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From: Michael Robinson on 4 May 2010 16:25

"amakyonin" <amakyonin-u1(a)yahoo.com> wrote in message
news:2deebac5-d838-4aba-85d5-c53609f816b2(a)i10g2000yqh.googlegroups.com...
> 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.

Is the load series or parallel RC?


From: gearhead on 4 May 2010 21:15
On May 2, 6:14 pm, amakyonin <amakyonin...(a)yahoo.com> wrote:
> 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.

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 response of the series RC
to the unit step function of voltage M*a at time t=a.
From: Michael Robinson on 4 May 2010 20:45

"amakyonin" <amakyonin-u1(a)yahoo.com> wrote in message
news:2deebac5-d838-4aba-85d5-c53609f816b2(a)i10g2000yqh.googlegroups.com...
> 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.



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.


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