# First-Order Transient Circuits

1. Apr 4, 2014

### rms5643

1. The problem statement, all variables and given/known data
Use the step-by-step method to find vo(t) for t > 0 in the circuit in the figure below.

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

3. 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.​

2. Apr 5, 2014

### Staff: Mentor

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).