In the following circuit, the current source has an exponential characteristic i

Question

In the following circuit, the current source has an exponential characteristic i(t) = 10e- 10muA. (a) Write the differential equation for the node v(t). (b) What is the time constant? Write the form of the complementary solution. Usually we would want to try a particular solution of the form vp(t) = Ke-10t, since the forcing function is exponential. Why won't that work here? Using the form up(t) = Kte-10t instead, find the particular solution. Then find the complete solution for v(t) by combining your solution with that from (b). Suppose that the sources in the following circuit both undergo step transitions at t = 0. Prior to this, both sources are off, and the capacitor is uncharged. For t = 0, t(t) = 1 mA and v(t) = 1 V. Use superposition to determine vo(t) for all time.

Explanation / Answer

convert the current source to voltage source so voltage =1v superposition=v(t)=0 soR=2 c=.5µf so we have v/2+cdv/dt=1-v/2 = integarting v=1/2(1-e^-2t/RC) 2/RC=2000 so v=1/2(1-e^-2000t) 1 similarly for v1=0 v/2+cdv/dt=1-v/2 2/RC=2000 integrating v=1/2(1-e^-2t/RC) v=1/2(1-e^-2000t) 2 add 1 and 2 v=1/2(1-e^-2000t)+1/2(1-e^-2000t)=(1-e^-2000t)

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