RC circuit current, large signal

I'd have to think about it a bit or write it out, but it seems wrong (but may be right) that there would be anything 2nd order (omega^2).

The solution seems reasonable otherwise: there is an exponential term for startup transient, and the driven function and its derivative, with coefficients (this is a nonhomogeneous system).

Tim

-- Seven Transistor Labs, LLC Electrical Engineering Consultation and Design Website:

The solution comes from here:

Wasn't sure myself but can't see where I went wrong in the original equation to make it that way. Total current thru the capacitor is the series current, that's I(t) = C*dv/dt, where v is the voltage across the capacitor. Voltage across cap is d/dt A*sin(omega*t) minus drop across the resistor which is V(t) = I(t)*R, dV/dt = dV/dI*dI/dt = R*I'(t)?

The omega squared only scales the amplitude so you can't get frequency multiplication out of it or anything (something would definitely be wrong, then.) If it's assumed C^2*R^2*omega^2 is large with respect to 1 the omega^2 term goes away

Using the Laplace transform of A*sin(omega*t) = A*omega/(s^2 + omega^2) times s domain RC transfer function 1/(1 + sCR) to find the voltage across the resistor, and the current would be dividing by R:

Looks similar, may be the same if you simplify it fully. there are definitely omega^2 terms

Well it's different because it doesn't incorporate the integration constant as in the ODE.

Not sure what you're trying to do.

You have a voltage source A*sin(wt) applied across a resistor and cap in series. The equation I see, using KVL is:

A*sin(wt) = Vc + Vr where Vc is the voltage across the cap, Vr across resistor

The current for the cap is I = C*dvc/dt, Vr = I*R

Substituting:

A*sin(wt) = Vc + RC*dVc/dt

And I agree with the other poster, you can't get second order anything out of it, it;s a linear system driven by a sinewave.

no, the voltage across the cap is just v or A*sin(omega*t) - voltage drop across resistor. voltage drop across resistor is R * I or RC dv/dt

A*sin(omega*t) = v +RC*dv/dt

which is V(t) = I(t)*R, dV/dt = dV/dI*dI/dt = R*I'(t)?

You certainly can't get freq multiplication out of it, it's a linear system driven by a sine wave. The only freq there is omega.

only difference in the solution I see is it flips the solution around such that the steady state periodic solution amplitudes are in terms of t^2 instead of angular frequency omega^2, and the transient solution exponential is in terms of omega instead of t. t^2 and (2*pi*f)^2 are multiplicative inverse of each other up to the factor four pi squared, t = 1/f. These equations are saying the same things

There's no squared term inside the trig functions either way. The steady state is a linear combination of sinusoids of the same angular frequency as the input. The t^2 or equivalently omega^2 terms only affects the amplitude.

still perfectly cromulent LTI system. There aren't any LTI-rules constraining what the amplitude coefficients can be of the linear sum of particular solutions as far as I know.

Nope. Redact that omega is a constant not a function of the variable t.

I still think it's fine to scale by a squared constant whatever it is because there's no frequency multiplication occuring. But I'm still confused as to why both the Laplace transform-solution and the ODE I wrote give the solution in terms of squared angular frequency while the above gives it in terms of squared time.

See:

Whoops. Wolfram Alpha was assuming omega was the independent variable and I didn't notice. Silly...

Anyway point is the equation A*sin(wt) = Vc + RC*dVc/dt, my equation, and the Laplace transform all agree the amplitudes are a function of omega^2.

Nevermind my follow-ups below, far as I can tell all these ODEs agree:

It's OK to scale the amplitude by omega^2. It's just a constant. No frequency multiplication occurs, sines and cosines are still functions of the same angular frequency as the input, everything is still LTI.

There is no way the amplitude of the current or voltages scales by the square of the frequency, not in a linear system. For example, the impedance of the cap is 1/SC. How do you get amplitude of the voltage on the cap or resistor or the current changing by the square of the freq out of that?

It's essentially a sine burst, so there is a DC component that fades out in time but complicates the math.

I see that a lot in my alternator-simulation Spice runs; the first few, sometimes many, cycles of a sine-forced system are different from the steady state.

Then all the original ODEs must be wrong somehow or not describe the actual physical behavior of a real circuit. I trust that the CAS has provided the correct solution to the ODE you provided as well and it includes a squared omega term, so where has this process gone wrong?

If it's impossible for the behavior of the real circuit to include an omega^2 term in the amplitude then what I'm curious about is where the math has gone wrong. The ODEs as written seem correct and they all return a solution including an omega^2 term. Solving in terms of current, in terms of voltage, and by taking the inverse Laplace transform, of the Laplace transform of a sine times the impulse response all seem to return congruent answers aside from arbitrary dependence on initial conditions.

I think it would be good to fully understand the large signal time-domain behavior of some of simple circuits, mathematically. Maybe make good interview question?

Perhaps once we understand RC circuits we can be engineers!!!

Tim I am also attempting to summon Jim Thompson back from beyond for the good of the group, and there is only one way to even theoretically accomplish that. One way. which is for a "leftist" to make a mathematical error in the description of a simple circuit.

I estimate the chances at 50/50

I mean, I got A's in my DE and signals classes. I have the tools to solve this problem, I just don't particularly care to.

JL might not be able to, but he went to school probably before DEs existed, so it's not entirely his fault.

Anyway, there are better tutors than us, elsewhere on the internet. This simple system has been solved time and again.

You can even find videos on it, probably if you don't mind that there's a hundred different Indian guys in the most recent videos that turn up in a YT or Google search...

Tim

I like to think of these things in phasor diagrams. Driven by a voltage your job is to find the current. There is some phase angle by which the voltage lags the current, and then the amplitude of the vector which is sqrt (R^2+ (1/wC)^2)

(Of course that doesn't include the transient response.)

George H.

I Spice stuff like that. I prefer waveforms to equations. And equations are pretty much useless for nonlinear circuits.

There's a visual learning feedback loop that Spice assists, to train my instincts... as long as the sims don't run too slow. What's that Skinner Box mouse training time constant? 30 seconds?