RC circuit current, large signal

[bitrex]

So without the Laplace transform just from first principles, apply a
sine of amplitude A, A*sin(omega*t) to a series RC to ground:

I(t) = C*dv/dt,

I(t) = C*(A*omega*cos(omega*t) - I'(t)*R)

I(t) = K*e^(-t/RC) + [A*C^2*R*omega^2*sin(omega*t)]/[C^2*R^2*omega^2 +
1] + [A*C*omega*cos(omega*t)]/[C^2*R^2*omega^2 + 1]

does that look right?

 

[bitrex]

On 9/5/19 5:56 PM, Tim Williams wrote:

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

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:

<https://www.wolframalpha.com/input/?i=inverse+Laplace+transform+%28A+%CF%89%29%2F%28s%5E2+%2B+C+R+s%5E3+%2B+%281+%2B+C+R+s%29+%CF%89%5E2%29>

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

 

[bitrex]

On 9/5/19 6:30 PM, bitrex wrote:

On 9/5/19 5:56 PM, Tim Williams wrote:
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


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:

https://www.wolframalpha.com/input/?i=inverse+Laplace+transform+%28A+%CF%89%29%2F%28s%5E2+%2B+C+R+s%5E3+%2B+%281+%2B+C+R+s%29+%CF%89%5E2%29


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.

 

[bitrex]

On 9/5/19 5:56 PM, Tim Williams wrote:

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

The solution comes from here:

<https://www.wolframalpha.com/input/?i=C*%28A*omega*cos%28omega*t%29+-+R*i%27%28t%29%29+%3D+i%28t%29&assumption=%22i%22+-%3E+%22Variable%22>

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

 

[Tim Williams]

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: https://www.seventransistorlabs.com/

"bitrex" <user@example.net> wrote in message
news:14ecF.533202$on8.156744@fx46.iad...

So without the Laplace transform just from first principles, apply a sine
of amplitude A, A*sin(omega*t) to a series RC to ground:

I(t) = C*dv/dt,

I(t) = C*(A*omega*cos(omega*t) - I'(t)*R)

I(t) = K*e^(-t/RC) + [A*C^2*R*omega^2*sin(omega*t)]/[C^2*R^2*omega^2 + 1]
+ [A*C*omega*cos(omega*t)]/[C^2*R^2*omega^2 + 1]

does that look right?

 

On Thursday, September 5, 2019 at 6:19:53 PM UTC-4, bitrex wrote:

On 9/5/19 5:56 PM, Tim Williams wrote:
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


The solution comes from here:

https://www.wolframalpha.com/input/?i=C*%28A*omega*cos%28omega*t%29+-+R*i%27%28t%29%29+%3D+i%28t%29&assumption=%22i%22+-%3E+%22Variable%22

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

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)?

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

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.

 

On Thursday, September 5, 2019 at 4:18:10 PM UTC-4, bitrex wrote:

So without the Laplace transform just from first principles, apply a
sine of amplitude A, A*sin(omega*t) to a series RC to ground:

I(t) = C*dv/dt,

I(t) = C*(A*omega*cos(omega*t) - I'(t)*R)

I(t) = K*e^(-t/RC) + [A*C^2*R*omega^2*sin(omega*t)]/[C^2*R^2*omega^2 +
1] + [A*C*omega*cos(omega*t)]/[C^2*R^2*omega^2 + 1]

does that look right?

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.

 

[bitrex]

On 9/5/19 9:46 PM, seagirt555@gmail.com wrote:

On Thursday, September 5, 2019 at 4:18:10 PM UTC-4, bitrex wrote:
So without the Laplace transform just from first principles, apply a
sine of amplitude A, A*sin(omega*t) to a series RC to ground:

I(t) = C*dv/dt,

I(t) = C*(A*omega*cos(omega*t) - I'(t)*R)

I(t) = K*e^(-t/RC) + [A*C^2*R*omega^2*sin(omega*t)]/[C^2*R^2*omega^2 +
1] + [A*C*omega*cos(omega*t)]/[C^2*R^2*omega^2 + 1]

does that look right?

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.

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.

 

[bitrex]

On 9/5/19 10:28 PM, bitrex wrote:

On 9/5/19 9:46 PM, seagirt555@gmail.com wrote:
On Thursday, September 5, 2019 at 4:18:10 PM UTC-4, bitrex wrote:
So without the Laplace transform just from first principles, apply a
sine of amplitude A, A*sin(omega*t) to a series RC to ground:

I(t) = C*dv/dt,

I(t) = C*(A*omega*cos(omega*t) - I'(t)*R)

I(t) = K*e^(-t/RC) + [A*C^2*R*omega^2*sin(omega*t)]/[C^2*R^2*omega^2 +
1] + [A*C*omega*cos(omega*t)]/[C^2*R^2*omega^2 + 1]

does that look right?

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.


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

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:
<https://www.wolframalpha.com/input/?i=inverse+Laplace+transform+%28A+%CF%89%29%2F%28s%5E2+%2B+C+R+s%5E3+%2B+%281+%2B+C+R+s%29+%CF%89%5E2%29>

 

[bitrex]

On 9/5/19 10:35 PM, bitrex wrote:

On 9/5/19 10:28 PM, bitrex wrote:
On 9/5/19 9:46 PM, seagirt555@gmail.com wrote:
On Thursday, September 5, 2019 at 4:18:10 PM UTC-4, bitrex wrote:
So without the Laplace transform just from first principles, apply a
sine of amplitude A, A*sin(omega*t) to a series RC to ground:

I(t) = C*dv/dt,

I(t) = C*(A*omega*cos(omega*t) - I'(t)*R)

I(t) = K*e^(-t/RC) + [A*C^2*R*omega^2*sin(omega*t)]/[C^2*R^2*omega^2 +
1] + [A*C*omega*cos(omega*t)]/[C^2*R^2*omega^2 + 1]

does that look right?

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.


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

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:
https://www.wolframalpha.com/input/?i=inverse+Laplace+transform+%28A+%CF%89%29%2F%28s%5E2+%2B+C+R+s%5E3+%2B+%281+%2B+C+R+s%29+%CF%89%5E2%29

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.

<https://www.wolframalpha.com/input/?i=sin%28omega*t%29+%3D+v%28t%29+%2B+RC*v%28t%29%27>

 

[bitrex]

On 9/5/19 9:52 PM, seagirt555@gmail.com wrote:

On Thursday, September 5, 2019 at 6:19:53 PM UTC-4, bitrex wrote:
On 9/5/19 5:56 PM, Tim Williams wrote:
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


The solution comes from here:

https://www.wolframalpha.com/input/?i=C*%28A*omega*cos%28omega*t%29+-+R*i%27%28t%29%29+%3D+i%28t%29&assumption=%22i%22+-%3E+%22Variable%22

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

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)?

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

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.

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

<https://www.wolframalpha.com/input/?i=sin%28omega*t%29+%3D+v%28t%29+%2B+RC*v%28t%29%27>

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.

 

On Thursday, September 5, 2019 at 10:46:55 PM UTC-4, bitrex wrote:

On 9/5/19 9:52 PM, seagirt555@gmail.com wrote:
On Thursday, September 5, 2019 at 6:19:53 PM UTC-4, bitrex wrote:
On 9/5/19 5:56 PM, Tim Williams wrote:
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


The solution comes from here:

https://www.wolframalpha.com/input/?i=C*%28A*omega*cos%28omega*t%29+-+R*i%27%28t%29%29+%3D+i%28t%29&assumption=%22i%22+-%3E+%22Variable%22

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

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)?

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

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.


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

https://www.wolframalpha.com/input/?i=sin%28omega*t%29+%3D+v%28t%29+%2B+RC*v%28t%29%27

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?

 

[bitrex]

On 9/6/19 12:08 PM, jlarkin@highlandsniptechnology.com wrote:

On Thu, 5 Sep 2019 18:52:53 -0700 (PDT), seagirt555@gmail.com wrote:

On Thursday, September 5, 2019 at 6:19:53 PM UTC-4, bitrex wrote:
On 9/5/19 5:56 PM, Tim Williams wrote:
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


The solution comes from here:

https://www.wolframalpha.com/input/?i=C*%28A*omega*cos%28omega*t%29+-+R*i%27%28t%29%29+%3D+i%28t%29&assumption=%22i%22+-%3E+%22Variable%22

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

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)?

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

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.

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.

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!!!

 

On Thu, 5 Sep 2019 18:52:53 -0700 (PDT), seagirt555@gmail.com wrote:

On Thursday, September 5, 2019 at 6:19:53 PM UTC-4, bitrex wrote:
On 9/5/19 5:56 PM, Tim Williams wrote:
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


The solution comes from here:

https://www.wolframalpha.com/input/?i=C*%28A*omega*cos%28omega*t%29+-+R*i%27%28t%29%29+%3D+i%28t%29&assumption=%22i%22+-%3E+%22Variable%22

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

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)?

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

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.

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.

 

[bitrex]

On 9/6/19 10:56 AM, seagirt555@gmail.com wrote:

On Thursday, September 5, 2019 at 10:46:55 PM UTC-4, bitrex wrote:
On 9/5/19 9:52 PM, seagirt555@gmail.com wrote:
On Thursday, September 5, 2019 at 6:19:53 PM UTC-4, bitrex wrote:
On 9/5/19 5:56 PM, Tim Williams wrote:
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


The solution comes from here:

https://www.wolframalpha.com/input/?i=C*%28A*omega*cos%28omega*t%29+-+R*i%27%28t%29%29+%3D+i%28t%29&assumption=%22i%22+-%3E+%22Variable%22

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

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)?

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

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.


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

https://www.wolframalpha.com/input/?i=sin%28omega*t%29+%3D+v%28t%29+%2B+RC*v%28t%29%27

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?

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?

 

[bitrex]

On 9/6/19 12:08 PM, jlarkin@highlandsniptechnology.com wrote:

On Thu, 5 Sep 2019 18:52:53 -0700 (PDT), seagirt555@gmail.com wrote:

On Thursday, September 5, 2019 at 6:19:53 PM UTC-4, bitrex wrote:
On 9/5/19 5:56 PM, Tim Williams wrote:
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


The solution comes from here:

https://www.wolframalpha.com/input/?i=C*%28A*omega*cos%28omega*t%29+-+R*i%27%28t%29%29+%3D+i%28t%29&assumption=%22i%22+-%3E+%22Variable%22

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

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)?

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

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.

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.

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.

 

[bitrex]

On 9/5/19 5:56 PM, Tim Williams wrote:

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

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

 

[John Larkin]

On Fri, 6 Sep 2019 10:17:33 -0700 (PDT), seagirt555@gmail.com wrote:

On Friday, September 6, 2019 at 12:08:15 PM UTC-4, jla...@highlandsniptechnology.com wrote:
On Thu, 5 Sep 2019 18:52:53 -0700 (PDT), seagirt555@gmail.com wrote:

On Thursday, September 5, 2019 at 6:19:53 PM UTC-4, bitrex wrote:
On 9/5/19 5:56 PM, Tim Williams wrote:
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


The solution comes from here:

https://www.wolframalpha.com/input/?i=C*%28A*omega*cos%28omega*t%29+-+R*i%27%28t%29%29+%3D+i%28t%29&assumption=%22i%22+-%3E+%22Variable%22

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

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)?

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

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.

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




It's only a sine burst if you include the case where the voltage source
is turned on, ie it's not steady state, continuous. Usually when dealing
with a sine wave source it's the steady state that we assume we're talking
about.

That's the dilemma of a differential equation solution: the sine has
to be turned on.

You see the same thing in Spice.

 

[Tim Williams]

"bitrex" <user@example.net> wrote in message
news:WUvcF.380363$AR4.33491@fx39.iad...

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!!!

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

--
Seven Transistor Labs, LLC
Electrical Engineering Consultation and Design
Website: https://www.seventransistorlabs.com/

 

[George Herold]

On Friday, September 6, 2019 at 12:35:06 PM UTC-4, bitrex wrote:

On 9/6/19 12:08 PM, jlarkin@highlandsniptechnology.com wrote:
On Thu, 5 Sep 2019 18:52:53 -0700 (PDT), seagirt555@gmail.com wrote:

On Thursday, September 5, 2019 at 6:19:53 PM UTC-4, bitrex wrote:
On 9/5/19 5:56 PM, Tim Williams wrote:
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


The solution comes from here:

https://www.wolframalpha.com/input/?i=C*%28A*omega*cos%28omega*t%29+-+R*i%27%28t%29%29+%3D+i%28t%29&assumption=%22i%22+-%3E+%22Variable%22

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

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)?

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

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.

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.


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!!!

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.