# EXPERIMENT # 3: Analog Simulation of a First Order System (RC Circuit)

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## EXPERIMENT # 3: Analog Simulation of a First Order System (RC Circuit)

OBJECTIVES:
The objective of this experiment is to simulate a first order system, an RC-circuit, and study its behavior.
INTRODUCTION
An electrical RC-circuit is the simplest example of a first order system. It comprises of a resistor and capacitor connected in series to a voltage supply as shown below on Figure 1. If the capacitor is initially uncharged at zero voltage when the circuit is switched on, it starts to charge due to the current i through the resistor until the voltage across it reaches the supply voltage. As soon as this happens, the current stops flowing or decays to zero, and the circuit becomes like an open circuit. However, if the supply voltage is removed, and the circuit Is closed, the capacitor will discharge the energy it stored again through the resistor. The time it takes the capacitor to charge depends on the time constant of the system, which is defined as the time taken by the voltage across the capacitor to rise to approximately 63% of the supply voltage. For a given RC-circuit, this time constant is . Hence its magnitude depends on the values of the circuit components.
The RC circuit will always behave in this way, no matter what the values of the components. That is, the voltage across the capacitor will never increase indefinitely. In this respect we will say that the system is passive and because of this property it is stable.

EQUIPMENT:

1. GP-6 Analog Computer

2. X-Y Plotter

BACKGROUND
For the RC-circuit as shown in Fig. 1, the equation governing its behavior is given by (1)
where is the voltage across the capacitor, R is the resistance and C is the capacitance. The constant is the time constant of the system and is defined as the time required by the system output i.e. to rise to 63% of its final value (which is E). Hence the above equation (1) can be expressed in terms of the time constant as: (1)

PROCEDURE
(1) First write the system equations in the form suitable for drawing the analog simulation diagram.

(ii) Now draw the analog block diagrams corresponding to the following values of R, C, and E:

given in Table 1.

Table 1 E=1.0 volts, Time R C 10 sec 1 0.2 20 sec 1 0.1

(iii) For the first set of values, connect the analog circuit on the GP-6 using one integrator. (iv) Set the Y/pot-address to GND/X and mode selector to OPR.

(v) Select the output amplifier from the X-address, press the OP push button and monitor the output of the output of the system v

(vi) Repeat the above procedure (iv) and plot the output of the system on the X-Y plot. The procedure of how to operate the X-Y plotter is explained in the attached sheet.

(vii) Now repeat the above procedures (i)-(vi) for the second set of values of R, C.
REPORT
Your report should include the following:

1. A plot of the capacitor output voltage for the two cases.

2. From the plots, estimate the time-constant of the system.

3. Compare the experimental values of the time-constant with the theoretical values from Table 1. ## EXPERIMENT # 4: Analog Simulation of a Second Order Mass-Spring Mechanical System

OBJECTIVES:
The objective of this experiment is to simulate a second order mass-spring system and study its behavior.
INTRODUCTION
The mass spring system is a very good example of a second order physical system. The equations governing the behavior of the system are easily derived from Newton’s law. Indeed all mechanical systems, right from the simple pendulum to the more complicated aircraft are of second order form; because of the fact that they obey Newton’s laws or Lagrange’s or Hamilton’s principle. Therefore, studying the behavior of the mass-spring system, will give us great insight into the behavior of many. mechanical systems without too much complicated mathematical analysis.

Furthermore, because of the frictional effects on the system, and an additional damping that may be introduced in the system, the mass-spring system is always a stable system. That is, it represents an energy dissipative system.

EQUIPMENT:

1. GP-6 Analog Computer

2. X-Y Plotter

BACKGROUND
The mass-spring system is shown in Fig.1, the equation governing its behavior is given by where f is the input force, x is the displacement, M is the mass of the system, B is friction and K is the spring constant.

PROCEDURE
(i) First write the system equations in the form suitable for drawing the analog simulation diagram.

(ii) Now draw the analog block diagrams corresponding to the following values of M, B, and K:

given in Table 1.

(iii) For the first set of values, connect the analog circuit on the GP-6 using two integrator.

(iv) Set the Y/pot-address to GND/X and mode selector to OPR.

(v) Select the output amplifier from the X-address, press the OP push button and monitor the output of the output of the system x(t).

(vi) Plot the response of the system on the X-Y plotter. Details of how to use the X-Y plotter are attached.
(vii) Now repeat the above procedures (i)-(vi) for the remaining set of values of M B, K.

Table 1.

 T (secs) M B K  10 1 0 2 0 0 10 1 1 2 2 0 10 1 2 2 1 1 10 1 3 2 0 0

REPORT
Your report should include the following:

1. A plot of the response (displacement x(t)) for the various values of the parameters.

2. Compare the response of the system for the various values of the parameters and make comments.

3. For what set of values of the parameters does the system has the fastest response?

4. For what set of values of the parameters does the system has the slowest response?

5. For what set of values of the parameters is the system oscillatory? Can you explain why? Figure 1: Mass-Spring-friction System