8.4 The Natural Response of a Series/Parallel RLC Circuit

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8.6 State EquationStep2 choose i and v as state variables.Step3 fundamental cutsetabout the capacitor tree branch.dvvC = i -dt 2Step4 fundamental loop for the inductor link.C.T. PandiL + v -12V +4 i = 0dt598.6 State EquationStep5⎡ 1 -1 ⎤⎡0⎤d ⎡v⎤ ⎢C 2C⎥⎡v⎤ = + ⎢1⎥ (12V)dt⎢i⎥ ⎢ ⎥-4 -1⎢i⎥⎣ ⎦ ⎢ ⎥⎣ ⎦ ⎢ ⎥⎢⎣L⎦⎣LL ⎥⎦[ A ] [ B]<strong>The</strong> desired solutions are v and i⎡v⎤ ⎡1 0⎤⎡v⎤ ⎡0⎤y = ⎢ = + (12V)i⎥ ⎢0 1⎥⎢i⎥ ⎢0⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦[ C ] [ D]with initial condition +v(0 ) = 12V+i(0 ) = 0AC.T. Pan 60
•Example 2C.T. Pan8.6 State Equation3 2d v d v dv+5 +4 +3 v=u(t)3 2dt dt dt⎧⎧ dx1⎪=xx21= v(t)⎪dt⎪⎪⎪ d(t) v ⎪ dx2Let ⎨x 2= ⇒ ⎨ =x3⎪ dt ⎪ dt2 3⎪ d v(t) ⎪dx3d v⎪x 3= =3 ⎪3⎩ dt ⎩ dt dt2d v dv= -5 - 4 -3 v+u(t)2dt dt61= -5x - 4x -3x +u(t)3 2 1State space representation8.6 State Equation⎡x1⎤ ⎡0 1 0⎤⎡x1⎤⎡0⎤d ⎢ x⎥ =⎢ 0 0 1⎥⎢ x⎥ +⎢ 0⎥ u(t)⎢ ⎥ ⎢ ⎥x ⎢⎣-3 -4 -5⎥⎦ x ⎢⎣1⎥⎦⎡x1⎤y = [ 1 0 0⎢] x⎥2+0u(t)⎢ ⎥[ ]⎢⎣x3⎥⎦∴2 2dt ⎢ ⎥ ⎢ ⎥⎢⎣ 3⎥⎦ ⎢⎣ 3⎥⎦A high order differential equation can be representedin the form <strong>of</strong> state equation.C.T. Pan62
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8.4 The Natural Response of a Series/Parallel RLC Circuit

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