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Q. Effect of feedback on stability?
Let us consider the direct transfer function of the unstable first-order system of Figure (a). The transfer function is given by
where p = d/dt is the differential operator. If a unit-step function is applied as the input quantity E, the output becomes
for which the corresponding time solution is given by
c(t) = K(1 - et/τ)
which is clearly unstable, since the response increases without limit as time passes. By placing a negative feedback path H around the direct transfer function, as shown in Figure (b), the closed-loop transfer function is then
where cf(t) denotes response with feedback. The system is now clearly stable with aK > τ . Thus, the insertion of feedback causes the unstable direct transmission system of Figure (a) to become stable. Such a technique is often used to stabilize space rockets and vehicles, which are inherently unstable because of their large length-to-diameter ratios. Note that the direct transfer function of Equation has a pole located in the right half p-plane at p = 1/τ , whereas the pole of the closed-loop transfer function of Equation with aK > τ is located in the left half p-plane.
Consider the circuit of Figure (a) and obtain the complete solution for the voltage v C (t) across the 5-F capacitor and the voltage v x (t) across the 5- resistor.
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