사용자:Dolicom/수학

전자공학자이므로 이와 관련된 수학에 관심이 많다. 따라서 생각나는데오 노트한다.

기초수학

자연로그

자연로그(natural logarithm)는 e를 밑으로 하는 로그를 뜻하며, $e^{x}=y$일 때, $\ln y=x$을 자연로그라 한다.

• 오일러의 공식과 자연로그

$re^{ix}=r(\cos x+i\sin x)$

양변에 자연로그를 씌우면

$re^{ix}=r(\cos x+i\sin x)$

$\therefore \ln r(\cos x+i\sin x)=ix+\ln r$

전자공학과 수학

라플라스 변환

라플라스 변환

정의

함수 $f(t)$의 라플라스 변환은 모든 실수 t ≥ 0 에 대해, 다음과 같은 함수 $F(s)$로 정의된다.

$F(s)={\mathcal {L}}\left\{f\right\}(s)=\int _{0^{-}}^{\infty }e^{-st}f(t)\,dt.$

여기서 $0^{-}$는 $\lim _{\epsilon \rightarrow +0}-\epsilon$를 간단히 나타낸 것이고 복소수 $s=\sigma +i\omega \,$, σ와 ω는 실수이다.

실제 사용시에는 엄밀히 정확하지는 않지만 $F(s)={\mathcal {L}}\left\{f(t)\right\}$로 표기하기도 한다.

변환 테이블

The following table provides Laplace transforms for many common functions of a single variable. 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.

${\mathcal {L}}\{f(t)+g(t)\}={\mathcal {L}}\{f(t)\}+{\mathcal {L}}\{g(t)\}$

• The Laplace transform of a multiple of a function is that multiple times the Laplace transformation of that function.

${\mathcal {L}}\{af(t)\}=a{\mathcal {L}}\{f(t)\}$

Function	Time domain $f(t)={\mathcal {L}}^{-1}\{F(s)\}$	Laplace s-domain $F(s)={\mathcal {L}}\{f(t)\}$	Region of convergence	Reference
unit impulse	$\delta (t)\$	$1$	all s	inspection
delayed impulse	$\delta (t-\tau )\$	$e^{-\tau s}\$		time shift of unit impulse
unit step	$u(t)\$	${1 \over s}$	Re(s) > 0	integrate unit impulse
delayed unit step	$u(t-\tau )\$	${\frac {1}{s}}e^{-\tau s}$	Re(s) > 0	time shift of unit step
ramp	$t\cdot u(t)\$	${\frac {1}{s^{2}}}$	Re(s) > 0	integrate unit impulse twice
nth power ( for integer n )	$t^{n}\cdot u(t)$	${n! \over s^{n+1}}$	Re(s) > 0 (n > −1)	Integrate unit step n times
qth power (for complex q)	$t^{q}\cdot u(t)$	${\Gamma (q+1) \over s^{q+1}}$	Re(s) > 0 Re(q) > −1
nth root	${\sqrt[{n}]{t}}\cdot u(t)$	${1 \over s^{{\frac {1}{n}}+1}}\Gamma \left({\frac {1}{n}}+1\right)$	Re(s) > 0	Set q = 1/n above.
nth power with frequency shift	$t^{n}e^{-\alpha t}\cdot u(t)$	${\frac {n!}{(s+\alpha )^{n+1}}}$	Re(s) > −α	Integrate unit step, apply frequency shift
delayed nth power with frequency shift	$(t-\tau )^{n}e^{-\alpha (t-\tau )}\cdot u(t-\tau )$	${\frac {n!\cdot e^{-\tau s}}{(s+\alpha )^{n+1}}}$	Re(s) > −α	Integrate unit step, apply frequency shift, apply time shift
exponential decay	$e^{-\alpha t}\cdot u(t)$	${1 \over s+\alpha }$	Re(s) > −α	Frequency shift of unit step
two-sided exponential decay (only for bilateral transform)	$e^{-\alpha |t|}\$	${2\alpha \over \alpha ^{2}-s^{2}}$	−α < Re(s) < α	Frequency shift of unit step
exponential approach	$(1-e^{-\alpha t})\cdot u(t)\$	${\frac {\alpha }{s(s+\alpha )}}$	Re(s) > 0	Unit step minus exponential decay
sine	$\sin(\omega t)\cdot u(t)\$	${\omega \over s^{2}+\omega ^{2}}$	Re(s) > 0	Bracewell 1978, 227쪽
cosine	$\cos(\omega t)\cdot u(t)\$	${s \over s^{2}+\omega ^{2}}$	Re(s) > 0	Bracewell 1978, 227쪽
hyperbolic sine	$\sinh(\alpha t)\cdot u(t)\$	${\alpha \over s^{2}-\alpha ^{2}}$	Re(s) > |α|	Williams 1973, 88쪽
hyperbolic cosine	$\cosh(\alpha t)\cdot u(t)\$	${s \over s^{2}-\alpha ^{2}}$	Re(s) > |α|	Williams 1973, 88쪽
exponentially decaying sine wave	$e^{-\alpha t}\sin(\omega t)\cdot u(t)\$	${\omega \over (s+\alpha )^{2}+\omega ^{2}}$	Re(s) > −α	Bracewell 1978, 227쪽
exponentially decaying cosine wave	$e^{-\alpha t}\cos(\omega t)\cdot u(t)\$	${s+\alpha \over (s+\alpha )^{2}+\omega ^{2}}$	Re(s) > −α	Bracewell 1978, 227쪽
natural logarithm	$\ln(t)\cdot u(t)$	$-{1 \over s}\,\left[\ln(s)+\gamma \right]$	Re(s) > 0	Williams 1973, 88쪽
Bessel function of the first kind, of order n	$J_{n}(\omega t)\cdot u(t)$	${\frac {\left({\sqrt {s^{2}+\omega ^{2}}}-s\right)^{n}}{\omega ^{n}{\sqrt {s^{2}+\omega ^{2}}}}}$	Re(s) > 0 (n > −1)	Williams 1973, 89쪽
Error function	$\mathrm {erf} (t)\cdot u(t)$	${\frac {1}{s}}e^{{\frac {1}{4}}s^{2}}\left(1-\operatorname {erf} {\frac {s}{2}}\right)$	Re(s) > 0	Williams 1973, 89쪽
Explanatory notes: u(t) represents the Heaviside step function. $\delta (t)\,$ represents the Dirac delta function. Γ(z) represents the Gamma function. γ is the Euler–Mascheroni constant. t, a real number, typically represents time, although it can represent any independent dimension. s is the complex angular frequency, and Re(s) is its real part. α, β, τ, and ω are real numbers. n is an integer.

s-도메인에서의 회로 등가모델과 임피던스(impedances)

The 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:

s-Domain equivalent circuits

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

• 기초수학

자연로그

• 전자공학 관련수학

라플라스 변환