P 8.7-6 Determine v(t) for the circuit shown in Figure P 8.7-6.

 

P8_7_6

Figure P 8.7-6

 

For t < 0, when the circuit is in steady state we have

At steady state the capacitor is acting like an open circuit so there is no current going through the 5 Ω resistor.  The voltage across the capacitor then is the voltage across the 2 Ω resistor.  We can find that by voltage division on the right loop of the circuit where current is flowing from the 30 V source.

 

 

After time t = 0 the circuit now looks like and we have annotated it to help in analyzing it.

Let’s write a current junction

 

 

 

We can find  by

 

Plugging this into the current junction equation

 

 

Rearranging

 

 

So this is the forcing equation.  Assume a forcing solution of

 

 

 

 

Collecting terms

 

Comparing coefficients

and

The first equation yields

And the second

So the forced solution is

 

For natural solution need Thevenin resistance.  Deactivating power source gives

4 Ω and 2 Ω resistors are in parallel so

 

 

This is now in series with the 5 Ω resistor, so the Thevenin resistance is

 

So the time constant is

 

So the natural solution is

So a full solution is

 

Now use value at t = 0

 

 

So

 

Or

 

 

Please send any comments or questions about this page to ddonovan@nmu.edu

This page last updated on July 26, 2019