P 8.7-5 A lossy integrator is shown in Figure P 8.7-5. The lossless capacitor of the ideal integrator circuit has been replaced with a model for the lossy capacitor, namely, a lossless capacitor in parallel with a 1-kΩ resistor. If

v_s = 15e^–2tu(t) V

and

v_o(0) = 10 V,

find v_o(t) for t > 0.

Figure P 8.7-5

Annotating the circuit with currents make the analysis easier so

Now write out a current junction equation

Note: relationship of the current going to capacitor shows that

simplifying

Assume a forced solution of

Then

Comparing coefficients

So

Now the capacitor discharges through the 1 kΩ resistor so the time constant is

So the natural solution looks like

So a total solution would be

Using the t = 0 condition

So

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This page last updated on July 25, 2019