17-First Order Circuits 1 - First Order Circuits First...

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First Order Circuits © Fred Terry Fall, 2008 1
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First Order Circuits Only One Energy Storage Element (one L or one C) Resistors, Independent and Dependent Sources Result with be a First Order Ordinary, Linear Differential Equation in t – First Order since we have one circuit element with with a d/dt element constraint 2
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First Order Ordinary Linear Differential First Order Ordinary , Differential Equations: The Short Story I Differential Equations of the form: dy ( ) ( ) P x y Q x + = dx where ( ), ( ) are functions of x only (possibly even constants). The GENERAL solution is made up of a particular solution ( ) P x Q x y x The GENERAL solution is made up of a particular solution which solves the total p equation and a homogeneous solution ( ) which solves the homogeneous equation: h y x dy ( ) 0 dx So the full, general solution is: P x y + = 3 ( ) ( ) ( ) p h y x y x y x = +
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First Order Ordinary Linear Differential First Order Ordinary , Differential Equations: The Short Story II The homogeneous solution ( ) to a 1st order, linear ODE will always have exactly 1 unknown constant, so the general solution: ( ) ( ) ( ) h y x y x y x y x = + will also always have exactly 1 unknown constant. We elimin p h ate that constant by matching to the initial condition (also called the boundary condition). We only match the solution to the initial condition. We will illustrate all this with circuits examples full We will illustrate all this with circuits examples.
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