Explanation Given data: Refer to Problem 16.12 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. Assume the inductor current i L and the capacitor voltage v C as the state variables. Therefore, x ˙ = [ v ˙ C i ˙ ] Apply Kirchhoff’s voltage laws for Figure 1. − 8 u ( t ) + i ( 1 Ω ) + v C + ( 1 H ) i ' = 0 − 8 u ( t ) + i + v C + i ' = 0 Simplify the above equation to find i ' . i ' = − v C − i + 8 u ( t ) (7) Write an expression to calculate the voltage across the capacitor. i = C v C ′ (8) Here, C is the value of capacitance, and v C ′ is the rate of change of capacitor voltage

6th Edition

ISBN: 9780078028229

Author: Charles K Alexander, Matthew Sadiku

Publisher: McGraw-Hill Education

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Chapter 16, Problem 86P

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Develop the state equations for Problem 16.12 in the textbook.

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Chapter 16, Problem 85P

Chapter 16, Problem 87P

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