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part3 - Part 3 System Modeling and Analysis Circuit Theory...

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Part 3: System Modeling and Analysis Circuit Theory z Voltage-Current Relationship z Resistor z Capacitor z Inductor R i v R R = C C i dt dv C = L L v dt di L =
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Kirchhoff’s voltage law (KVL) z Conservation of electrostatic field z Apply to a loop – a closed path z The algebraic sum of the voltage dropped in a closed path circuit is equal to the algebraic sum of the source voltage applied. 3 2 1 v v v v S + + = Kirchhoff’s current law (KCL) z Conservation of charge z Apply to a junction or a node in a circuit - a point in the circuit where charge has several possible paths to travel z The sum of the currents flowing into a junction is equal to the sum of the currents flowing out of that same point. 3 2 1 i i i i R + + =
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Something you should know z Mathematics Complex Number Exponential Functions L + + + + = ! 3 ! 2 1 3 2 x x x e x x x e dx de = 1 = j 1 2 = j ( ) ( ) t j t e t j ω ω ω sin cos + = t j t j e j dt de ω ω ω = c e j dt e t j t j + = ω ω ω 1 ( )( ) 2 2 1 b a jb a jb a jb a jb a jb a + = + = + ( ) a b j e b a jb a 1 tan 2 2 + = + System Modeling - Ordinary Differential Equation z A linear time-invariant system is usually modeled as an ordinary differential equation: where x(t) and y(t) are the input and output of the system, respectively; all the coefficients are constant. z Let p be the differentiation operation z ( ) can be rewritten as or where D(p) and N(p) are polynomials of p; ( ) ( ) ( ) ( ) t x p N t y p D = x b dt x d b y a dt dy a dt y d a dt y d a o m m m n n n n n n + + = + + + + L L 0 1 1 1 1 dt d p = ( ) ( ) ( ) ( ) t x b p b p b t y a p a p a p a m m n n n n 0 1 0 1 1 1 + + + = + + + + L L ( ) k k k dt y d t y p = ( )
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How to find the system output if the input is a trigonometric function? Approach 1. Assume a complex exponential input. Eg. 2. Refer to ( ), the frequency remains unchanged for an ODE. Assume the output function as 3. Replace x and y in ( ) with the corresponding exponential functions. 4. Solve the unknown variables 5. Decompose the function into its real part (or imaginary part) if the actual input is real (or imaginary).
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