Explanation Problem design: Develop the state equation for Figure 16.36 in the textbook by assuming the value of resistance R 1 as 2 Ω , R 2 as 4 Ω , the value of inductance L as 1 H and the value of capacitance C as 1 8 H and the input voltage v s as 4 u ( t ) V . Given data: Refer to Problem 16.13 and Figure 16.36 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 ˙ = [ v ˙ C i ˙ L ] Apply Kirchhoff’s current law for Figure 1. i L − v x 2 − 1 8 v ˙ c = 0 8 i L − 4 v x − v ˙ c 8 = 0 8 i L − 4 v x − v ˙ c = 0 Simplify the above equation to find v ˙ c . v ˙ c = 8 i L − 4 v x (7) Refer to Figure 1, the voltage across inductor is calculated as follows: ( 1 ) i ˙ L = v s − v x i ˙ L = v s − v x (8) Apply Kirchhoff’s voltage law for Loop 2 in Figure 1. v x − v C − 4 ( 1 8 v ˙ C ) = 0 v x − v C − 1 2 v ˙ C = 0 Simplify the above equation to find v x . v x = v C + 1 2 v ˙ C (9) Substitute equation (7) in equation (9) to find v x . v x = v C + 1 2 ( 8 i L − 4 v x ) v x = v c + 4 i L − 2 v x v x + 2 v x = v C + 4 i L 3 v x = v C + 4 i L Simplify the above equation to find v x

6th Edition

ISBN: 9780078028229

Author: Charles K Alexander, Matthew Sadiku

Publisher: McGraw-Hill Education

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Chapter 16, Problem 87P

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Develop the state equation for the problem designed in problem 16.13.

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Chapter 16, Problem 86P

Chapter 16, Problem 88P

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