## 1. Pre-Lab Introduction

Figures 7-1 and 7-2 illustrate two op-amp based circuits designed to perform differentiation and integration respectively. The operations are performed "real-time" and can be helpful in observing both initial transients and steady state response. The analysis of the circuits is based on the "ideal" op-amp assumptions and performed in the time domain. The resistor RI shown in the two circuits is included to help with stability and for general circuit protection. The value for RI is nominally set equal to the feedback resistor (Figure 7-1) or the input resistor (Figure 7-2). The purpose of the optional resistor is left for student investigation in conjunction with the summary questions.
The differentiator and integrator circuits may be combined with "standard" inverting and non-inverting op-amp circuits to provide the building blocks for analog computers. The resultant analog computer circuits are designed to solve differential and/or integral/differential equations in a real time environment. The ability to easily include, and change, initial conditions and forcing functions are additional benefits of the analog computers. Figure 3-3 illustrates a circuit designed to solve the second order differential equation KY - Y = 0 with the initial condition Y(0) = - VX and K = R1R2C1C2. The initial condition is "set" by using the momentary contact switch to force the output to equal the applied voltage at t = 0 (the time the switch is closed).

While the major advances in digital computers and digital signal processing have reduced the use of these three circuits, they are still a fast and relatively inexpensive method for process control and stability/operation analysis for systems that can be represented in terms of differential equations.

## 2. Design

1. Derive the expressions relating the input and output signals for the circuits shown in Figures 7-1 and 7-2.
2. Design an analog computer to solve with y(0) = 2. Solve the differential equation when f(t) = 0 and verify your results using PSpice®.

Figure 7 - 1: Differentiator

Figure 7 - 2: Integrator

Figure 7 - 3: Analog Computer (linear, second order, homogenous differential equation)

## 3. Lab Procedure

1. Construct the circuit shown in Figure 7-1. Use ± 15 V supplies for the op-amp and a load resistance of 2.4 k-Ohms.
2. Verify the operation of the circuit using a 500 mV peak, 50 Hz sinewave as the input signal. Be sure to design the "gain" such that the output does not saturate.

3. Repeat step 2 with a sinewave frequency of 500 Hz. Does the circuit still operate correctly? What changes need to be made to prevent output saturation?

4. Repeat steps 2 and 3 using a triangle wave and then using a square wave with the same magnitudes and frequencies as used in steps 2 and 3.

5. Construct the circuit shown in Figure 7-2. Again, use ± 15 V supplies for the op-amp and a load resistance of 2.4 k-Ohms.

6. Repeat steps 2 through 4 for this circuit. Be sure to adjust your "gain" as necessary to maintain an output signal within the saturation limits of ± 12 V.

7. Construct the circuit designed to solve the differential equation in part 2 of the design section. Verify the operation of the design using three different input waves (sine, triangle, and square). Determine the operation for at least three different

frequencies -- 10 Hz, 1 kHz, and 100 kHz. Explain any differences in operation of the circuit. What affect does the initial condition have on the result?

1. How could you use the differentiator to obtain an estimate of the slew rate for the op-amp?
2. Why should you include a resistor in parallel with the capacitor in the integrator?

3. What is the purpose of the resistor in series with the input capacitor in the differentiator?

4. Is it possible to design a circuit to perform the differentiation and integration functions using the non-inverting input? Explain your answer.

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