L5: OpAmp Dynamics, Integrators Differentiators

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L5 Op-Amp Dynamics,IntegratorsDifferentiators

• ICB EngineeringModeling and ControlEngineering
  of Compartment Systems
• Fall, 2006

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Review Our final Model of an Op-Amp
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Last TimeUsing Op-amps for Gain
Out Converges to 2 X In
V Out / 2
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Last TimeWhat does this do?
Out Converges to -In
Converges to 0
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Take Away Lessons from Last Time

• Negative Feedback around a integrator loop causes
  the integrator output to converge to a value
  where the integrator input is 0.
• i.e. negative feedback drives error to 0.

Error
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What Determines How Fast We Converge?
The Gain! Low Gain Slow convergence High Gain
Fast convergence
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Can we figure out exactly how fast?
What functions looks like themselves after
integration or differentiation?
A Exponentials!
What are the units?
What is ? The Time Constant
A Seconds
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How far do we converge in one time constant?
This is
of the distance from out(0) to 0
or 63 of the way
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How far do we converge in two time constants?
This is
of the distance from out(0) to 0
or 82 of the way
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How far do we converge in three time constants?
This is
of the distance from out(0) to 0
or 88 of the way
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Big Idea
of the way there
Each additional time constant gets us another
Self-Similarity All of the state at any time is
captured in the value of the single integrator,
so we can arbitrarily pick t0 at any point along
the way, and see the same (scaled) exponential
convergence with the SAME time constant.
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What it the Gain is very High (e.g. 107)?
Then the time constant is very small! e.g. 10-7
seconds (100 ns).
Result System converges very quickly. We can now
shift out attention to the
quasi-static equilibrium, and assume the
convergence happens really really
fast (i.e. on a time scale finer than we care
about).
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New Circuit
What does this do?
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Review Capacitors
Ic
Capacitors are integrators of current.

Vc
C
C Capacitance (Farads)
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Just Like Bathtubs!
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New Circuit The Integrator
RC inverse gain of the integration
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Whats going on here?
A Leak in the Tub !
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What does this do?
Differentiate !