Lect06a-FirstOrderSystems

# Lect06a-FirstOrderSystems - M.D Bryant ME 340 notes First...

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M.D. Bryant ME 340 notes 6/15/10 1 First Order Systems Simplest dynamic system 1 energy storage element Example : RC circuit, switch closes t = 0 KVL: V + iR + q C = 0 , t > 0 Series elements: i = dq dt = ˙ q

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M.D. Bryant ME 340 notes 6/15/10 2 Differential equation describes t > 0: RC dq dt + q = CV , t > 0 (ON) Initial condition describes t = 0: q(0) = Q = C v C (0) Rearrange DE into standard form : τ ˙ q + q = f ( t ) time constant τ Example: R = 10 , C = 1 μF, v = 10 V, v C (0) = 2 V τ = RC = 10 μs f(t) = CV = 1 μF (10 V) = 10 μC q(0) = 1 μF (2 V) = 2 μC
M.D. Bryant ME 340 notes 6/15/10 3 Solution: Differential Equations Definition : Given differential equations (i.e., eqns with derivative term(s)) that relate dependent variables x to an independent variable t . A solution x(t) , when substituted into the equation, produces an identity devoid of (without) derivatives. Solution method: Guess & Test Guess: obtain candidate solution any way possible Test: Substitute candidate into differential equation. If equation reduces to identity without derivatives, i.e. derivatives disappear, solution = candidate .

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M.D. Bryant
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