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Graphical Analysis of Integrator Responses to Impulse and Step Functions in System Dynamics

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This article explores the graphical representations of integrator circuit responses to various standard inputs, including the unit impulse function, unit step function, unit ramp input, and rectangular pulse. Each response is examined with detailed illustrations to facilitate understanding of how integrators process these signals over time. Key concepts such as the impulse response, integration boundaries, and the effects of different signal types are critically analyzed, providing insight into the behavior of linear time-invariant systems.

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Graphical Analysis of Integrator Responses to Impulse and Step Functions in System Dynamics

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  1. Examples of Convolution 3.2

  2. Graphical Examples • Impulse Response of an Integrator circuit due to the unit impulse function • Response of an integrator due to aunit step function • Response of an Integrator circuit due to the unit ramp input • Response of an integrator due to a rectangular pulse

  3. Impulse Response of an Integrator =δ(t) d(τ) d(τ) 0 0 interval of integration interval of integration

  4. Graphical Examples • Impulse Response of an Integrator circuit due to the unit impulse function • Response of an integrator due to aunit step function • Response of an Integrator circuit due to the unit ramp input • Response of an integrator due to a rectangular pulse

  5. Response of an Integrator Due to a Unit Ramp Function =tu(t) Why is h(t)=u(t)?

  6. Response of an Integrator Due to a Unit Step Function =tu(t) Constant!

  7. Graphical Illustration (t=-1) No area of intersection

  8. Graphical Illustration (t=1)

  9. Response of an Integrator Due to a Unit Step Function =tu(t) Unit step function is only 1 when the argument is greater than 0. It does not make sense to integrate all the way to infinity.

  10. A system with rectangular impulse response u(t) δ(t) u(t-2)

  11. Example (why not take advatage of linearity?)

  12. Focus on x1(t) δ(t)→h(t) δ(t+3)→h(t+3)

  13. Understanding h(t-τ)

  14. h(t-τ) when t=-1 h(τ) h(-1+τ) h(-1-τ)

  15. Integration when t<0 h(t-τ) for t<0

  16. h(t-τ) when t=0.5 h(τ) h(0.5-τ) h(0.5+τ)

  17. Integration when 0<t<2 h(t-τ) for 0<t<2 (move to the right as t increases)

  18. h(t-τ) when t=3.5 h(τ) h(3.5+τ) h(3.5-τ) 3.5 3.5-2

  19. Integration when t>2 t-2 t

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