Unit 1                                                       

2270                              PRACTICE EXAM SOLUTION Prob 3                                     

 

3.      (50 points)

           

 

         a.      After being closed for a long time, the switch is opened at t = 0. Write a numerical time-domain expression for i(t), the current through the capacitance. This expression must not contain any complex numbers.

         b.      State whether i(t) is underdamped, overdamped, or critically damped.

 

 

 

ans:  a)  

         b)   Underdamped.

 

 

 

sol'n:  (a)   When the switch is open, we have series RLC.

Now find initial condition (i.e. i and di/dt, or v and dv/dt at t = 0+).

Circuit for t = 0-: L's = wires, C's = open circuits.

iL(t=0-) = A (current divider)

,       

After the switch is open, i = -iL since C and L are in series.

∴ Solve for iL and then change the sign. Note that iL is the variable in the differential equation for a series RLC. Thus, we know how to find it.

We also need

Use V-loop for RLC at t = 0+:

Note that at t = 0+, (iR = iL since R, L in series),

Now use general underdamped solution:

sol'n:  (b)  ωo > α ⇒ underdamped