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Unformatted text preview: ECE 209: Circuits and Electronics Laboratory Review of Circuits as LTI Systems * Math Background: ODEs, LTI Systems, and Laplace Transforms Engineers must have analytical machinery to understand how systems change over time. For example, springs and dampers in car suspension systems absorb kinetic energy after road disturbances and dissipate the energy gently over time. Electronic filters protect circuits from large input transients in a similar way. Hence, engineers need to know how to choose components in dynamical systems. LTI Systems and ODEs: The motion of many physical systems can be modeled knowing only a few of its derivatives. For example, a cars position over time can be described well knowing its velocity or acceleration. So we focus on the linear time-invariant (LTI) system x ( t ) LTI System y ( t ) which has input x ( t ) (e.g., height of terrain under the car) and output y ( t ) (e.g., height of passengers seat). By our assumption that all we need are a few derivatives of the input and the output, then a typical ordinary differential equation (ODE) that might model such a system is y + 3 y + 2 y = x x (1) where y and y are the first and second derivatives of signal y ( t ). Given a known input x ( t ), we would like to integrate this differential equation to find an expression of y ( t ) that is only in terms of t . Then we can see how to adjust our design parameters so that y ( t ) has a desirable shape (e.g., even with a bumpy x ( t ) road surface, the car seat y ( t ) stays in one place). So we need a convenient way to solve this differential equation. Linear Decomposition: Assume that x can be broken into two parts so that x ( t ) = x 1 ( t ) + x 2 ( t ). By inspection, if I can solve Equation (1) for the response y 1 to x 1 alone and the response y 2 to x 2 alone, then the response y to x would be y 1 + y 2 . Further, imagine that I could find a set of functions such that: When a function in the set is an input to Equation (1) , the output is simple to find. Every useful input x ( t ) can be expressed as a sum (or integral) of functions from this set. In this case, solving ODEs like Equation (1) would be trivial. Before even knowing the input x , I could find the solution to the ODE for every function in this special set. Then when I finally do know my input, I just sum up the relevant prototypical solutions. Consider functions of the form t mapsto e st where s = + j is a complex number with real part and imaginary part . Notice that for a function f ( t ) = Ae st (where A is any constant complex number), f ( t ) = sAe st = sf ( t ) and integraldisplay f ( t )d t = 1 s Ae st = 1 s f ( t ) . (2) That is, for a complex exponential , differentiation is identical to multiplication by s and integration is identical to division by s . So complex exponentials turn ODE calculus into simple algebra....
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This note was uploaded on 08/05/2011 for the course ECE 209 taught by Professor Staff during the Fall '08 term at Ohio State.
- Fall '08