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

See similar textbooks

Videos

Question

Chapter 16, Problem 90P

To determine

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

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 16, Problem 89P

Chapter 16, Problem 91P

Students have asked these similar questions

Figure 2 3) *** The circuit of Figure 3 is designed with W/L = 20/0.18, λ= 0, and ID = 0.25 mA. (Optional- 20 points) a) Compute the required gate bias voltage. b) With such a gate voltage, how much can W/L be increased while M1 remains in saturation? What is the maximum voltage gain that can be achieved as W/L increases? VDD = 1.8 V RD 2k - Vout Vin M₁ Figure 3

1) Rs = 4kQ, R₁ = 850 kQ, R₂ = 350 kQ, and R₁ = 4 kQ. The transistor parameters are VTP = -12 V, K'p = 40 µA / V², W/L = 80, and λ = 0.05 V-1. (50 Points) a) Determine IDQ and VSDQ. b) Find the small signal voltage gain. (Av) c) Determine the small signal circuit transconductance gain. (Ag = io/vi) d) Find the small signal output resistance. VDD = 10 V 2'; www www Figure 1 Сс

Q11

Chapter 16 Solutions

Fundamentals of Electric Circuits

Ch. 16.2 - Determine vo(t) in the circuit of Fig. 16.6,...Ch. 16.2 - Prob. 2PP Ch. 16.2 - Prob. 3PP Ch. 16.3 - For the circuit shown in Fig. 16.12 with the same...Ch. 16.3 - Prob. 5PP Ch. 16.3 - The initial energy in the circuit of Fig. 16.17 is...Ch. 16.4 - Prob. 7PP Ch. 16.4 - Prob. 8PP Ch. 16.4 - Prob. 9PP Ch. 16.5 - Obtain the state variable model for the circuit...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, electrical-engineering and related others by exploring similar questions and additional content below.

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.

Videos

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

Want to see the full answer?

Chapter 16 Solutions