![Page 1: Laplace 변환 - KNUbh.knu.ac.kr/~ilrhee/lecture/mibun/5-laplace.pdf · 2011-10-25 · Laplace 변환 1. Laplace 변환 및 특성 정의: ℒ ∞ ∞ lim →∞ Ex1) ℒ ∞ ∙](https://reader033.vdocuments.mx/reader033/viewer/2022041813/5e595ad60d0d731b9c763180/html5/thumbnails/1.jpg)
Laplace 변환
1. Laplace 변환 및 특성
정의: ℒ
∞
∞
lim→∞
Ex1)
ℒ
∞
∙ lim→∞
lim→∞
∞
lim→∞
← 인 경우
Ex2)
ℒ
∞
lim→∞
lim→∞
∞
← 인 경우
![Page 2: Laplace 변환 - KNUbh.knu.ac.kr/~ilrhee/lecture/mibun/5-laplace.pdf · 2011-10-25 · Laplace 변환 1. Laplace 변환 및 특성 정의: ℒ ∞ ∞ lim →∞ Ex1) ℒ ∞ ∙](https://reader033.vdocuments.mx/reader033/viewer/2022041813/5e595ad60d0d731b9c763180/html5/thumbnails/2.jpg)
Ex3)
ℒ
←인 경우
(1) 선형성(linearity)
ℒ ℒ ℒ
Ex1) cosh
ℒcosh ℒ
ℒ
동일하게 ℒsinh
Ex2) cos sin
cos sin
ℒ ℒcos ℒsin ←ℒ
∴ℒ cos ℒsin
![Page 3: Laplace 변환 - KNUbh.knu.ac.kr/~ilrhee/lecture/mibun/5-laplace.pdf · 2011-10-25 · Laplace 변환 1. Laplace 변환 및 특성 정의: ℒ ∞ ∞ lim →∞ Ex1) ℒ ∞ ∙](https://reader033.vdocuments.mx/reader033/viewer/2022041813/5e595ad60d0d731b9c763180/html5/thumbnails/3.jpg)
(2) 변수에 따른 변환
ℒ
←ℒ
pf) ℒ
∞
∞
←
Ex) ℒ →ℒ
ℒ
(3) 이동성질
ℒ ←ℒ
pf) ℒ
∞
∞
![Page 4: Laplace 변환 - KNUbh.knu.ac.kr/~ilrhee/lecture/mibun/5-laplace.pdf · 2011-10-25 · Laplace 변환 1. Laplace 변환 및 특성 정의: ℒ ∞ ∞ lim →∞ Ex1) ℒ ∞ ∙](https://reader033.vdocuments.mx/reader033/viewer/2022041813/5e595ad60d0d731b9c763180/html5/thumbnails/4.jpg)
(4) 의 Laplace 변환
ℒ
; ≥
pf) ℒ
∞
∞
∞
lim→∞
∞
lim→∞
ℒ ←번∞
∞정리적용
ℒ
계속 적용하면
ℒ
ℒ
ℒ
⋯
⋯
ℒ
![Page 5: Laplace 변환 - KNUbh.knu.ac.kr/~ilrhee/lecture/mibun/5-laplace.pdf · 2011-10-25 · Laplace 변환 1. Laplace 변환 및 특성 정의: ℒ ∞ ∞ lim →∞ Ex1) ℒ ∞ ∙](https://reader033.vdocuments.mx/reader033/viewer/2022041813/5e595ad60d0d731b9c763180/html5/thumbnails/5.jpg)
(5) 의 Laplace 변환
ℒ
←ℒ
pf)
∞
∞
∞
∞
ℒ
위 결과를 다시 적용하면
ℒ
ℒ
이를 일반화시키면
ℒ
![Page 6: Laplace 변환 - KNUbh.knu.ac.kr/~ilrhee/lecture/mibun/5-laplace.pdf · 2011-10-25 · Laplace 변환 1. Laplace 변환 및 특성 정의: ℒ ∞ ∞ lim →∞ Ex1) ℒ ∞ ∙](https://reader033.vdocuments.mx/reader033/viewer/2022041813/5e595ad60d0d731b9c763180/html5/thumbnails/6.jpg)
Ex) ℒ ℒℒ
ℒ
ℒ
Ex) ℒsin ℒsin
Ex)
ℒ ℒ ℒ ℒℒ
Ex) coscos
ℒ ℒℒcosℒcos
![Page 7: Laplace 변환 - KNUbh.knu.ac.kr/~ilrhee/lecture/mibun/5-laplace.pdf · 2011-10-25 · Laplace 변환 1. Laplace 변환 및 특성 정의: ℒ ∞ ∞ lim →∞ Ex1) ℒ ∞ ∙](https://reader033.vdocuments.mx/reader033/viewer/2022041813/5e595ad60d0d731b9c763180/html5/thumbnails/7.jpg)
Ex) sin
ℒ ℒsin
2. Laplace 변환을 이용한 미분방정식의 해법
(1) 도함수의 Laplace 변환
(i) ℒ ′ pf) ℒ ′
∞
′
∞
∞
ℒ
(ii) ℒ ℒ ′
⋯
pf) ℒ ″ ℒ ′′ ℒ ′ ′ ℒ ′ ℒ ′
![Page 8: Laplace 변환 - KNUbh.knu.ac.kr/~ilrhee/lecture/mibun/5-laplace.pdf · 2011-10-25 · Laplace 변환 1. Laplace 변환 및 특성 정의: ℒ ∞ ∞ lim →∞ Ex1) ℒ ∞ ∙](https://reader033.vdocuments.mx/reader033/viewer/2022041813/5e595ad60d0d731b9c763180/html5/thumbnails/8.jpg)
ℒ ″′ ℒ ′′′ ℒ ′′ ′′ ℒ ′ ′
⋮
ℒ ℒ ′
⋯
(2) 역변환
ℒ
Ex) ℒ
ℒ
←ℒ
Ex) ℒ
ℒ
sin
←ℒsin
Ex)
ℒ ℒ
ℒ
ℒ
cossin←ℒsin
ℒ cos
![Page 9: Laplace 변환 - KNUbh.knu.ac.kr/~ilrhee/lecture/mibun/5-laplace.pdf · 2011-10-25 · Laplace 변환 1. Laplace 변환 및 특성 정의: ℒ ∞ ∞ lim →∞ Ex1) ℒ ∞ ∙](https://reader033.vdocuments.mx/reader033/viewer/2022041813/5e595ad60d0d731b9c763180/html5/thumbnails/9.jpg)
Ex) ℒ
×
∴ →
×
∴ →
×
∴
ℒ
ℒ
ℒ
ℒ
![Page 10: Laplace 변환 - KNUbh.knu.ac.kr/~ilrhee/lecture/mibun/5-laplace.pdf · 2011-10-25 · Laplace 변환 1. Laplace 변환 및 특성 정의: ℒ ∞ ∞ lim →∞ Ex1) ℒ ∞ ∙](https://reader033.vdocuments.mx/reader033/viewer/2022041813/5e595ad60d0d731b9c763180/html5/thumbnails/10.jpg)
(3) 미분방정식의 예
Ex) ″′ ′
ℒ″ℒ′ ℒ ℒ ′
초기조건을 이용하면
∴
→
∴
![Page 11: Laplace 변환 - KNUbh.knu.ac.kr/~ilrhee/lecture/mibun/5-laplace.pdf · 2011-10-25 · Laplace 변환 1. Laplace 변환 및 특성 정의: ℒ ∞ ∞ lim →∞ Ex1) ℒ ∞ ∙](https://reader033.vdocuments.mx/reader033/viewer/2022041813/5e595ad60d0d731b9c763180/html5/thumbnails/11.jpg)
ℒ
ℒ
ℒ
ℒ
Ex) ′ sin
ℒ′ ℒ ℒsin
초기조건을 이용하면
∴
→
항: →
상수항: →
∴
![Page 12: Laplace 변환 - KNUbh.knu.ac.kr/~ilrhee/lecture/mibun/5-laplace.pdf · 2011-10-25 · Laplace 변환 1. Laplace 변환 및 특성 정의: ℒ ∞ ∞ lim →∞ Ex1) ℒ ∞ ∙](https://reader033.vdocuments.mx/reader033/viewer/2022041813/5e595ad60d0d731b9c763180/html5/thumbnails/12.jpg)
ℒ ℒ ℒ
ℒ
cossin
Ex) ′′ cos ′
ℒ′′ℒ ℒcos ′
초기조건을 이용하면
∴
ℒ ℒ
ℒ
sin sin
←ℒ ℒ
sin ℒsin
ℒ
sin
![Page 13: Laplace 변환 - KNUbh.knu.ac.kr/~ilrhee/lecture/mibun/5-laplace.pdf · 2011-10-25 · Laplace 변환 1. Laplace 변환 및 특성 정의: ℒ ∞ ∞ lim →∞ Ex1) ℒ ∞ ∙](https://reader033.vdocuments.mx/reader033/viewer/2022041813/5e595ad60d0d731b9c763180/html5/thumbnails/13.jpg)
Ex) ′′′ ′
ℒ′′ℒ′ ℒ ℒ ′
초기조건을 이용하면
∴
ℒ ℒ
ℒ
←ℒ
Ex) ′′′ ′
ℒ′′ℒ′ ℒ ℒ ′
초기조건을 이용하면
![Page 14: Laplace 변환 - KNUbh.knu.ac.kr/~ilrhee/lecture/mibun/5-laplace.pdf · 2011-10-25 · Laplace 변환 1. Laplace 변환 및 특성 정의: ℒ ∞ ∞ lim →∞ Ex1) ℒ ∞ ∙](https://reader033.vdocuments.mx/reader033/viewer/2022041813/5e595ad60d0d731b9c763180/html5/thumbnails/14.jpg)
∴
: →
항: →
: →
∴
ℒ
ℒ
ℒ
ℒ
cos
sin
![Page 15: Laplace 변환 - KNUbh.knu.ac.kr/~ilrhee/lecture/mibun/5-laplace.pdf · 2011-10-25 · Laplace 변환 1. Laplace 변환 및 특성 정의: ℒ ∞ ∞ lim →∞ Ex1) ℒ ∞ ∙](https://reader033.vdocuments.mx/reader033/viewer/2022041813/5e595ad60d0d731b9c763180/html5/thumbnails/15.jpg)
Ex) 계수에 변수 포함
″′ ′
ℒ′ ℒ′
′
ℒ′′ ℒ′′
′∴ ′ ′
초기조건을 이용하면
∴ ′
Integrating factor;
ln
위 미분방정식에 이 factor를 곱하면
′ →′ → ′′
→∴
ℒ ℒ ℒ
′ 를 이용하면
![Page 16: Laplace 변환 - KNUbh.knu.ac.kr/~ilrhee/lecture/mibun/5-laplace.pdf · 2011-10-25 · Laplace 변환 1. Laplace 변환 및 특성 정의: ℒ ∞ ∞ lim →∞ Ex1) ℒ ∞ ∙](https://reader033.vdocuments.mx/reader033/viewer/2022041813/5e595ad60d0d731b9c763180/html5/thumbnails/16.jpg)
Ex) 스프링 시스템
초기조건 ↓
↑
(1) →ℒ″ ℒℒ ′
초기조건을 이용하면
![Page 17: Laplace 변환 - KNUbh.knu.ac.kr/~ilrhee/lecture/mibun/5-laplace.pdf · 2011-10-25 · Laplace 변환 1. Laplace 변환 및 특성 정의: ℒ ∞ ∞ lim →∞ Ex1) ℒ ∞ ∙](https://reader033.vdocuments.mx/reader033/viewer/2022041813/5e595ad60d0d731b9c763180/html5/thumbnails/17.jpg)
(2) →ℒ″ℒℒ ′
초기조건을 이용하면
× × 를 하면
ℒ
sin sin
ℒ
sin sin
![Page 18: Laplace 변환 - KNUbh.knu.ac.kr/~ilrhee/lecture/mibun/5-laplace.pdf · 2011-10-25 · Laplace 변환 1. Laplace 변환 및 특성 정의: ℒ ∞ ∞ lim →∞ Ex1) ℒ ∞ ∙](https://reader033.vdocuments.mx/reader033/viewer/2022041813/5e595ad60d0d731b9c763180/html5/thumbnails/18.jpg)
Ex2) Coupled oscillator
(1)
초기조건
Laplace 변환을 하면
(2)
위 식에서
라고 두면
(3)
![Page 19: Laplace 변환 - KNUbh.knu.ac.kr/~ilrhee/lecture/mibun/5-laplace.pdf · 2011-10-25 · Laplace 변환 1. Laplace 변환 및 특성 정의: ℒ ∞ ∞ lim →∞ Ex1) ℒ ∞ ∙](https://reader033.vdocuments.mx/reader033/viewer/2022041813/5e595ad60d0d731b9c763180/html5/thumbnails/19.jpg)
위 두식을 더하고 빼면
(4)
이다. 따라서
(5)
위식을 역 Laplace변환을 하면
cos cos
(6)
을 얻을 수 있다. 위 두식을 더하고 빼면
cos cos
cos cos
을 구할 수 있다.
(다른 방법) 식 1에서
라고 두면
![Page 20: Laplace 변환 - KNUbh.knu.ac.kr/~ilrhee/lecture/mibun/5-laplace.pdf · 2011-10-25 · Laplace 변환 1. Laplace 변환 및 특성 정의: ℒ ∞ ∞ lim →∞ Ex1) ℒ ∞ ∙](https://reader033.vdocuments.mx/reader033/viewer/2022041813/5e595ad60d0d731b9c763180/html5/thumbnails/20.jpg)
두 식을 더하고 빼면
을 구할 수 있다. 라고 두면
→ cossin
→ cossin
을 얻는다. 두 식을 더하고 빼면
cossincossin cossincossin
초기조건을 대입하여 상수를 결정한다.
![Page 21: Laplace 변환 - KNUbh.knu.ac.kr/~ilrhee/lecture/mibun/5-laplace.pdf · 2011-10-25 · Laplace 변환 1. Laplace 변환 및 특성 정의: ℒ ∞ ∞ lim →∞ Ex1) ℒ ∞ ∙](https://reader033.vdocuments.mx/reader033/viewer/2022041813/5e595ad60d0d731b9c763180/html5/thumbnails/21.jpg)
Ex) 회로에의 적용
→
를 이용하면
(1)에서
![Page 22: Laplace 변환 - KNUbh.knu.ac.kr/~ilrhee/lecture/mibun/5-laplace.pdf · 2011-10-25 · Laplace 변환 1. Laplace 변환 및 특성 정의: ℒ ∞ ∞ lim →∞ Ex1) ℒ ∞ ∙](https://reader033.vdocuments.mx/reader033/viewer/2022041813/5e595ad60d0d731b9c763180/html5/thumbnails/22.jpg)
(2)에서
(3)에서 ℒ′ ℒ ℒ
(4)에서 ℒ′ ℒℒ
(5)에서
(6)에서
× ×에서
→
∴
ℒ
![Page 23: Laplace 변환 - KNUbh.knu.ac.kr/~ilrhee/lecture/mibun/5-laplace.pdf · 2011-10-25 · Laplace 변환 1. Laplace 변환 및 특성 정의: ℒ ∞ ∞ lim →∞ Ex1) ℒ ∞ ∙](https://reader033.vdocuments.mx/reader033/viewer/2022041813/5e595ad60d0d731b9c763180/html5/thumbnails/23.jpg)
ℒ
→∞에서
←
(오랜 시간 후엔 인덕터는 도선)
(모든 전압이 저항에 떨어짐)
(축전기는 완전히 충전)