Explanation Given data: Refer to Problem 16.104 in the textbook. Formula used: Write a standard expression to represent the state equation. x ˙ = A x + B z (1) Here, A is the system matrix, B is the input coupling matrix, x is the state vector , and z is the input vector. Write a general form to calculate the state vector representing n state vectors. x = [ x 1 x 2 ⋮ x n ] (2) Write a general form to calculate the input vector representing m inputs. z = [ z 1 z 2 ⋮ z m ] (3) Write a general form of output equation. y = C x + D z (4) Here, C is the output matrix, and D is the feedforward matrix. Substitute the equations (2) and (3) in equation (1) to find x ˙ . x ˙ = A [ x 1 x 2 ⋮ x n ] + B [ z 1 z 2 ⋮ z m ] (5) Substitute the equations (2) and (3) in equation (4) to find y . y = C [ x 1 x 2 ⋮ x n ] + D [ z 1 z 2 ⋮ z m ] (6) Calculation: The given circuit is drawn as shown in Figure 1 with the assumed values. Assume the inductor current i L and the capacitor voltage v C as the state variables. Therefore, x ˙ = [ i ˙ L v ˙ C ] Let current through Loop 1 in Figure 1 is current through capacitor C . i 1 ( t ) = 1 4 v ˙ C . (7) The current through Loop 2 in Figure 1 is current through inductor L . i 2 ( t ) = i L (8) Apply Kirchhoff’s voltage law for Loop 1 in Figure 1. − v 1 ( t ) + v C + 2 i 1 ( t ) − 2 i 2 ( t ) = 0 Substitute equation (7) and (8) in above equation. − v 1 ( t ) + v C + 2 ( 1 4 v ˙ C ) − 2 i L = 0 1 2 v ˙ C = v 1 ( t ) − v C + 2 i L Simplify the above equation to find v ˙ c . v ˙ C = 2 v 1 ( t ) − 2 v C + 4 i L (9) Apply Kirchhoff’s voltage law for Loop 2 in Figure 1. ( 1 ) i 2 + v 2 ( t ) + 2 i 2 ( t ) − 2 i 1 ( t ) = 0 Substitute equation (7) and (8) in above equation ( 1 ) i ˙ L + v 2 ( t ) + 2 i L − 2 ( 1 4 v ˙ C ) = 0 i ˙ L + v 2 ( t ) + 2 i L − 1 2 v ˙ C = 0 Simplify the above equation to find i ˙ L . i ˙ L = − v 2 ( t ) − 2 i L + 1 2 v ˙ C Substitute equation (9) in above equation to find i ˙ L . i ˙ L = − v 2 ( t ) − 2 i L + 1 2 ( 2 v 1 ( t ) − 2 v C + 4 i L ) = − v 2 ( t ) − 2 i L + v 1 ( t ) − v C + 2 i L i ˙ L = − v C + v 1 − v 2 (10) Substitute equation (9) in equation (7) to find i 1 ( t )

6th Edition

ISBN: 9780078028229

Author: Charles K Alexander, Matthew Sadiku

Publisher: McGraw-Hill Education

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Chapter 16, Problem 90P

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Develop the state equation for the problem designed in problem 16.104.

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Chapter 16 Solutions

Fundamentals of Electric Circuits

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Develop the state equation for the problem designed in problem 16.104.

Solution Summary: The author explains how to develop the state equation for the problem designed in problem 16.104 in the book.

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Develop the state equation for the problem designed in problem 16.104.

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Chapter 16 Solutions