Assignment #18

ECE 2004

  1. For the first-order operational amplifier circuit shown below:
  2. Draw two circuits – one before the voltage source has changed magnitude and the second after the voltage source has changed and the voltages and currents in the circuit have reached steady state.

Prior to t=10 ms, the capacitor has reached steady state, in this case fully discharged to Vc=0 V (and Ic=0 A). Thus, the capacitor appears as an open circuit with 0 V dropped across it.

After the voltage supply asserts to 5V and the capacitor voltage (and current) reach steady state, the capacitor reaches full charge with no current flowing through it. Thus, it appears as an open circuit yielding:

  1. Determine the initial and final values of vL.

From the first figure above (t<10ms), we see that:

From the second schematic, with 5 V applied:

  1. Write an equation for the output voltage when t ≥ to after solving for the relationship between the voltage across the capacitor and the current flowing through it.

Note that , as this denotes the time when the circuit changes states. We can implement KCL at the inverting terminal of the opamp giving:

Thus, the forced response of the first order, differential equation is simply:

The natural (homogeneous) response of the circuit is:

This gives a complete response of:

Utilizing the boundary condition allows us to find the unknown constant, A.

Giving:

Finally:

  1. At what minimum time, t, do we assume that the circuit has reached steady state after the magnitude of the voltage source has changed?

Steady state occurs at: = 5

  1. For the first-order operational amplifier circuit shown below,
  2. Draw two circuits – one before the voltage source has changed magnitude and the second after the voltage source has changed and the voltages and currents in the circuit have reached steady state.

  1. Determine the initial and final values of vL.
  1. Write an equation for the output voltage when t ≥ to after solving for the relationship between the voltage across the inductor and the current flowing through it.

Notice that an inductor requires that: for the derivative to be continuous. However, does not, strictly speaking, exist at t=-60s. However, for t>-60, the inductor voltage returns to 0 V, making .

  1. What is the minimum value of the power supplies for the op amp (V+ and V-) that would allow the circuit to operate as predicted by the equation that you found in part c? The discontinuity of the inductor current, and the resulting infinite voltage spike due to the derivative at t=-60s, would dictate an infinite voltage supply. However, for t>-60s,

  1. Use superposition to determine an equation for the 1st order op amp circuit shown below.

Implementing super-position:

V1:

where 1 denotes the 1st step in solving using super-position.

So that:

V2:

Finally: