2, 1 Step

Consider the circuit below:

To start, consider the case where the current source has been on with value I_o for a long time and the system has reached the steady state. After this, at t=0, the current source changes from I_o to I_1, where I_o > I_1 > 0, so that I(t) follows the general shape shown below:

Part 1

Find the values of i_{\rm R}, i_{\rm C}, i_{\rm L}, and v immediately before and after the step (i.e., at t=0_- and t=0_+), as well as in the limit as t\to\infty. Express each of your answers in terms of R, L, C, I_o, and/or I_1.

Part 2

Now find the first derivatives of the signals part 1, each evaluated at time t=0_+. Express your answers in terms of R, L, C, I_o, and/or I_1.

• What is the value of \displaystyle{{d\over dt}i_{\rm R}(t)} just after time t=0?

• What is the value of \displaystyle{{d\over dt}i_{\rm C}(t)} just after time t=0?

• What is the value of \displaystyle{{d\over dt}i_{\rm L}(t)} just after time t=0?

• What is the value of \displaystyle{{d\over dt}v(t)} just after time t=0?

Part 3

Assuming R > \sqrt{L/C}, i.e., an underdamped system, determine which of the following graphs correctly describes the value of i_{\rm R}, i_{\rm C}, and i_{\rm L} for t > 0. Indicate one graph for each current and explain your reasoning. The scales on the vertical axes are not necessarily consistent from plot to plot (nor with the graph of I(t) from earlier in the problem), but in each graph you may assume that the axes cross at the origin.

                    

                    

                    

                    

 
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