f(t)	1	t^n	e^at	sin(at)	cos(at)	sinh(at)	cosh(at)	e^att^n	e^atsin(bt)	e^atcos(bt)
F(s)	$$\frac{1}{s}$$	$$\frac{n!}{s^{n + 1}}$$	$$\frac{1}{s - a}$$	$$\frac{a}{s^{2} + a^{2}}$$	$$\frac{s}{s^{2} + a^{2}}$$	$$\frac{a}{s^{2} - a^{2}}$$	$$\frac{s}{s^{2} - a^{2}}$$	$$\frac{n!}{\left( s - a \right)^{n + 1}}$$	$$\frac{b}{\left( s - a \right)^{2} + b^{2}}$$	$$\frac{s - a}{\left( s - a \right)^{2} + b^{2}}$$

L[f(t)] = F(s) ; $L\left\lbrack \frac{d^{n}f\left( t \right)}{\text{dt}^{2}} \right\rbrack = s^{n}F(s)$ ; $L\left\lbrack \int_{0}^{- t}{f\left( \tau \right)\text{dτ}} \right\rbrack = \frac{F(s)}{s}$ ; L[e^αtf(t)] = F(s − ∝) ; L[f(t−τ)] = F(s)e^−τs ; $1|\left( t \right) = \left\{ \begin{matrix} 0,t \leq 0 \\ 1,t > 0 \\ \end{matrix} \right.\ \text{\ \ \ \ L}\left\lbrack 1 \middle| \left( t \right) \right\rbrack = \frac{1}{s}$ ; $f\left( t \right) = \left\{ \begin{matrix} 0,t < 0 \\ at,t > 0 \\ \end{matrix} \right.\ \text{\ \ \ \ L}\left\lbrack \text{at} \right\rbrack = L\left\lbrack a\int_{0}^{t}{1|\left( t \right)\text{dt}} \right\rbrack = \frac{a}{s^{2}}$ ;

$L\left\lbrack t^{n} \right\rbrack = \frac{n!}{s^{n + 1}}$ ; $f\left( t \right) = \left\{ \begin{matrix} 0,t \leq 0 \\ e^{\text{αt}},t > 0 \\ \end{matrix} \right.\ \text{\ \ \ L}\left\lbrack e^{\text{αt}} \right\rbrack = \frac{1}{s - \propto}\text{\ \ \ \ L}\left\lbrack te^{\text{αt}} \right\rbrack = \frac{1}{\left( s - \propto \right)^{2}}\text{\ \ \ \ L}\left\lbrack t^{n}e^{\text{αt}} \right\rbrack = \frac{n!}{\left( s - \propto \right)^{n + 1}}\text{\ \ \ \ }$ ; $L\left\lbrack \cos\left( \text{ωt} \right) \right\rbrack = \frac{s}{s^{2} + \omega^{2}}\text{\ \ \ \ \ L}\left\lbrack \sin\left( \text{ωt} \right) \right\rbrack = \frac{\omega}{s^{2} + \omega^{2}}\text{\ \ \ \ \ L}\left\lbrack e^{\text{αt}}\cos\left( \text{ωt} \right) \right\rbrack = \frac{s - \propto}{\left( s - \propto \right)^{2} + \omega^{2}}\text{\ \ \ \ \ L}\left\lbrack \sin\left( \text{ωt} \right) \right\rbrack = \frac{\omega}{\left( s - \propto \right)^{2} + \omega^{2}}\ $ ;

$\left\{ \begin{matrix} 0,t < 0 \\ \infty,t = 0 \\ 0,t > 0 \\ \end{matrix} \right.\ \text{\ \ \ L}\left\lbrack \delta(f) \right\rbrack = L\left\lbrack \frac{d1|(t)}{\text{dt}} \right\rbrack = 1$ $s_{1,2} - liczymy\ z\ dlety\ z\ mianownika\ \ \ X\left( t \right) = Ae^{\propto t} + Be^{\propto t}\text{\ \ \ \ }B_{p - j} = \frac{1}{j!}\frac{d^{j}}{\text{ds}^{j}}\left( \frac{L(s)}{N(s)}\left( s - s_{m} \right)^{p} \right)\ dla\ s = sm$

123456

Zadania

1 $\left\{ \begin{matrix} u\left( t \right) = R_{1}\left( i_{1}(t) - i_{2}(t) \right) + R_{2}i_{1}(t) \\ 0 = R_{1}\left( i_{2}(t) - i_{1}(t) \right) + \frac{1}{C}\int_{0}^{t}{i_{2}\left( \tau \right)\text{dτ}} \\ y\left( t \right) = R_{2}i_{1}(t) \\ \end{matrix} \right.\ $ WP = 0   L[u(t)] = U(s)L[y(t)] = Y(s); L[i₁(t)] = I₁(s); L[i₂(t)] = I₂(s) $\left\{ \begin{matrix} U\left( s \right) = R_{1}\left( I_{1}(s) - I_{2}(s) \right) + R_{2}I_{1}(s) \\ 0 = R_{1}\left( I_{2}(s) - I_{1}(s) \right) + \frac{1}{C}\frac{I_{2}(s)}{s}\ \ ,\ \\ Y\left( s \right) = R_{2}I_{1}(s) \\ \end{matrix} \right.\ \text{\ \ \ }$

$\text{z\ drugiego\ }R_{1}I_{1}\left( s \right) = R_{1}I_{2}\left( s \right) + \frac{1}{\text{Cs}}I_{2}\left( s \right) = > I_{1}\left( s \right) = I_{2}\left( s \right)\left( 1 + \frac{1}{R_{1}\text{Cs}} \right) = > \ I_{1}(s) = I_{2}(s)\frac{R_{1}Cs + 1}{R_{1}\text{Cs}}$ $\left\{ \begin{matrix} U\left( s \right) = R_{1}I_{2}\left( s \right)\frac{R_{1}Cs + 1}{R_{1}\text{Cs}} - R_{1}I_{2}(s) + R_{2}I_{2}(s)\frac{R_{1}Cs + 1}{R_{1}\text{Cs}} \\ Y\left( s \right) = R_{2}I_{2}(s)\frac{R_{1}Cs + 1}{R_{1}\text{Cs}} \\ \end{matrix} \right.\ $

$\frac{Y(s)}{U(s)} = \frac{I_{2}(s)\left( \frac{R_{1}R_{2}Cs + R_{2}}{R_{1}\text{Cs}} \right)}{I_{2}(s)\left( \frac{R_{1}Cs + 1}{\text{Cs}} - R_{1} + \frac{R_{1}R_{2}Cs + R_{2}}{R_{1}\text{Cs}} \right)} = \frac{\frac{R_{1}R_{2}Cs + R_{2}}{R_{1}\text{Cs}}}{\frac{R_{1}^{2}Cs + R_{1} - R_{1}^{2}Cs + R_{1}R_{2}Cs + R_{2}}{R_{1}\text{Cs}}} = \frac{R_{1}R_{2}Cs + R_{2}}{R_{1} + R_{1}R_{2}Cs + R_{2}}$

2 $\left\{ \begin{matrix} u\left( t \right) = L\frac{di_{1}(t)}{\text{dt}} + \frac{1}{C_{1}}\int_{0}^{t}{\left( i_{1}(t) - i_{2}(t) \right)\text{dt}} \\ 0 = \frac{1}{C_{1}}\int_{0}^{t}{\left( i_{2}(t) - i_{1}(t) \right)\text{dt}} + Ri_{2}\left( t \right) + \frac{1}{C_{2}}\int_{0}^{t}{i_{2}(t)dt} \\ y\left( t \right) = \frac{1}{C_{2}}\int_{0}^{t}{i_{2}(t)dt} \\ \end{matrix} \right.\ $ WP = 0   L[u(t)] = U(s) L[y(t)] = Y(s); L[i₁(t)] = I₁(s); L[i₂(t)] = I₂(s)

$\left\{ \begin{matrix} U\left( s \right) = LsI_{1} + \frac{1}{C_{1}}\frac{I_{1}(s) - I_{2}(s)}{s} \\ 0 = \frac{1}{C_{1}}\frac{I_{1}(s) - I_{2}(s)}{s} + RI_{2}\left( s \right) + \frac{1}{C_{2}}\frac{I_{2}}{s} \\ Y\left( s \right) = \frac{1}{C_{2}}\frac{I_{2}}{s} \\ \end{matrix} \right.\ $ $z\ drugiego\backslash*C_{1}s = > 0 = I_{1}\left( s \right) - I_{2}\left( s \right) + RI_{2}\left( s \right)C_{1}s + \frac{C_{1}I_{2}}{C_{2}} = > \ I_{1} = I_{2}\left( 1 + RC_{1}s + \frac{C_{1}}{C_{2}} \right)$

$\left\{ \begin{matrix} U\left( s \right) = LsI_{2}\left( 1 + RC_{1}s + \frac{C_{1}}{C_{2}} \right) + \frac{1}{C_{1}s}I_{2}\left( 1 + RC_{1}s + \frac{C_{1}}{C_{2}} \right) - \frac{1}{C_{1}s}I_{2} \\ Y\left( s \right) = \frac{I_{2}}{C_{2}s} \\ \end{matrix} \right.\ $ $\frac{Y(s)}{U(s)} = \frac{\frac{1}{C_{2}s}}{Ls + LRC_{1}s^{2} + Ls\frac{C_{1}}{C_{2}} + \frac{1}{C_{1}s} + R + \frac{1}{C_{2}s} - \frac{1}{C_{1}s}}\ \ \backslash*C_{2}s$ $\frac{Y(s)}{U(s)} = \frac{1}{LC_{2}s^{2} + LRC_{2}C_{1}s^{3} + Ls^{2}C_{1} + RC_{2}s + 1}$

3 $\left\{ \begin{matrix} F\left( t \right) = u\left( t \right) = k_{1}\left( y\left( t \right) - x_{2}(t) \right) \\ 0 = k_{1}\left( x_{2}\left( t \right) - y(t) \right) + B_{1}\frac{d\left( x_{2}\left( t \right) - x_{1}(t) \right)}{\text{dt}} \\ 0 = B_{1}\frac{d\left( x_{1}\left( t \right) - x_{2}(t) \right)}{\text{dt}} + B_{2}\frac{d\left( x_{1}\left( t \right) - 0 \right)}{\text{dt}} + k_{2}x_{1}(t) \\ \end{matrix} \right.\ $ WP = 0   L[u(t)] = U(s) L[y(t)] = Y(s); L[x₁(t)] = X₁(s); L[x₂(t)] = X₂(s)

$\left\{ \begin{matrix} U\left( \text{st} \right) = k_{1}Y\left( s \right) - k_{1}X_{2}\left( s \right) \\ 0 = k_{1}X_{2}\left( s \right) - k_{1}Y\left( s \right) + B_{1}sX_{2}\left( s \right) - B_{1}sX_{1}\left( s \right) \\ 0 = B_{1}sX_{1}\left( s \right) - B_{1}sX_{2}\left( s \right) + B_{2}sX_{1}\left( s \right) + k_{2}X_{1}\left( s \right) \\ \end{matrix} \right.\ $

4 $\left\{ \begin{matrix} 0 = B\frac{d\left( x\left( t \right) - y(t) \right)}{\text{dt}} + k_{2}x(t) \\ 0 = B\frac{d\left( y\left( t \right) - x(t) \right)}{\text{dt}} + k_{1}\left( y\left( t \right) - u(t) \right) + m\frac{d^{2}y}{dt^{2}} \\ \end{matrix} \right.\ $ WP = 0   L[u(t)] = U(s) L[y(t)] = Y(s); L[x(t)] = X(s); $\left\{ \begin{matrix} 0 = Bs(X(s) - Y(s)) + k_{2}X(s) \\ 0 = Bs(Y(s) - X(s)) + k_{1}(Y\left( s \right) - U\left( s \right)) + ms^{2}Y(s) \\ \end{matrix} \right.\ $

$z\ pierwszego\ 0 = BsX\left( s \right) - BsY\left( s \right) + k_{2}X\left( s \right) = > BsY\left( s \right) = X\left( s \right)\left( Bs + k_{2} \right)\backslash:\left( Bs + k_{2} \right) = > X\left( s \right) = \frac{\text{BsY}\left( s \right)}{\left( Bs + k_{2} \right)}$

$0 = BsY\left( s \right) - \frac{B^{2}s^{2}Y\left( s \right)}{\left( Bs + k_{2} \right)} + k_{1}Y\left( s \right) - k_{1}U\left( s \right) + ms^{2}Y\left( s \right) = > k_{1}U\left( s \right) = BsY\left( s \right) - \frac{B^{2}s^{2}Y\left( s \right)}{\left( Bs + k_{2} \right)} + k_{1}Y\left( s \right) + ms^{2}Y\left( s \right)\backslash*\left( Bs + k_{2} \right)$

U(s)(k₁Bs+k₁k₂) = BsY(s)(Bs+k₂) − B²s²Y(s) + k₁Y(s)(Bs+k₂) + ms²Y(s)(Bs+k₂)

U(s)(k₁Bs+k₁k₂) = Y(s)(B²s²+Bsk₂−B²s²+k₁Bs+k₁k₂+mBs³+ms²k₂) $\frac{Y(s)}{U(s)} = \frac{\left( k_{1}Bs + k_{1}k_{2} \right)}{\left( \text{Bs}k_{2} + k_{1}Bs + k_{1}k_{2} + mBs^{3} + ms^{2}k_{2} \right)}\backslash*\frac{1}{k_{1}k_{2}}$ $\text{\ \ }\frac{Y(s)}{U(s)} = \frac{\frac{B}{k_{2}}s + 1}{\frac{\text{mB}}{k_{1}k_{2}}s^{3} + \frac{m}{k_{1}}s^{2} + \frac{B}{k_{2}}s + \frac{B}{k_{1}}s + 1}$

5 $\left\{ \begin{matrix} u\left( t \right) = R_{1}\left( i_{1}(t) - i_{2}(t) \right) + \frac{1}{C}\int_{0}^{t}{i_{1}(t)dt} \\ 0 = R_{1}\left( i_{2}(t) - i_{1}(t) \right) + L\frac{di_{2}(t)}{\text{dt}} + R_{2}i_{2}\left( t \right) \\ y\left( t \right) = R_{1}i_{2}\left( t \right) + \frac{1}{C}\int_{0}^{t}{i_{1}(t)dt} \\ \end{matrix} \right.\ $ WP = 0   L[u(t)] = U(s)L[y(t)] = Y(s); L[i₁(t)] = I₁(s); L[i₂(t)] = I₂(s) $\left\{ \begin{matrix} u\left( t \right) = R_{1}\left( I_{1}(s) - I_{2}(s) \right) + \frac{1}{C}\frac{I_{1}}{s} \\ 0 = R_{1}\left( I_{2}(s) - I_{1}(s) \right) + LsI_{2}(s) + R_{2}I_{2}(s) \\ y\left( t \right) = R_{1}I_{2}(s) + \frac{1}{C}\frac{I_{1}}{s} \\ \end{matrix} \right.\ $

$\text{z\ drugiego}\text{\ \ \ \ I}_{1}\left( s \right) = I_{2}\left( s \right) + \frac{l}{R_{1}}\text{s\ }I_{2}\left( s \right) + \frac{R_{2}}{R_{1}}I_{2}\left( s \right) = > I_{1}\left( s \right) = I_{2}\left( s \right)\left( 1 + \frac{l}{R_{1}}s\ + \frac{R_{2}}{R_{1}} \right)$ $I_{1}\left( s \right) = I_{2}\left( s \right)\left( \frac{R_{1} + ls\ + R_{2}}{R_{1}} \right)$ $\left\{ \begin{matrix} U(s) = R_{1}I_{2}\left( s \right)\left( \frac{R_{1} + ls\ + R_{2}}{R_{1}} \right) - R_{1}I_{2}\left( s \right) + \frac{1}{\text{Cs}}I_{2}\left( s \right)\left( \frac{R_{1} + ls\ + R_{2}}{R_{1}} \right) \\ Y\left( s \right) = R_{2}I_{2}\left( s \right) + \frac{1}{\text{Cs}}I_{2}\left( s \right)\left( \frac{R_{1} + ls\ + R_{2}}{R_{1}} \right) \\ \end{matrix} \right.\ $

$\frac{Y(s)}{U(s)} = \frac{I_{2}\left( s \right)\left( R_{2} + \frac{1}{\text{Cs}}\left( \frac{R_{1} + ls\ + R_{2}}{R_{1}} \right) \right)}{I_{2}\left( s \right)\left( R_{1} + ls\ + R_{2} - R_{1} + \left( \frac{R_{1} + ls\ + R_{2}}{R_{1}\text{Cs}} \right) \right)}\backslash*R_{1}\text{Cs}$ $\frac{Y(s)}{U(s)} = \frac{R_{2}R_{1}\text{Cs} + R_{1} + ls\ + R_{2}}{lR_{1}Cs^{2}\ + R_{2}R_{1}Cs + R_{1} + ls\ + R_{2}}$

6 $\left\{ \begin{matrix} 0 = B_{1}\frac{dx_{1}\left( t \right)}{\text{dt}} + k_{1}\left( x_{1}\left( t \right) - y(t) \right) \\ 0 = k_{2}x_{2}\left( t \right) + B_{2}\frac{d\left( x_{2}\left( t \right) - y\left( t \right) \right)}{\text{dt}} + k_{3}\left( x_{2}\left( t \right) - y(t) \right) \\ F\left( t \right) = u\left( t \right) = k_{1}\left( y\left( t \right) - x_{1}(t) \right) + B_{2}\frac{d\left( y\left( t \right) - x_{2}(t) \right)}{\text{dt}} + k_{3}\left( y\left( t \right) - x_{2}(t) \right) \\ \end{matrix} \right.\ $ WP = 0   L[u(t)] = U(s) L[y(t)] = Y(s); L[x₁(t)] = X₁(s); L[x₂(t)] = X₂(s)

$\left\{ \begin{matrix} 0 = B_{1}sX_{1}(s) + k_{1}\left( X_{1}\left( s \right) - Y(s) \right) \\ 0 = k_{2}X_{2}\left( \text{st} \right) + B_{2}s\left( X_{2}(s) - Y(s) \right) + k_{3}\left( X_{2}\left( s \right) - Y(s) \right) \\ F\left( t \right) = u\left( t \right) = k_{1}\left( Y\left( s \right) - X_{1}(s) \right) + B_{2}s\left( Y(s) - X_{2}(s) \right) + k_{3}\left( Y\left( s \right) - X_{2}(s) \right) \\ \end{matrix} \right.\ $ 0 = X₁(s)(B₁s+k₁) − k₁Y(s) $X_{1} = \frac{k_{1}Y(s)}{B_{1}s + k_{1}}$ $X_{2} = \frac{B_{2}\text{sY}(s) + k_{3}Y(s)}{k_{2} + B_{2}s + k_{3}}$

$U\left( s \right) = k_{1}Y\left( s \right) - Y(s)\frac{k_{1}^{2}}{B_{1}s + k_{1}} - Y(s)\frac{B_{2}^{2}s^{2} + B_{2}sk_{3}}{k_{2} + B_{2}s + k_{3}} + B_{2}\text{sY}\left( s \right) + k_{3}Y\left( s \right) - Y(s)\frac{B_{2}sk_{3} + k_{3}Y(s)}{k_{2} + B_{2}s + k_{3}}$

$U\left( s \right) = Y\left( s \right)\frac{B_{1}sk_{1} + k_{1}^{2} - k_{1}^{2}}{B_{1}s + k_{1}} + Y\left( s \right)\frac{k_{2}B_{2}s + B_{2}sk_{3} + B_{2}^{2}s^{2} - B_{2}^{2}s^{2} - B_{2}sk_{3}}{k_{2} + B_{2}s + k_{3}} + Y\left( s \right)\frac{k_{2}k_{3} + B_{2}sk_{3} + k_{3}^{2} - B_{2}sk_{3}}{k_{2} + B_{2}s + k_{3}} = Y\left( s \right)\left( \frac{B_{1}sk_{1}}{B_{1}s + k_{1}} + \frac{k_{2}B_{2}s + k_{2}k_{3}}{k_{2} + B_{2}s + k_{3}} \right) = Y(s)\left( \frac{k_{1}k_{2}B_{1}s + k_{1}B_{1}B_{2}s^{2} + k_{1}k_{3}B_{1}s + k_{2}B_{1}B_{2}s^{2} + k_{2}k_{3}B_{1}s + k_{1}k_{2}B_{2}s + k_{1}{k_{2}k}_{3}}{k_{2}B_{1}s + B_{1}B_{2}s^{2} + k_{3}B_{1}s + k_{1}k_{2} + k_{1}B_{2}s + k_{1}k_{3}} \right)$

$\frac{\mathrm{Y}(s)}{U(s)} = \frac{B_{1}B_{2}s^{2} + s\left( k_{2}B_{1} + k_{3}B_{1} + k_{1}B_{2} \right) + k_{1}k_{3} + k_{1}k_{2}}{s^{2}\left( k_{1}B_{1}B_{2} + k_{2}B_{1}B_{2} \right) + s\left( k_{1}k_{2}B_{1} + k_{1}k_{3}B_{1} + k_{2}k_{3}B_{1} + k_{1}k_{2}B_{2} \right) + k_{1}{k_{2}k}_{3}}$