사용자:Dolicom/수학
전자공학자이므로 이와 관련된 수학에 관심이 많다. 따라서 생각나는데오 노트한다.
기초수학
편집- 오일러의 공식과 자연로그
양변에 자연로그를 씌우면
전자공학과 수학
편집라플라스 변환
편집정의
편집함수 의 라플라스 변환은 모든 실수 t ≥ 0 에 대해, 다음과 같은 함수 로 정의된다[1].
여기서 는 를 간단히 나타낸 것이고 복소수 , σ와 ω는 실수이다.
실제 사용시에는 엄밀히 정확하지는 않지만 로 표기하기도 한다.
변환 테이블
편집The following table provides Laplace transforms for many common functions of a single variable.[2][3] For definitions and explanations, see the Explanatory Notes at the end of the table.
Because the Laplace transform is a linear operator:
- The Laplace transform of a sum is the sum of Laplace transforms of each term.
- The Laplace transform of a multiple of a function is that multiple times the Laplace transformation of that function.
Function | Time domain |
Laplace s-domain |
Region of convergence | Reference | ||
---|---|---|---|---|---|---|
unit impulse | all s | inspection | ||||
delayed impulse | time shift of unit impulse | |||||
unit step | Re(s) > 0 | integrate unit impulse | ||||
delayed unit step | Re(s) > 0 | time shift of unit step | ||||
ramp | Re(s) > 0 | integrate unit impulse twice | ||||
nth power ( for integer n ) |
Re(s) > 0 (n > −1) |
Integrate unit step n times | ||||
qth power (for complex q) |
Re(s) > 0 Re(q) > −1 |
[4][5] | ||||
nth root | Re(s) > 0 | Set q = 1/n above. | ||||
nth power with frequency shift | Re(s) > −α | Integrate unit step, apply frequency shift | ||||
delayed nth power with frequency shift |
Re(s) > −α | Integrate unit step, apply frequency shift, apply time shift | ||||
exponential decay | Re(s) > −α | Frequency shift of unit step | ||||
two-sided exponential decay (only for bilateral transform) |
−α < Re(s) < α | Frequency shift of unit step | ||||
exponential approach | Re(s) > 0 | Unit step minus exponential decay | ||||
sine | Re(s) > 0 | Bracewell 1978, 227쪽 | ||||
cosine | Re(s) > 0 | Bracewell 1978, 227쪽 | ||||
hyperbolic sine | Re(s) > |α| | Williams 1973, 88쪽 | ||||
hyperbolic cosine | Re(s) > |α| | Williams 1973, 88쪽 | ||||
exponentially decaying sine wave |
Re(s) > −α | Bracewell 1978, 227쪽 | ||||
exponentially decaying cosine wave |
Re(s) > −α | Bracewell 1978, 227쪽 | ||||
natural logarithm | Re(s) > 0 | Williams 1973, 88쪽 | ||||
Bessel function of the first kind, of order n |
Re(s) > 0 (n > −1) |
Williams 1973, 89쪽 | ||||
Error function | Re(s) > 0 | Williams 1973, 89쪽 | ||||
Explanatory notes:
|
s-도메인에서의 회로 등가모델과 임피던스(impedances)
편집The Laplace transform is often used in circuit analysis, and simple conversions to the s-Domain of circuit elements can be made. Circuit elements can be transformed into impedances, very similar to phasor impedances.
Here is a summary of equivalents:
Note that the resistor is exactly the same in the time domain and the s-Domain. The sources are put in if there are initial conditions on the circuit elements. For example, if a capacitor has an initial voltage across it, or if the inductor has an initial current through it, the sources inserted in the s-Domain account for that.
The equivalents for current and voltage sources are simply derived from the transformations in the table above.
같이 보기
편집- 기초수학
- 전자공학 관련수학
각주
편집- ↑ Kreyszig, E. (2006). 《Advanced Engineering Mathematics》 9판. John Wiley & Sons. ISBN 978-0-471-72897-9.
- ↑ Riley, K. F.; Hobson, M. P.; Bence, S. J. (2010), 《Mathematical methods for physics and engineering》 3판, Cambridge University Press, 455쪽, ISBN 978-0-521-86153-3
- ↑ Distefano, J. J.; Stubberud, A. R.; Williams, I. J. (1995), 《Feedback systems and control》, Schaum's outlines 2판, McGraw-Hill, 78쪽, ISBN 0-07-017052-5
- ↑ Lipschutz, S.; Spiegel, M. R.; Liu, J. (2009), 《Mathematical Handbook of Formulas and Tables》, Schaum's Outline Series 3판, McGraw-Hill, 183쪽, ISBN 978-0-07-154855-7 - provides the case for real q.
- ↑ http://mathworld.wolfram.com/LaplaceTransform.html - Wolfram Mathword provides case for complex q