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Chap 8 First_Order_Circuits

Chap 8 First_Order_Circuits - First Order Circuits RC and...

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First Order Circuits: RC and RL Circuits Circuits that contain energy storage elements are solved using differential equations. The “order” of the circuit is specified by the order of the differential equation that solves it. A zero order circuit has zero energy storage elements. (Called a “purely resistive” circuit.) The equations that solve it are zero-order differential equations. (i.e. purely algebraic.) A first order circuit has one (irreducible) energy storage element. The equations that solve it are first order differential equations. A second order circuit has two (irreducible) energy storage elements. The equations that solve it are second order differential equations. etc. Let’s consider a circuit with just one capacitor or one inductor, i.e. a first order circuit . If you considering viewing the circuit from the perspective of the energy storagy element, then by Thevenin’s and Norton’s Theorems we can always reduce a first order circuit to one of these: To solve for the capacitor voltage or the inductor current we need to find solve the equation shown below each of these circuits. These equations merely came from applying KVL (to the circuit with the capacitor) and KCL (to the circuit with the inductor). How do we solve these equations? Note that they all have the same mathematical form! The form is a homogeneous, (i.e. one variable), linear, (i.e. first order), differential equation. The solution to these differential equations are functions of time: v(t) or i(t) We can re-write both equations above in a general way as follows: where o x(t) is either the capacitor voltage,v(t), or the inductor current, i(t) (both are functions of time) o τ is the constant RC or L/R (note: C might be C eq , L might be L eq and R = R T ) o y(t) is the forcing function (related to the voltage source or current source) Page 1 of 11

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So to solve a first order circuit (i.e. solve for v(t) or i(t) ), we need to solve this differential equation. After some work (and taking a class in differential equations) we get the solution: The solution to a differential equation always has two parts: A forced response , x f (t), and a natural response , x n (t). (They have other names too.) Together they form the complete response , x(t). K is an arbitrary constant of integration that shows up in the natural response and is determined from the initial conditions. (There will be the same number of arbitrary constants as the order of the equation. In this case there is only one because it is a first order differential equation.)

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Chap 8 First_Order_Circuits - First Order Circuits RC and...

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