8.4 The Natural Response of a Series/Parallel RLC Circuit

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8.6 State EquationR∴v = e - ( i + i )RR43 4R4 s L1 L2R+R3 4R+R3 4C.T. Pan67Step5⎡1⎤⎢0 0 0C ⎥10⎢⎥⎡ ⎤⎡vC1⎤ ⎢1 1 ⎥ ⎡vC1⎤⎢0⎥⎢ 0 0d v⎥ ⎢C2 C2 C⎥ ⎢2 v⎥⎢ ⎥⎢ ⎥ C2 1 R= ⎢⎥ ⎢ ⎥⎢ ⎥+ 4 esdt ⎢iL1⎥ ⎢⎢1 1 (R1 R 2)R iL1L⎥− − − + − ⎥ ⎢ ⎥ 1 R 3+R4⎢ ⎥ ⎢⎥ ⎢ ⎥⎢ ⎥⎣iL2⎦ ⎢L1 L1 L1 L1⎥ ⎣iL2⎦ ⎢ 1 ⎥⎢⎢ ⎥−1 −R− (R2+ R) ⎥ L0⎣ 2 ⎦⎢⎥⎣ L2 L2 L2⎦C.T. Pan8.6 State Equation⎡vC1⎤⎡ -R3R4 -R3R4 ⎤⎢v⎥C2 Rv4R4 = ⎢0 0 ⎥ ⎢ ⎥ + es⎣ R3 + R4 R3 + R4⎦ ⎢ iL1⎥ R3 + R4⎢ ⎥⎣iL2⎦RRwhere R @ R 3 45 + R 3 +R 468
Special case8.6 State Equation(a)i 1 i 3i 2From KCLi 1 +i 2 +i 3 = 0∴ i 3 = -i 1 -i 2Inductor current i 3 is dependent on i 1 and i 2 andis no longer a state variable.One can choose only n-1 (here 2) inductorcurrents as state variables.C.T. Pan698.6 State Equation(b)From KVLv C1 +v C2 = v C3One can choose n-1 (here 2) capacitorvoltages as state variables.C.T. Pan70
Page 2 and 3: 8.1 Linear Second Order Circuits•
Page 5: 8.3 Finding Initial ValuesUnder tra
Page 8 and 9: 8.3 Finding Initial Values• Examp
Page 14 and 15: 8.4 The Natural Response of a Serie
Page 20 and 21: 8.5 The Step Response of a Series/P
Page 28 and 29: 8.6 State EquationStep1. Pick a tre
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8.4 The Natural Response of a Series/Parallel RLC Circuit

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