Second Order Op Amp circuit: Find Vo for t > 0

[VinnyCee]

Homework Statement

In the circuit below, determine $v_o(t)$ for t > 0. Let $V_{IN}\,=\,u(t)\,V$, $R_!\,=\,R_2\,=\,10\,k\Omega$, $C_1\,=\,C_2\,=\,100\,\muF$.

http://img249.imageshack.us/img249/3840/problem867cg5.jpg

Homework Equations

$$i_c\,=\,C\,\frac{dv_c}{dt}$$

The Attempt at a Solution

Add some node voltages:

http://img413.imageshack.us/img413/2935/problem867part2vn2.jpg

KCL @ $V_1$:

$$\frac{V_{IN}\,-\,V_1}{R_1}\,=\,C_2\,\frac{dv_2}{dt}\,+\,C_1\,\frac{dv_1}{dt}$$

KCL @ $V_2$:

$$C_2\,\frac{dv_2}{dt}\,=\,\frac{V_2\,-\,V_o}{R_2}$$

Substituting for $\frac{dv_2}{dt}$ in the KCL @ $V_1$ equation:

$$V_o\,=\,-R_2C_2\left(\frac{V_{IN}\,-\,V_1}{R_1C_2}\,-\,\frac{C_1}{C_2}\,\frac{dv_1}{dt}\right)$$

Now what do I do to solve?

 

[berkeman]

Set V2 and dV2/dt = 0. Does that help?

 

[m2003]

First, it is important to note that this is a second order op amp circuit, meaning that it contains two energy-storing elements (the capacitors) and will require a second order differential equation to solve. The first step in solving this problem would be to write out the two KCL equations for V1 and V2, as you have done.

Next, you can use the fact that V2 = V1 + Vo (since they are connected in series) to eliminate V2 from the equations and solve for Vo. This will give you a second order differential equation in terms of V1 and Vo.

To solve this equation, you can use standard techniques such as Laplace transforms or circuit analysis methods like the node-voltage method. Once you have solved for V1 and Vo, you can use the relationship Vo = V2 - V1 to find the value of Vo for t > 0.

It is also important to note that since the input voltage is a step function u(t), the output voltage will also be a step function with a time delay due to the capacitors in the circuit. This means that the output voltage will have a transient response before reaching its steady state value.