial conditions.
Finally, the particular solution of the second order equation can also be determined using Laplace transform method. Reference [1] E. W. Kryszig, Advanced Engineering Mathematics, Wiley International Edition, 1992. EXPERIMENT # 2: Basic Operations of the Analog ComputerOBJECTIVES: The objective of this experiment is to learn basic analog computer operations, namely, summing and integration operations. INTRODUCTION The operation of summation of two quantities (signals) can be performed using the analog computer connected as a summer. This is essentially an Op amp. with a resistor in the feedback path (the feedback resistor) and two resistors connected in the forward path through which the signals are summed. Similarly, an integration of a signal can be performed with the analog computer by connecting it as an integrator. In this case, a capacitor is connected in the feedback path and a resistor in the forward path. Moreover, a number of signals can be integrated at the same time and the sum of their integrals be taken using a summing integrator. EQUIPMENT: GP-6 Analog computer with power supply and connecting wires. BACKGROUND The operation of summation of any number of input voltages, say, E1, E2, …En is given by the output voltage:  (1)
where Rf is the feedback resistor in the summer configuration and R1,R2, . . . , Rn are the resistors in the forward path. The minus sign appearing in the expression (1), is due to the inversion of the signals by the OP amp. in the inverting configuration. Fig. 1. shows how the summer is connected. Similarly, an integration operation results in the following output voltage  (2)
where E is the input voltage, R is the value of the resistor in the forward path and C is the value of the capacitor in the feedback path. For n input voltages, E1,E2,..,En connected through n resistors R1,R2,..,Rn the sum of their integrals is given by the output voltage:  (3)
Finally, the operation of the summing integrator and how it is connected are also shown in Fig. 2. PROCEDURE: 1. To verify Summing Operation (i) Turn on the power ON switch using the COMPUTE TIME knob. (ii) Check to see that all Op amps. (six of them) are connected either through a resistor or a capacitor in the feedback path. (iii) Now Connect the summing amplifier as shown in Fig: 1(a). (iv) To set the potentiometers, set the Mode selector to Pot set and select the desired potentiometer from the Y/pot address. Now connect the potentiometer input to the supply and adjust its setting using the corresponding potentiometer knob and the display. While you are doing this, check the rear of the panel to make sure that the meter is connected to the Y/address. (v) Complete the connections for the summing amplifier as shown in Fig. 2. (Patch panel operation). Once you are finished with the connections, call the instructor to check it for you and make sure that it is correct. (vi) Now monitor the output of the summer on the display panel and record the result. To do this, set the Y/pot address to GND/X and the X/address to the desired amplifier (1-6). Set the Mode selector to OPR and press the OP push button. Also check the rear of the GP-6 to see that the X-address is connected to the meter. 11 .To verify Integrator Operation (i) Similarly, connect the integrator as shown in Fig. 1(b) to generate a ramp. (ii) Increase the compute time to 100 using the compute-time knob. This allows the integrator to integrate slowly. (iii) Now monitor the output of the corresponding amplifier on the display panel, and see that it overloads after a short time. This confirms that the output of the amplifier is the integral of the input, and reaches the maximum value of 10 volts as the input is being summed. Report Your report should include the following:
 
Figure 1: A Summer with Four inputs 
Figure 2 An Integrator with one input Download 184.48 Kb. Share with your friends:
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