diffeqns_06 - Difference and Differential Equations 1....

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Difference and Differential Equations 1. Differential Equations – Continuous-Time Systems Consider the following simple circuit. In this circuit, the voltage source x(t) will be assumed to be the input and the voltage across the resistor, y(t) , the output. A voltage equation involving the source and the voltage drops across the circuit elements can be written as follows. () () () () 1 Integro-differential Equation t di t xt L i d R it dt C ττ −∞ =+ + () () () ( ) y t yt R R =⇒ = Substituting this last expression for i(t) into the integro-differential equation, 1 t Ldyt xt y d y t Rd t R C −∞ + This is a new integro-differential equation involving the input x(t) and the output y(t) . This equation can be turned into a differential equation by differentiating both sides of the above equation. ( ) ( ) ( ) 2 2 1 dx t d y t dy t L yt dt R dt RC dt + Now assume x(t) is a known input signal, and it is desired to determine the output signal, y(t) . Rewrite the differential equation in standard form,
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2 ( ) ( ) () ( ) 2 2 1 d y td y x t RR yt L dt LC L dt dt ++ = This is a second order differential equation with constant coefficients. It directly relates the input and output. For electrical circuits composed of resistors, inductors, and capacitors, no matter how complicated, the relationship between any two variables in the circuit is given by a differential equation with constant coefficients. The order of the difference equation will be equal to the number of independent storage elements in the circuit (inductors and capacitors). It can be shown that the system described by differential equations with constant coefficients are always linear and time-invariant. Previously it was shown that one method of determining the output of a LTI system was to use convolution. A second method of determining the output of a time-continuous LTI system is to solve the differential equation relating the output and the input. 2. Difference Equations – Discrete-Time Systems The relationship between the input and output of discrete-time LTI systems can be described by a difference equation relating the input and output. Ex: Consider the following discrete-time system. In this diagram the boxes with a D in them are delay elements. Their output is equal to the input delayed by one unit. The triangular shaped elements produce an output which is the input multiplied by the constant value indicated. The circular elements with a plus sign produce an output which is the sum of their inputs. This is an example of a LTI, discrete-time system. The relationship between the output and input can now be obtained easily. ( ) ( ) ( ) ( ) 12 1 1 yn ayn bxn =− − − − +
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3 This can be written in a standard form ( ) ( ) ( ) ( ) 12 1 1 yn ayn ayn bxn +− + =− This is a 2 nd order difference equation. The order is determined by the difference between the largest and smallest index appearing in the equation.
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This note was uploaded on 10/24/2009 for the course ECE 108 taught by Professor Li during the Spring '08 term at Lehigh University .

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diffeqns_06 - Difference and Differential Equations 1....

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