modeling12 - ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 2 2...

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Modeling Q UESTION 12 Find the transfer function ( 29 ( 29 ( 29 o i E s G s E s = for the circuit shown in Figure 12. L 1 R 1 L 2 e i (t) + - e o (t) + - R 2 Figure 12 The circuit loops and nodes are defined in Figure 12.1 below. The constitutive law for the first resistor is ( 29 ( 29 1 3 A e t R I t = (1) The constitutive law for the first inductor is ( 29 ( 29 ( 29 1 1 i A e t e t L I t - = (2) The constitutive law for the second inductor is ( 29 ( 29 2 2 o e t L I t = (3) The constitutive law for the second resistor is ( 29 ( 29 ( 29 2 2 A o e t e t R I t - = (4) Applying KCL to node A
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Modeling ( 29 ( 29 ( 29 1 2 3 I t I t I t = + (5) Combining equation (3) with the derivative of equation (5) ( 29 ( 29 ( 29 2 1 3 o e t L I t I t = - (6) Combining equation (2), the derivative of equation (1), and equation (6) ( 29 ( 29 ( 29 ( 29 2 1 1 i A A o e t e t e t e t L L R - = - (7) Substituting equation (4) and its derivative into equation (7)
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Unformatted text preview: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 2 2 1 1 1 1 o i o o L L L R e t e t e t R I t e t L I t L L R R =-+-- & & (8) Taking the derivative of equation (8) and substituting equation (3) and its derivative into the resulting equation ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 1 1 1 1 1 1 o o o i L R L R L e t e t e t e t R R L L L + + + + = && & & (9) Taking the Laplace Transform of equation (9), assuming zero initial conditions, and rearranging ( 29 ( 29 ( 29 1 2 2 1 2 1 1 1 2 1 2 1 2 o i E s R L s E s L L s R L L R R L s R R = + + + + (10) I 3 I 1 I 2 L 1 R 1 L 2 e i (t) +-e o (t) +-R 2 A Figure 12.1 2...
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modeling12 - ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 2 2...

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