Differential Equation Corresponding To An Analog Computer Circuit

Differential Equation Corresponding To An Analog Computer Circuit
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Figure 15.1 illustrates an analog computer circuit. Digital computers have replaced analog computers in many functionalities. However, the replacement is not total as there are still some important functions being carried out by analog computers. An analog computer is an electric/electronic device that comprises three operational amplifiers: the inverting amplifier, the summer and the integrator. These three op-amps form the building blocks for analog computers. Their circuits can be integrated to solve differential equations and to simulate dynamic systems.
Find the differential equation corresponding to the analog computer circuit illustrated in figure 15.1. Assume each op-amp is an ideal op-amp.

The strings: S₇P₅A₅₁ (change - physical).

The math:
Pj Problem of Interest is rate of change and so is of type change (physical - change).

Differential Equation Corresponding To An Analog Computer Circuit

Consider figure 15.1. The rightmost (third) op-amp is an inverting op-amp; the middle op-amp (second) is an integrator and the leftmost (first) op-amp is a combination of an integrator and a summer.
In order to find the differential equation correspondng to the analog computer circuit, we start from the output of the rightmost op-amp and work our way back to the output of the leftmost op-amp:
Assume output of middle op-amp = z and output of leftmost op-amp is y.
Output of rightmost op-amp, x = -1/1(z) = -z -------(1)
So, dx/dt = -dz/dt -------(2)
Output of middle op-amp, z = (1/RC)∫(y(t))dt) = (1/1)∫(y(t))dt
So, dz/dt = y -------(3)
Output of leftmost op-amp, y = -(1/0.5)∫(x(t))dt - (1/0.1)∫(f(t))dt
So, y = -2∫(x(t))dt - 10∫(f(t))dt
So, dy/dt = -2x -10f
So, dx/dt = y (substitute (3) into (2)).
So, d²x/dt² = dy/dt = -2x - 10f
So, d²x/dt² + 2x = - 10f -------(4)
So, equation (4) is the differential equation corresponding to the analog computer of figure 15.1.

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