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Currently I am trying to build an active integrator.

The circuit is similar to this one...

http://www.art-sci.udel.edu/ghw/phys245/05S/classes/images/opamp-integrator.gif

However, the voltage source is an inductor, and the voltage out goes to an oscilloscope (50 Ohms). Using a Marx Generator, a pulse current induces a voltage across my inductor. Using Lenz's Law, the voltage to abhor the flux is of the equation

V_in = L*di/dt + R*i + V_op_amp

and the Voltage Out

V_out = -V_capacitor + V_op_amp

After applying the ideal OpAmp we get

V_in = L*di/dt + R*i & V_out = -V_capacitor

The capacitor is set up the same across the inverting input. My challenge is trying to solve an integral across the capacitor in order to find the integrator time constant.

For now, solving the difference equation for i(t) is...

i(t)= (V_in(t))/r*(1-exp(-(r*t)/l))

When I integrate this equation through the capacitor, I am having trouble with the integral

int(-infinity to t) (1/(r*c))*V_in(t')*exp(-(r*t')/l) w.r.t. t'

If I could somehow preserve the integral of V_in and pull out that exponential function then I could find the integrating factor! I know that with the current being generated (1MA) and the pulse duration (risetime of 90ns) the circuit will have a lot of problems, but the practice of the circuit is not a concern; the theory part is my main challenge. If you want to know the V_in(t) it is...

V_in(t) = (l/n)*A*sin(w*t)*sin(w*t)

where

l=inductance (60 nh)

n= number of turns on inductor (4)

A= Amplitude (1MA)

w= angular frequency (Pi/180ns)

t= time (t = [0, 90ns])

If I may put my problem aptly, it is what technique should I go through to solve that integral with the product of the voltage source and exponential function? Laplace? Fourier? Nevermind the back emf, or saturation across the OpAmp; I just need to know how to solve that integral.

Thank you for your time and any help and/or advice would be appreciated!

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# Trying to solve an integral involving an LRC circuit

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