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Transient response to ramp input

Okay. I'll start out by saying that this is *not* a homework problem. I'm trying to recreate some work I did many years ago which I subsequently lost the notes and spreadsheet for.

My ultimate problem is to compute a reasonable estimate of the power dissipated in a termination resistor as used in a typical digital circuit. This is critical in determining an appropriate minimum package size after consideration of derating requirements. My original solution was applicable to both a source termination driving a CMOS capacitive load or an RC termination at an input.

To this end I'm trying to determine the formula for the transient response of an RC circuit when driven by a ramp function up until a specified rise time. The typical textbook analysis only covers the step response which produces an overly pessimistic estimate of power dissipation. I have found some discussion online involving the idealized unit ramp function but the formula presented have been simplified due to the unitless ramp and don't provide any indication on how to incorporate the rise time of the ramp for a real world analysis.

With the voltage across the resistor described for both the ramp and level portion of the input signal I can integrate the curves to get the total power dissipated in the switching event. My original analysis carried this forward to derive a formula that described the maximum capacitance for various resistances and power limits.

While a relatively simple matter, my skills have unfortunately eroded and for some reason there is no readily available discussion of this topic. I would appreciate any assistance in resolving this problem.

Reply to
amakyonin
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How are you with Laplace domain stuff? I'll show you how I'd do it, and you can tell me if it's over your head (I'm pretty facile with it, this will look oddball to a lot of folks) or if it all makes sense now.

Unit ramp:

x(t) = t * u(t) ==> 1/s^2 (note that u(t) is the unit step function)

Rise to voltage V in time tr and keep right on going:

x(t) = (V/tr) * t * u(t) ==> (V/tr)/s^2

Rise to voltage V in time tr then stay steady:

x(t) = (V/tr) * (t * u(t) - (t - tr) * u(t - tr)) ==> (V/tr) * (1 - e^(-tr * s)) / s^2)

Note that there's a freaking exponential in the numerator of our signal! Don't let this confuse you -- what you want to do is calculate the current through your resistor for the "rise and keep on rising" case, then subtract a delayed version of it (which is what that exponential is describing).

--
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com
Reply to
Tim Wescott

I'm not sure I get what you are trying to do. Is this correct? You want to know the power dissipation in an RC circuit that is driven with a step, but the step has a known slope. (your ramp time)

If that is correct then, For ramp times that are much longer than the RC time the voltage across the R will just follow the ramp. (you can ignore the C.) For ramp times that are much less than the RC time then it's just the step response and you can ignore the ramp time. You are then left with only the intermediate case. For which you may just want to use the ramp time... and ignore the slightly longer time it takes to charge up the cap.

George H.

Reply to
George Herold

Heh. I had my head stuck so deeply up my mathematics that the simple solution didn't occur to me in my rather esoteric answer.

Of course this makes sense! Moreover, if you want more precision for your "long ramp" case, the current in the resistor will simply rise exponentially (per the time constant) then fall the same way -- and probably won't make a significant difference.

--
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com
Reply to
Tim Wescott

u

e quoted text -

Gee thanks Tim, When the two times are equal you might guess the total time constant is something like twice.. If you need to know the details then you need the hairy math. Though if this is used to pick resistor sizes I'm a bit worried. I want all my resistors to be 2 or

3 times over-sized. 50% effects hardly matter.

George H.

Reply to
George Herold

Is the load series or parallel RC?

Reply to
Michael Robinson

For a voltage ramp that terminates at time t=a and drops to zero, has a ramp rate of M, and drives a series RC load, here is the expression for current as a function of time:

i(t)=MC((1-(1/RC))exp(a/RC)-1)exp(-t/RC)

That was the hard part. It should be easy to finish the job off: to get the full expression for a ramp that stops rising at time t=a and levels off to a constant voltage of M*a that it holds permanently, you can take the above relation and add to it the the series RC response to the unit step function of voltage M*a at time t=a.

Reply to
Michael Robinson

For a voltage ramp that terminates at time t=3Da and drops to zero, has a ramp rate of M, and drives a series RC load, here is the expression for current as a function of time: i(t)=3DMC((1-(1/RC))exp(a/RC)-1)exp(-t/RC) That was the hard part. It should be easy to finish the job off: to get the full expression for a ramp that stops rising at time t=3Da and levels off to a constant voltage of M*a that it holds permanently, you can take the above relation and add to it the response of the series RC to the unit step function of voltage M*a at time t=3Da.

Reply to
gearhead

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