How Do You Solve an Op-Amp Integrator Circuit with a DC Offset?

[rhysticlight]

Homework Statement

The problem is shown in the attached PDF, it is basically applying a square wave signal to an op-amp integrator which has a resistor across the capacitor to correct for small DC signal.

In this case it is given that the capacitor has an initial charge of 1 V on it.

Homework Equations

Relevant equations are the transfer function of such an op-amp:
H(S) = \frac{-R2}{R1} \frac{1}{1+R2_{2}SC}

As well as the time domain equation for the output voltage:
V_out = (-1/R1C)*[ integral from 0 to t (V_indt] - V_c

where V_c is the initial voltage across the capacitor

The Attempt at a Solution

So basically I have found three separate ways to solve this problem, but am not sure which one is correct:

First Way
Recognize the fact that the corner frequency of the low pass filter is 0.5 rad/s while the frequency of our input waveform (from 0 to 500 ms) is 4 pi, which is much greater than the corner frequency and we can therefore neglect the shunt resistor and treat this as a simple integrator.
This would leave the same 1 V charge on the capacitor at the end of the waveform, and for 500 ms < t < inifinity, we simply have an exponential decay of this 1 V charge with a time constant of 2 seconds.

Second Way
Draw the circuit in the Laplace domain, which results in us having to add a voltage source in series with the capacitor to account for the initial charge on the capacitor. The expression for the output in the s-domain then becomes:
(see attachment 2)

This reduces to the same expression for my case of 500 ms < t < inifinity, but I don't think it is correct for the other two intervals, but am not quite sure why (I feel like this expression isn't properly accounting for the fact that the capacitor is charging)

Third Way
Use a formula that I got out of the textbook that basically has the form of the final equation in the attached thumbnail. This basically treats the circuit as a simple STC circuit, I feel like the answers I get with this equation should match up with my second method above, but so far they are not.

Clearly I am pretty confused as to how to best approach this problem and would appreciate any advice you can give as everything I have found online or in books doesn't really examine this type of op-amp in this manner (generally just stops after getting the transfer function and noting its characteristics). Sorry for the long and possibly confusing post!

Thanks!

 

[berkeman]

Homework Statement

The problem is shown in the attached PDF, it is basically applying a square wave signal to an op-amp integrator which has a resistor across the capacitor to correct for small DC signal.

In this case it is given that the capacitor has an initial charge of 1 V on it.

Homework Equations

Relevant equations are the transfer function of such an op-amp:
H(S) = \frac{-R2}{R1} \frac{1}{1+R2_{2}SC}

As well as the time domain equation for the output voltage:
V_out = (-1/R1C)*[ integral from 0 to t (V_indt] - V_c

where V_c is the initial voltage across the capacitor

The Attempt at a Solution

So basically I have found three separate ways to solve this problem, but am not sure which one is correct:

First Way
Recognize the fact that the corner frequency of the low pass filter is 0.5 rad/s while the frequency of our input waveform (from 0 to 500 ms) is 4 pi, which is much greater than the corner frequency and we can therefore neglect the shunt resistor and treat this as a simple integrator.
This would leave the same 1 V charge on the capacitor at the end of the waveform, and for 500 ms < t < inifinity, we simply have an exponential decay of this 1 V charge with a time constant of 2 seconds.

Second Way
Draw the circuit in the Laplace domain, which results in us having to add a voltage source in series with the capacitor to account for the initial charge on the capacitor. The expression for the output in the s-domain then becomes:
(see attachment 2)

This reduces to the same expression for my case of 500 ms < t < inifinity, but I don't think it is correct for the other two intervals, but am not quite sure why (I feel like this expression isn't properly accounting for the fact that the capacitor is charging)

Third Way
Use a formula that I got out of the textbook that basically has the form of the final equation in the attached thumbnail. This basically treats the circuit as a simple STC circuit, I feel like the answers I get with this equation should match up with my second method above, but so far they are not.

Clearly I am pretty confused as to how to best approach this problem and would appreciate any advice you can give as everything I have found online or in books doesn't really examine this type of op-amp in this manner (generally just stops after getting the transfer function and noting its characteristics). Sorry for the long and possibly confusing post!

Thanks!

First of all, the resistor across the integration cap forms a highpass filter, not a lowpass filter, IMO.

Second, it would seem to be simpler for this problem to just write an equation for the voltage across the cap based on the virtual ground property of the opamp. You are given the input voltage as a function of time, and you know that the left side of the cap is "grounded", and the current through the source resistor has to all flow through the cap and resistor. You should be able to write a KCL or similar equation, for the voltage across the R//C as a function of time... Maybe try that approach to see if it works for you.

Remember, the circuit is just an integrator, with some leakage off of the cap afforded by the parallel resistor. So the output waveform will look like parts of triangle waves (ramps), with some leakage toward zero volts opposing the ramps away from zero volts (and helping the ramps toward zero volts...).