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107_notes01

# 107_notes01 - MAE107 Introduction of Modeling and Analysis...

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MAE107 Introduction of Modeling and Analysis of Dynamic Systems Lecture Notes #1 Prof. M’Closkey First order differential equation review Simple circuit, mass, and fluid examples Block diagrams Textbook reading: all of Chapter 1, especially concentrating on the block diagram concepts; review EE100/EE110L material on modeling simple passive circuits

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First Order Linear Ordinary Differential Equations (ODE) The first order linear ODE is given by: d dt ( x ( t )) = ax ( t ) + bu ( t ) (1) where x = dependent variable that is determined by solving the ODE t = the independent variable, time u = the “forcing” or “nonhomogeneous” term that is known or specified a, b = coefficients that are typically constant . This ODE is linear because x and its time derivative appear linearly in the expression. Another shorthand way of writing this expression is: ˙ x = ax + bu. Initial Value Problem. A common problem is the following: given an initial condition x 0 specified at time t 0 (in other words, x 0 = x ( t 0 ) is specified) and the forcing function u is known for time in the interval t [ t 0 , t 1 ] , then find x over the same time interval. Notes: the “starting time” t 0 is often 0 t 1 may be 1
First Order ODE Circuit Examples Basic Passive Circuit Elements: Equations for these elements: Resistor R 1 + V - Capacitor C 1 + V - Inductor I 1 + V - Resistor: V ( t ) = R 1 i ( t ) Capacitor: i ( t ) = C 1 d dt ( V ( t )) Inductor: V ( t ) = I 1 d dt ( i ( t )) Also recall Kirchoff voltage and current laws: X node i = 0 X closed path V = 0 Example: Low-pass filter circuit example: R 1 C 1 + V in - + V out - V in ( t ) = R 1 i ( t ) + V out ( t ) i ( t ) = C 1 ˙ V out ( t ) ) = ˙ V out ( t ) + 1 R 1 C 1 V out ( t ) = 1 R 1 C 1 V in ( t ) . 2

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First order linear ODE relating V in and V out : ˙ V out ( t ) = - 1 R 1 C 1 | {z } a V out ( t ) + 1 R 1 C 1 | {z } b V in ( t ) (2) Why is this called a “low pass” filter? Let V in = cos( ωt ) , where ω is the frequency in radians per second. Stable linear ODE’s driven with a sinusoidal “forcing” function will posses a sinusoidal response after some initial transient. Thus, we can assume the sinusoidal response of V out to be V out ( t ) = α cos( ωt ) + β sin( ωt ) , where α and β are to be determined (they are functions of ω ). Substituting this expres- sion into (2) yields ω ( - α sin( ωt ) + β cos( ωt )) = - 1 R 1 C 1 ( α cos( ωt ) + β sin( ωt )) + 1 R 1 C 1 cos( ωt ) . Suppose ω is chosen such that ω << 1 R 1 C 1 , then the terms on the left-hand side are much smaller than the terms on the right-hand side so that we find α 1 and β 0 . In other words, V out ( t ) V in ( t ) . On the other hand, if ω >> 1 R 1 C 1 , then α 0 and β 1 R 1 C 1 ω << 1 . Thus, V out is much smaller in amplitude than V in . When ω 1 / ( R 1 C 1 ) the relationship between α and β and ω is more complicated. This circuit is called a low-pass filter precisely because V out ( t ) V in ( t ) when ω << 1 / ( R 1 C 1 ) and because the amplitude of V out is much less than the amplitude of V in when ω >> 1 / ( R 1 C 1 ) . This circuit “passes” sinusoids with low frequency and attenuates sinusoids with high frequency, where low frequency versus high frequency is determined by the parameter 1 / ( R 1 C 1 ) .
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107_notes01 - MAE107 Introduction of Modeling and Analysis...

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