Using Laplace to solve differential equations'

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Lecture 9

• Using Laplace to solve differential equations.
• System defined by a differential equation.
• Transfer function of an LTI, causal system.
• Cascaded systems and other block diagram
  interconnections.

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Using Laplace to solve differential equations.
Linear, non-homogeneous, order n, constant
coefficients.
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General solution, homogeneous case.
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System defined by a differential equation
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In the Laplace domain, the input and output are
related by a simple multiplication by a certain
function H(s). This is a first example of a
transfer function of an LTI system.
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Transfer function of an LTI, causal system
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Proof
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Remarks
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More generally, we can build block-diagrams

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Feedback interconnection

Main lesson Use simple algebra to study
complex systems.
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Example build a transfer function from simple
blocks.
There exist circuits (e.g., based on OP-AMPs)
that approximately implement these basic
functions.
Now, we use them to build a more complicated
transfer function. An analog computer.
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