First-Order Transient Circuits

[rms5643]

Homework Statement

Use the step-by-step method to find vo(t) for t > 0 in the circuit in the figure below.

Homework Equations

V=IR, KVL, Mesh Analysis, Voltage Division, Solution form of first order equations

The Attempt at a Solution

1. Finding the current through the inductor before the switch is thrown:
   • Since the circuit has reached steady state, the inductor can be replaced with a short circuit, circumventing the 4Ω resistor.
   • Mesh Analysis: With Y being the current in the left loop and X being the current in the right loop, both clockwise
   • -12+(Y-X)*2-12=0,

     12+(X-Y)*2+X*2=0
     [*]X=6, Y=18
     [*]Current through inductor is 6A at t=0-
     ​
   
   [*]Finding the voltage drop across the 2Ω resistor after the switch has been thrown:
   

   • At the moment the switch is thrown, the inductor can be replaced by a 6A current source. The voltage source on the left is thrown out of the circuit.
   • Loop Analysis clockwise:
   • 12+X*2+(X-6)*4+X*2=0
   • X=1.5
   • 1.5A*2Ω=3V
   • Voltage drop across the resistor in question is 3V at t=0+

   [*]Finding the voltage drop across the 2Ω resistor after the switch has been thrown for a very long time:
   

   • The inductor acts a short circuit, thus circumventing the 4Ω resistor at the top
   • Voltage division:
   • -12*2/4=-6V
   • Voltage drop across the resistor in question is -6V at t=infinity

   [*]Finding the Thevenin equivalent circuit & calculating the time constant τ
   

   • Since the 12V source becomes an open circuit, we only have the 3 resistors
   • 2Ω is in series with the other 2Ω, and the 4Ω equivalent resistor is in parallel with the other 4Ω resistor at the top.
   • The Thevenin resistance is 2Ω
   • Then, τ=(1/3H)/2Ω => 0.1666

   [*]Therefore, the equation modeling this circuit is:
   [*]-6+[3--6]*e^(-t/0.166)
   [*]-6+9*e^(-t/0.166)
   ​

Does this look correct? I've redone my steps and I haven't found an error except in the calculation in my time constant, as you can see the change from 0.08333 to 0.1666. Though, I have one attempt left at this and would like to make sure the previous steps are correct.​

 

[gneill]

Your calculations look good to me.

Note that your time constant works out to 0.1666 which is to say, 1/6. So you could write $\left( -\frac{t}{\tau} \right)$ as (-6t).