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S. Awad, Ph.D. M. Corless, M.S.E.E. E.C.E. Department University of Michigan-Dearborn Differential Equations Math Review with Matlab: First Order Constant Coefficient Linear Differential Equations

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Differential Equations:First Order Systems 2 First Order Constant Coefficient Linear Differential Equations n First Order Differential Equations First Order Differential Equations n General Solution of a First Order Constant Coefficient Differential Equation General Solution of a First Order Constant Coefficient Differential Equation n Electrical Applications Electrical Applications n RC Application Example RC Application Example

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Differential Equations:First Order Systems 3 First Order D.E. A General First Order Linear Constant Coefficient Differential Equation of x(t) has the form: Where is a constant and the function f(t) is given

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Differential Equations:First Order Systems 4 In general the coefficient of dx/dt is normalized to 1 Properties n The DE is a linear combination of x(t) and its derivative n x(t) and its derivative are multiplied by constants A General First Order Linear Constant Coefficient DE of x(t) has the properties: n There are no cross products

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Differential Equations:First Order Systems 5 SOLUTION Fundamental Theorem n A fundamental theorem of differential equations states that given a differential equation of the form below where x(t)=x p (t) is any solution to: n and x(t)=x c (t) is any solution to the homogenous equation n Then x(t) = x p (t)+x c (t) is also a solution to the original DE

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Differential Equations:First Order Systems 6 f(t) = Constant Solution If f(t) = (some constant) the general solution to the differential equation consists of two parts that are obtained by solving the two equations: x p (t) = Particular Integral Solution x c (t) = Complementary Solution

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Differential Equations:First Order Systems 7 Particular Integral Solution n Since the right-hand side is a constant, it is reasonable to assume that x p (t) must also be a constant n Substituting yields:

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Differential Equations:First Order Systems 8 Complementary Solution n To solve for x c (t) rearrange terms n Which is equivalent to: n Integrating both sides: n Taking the exponential of both sides: n Resulting in:

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Differential Equations:First Order Systems 9 First Order Solution Summary n A General First-Order Constant Coefficient Differential Equation of the form: and are constants n Has a General Solution of the form and are constants

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Differential Equations:First Order Systems 10 Particular and Complementary Solutions Particular Integral SolutionComplementary Solution

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Differential Equations:First Order Systems 11 Determining K 1 and K 2 n In certain applications it may be possible to directly determine the constants K 1 and K 2 n The second by taking the limit as t approaches infinity n The first relationship can be seen by evaluating for t=0

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Differential Equations:First Order Systems 12 Solution Summary n By rearranging terms, we see that given particular conditions, the solution to: and are constants n Takes the form:

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Differential Equations:First Order Systems 13 n A Resistor has a linear relationship between voltage and current governed by Ohms Law Electrical Applications n Basic electrical elements such as resistors (R), capacitors (C), and inductors (L) are defined by their voltage and current relationships

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Differential Equations:First Order Systems 14 Capacitors and Inductors n The current and voltage relationship for a capacitor C is given by: n The current and voltage relationship for an inductor L is given by: n A first-order differential equation is used to describe electrical circuits containing a single memory storage elements like a capacitors or inductor

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Differential Equations:First Order Systems 15 RC Application Example n Example: For the circuit below, determine an equation for the voltage across the capacitor for t>0. Assume that the capacitor is initially discharged and the switch closes at time t=0

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Differential Equations:First Order Systems 16 Plan of Attack n Write a first-order differential equation for the circuit for time t>0 The solution will be of the form K 1 +K 2 e - t n These constants can be found by: Determining u Determining v c (0) Determining v c ( ) n Finally graph the resulting v c (t)

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Differential Equations:First Order Systems 17 Equation for t > 0 n Use KVL and Ohms Law to write an equation describing the circuit after the switch closes n Kirchhoffs Voltage Law (KVL) states that the sum of the voltages around a closed loop must equal zero n Ohms Law states that the voltage across a resistor is directly proportional to the current through it, V=IR

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Differential Equations:First Order Systems 18 Differential Equation n Since we want to solve for v c (t), write the differential equation for the circuit in terms of v c (t) Replace i = Cdv/dt for capacitor current voltage relationship n Rearrange terms to put DE in Standard Form

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Differential Equations:First Order Systems 19 General Solution n The solution will now take the standard form: can be directly determined K 1 and K 2 depend on v c (0) and v c ( )

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Differential Equations:First Order Systems 20 Initial Condition n A physical property of a capacitor is that voltage cannot change instantaneously across it n Before the switch closes, the capacitor was initially discharged, therefore: n Therefore voltage is a continuous function of time and the limit as t approaches 0 from the right v c (0 - ) is the same as t approaching from the left v c (0 + ) n Substituting gives:

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Differential Equations:First Order Systems 21 Steady State Condition n As t approaches infinity, the capacitor will fully charge to the source V DC voltage No current will flow in the circuit because there will be no potential difference across the resistor, v R ( ) = 0 V

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Differential Equations:First Order Systems 22 Solve Differential Equation Now solve for K 1 and K 2 n Replace to solve differential equation for v c (t)

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Differential Equations:First Order Systems 23 Time Constant When analyzing electrical circuits the constant 1/ is called the Time Constant n The time constant determines the rate at which the decaying exponential goes to zero K 1 = Steady State Solution = Time Constant Hence the time constant determines how long it takes to reach the steady state constant value of K 1

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Differential Equations:First Order Systems 24 Plot Capacitor Voltage For First-order RC circuits the Time Constant = 1/RC

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Differential Equations:First Order Systems 25 Summary n Discussed general form of a first order constant coefficient differential equation n Proved general solution to a first order constant coefficient differential equation n Applied general solution to analyze a resistor and capacitor electrical circuit

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