자동제어공학 2. 물리적 시스템의 전달함수 정 우 용

제2장 물리적 시스템의 전달함수 (Transfer Functions of Physical Systems) 개요 Laplace 변환 복습 전달 함수 전달함수 예 비선형성과 선형화

개 요 물리적 시스템의 구조로부터 수학적 모델을 얻는 방법 1. 주파수 영역에서의 전달함수 (2장) 개 요 물리적 시스템의 구조로부터 수학적 모델을 얻는 방법 1. 주파수 영역에서의 전달함수 (2장) 2. 시간 영역에서의 상태방정식 (3장) 시스템 블록도 부 시스템들의 상호 접속 블록도

라플라스 프랑스의 천문학자 ·수학자. 국적 : 프랑스 활동분야 : 천문학 ·수학 출생지 : 프랑스 칼바도스 보콩타노주 주요저서 : 《천체역학》(전 5권, 1799~1825), 《세계계도설(世界系圖說)》(1796)

Laplace 변환 복습 Laplace 변환 특징 미분 방정식으로 된 시스템은 블록선도로 모델링 하기 힘듬 표현할 수 있다. 복소수 u(t) = 1 t>0 : unit step function = 0 t<0

EX 2.1 Laplace transform of a time function Prob. Find the Laplace transform of

Inverse transform of EX 2.2 Inverse Laplace transform Prob. Find the inverse Laplace transform of F(s)=1/(s+3)2 Inverse transform of

부분 분수 전개 (Partial fraction expansion ) 1. 함수 F(s)의 분모가 서로 다른 실근 2. 함수 F(s)의 분모항이 실수인 중근을 포함 3. 함수 F(s)의 분모항이 복소수나 순 허근을 포함 Euler 공식

1. 함수 F(s)의 분모가 서로 다른 실근

2. 함수 F(s)의 분모항이 실수인 중근을 포함

3. 함수 F(s)의 분모항이 복소수나 순 허근을 포함

Symbolic Math Laplace transform과 inverse Laplace transform 실행 LT I-LT

Symbolic Math Partial fraction expansion 실행 r – 분자 p – 분모의 근 k - 몫

전달함수 예

b. transformed two-loop electrical network; EX 2.10 Figure 2.6 a. Two-loop electrical network b. transformed two-loop electrical network; c. block diagram Mesh 1 Mesh 2

2.83(a) 2.83(b)

EX 2.12

2.90(a) 2.90(b)

Figure 2.9 Three-loop electrical network EX 2.13 Mesh 3 Figure 2.9 Three-loop electrical network Mesh 1 Mesh 2

2.91 2.92 2.93

Figure 2. 10 a. Operational amplifier; b Figure 2.10 a. Operational amplifier; b. schematic for an inverting operational amplifier; c. inverting operational amplifier configured for transfer function realization. Typically, the amplifier gain, A, is omitted

EX 2.14 Transfer function – inverting operational amplifier circuit Problem Find the transfer function, , for the circuit given in figure 2.11

EX 2.15 Transfer function – noninverting operational amplifier circuit Problem Find the transfer function, , for the circuit given in figure 2.13 Figure 2.13 Noninverting operational amplifier circuit for Example 2.15

Table 2.4 Force-velocity, force-displacement, and impedance translational relationships for springs, viscous dampers, and mass

Figure 2.15 a. Mass, spring, and damper system b. block diagram

Figure 2. 16 a. Free-body diagram of mass, spring, and damper system b Figure 2.16 a. Free-body diagram of mass, spring, and damper system b. transformed free-body diagram

Figure 2.17 a. Two-degrees-of-freedom translational mechanical system8; b. block diagram

2.120(a) 2.120(b)

Figure 2.20 Three-degrees-of-freedom translational mechanical system

2.121 2.122 2.123

Table 2.5 Torque-angular velocity, torque-angular displacement, and impedance rotational relationships for springs, viscous dampers, and inertia

Figure 2.22 a. Physical system; b. schematic; c. block diagram

Figure 2. 23 a. Torques on J1 due only to the motion of J1 b Figure 2.23 a. Torques on J1 due only to the motion of J1 b. torques on J1 due only to the motion of J2 c. final free-body diagram for J1

Figure 2. 24 a. Torques on J2 due only to the motion of J2; b Figure 2.24 a. Torques on J2 due only to the motion of J2; b. torques on J2 due only to the motion of J1 c. final free-body diagram for J2

Figure 2.25 Three-degrees-of-freedom rotational system

SKILL-ASSESSMENT EXERCISE 2.9 Problem: Find the transfer function, G(s)=θ2/T(s), for the rotational mechanical system shown in Figure 2.26. FIGURE 2.26 Rotational mechanical system for Skill-assessment exercise 2.9 ANSWER:

Figure 2.27 A gear system

Figure 2. 29 a. Rotational system driven by gears; b Figure 2.29 a. Rotational system driven by gears; b. equivalent system at the output after reflection of input torque; c. equivalent system at the input after reflection of impedances

Figure 2. 30 a. Rotational mechanical system with gears; b Figure 2.30 a. Rotational mechanical system with gears; b. system after reflection of torques and impedances to the output shaft; c. block diagram

Figure 2.36 Typical equivalent mechanical loading on a motor 전기 기계 시스템 (Electromechanical systems) Figure 2.34 NASA flight simulator robot arm with electromechanical control system components © Debra Lex. Figure 2.36 Typical equivalent mechanical loading on a motor