Solved: Design a circuit with one input, x(t), and one output, y(t), that are related by

Chapter 7, Problem P7.9-4

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QUESTION:

Design a circuit with one input, x(t), and one output, y(t), that are related by this differential equation:

\(\frac{d^3}{d t^3} y(t)+16 \frac{d^2}{d t^2} y(t)+8 \frac{d}{d t} y(t)+10 y(t)=4 x(t)\)

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QUESTION:

Design a circuit with one input, x(t), and one output, y(t), that are related by this differential equation:

\(\frac{d^3}{d t^3} y(t)+16 \frac{d^2}{d t^2} y(t)+8 \frac{d}{d t} y(t)+10 y(t)=4 x(t)\)

ANSWER:

Step 1 of 9:

For first order circuit, voltage of the capacitor equals:

In this case the output voltage is known, and equals:

When comparing these two equations we can conclude the following:

                                                                                                                                    (1)

                                                                                                                                     (2)

                                                                                                                     (3)

We can solve equations 2 and 3 to find  and :

=                                                                                                                                              (4)

=                                                                                                                                           (5)

For  the switch is closed, the capacitor is at steady state, meaning it acts as an open loop. The circuit looks as shown below.

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