Differential Equations

# Differential Equations - Dierential Equations The behaviour...

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Diferential Equations The behaviour oF RLC circuits is described by diferential equations. The relationship between any voltage or current in the circuit is related to any other voltage or current in the circuit by a diferential equation.

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These diferential equations have a special structure. For circuits with one voltage source or one current source, they are always o± the ±orm a n d n y dt n + a n 1 d n 1 y dt n 1 + ... + a 1 dy dt + a 0 y = b n d m u dt m + b m 1 d m 1 u dt m 1 + + b 1 du dt + b 0 u (1)
They are ordinary diferential equations i.e. no partial derivatives arise. They are linear o.d.e.’s i.e. each term in the equation involves only one signal (i.e. voltage or current). The coeﬃcients are constants. All in all, equations oF this Form are called linear ordinary diferential equations with constant coeﬃcients.

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Operator Notation When solving the equations given by nodal or mesh analysis, there is a formalism which makes the process much easier. Let multiplication by s denote diFerentiation. Let multiplication by 1 s denote integration. Then, multiplication by s 2 denotes diFerentiating twice, etc. C d dt ( v 3 v 2 ) Cs ( v 3 v 2 ) L ° ( v 4 v 3 ) L 1 s ( v 4 v 3 )
Example ✫✪ ✬✩ i s 2 F 3Ω 6Ω 4 F v o + How does the voltage v o depend on the applied current i s ?

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Node v 1 : v 1 3 +2 s ( v 1 v o )+ v 1 v o 6 i s =0 2 v 1 + 12 sv 1 12 sv o + v 1 v o 6 i s 3 v 1 + 12 sv 1 12 sv o v o 6 i s (12 s + 3) v 1 (12 s + 1) v o 6 i s Node v 0 : v o v 1 6 +4 sv o s ( v o v 1 )=0 v o v 1 + 24 sv o + 12 sv o 12 sv 1 v o v 1 + 36 sv o 12 sv 1 (36 s + 1) v o (12 s + 1) v 1 Eliminate v 1 : v 1 = 12 s +1 12 s +3 v o + 6 12 s i s v 1 = 36 s 12 s v o 12 s 12 s v o + 6 12 s i s = 36 s 12 s v o
12 s +1 12 s +3 v o + 6 12 s i s = 36 s 12 s v o Clear denominators (12 s + 1)(12 s + 1) v o + 6(12 s + 1) i s = (36 s + 1)(12 s + 3) v o 6(12 s + 1) i s = (36 s +1)(12 s +3) v o (12 s s +1) v o (72 s + 6) i s = (288 s 2 + 96 s + 2) v o (36 s + 3) i s = (144 s 2 + 48 s + 1) v o Equivalently, 36 di s dt i s = 144 d 2 v o dt + 48 dv o dt + v o The transfer function is v o i s = 36 s 144 s 2 + 48 s

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Transfer Functions Eqn (1) in operator notation is . . . ( a n s n + a n 1 s n 1 + ... + a 1 s 1 + a 0 ) y =( b m s m + b m 1 s m 1 + + b 1 s 1 + b 0 ) u y u = ° m i =0 b i s i ° n i =0 a i s i = G ( s ) This is called the system’s transfer function.
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Differential Equations - Dierential Equations The behaviour...

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