Block diagram

X(s) Y(s)

where:

X – input value

y – respose

G – transfer function

1. Proportional P

$$G\left( s \right) = \frac{Y(s)}{X(s)} = k$$

where:

k – gain

Response for input - step function x(t)=1(t)

y(t) = k • 1(t)

Response for input - impulse function x(t)=δ(t)

y(t) = k • (t)

2. Inertial first order

$$G\left( s \right) = \frac{Y(s)}{X(s)} = \frac{\text{k\ }}{Ts + 1}$$

where:

k – gain

T – inertial time

Response for input - step function x(t)=1(t) , $X\left( s \right) = \frac{1}{s}$

$$Y\left( s \right) = \frac{\text{k\ }}{Ts + 1}\ \bullet \frac{1}{s}$$

$$Y = \frac{A}{s} + \frac{B}{\left( Ts + 1 \right)}$$

$$Y = \frac{ATs + Bs + A}{s\left( Ts + 1 \right)}$$

$$\left\{ \begin{matrix} A = k \\ B = - kT \\ \end{matrix} \right.\ $$

$$Y = \frac{k}{s} - kT\frac{1}{\left( Ts + 1 \right)}$$

$$Y = \frac{k}{s} - k\frac{1}{\left( s + \frac{1}{T} \right)}$$

$$y = k - ke^{- \frac{t}{T}}$$

Response for input - impulse function x(t)=δ(t) , X(s) = 1

$$Y\left( s \right) = \frac{\text{k\ }}{Ts + 1}\ \bullet 1$$

$$Y = \frac{k}{T}\frac{1}{\left( s + \frac{1}{T} \right)}$$

$$y = \frac{k}{T}e^{- \frac{t}{T}}$$

3. Inertial second order

$$G\left( s \right) = \frac{Y(s)}{X(s)} = \frac{\text{k\ }}{\left( T_{1}s + 1 \right)\left( T_{2}s + 1 \right)}$$

where:

k – gain

T₁ – first inertial time

T₂ – second inertial time

Response for input - step function x(t)=1(t) , $X\left( s \right) = \frac{1}{s}$

$$Y\left( s \right) = \frac{\text{k\ }}{\left( T_{1}s + 1 \right)\left( T_{2}s + 1 \right)}\ \bullet \frac{1}{s}$$

$$Y = \frac{A}{s} + \frac{B}{\left( T_{1}s + 1 \right)} + \frac{C}{\left( T_{2}s + 1 \right)}$$

$$\left\{ \begin{matrix} A = k \\ B = \frac{- T_{1}^{2}}{T_{1} - T_{2}}k \\ C = \frac{T_{2}^{2}}{T_{1} - T_{2}}k \\ \end{matrix} \right.\ $$

$$Y = \frac{k}{s} - \frac{T_{1}^{2}}{T_{1} - T_{2}}\frac{k}{\left( T_{1}s + 1 \right)} + \frac{T_{2}^{2}}{T_{1} - T_{2}}\frac{k}{\left( T_{2}s + 1 \right)}$$

$$Y = \frac{k}{s} - \frac{k}{T_{1} - T_{2}}\frac{T_{1}}{\left( s + \frac{1}{T_{1}} \right)} + \frac{k}{T_{1} - T_{2}}\frac{T_{2}}{\left( s + \frac{1}{T_{2}} \right)}$$

$$y = k + \frac{T_{1}}{T_{2} - T_{1}}\text{ke}^{- \frac{t}{T_{1}}} - \frac{T_{2}}{T_{2} - T_{1}}ke^{- \frac{t}{T_{2}}}$$

$$G\left( s \right) = \frac{Y(s)}{X(s)} = \frac{ke^{- \tau s}\ }{T_{z}s + 1}$$

Response for input - impulse function x(t)=δ(t) , X(s) = 1

$$Y\left( s \right) = \frac{\text{k\ }}{\left( T_{1}s + 1 \right)\left( T_{2}s + 1 \right)}\ \bullet 1$$

$$Y = \frac{A}{\left( T_{1}s + 1 \right)} + \frac{B}{\left( T_{2}s + 1 \right)}$$

$$\left\{ \begin{matrix} A = \frac{T_{1}}{T_{1} - T_{2}}k \\ B = \frac{T_{2}}{T_{2} - T_{1}}k \\ \end{matrix} \right.\ $$

$$Y = \frac{T_{1}}{T_{1} - T_{2}} \bullet \frac{k}{\left( T_{1}s + 1 \right)} + \frac{T_{2}}{T_{2} - T_{1}} \bullet \frac{k}{\left( T_{2}s + 1 \right)}$$

$$Y = \frac{k}{T_{1} - T_{2}}\frac{1}{\left( s + \frac{1}{T_{1}} \right)} + \frac{k}{T_{2} - T_{1}}\frac{1}{\left( s + \frac{1}{T_{2}} \right)}$$

$$y = \frac{k}{T_{1} - T_{2}}\left( e^{- \frac{t}{T_{1}}} - e^{- \frac{t}{T_{2}}} \right)$$

4. Inertial third order

$$G\left( s \right) = \frac{Y(s)}{X(s)} = \frac{\text{k\ }}{\left( T_{1}s + 1 \right)\left( T_{2}s + 1 \right)\left( T_{3}s + 1 \right)}$$

where:

k – gain

T₁ – first inertial time

T₂ – second inertial time

T₃ – third inertial time

Response for input - step function x(t)=1(t) , $X\left( s \right) = \frac{1}{s}$

$$Y\left( s \right) = \frac{\text{k\ }}{\left( T_{1}s + 1 \right)\left( T_{2}s + 1 \right)\left( T_{3}s + 1 \right)}\ \bullet \frac{1}{s}$$

$$Y = \frac{\frac{k}{T_{1}T_{2}T_{3}}}{\left( s + \frac{1}{T_{1}} \right)\left( s + \frac{1}{T_{2}} \right)\left( s + \frac{1}{T_{3}} \right)} \bullet \frac{1}{s}$$

$$Y = \frac{A}{s} + \frac{B}{\left( s + \frac{1}{T_{1}} \right)} + \frac{C}{\left( s + \frac{1}{T_{2}} \right)} + \frac{D}{\left( s + \frac{1}{T_{3}} \right)}$$

$$\left\{ \begin{matrix} A = k \\ B = \frac{kT_{1}^{2}}{\left( T_{3} - T_{1} \right)\left( T_{1} - T_{2} \right)} \\ \begin{matrix} C = \frac{kT_{2}^{2}}{\left( T_{3} - T_{2} \right)\left( T_{2} - T_{1} \right)} \\ D = \frac{kT_{3}^{2}}{\left( T_{1} - T_{3} \right)\left( T_{3} - T_{2} \right)} \\ \end{matrix} \\ \end{matrix} \right.\ $$

$$Y = \frac{k}{s} + \frac{T_{1}^{2}}{\left( T_{3} - T_{1} \right)\left( T_{1} - T_{2} \right)}\frac{k}{\left( s + \frac{1}{T_{1}} \right)} + \frac{T_{2}^{2}}{\left( T_{3} - T_{2} \right)\left( T_{2} - T_{1} \right)}\frac{k}{\left( s + \frac{1}{T_{2}} \right)} + \frac{T_{3}^{2}}{\left( T_{1} - T_{3} \right)\left( T_{3} - T_{2} \right)}\frac{k}{\left( s + \frac{1}{T_{3}} \right)}$$

$$y = k + \frac{kT_{1}^{2}}{\left( T_{3} - T_{1} \right)\left( T_{1} - T_{2} \right)}e^{- \frac{t}{T_{1}}} + \frac{kT_{2}^{2}}{\left( T_{3} - T_{2} \right)\left( T_{2} - T_{1} \right)}e^{- \frac{t}{T_{2}}} + \frac{kT_{3}^{2}}{\left( T_{1} - T_{3} \right)\left( T_{3} - T_{2} \right)}e^{- \frac{t}{T_{3}}}$$

$$G\left( s \right) = \frac{Y(s)}{X(s)} = \frac{ke^{- \tau s}\ }{T_{z}s + 1}$$

Response for input - impulse function x(t)=δ(t) , X(s) = 1

$$Y\left( s \right) = \frac{\text{k\ }}{\left( T_{1}s + 1 \right)\left( T_{2}s + 1 \right)\left( T_{3}s + 1 \right)}\ \bullet 1$$

$$Y = \frac{\frac{k}{T_{1}T_{2}T_{3}}}{\left( s + \frac{1}{T_{1}} \right)\left( s + \frac{1}{T_{2}} \right)\left( s + \frac{1}{T_{3}} \right)}$$

$$Y = \frac{A}{\left( s + \frac{1}{T_{1}} \right)} + \frac{B}{\left( s + \frac{1}{T_{2}} \right)} + \frac{C}{\left( s + \frac{1}{T_{3}} \right)}$$

$$\left\{ \begin{matrix} \begin{matrix} A = \frac{T_{1}}{\left( T_{1} - T_{2} \right)\left( T_{1} - T_{3} \right)}k \\ B = \frac{T_{2}}{\left( T_{2} - T_{1} \right)\left( T_{2} - T_{3} \right)}k \\ \end{matrix} \\ C = \frac{T_{3}}{\left( T_{3} - T_{1} \right)\left( T_{3} - T_{2} \right)}k \\ \end{matrix} \right.\ $$

$$Y = \frac{T_{1}}{\left( T_{1} - T_{2} \right)\left( T_{1} - T_{3} \right)}\frac{k}{\left( s + \frac{1}{T_{1}} \right)} + \frac{T_{2}}{\left( T_{2} - T_{1} \right)\left( T_{2} - T_{3} \right)}\frac{k}{\left( s + \frac{1}{T_{2}} \right)} + \frac{T_{3}}{\left( T_{3} - T_{1} \right)\left( T_{3} - T_{2} \right)}\frac{k}{\left( s + \frac{1}{T_{3}} \right)}$$

$$y = \frac{\text{kT}_{1}}{\left( T_{1} - T_{2} \right)\left( T_{1} - T_{3} \right)}e^{- \frac{t}{T_{1}}} + \frac{kT_{2}}{\left( T_{2} - T_{1} \right)\left( T_{2} - T_{3} \right)}e^{- \frac{t}{T_{2}}} + \frac{kT_{3}}{\left( T_{3} - T_{1} \right)\left( T_{3} - T_{2} \right)}e^{- \frac{t}{T_{3}}}$$

Comparison

5. Oscillation - component second order

$$G\left( s \right) = \frac{Y(s)}{X(s)} = \frac{k}{T_{0}^{2}s^{2} + 2\xi T_{0}s + 1}\ ;\ \ \ \ \ \ \ \ \ \ < 0$$

$$G\left( s \right) = \frac{k\omega_{0}^{2}}{{s^{2} + 2\xi\omega}_{0}s + \omega_{0}^{2}}$$

where:

k – gain

ω₀ – owner frequency

ξ – coefficient damping

$$\begin{matrix} 0 < \xi < 1\ \ \ \ \ \ oscillatian\ component \\ \xi > 1\ \ \ \ \ \ \ \ inertial\ component \\ \end{matrix}$$

ω – real frequency $\omega = \omega_{0}\sqrt{1 - \xi^{2}}$

T – real period time $T = \frac{2\pi}{\omega} = \frac{2\pi}{\omega_{0}\sqrt{1 - \xi^{2}}}$

Response for input - step function x(t)=1(t) , $X\left( s \right) = \frac{1}{s}$

$$Y = \frac{k\omega_{0}^{2}}{{s^{2} + 2\xi\omega}_{0}s + \omega_{0}^{2}} \bullet \frac{1}{s}$$

$$Y = \frac{A}{s} + \frac{Bs + C}{{s^{2} + 2\xi\omega}_{0}s + \omega_{0}^{2}}$$

$$\left\{ \begin{matrix} A = k \\ B = - k \\ C = {- 2k\xi\omega}_{0} \\ \end{matrix} \right.\ $$

$$Y = \frac{k}{s} - k\frac{s + {2\xi\omega}_{0}}{{s^{2} + 2\xi\omega}_{0}s + \omega_{0}^{2}}$$

$$Y = \frac{k}{s} - k\frac{s + {2\xi\omega}_{0}}{\left( {s + \xi\omega}_{0} \right)^{2} - {\xi^{2}\omega}_{0}^{2} + \omega_{0}^{2}}$$

$$Y = \frac{k}{s} - k\frac{s + {2\xi\omega}_{0}}{\left( {s + \xi\omega}_{0} \right)^{2} + \omega_{0}^{2}\left( 1 - \xi^{2} \right)}$$

$$Y = \frac{k}{s} - k\frac{s + \text{ξω}_{0}}{\left( {s + \xi\omega}_{0} \right)^{2} + \omega_{0}^{2}\left( 1 - \xi^{2} \right)} - k\frac{\text{ξω}_{0}}{\left( {s + \xi\omega}_{0} \right)^{2} + \omega_{0}^{2}\left( 1 - \xi^{2} \right)}$$

$$Y = \frac{k}{s} - k\frac{s + \text{ξω}_{0}}{\left( {s + \xi\omega}_{0} \right)^{2} + \omega_{0}^{2}\left( 1 - \xi^{2} \right)} - \frac{k\xi}{\sqrt{1 - \xi^{2}}}\frac{\omega_{0}\sqrt{1 - \xi^{2}}}{\left( {s + \xi\omega}_{0} \right)^{2} + \omega_{0}^{2}\left( 1 - \xi^{2} \right)}$$

$$y = k - ke^{- \text{ξω}_{0}t}\cos\left( \omega_{0}\sqrt{1 - \xi^{2}}t \right) - \frac{k\xi}{\sqrt{1 - \xi^{2}}}e^{- \text{ξω}_{0}t}\sin\left( \omega_{0}\sqrt{1 - \xi^{2}}t \right)$$

$$y = k - ke^{- \text{ξω}_{0}t}\left\lbrack \cos\left( \omega_{0}\sqrt{1 - \xi^{2}}t \right) + \frac{\xi}{\sqrt{1 - \xi^{2}}}\sin\left( \omega_{0}\sqrt{1 - \xi^{2}}t \right) \right\rbrack$$

Asin(α+β) = Asinαcosβ + Acosαsinβ

$$\left\{ \begin{matrix} \beta = \omega_{0}\sqrt{1 - \xi^{2}}t \\ A\sin\alpha = 1 \\ \operatorname{Acos}\alpha = \frac{\xi}{\sqrt{1 - \xi^{2}}} \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left\{ \begin{matrix} \beta = \omega_{0}\sqrt{1 - \xi^{2}}t \\ A^{2}{\sin\alpha}^{2} = 1 \\ {\cos\alpha}^{2} = \frac{\xi^{2}}{1 - \xi^{2}} \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ \ }\left\{ \begin{matrix} \beta = \omega_{0}\sqrt{1 - \xi^{2}}t \\ A^{2}\left( {\sin\alpha}^{2} + {\cos\alpha}^{2} \right) = \frac{1}{1 - \xi^{2}} \\ \tan\alpha = \frac{\sqrt{1 - \xi^{2}}}{\xi} \\ \end{matrix} \right.\ \ \ \ \ \ \ \ \ A = \frac{1}{\sqrt{1 - \xi^{2}}}$$

$$y = k - \frac{k}{\sqrt{1 - \xi^{2}}}e^{- \text{ξω}_{0}t}\sin\left( \omega_{0}\sqrt{1 - \xi^{2}}t + \alpha \right)\ $$

$$\xi = \frac{\ln\frac{e_{1}}{e_{2}}}{\sqrt{\pi^{2} + \left( \ln\frac{e_{1}}{e_{2}} \right)^{2}}}$$

Response for input - impulse function x(t)=δ(t) , X(s) = 1

$$Y = \frac{k\omega_{0}^{2}}{{s^{2} + 2\xi\omega}_{0}s + \omega_{0}^{2}} \bullet 1$$

$$Y = \frac{k\omega_{0}^{2}}{\left( {s + \xi\omega}_{0} \right)^{2} - {\xi^{2}\omega}_{0}^{2} + \omega_{0}^{2}}$$

$$Y = \frac{k\omega_{0}^{2}}{\left( {s + \xi\omega}_{0} \right)^{2} + \omega_{0}^{2}\left( 1 - \xi^{2} \right)}$$

$$Y = \frac{k\omega_{0}}{\sqrt{1 - \xi^{2}}}\frac{\omega_{0}\sqrt{1 - \xi^{2}}}{\left( {s + \xi\omega}_{0} \right)^{2} + \omega_{0}^{2}\left( 1 - \xi^{2} \right)}$$

$$y = \frac{k\omega_{0}}{\sqrt{1 - \xi^{2}}}e^{- \text{ξω}_{0}t}\sin\left( \omega_{0}\sqrt{1 - \xi^{2}}t \right)$$

6. Integral I

$$G\left( s \right) = \frac{Y(s)}{X(s)} = \frac{1}{T_{i}s} = \frac{k}{s}$$

where:

T_i – integral time

Response for input - step function x(t)=1(t) , $X\left( s \right) = \frac{1}{s}$

$$Y\left( s \right) = \frac{1}{T_{i}s} \bullet \frac{1}{s}$$

$$y\left( t \right) = \frac{1}{T_{i}} \bullet t$$

Response for input - impulse function x(t)=δ(t) , X(s) = 1

$$Y\left( s \right) = \frac{1}{T_{i}s} \bullet 1$$

$$y\left( t \right) = \frac{1}{T_{i}} \bullet 1(t)$$

7. Integral I real

$$G\left( s \right) = \frac{Y(s)}{X(s)} = \frac{1\ }{T_{i}s\left( Ts + 1 \right)}$$

where:

T –inertial time

T_i –integral time

Response for input - step function x(t)=1(t) , $X\left( s \right) = \frac{1}{s}$

$$Y\left( s \right) = \frac{1\ }{T_{i}s\left( Ts + 1 \right)}\ \bullet \frac{1}{s}$$

$$Y = \frac{A}{s^{2}} + \frac{B}{s} + \frac{C}{\left( Ts + 1 \right)}$$

$$\left\{ \begin{matrix} A = \frac{1}{T_{i}} \\ B = - \frac{T}{T_{i}} \\ C = \frac{T^{2}}{T_{i}} \\ \end{matrix} \right.\ $$

$$Y = \frac{1}{T_{i}s^{2}} - \frac{T}{T_{i}s} + \frac{T^{2}}{T_{i}}\frac{1}{\left( Ts + 1 \right)}$$

$$Y = \frac{1}{T_{i}s^{2}} - \frac{T}{T_{i}s} + \frac{T}{T_{i}}\frac{1}{\left( s + \frac{1}{T} \right)}$$

$$y = \frac{1}{T_{i}}t - \frac{T}{T_{i}}1(t) + \frac{T}{T_{i}}e^{- \frac{t}{T}}$$

$$y = \frac{1}{T_{i}}\left( t - T + Te^{- \frac{t}{T}} \right)$$

Response for input - impulse function x(t)=δ(t) , X(s) = 1

$$Y\left( s \right) = \frac{1\ }{T_{i}s\left( Ts + 1 \right)}\ \bullet 1$$

$$Y = \frac{A}{s} + \frac{B}{\left( Ts + 1 \right)}$$

$$\left\{ \begin{matrix} A = \frac{1}{T_{i}} \\ B = - \frac{T}{T_{i}} \\ \end{matrix} \right.\ $$

$$Y = \frac{1}{T_{i}s} - \frac{T}{T_{i}}\frac{1}{\left( Ts + 1 \right)}$$

$$Y = \frac{1}{T_{i}s} - \frac{1}{T_{i}}\frac{1}{\left( s + \frac{1}{T} \right)}$$

$$y = \frac{1}{T_{i}} - \frac{1}{T_{i}}e^{- \frac{t}{T}}$$

$$y = \frac{1}{T_{i}}\left( 1 - e^{- \frac{t}{T}} \right)$$

8. Differential D

$$G\left( s \right) = \frac{Y(s)}{X(s)} = T_{d}\text{s\ }$$

where:

T_d – differential time

Response for input - step function x(t)=1(t) , $X\left( s \right) = \frac{1}{s}$

$$Y\left( s \right) = T_{d}s\ \bullet \frac{1}{s}$$

y(t) = T_d • δ(t)

9. Differential D real

$$G\left( s \right) = \frac{Y(s)}{X(s)} = \frac{T_{d}\text{s\ }}{Ts + 1}$$

where:

$\frac{T_{d}\ }{T} = k_{d}$ – dynamic gain

T_d – differential time

T – inertial time of diferential component

Response for input - step function x(t)=1(t) , $X\left( s \right) = \frac{1}{s}$

$$Y\left( s \right) = \frac{T_{d}\text{s\ }}{Ts + 1}\ \bullet \frac{1}{s}$$

$$y\left( t \right) = \frac{T_{d}}{T}{\bullet e}^{- \frac{t}{T}}$$

10. Delay

$$G\left( s \right) = \frac{Y(s)}{X(s)} = e^{- \tau s}$$

where:

τ – delay time

Response for input - step function x(t)=1(t) , $X\left( s \right) = \frac{1}{s}$

$$Y\left( s \right) = e^{- \tau s} \bullet \frac{1}{s}$$

y(t) = 1(t − τ)

Response for input - impulse function x(t)=δ(t) X(s) = 1

Y(s) = e^−τs • 1

y(t) = (t − τ)

Spectrum transfer function and characteristic

1. Proportional P

G(s) = k_p

P(ω) = k_p;                           Q(ω) = 0

M(ω) = k_p;                           φ(ω) = 0

1. Inertial first order

$$G\left( s \right) = \frac{k}{Ts + 1}$$

$$G\left( \text{jω} \right) = \frac{k}{Tj\omega + 1}$$

$$G\left( \text{jω} \right) = \frac{k}{1 + Tj\omega} \bullet \frac{1 - Tj\omega}{1 - Tj\omega}$$

$$G\left( \text{jω} \right) = \frac{k - kTj\omega}{1 + {T^{2}\omega}^{2}}$$

$$G\left( \text{jω} \right) = \frac{k}{1 + {T^{2}\omega}^{2}} - j\frac{\text{kTω}}{1 + {T^{2}\omega}^{2}}$$

G(jω) = P + jQ

$$P = \frac{k}{1 + {T^{2}\omega}^{2}};\ \ \ \ \ \ \ \ \ \ Q = - \frac{\text{kTω}}{1 + {T^{2}\omega}^{2}}$$

$$G\left( \text{jω} \right) = \sqrt{P^{2} + Q^{2}}e^{- j\varphi} = Me^{- j\varphi}$$

$$M = \frac{A_{y}}{A_{x}} = \sqrt{P^{2} + Q^{2}}$$

$$M = \sqrt{\frac{k^{2}}{\left( 1 + {T^{2}\omega}^{2} \right)^{2}} + \frac{k^{2}{T^{2}\omega}^{2}}{\left( 1 + {T^{2}\omega}^{2} \right)^{2}}}$$

$$M = \sqrt{\frac{k^{2}\left( 1 + {T^{2}\omega}^{2} \right)}{\left( 1 + {T^{2}\omega}^{2} \right)^{2}}}$$

$$M = \sqrt{\frac{k^{2}}{1 + {T^{2}\omega}^{2}}}$$

$$M = \frac{A_{y}}{A_{x}} = \frac{k}{\sqrt{1 + {T^{2}\omega}^{2}}}$$

$$\tan\varphi = \frac{Q}{P} = - \frac{\frac{\text{kTω}}{1 + {T^{2}\omega}^{2}}}{\frac{k}{1 + {T^{2}\omega}^{2}}} = - T\omega$$

tan(−φ) = Tω;        φ = −arctan(Tω)

$$G\left( \text{jω} \right) = \frac{k}{\sqrt{1 + {T^{2}\omega}^{2}}}e^{- j\varphi}$$

1. Inertial second order

$$G\left( s \right) = \frac{k}{\left( T_{1}s + 1 \right)\left( T_{2}s + 1 \right)}$$

$$G\left( \text{jω} \right) = \frac{k}{\left( T_{1}j\omega + 1 \right)\left( T_{2}j\omega + 1 \right)}$$

$$G\left( \text{jω} \right) = \frac{k}{1 - T_{1}T_{2}\omega^{2} + j\omega\left( T_{1} + T_{2} \right)}$$

$$G\left( \text{jω} \right) = \frac{k}{1 - T_{1}T_{2}\omega^{2} + j\omega\left( T_{1} + T_{2} \right)} \bullet \frac{1 - T_{1}T_{2}\omega^{2} - j\omega\left( T_{1} + T_{2} \right)}{1 - T_{1}T_{2}\omega^{2} - j\omega\left( T_{1} + T_{2} \right)}$$

$$G\left( \text{jω} \right) = \frac{k\left( 1 - T_{1}T_{2}\omega^{2} \right) - j\omega k\left( T_{1} + T_{2} \right)}{\left( 1 - T_{1}T_{2}\omega^{2} \right)^{2} + \omega^{2}\left( T_{1} + T_{2} \right)^{2}}$$

$$G\left( \text{jω} \right) = \frac{k\left( 1 - T_{1}T_{2}\omega^{2} \right)}{\left( 1 - T_{1}T_{2}\omega^{2} \right)^{2} + \omega^{2}\left( T_{1} + T_{2} \right)^{2}} - j\frac{\text{ωk}\left( T_{1} + T_{2} \right)}{\left( 1 - T_{1}T_{2}\omega^{2} \right)^{2} + \omega^{2}\left( T_{1} + T_{2} \right)^{2}}$$

G(jω) = P + jQ

$$P = \frac{k\left( 1 - T_{1}T_{2}\omega^{2} \right)}{\left( 1 - T_{1}T_{2}\omega^{2} \right)^{2} + \omega^{2}\left( T_{1} + T_{2} \right)^{2}};\ \ \ \ \ \ \ \ \ \ Q = - \frac{\text{ωk}\left( T_{1} + T_{2} \right)}{\left( 1 - T_{1}T_{2}\omega^{2} \right)^{2} + \omega^{2}\left( T_{1} + T_{2} \right)^{2}}$$

$$G\left( \text{jω} \right) = \sqrt{P^{2} + Q^{2}}e^{- j\varphi} = Me^{- j\varphi}$$

$$M = \frac{A_{y}}{A_{x}} = \sqrt{P^{2} + Q^{2}}$$

$$M = \sqrt{\frac{k^{2}\left( 1 - T_{1}T_{2}\omega^{2} \right)^{2}}{\left( \left( 1 - T_{1}T_{2}\omega^{2} \right)^{2} + \omega^{2}\left( T_{1} + T_{2} \right)^{2} \right)^{2}} + \frac{k^{2}\omega^{2}\left( T_{1} + T_{2} \right)^{2}}{\left( \left( 1 - T_{1}T_{2}\omega^{2} \right)^{2} + \omega^{2}\left( T_{1} + T_{2} \right)^{2} \right)^{2}}}$$

$$M = \frac{A_{y}}{A_{x}} = \frac{k}{\sqrt{\left( 1 - T_{1}T_{2}\omega^{2} \right)^{2} + \omega^{2}\left( T_{1} + T_{2} \right)^{2}}}$$

$$\tan\varphi = \frac{Q}{P} = - \frac{\omega\left( T_{1} + T_{2} \right)}{1 - T_{1}T_{2}\omega^{2}}$$

$$\varphi = - arc\tan\left( \frac{\omega\left( T_{1} + T_{2} \right)}{1 - T_{1}T_{2}\omega^{2}} \right)$$

$$G\left( \text{jω} \right) = \frac{k}{\sqrt{\left( 1 - T_{1}T_{2}\omega^{2} \right)^{2} + \omega^{2}\left( T_{1} + T_{2} \right)^{2}}}e^{- j\varphi}$$

1. Inertial third order

$$G\left( s \right) = \frac{k}{\left( T_{1}s + 1 \right)\left( T_{2}s + 1 \right)\left( T_{3}s + 1 \right)}$$

$$G\left( \text{jω} \right) = \frac{k}{\left( T_{1}j\omega + 1 \right)\left( T_{2}j\omega + 1 \right)\left( T_{3}j\omega + 1 \right)}$$

$$G\left( \text{jω} \right) = \frac{k}{1 - \omega^{2}\left( T_{1}T_{2} + T_{1}T_{3} + T_{2}T_{3} \right) + j\omega\left( T_{1} + T_{2} + T_{3} \right) - jT_{1}T_{2}T_{3}\omega^{3}}$$

$$G\left( \text{jω} \right) = \frac{k}{1 - \omega^{2}\left( T_{1}T_{2} + T_{1}T_{3} + T_{2}T_{3} \right) + j\omega\left( T_{1} + T_{2} + T_{3} \right) - jT_{1}T_{2}T_{3}\omega^{3}} \bullet \frac{1 - \omega^{2}\left( T_{1}T_{2} + T_{1}T_{3} + T_{2}T_{3} \right) - j\omega\left( T_{1} + T_{2} + T_{3} \right) + jT_{1}T_{2}T_{3}\omega^{3}}{1 - \omega^{2}\left( T_{1}T_{2} + T_{1}T_{3} + T_{2}T_{3} \right) - j\omega\left( T_{1} + T_{2} + T_{3} \right) + jT_{1}T_{2}T_{3}\omega^{3}}$$

$$G\left( \text{jω} \right) = \frac{k\left( 1 - \omega^{2}\left( T_{1}T_{2} + T_{1}T_{3} + T_{2}T_{3} \right) \right) - jk\left( \omega\left( T_{1} + T_{2} + T_{3} \right) - T_{1}T_{2}T_{3}\omega^{3} \right)}{\left( 1 - \omega^{2}\left( T_{1}T_{2} + T_{1}T_{3} + T_{2}T_{3} \right) \right)^{2} + \left( \omega\left( T_{1} + T_{2} + T_{3} \right) - T_{1}T_{2}T_{3}\omega^{3} \right)^{2}}$$

G(jω) = P + jQ

$$P = \frac{k\left( 1 - \omega^{2}\left( T_{1}T_{2} + T_{1}T_{3} + T_{2}T_{3} \right) \right)}{\left( 1 - \omega^{2}\left( T_{1}T_{2} + T_{1}T_{3} + T_{2}T_{3} \right) \right)^{2} + \left( \omega\left( T_{1} + T_{2} + T_{3} \right) - T_{1}T_{2}T_{3}\omega^{3} \right)^{2}}$$

$$Q = - \frac{k\left( \omega\left( T_{1} + T_{2} + T_{3} \right) - T_{1}T_{2}T_{3}\omega^{3} \right)}{\left( 1 - \omega^{2}\left( T_{1}T_{2} + T_{1}T_{3} + T_{2}T_{3} \right) \right)^{2} + \left( \omega\left( T_{1} + T_{2} + T_{3} \right) - T_{1}T_{2}T_{3}\omega^{3} \right)^{2}}$$

$$G\left( \text{jω} \right) = \sqrt{P^{2} + Q^{2}}e^{- j\varphi} = Me^{- j\varphi}$$

$$M = \frac{A_{y}}{A_{x}} = \sqrt{P^{2} + Q^{2}}$$

$$M = \frac{A_{y}}{A_{x}} = \frac{k}{\sqrt{\left( 1 - \omega^{2}\left( T_{1}T_{2} + T_{1}T_{3} + T_{2}T_{3} \right) \right)^{2} + \left( \omega\left( T_{1} + T_{2} + T_{3} \right) - T_{1}T_{2}T_{3}\omega^{3} \right)^{2}}}$$

$$\tan\varphi = \frac{Q}{P} = - \frac{\omega\left( T_{1} + T_{2} + T_{3} \right) - T_{1}T_{2}T_{3}\omega^{3}}{1 - \omega^{2}\left( T_{1}T_{2} + T_{1}T_{3} + T_{2}T_{3} \right)}$$

$$\varphi = - arc\tan\left( \frac{\omega\left( T_{1} + T_{2} + T_{3} \right) - T_{1}T_{2}T_{3}\omega^{3}}{1 - \omega^{2}\left( T_{1}T_{2} + T_{1}T_{3} + T_{2}T_{3} \right)} \right)$$

$$G\left( \text{jω} \right) = \frac{k}{\sqrt{\left( 1 - \omega^{2}\left( T_{1}T_{2} + T_{1}T_{3} + T_{2}T_{3} \right) \right)^{2} + \left( \omega\left( T_{1} + T_{2} + T_{3} \right) - T_{1}T_{2}T_{3}\omega^{3} \right)^{2}}}e^{- j\varphi}$$

5. Oscillation - component second order

$$G\left( s \right) = \frac{k\omega_{0}^{2}}{{s^{2} + 2\xi\omega}_{0}s + \omega_{0}^{2}}$$

$$G\left( \text{jω} \right) = \frac{k\omega_{0}^{2}}{{\left( \text{jω} \right)^{2} + 2\xi\omega}_{0}\text{jω} + \omega_{0}^{2}}$$

$$G\left( \text{jω} \right) = \frac{k\omega_{0}^{2}}{\omega_{0}^{2}{{- \omega}^{2} + j2\xi\omega}_{0}\omega}$$

$$G\left( \text{jω} \right) = \frac{k\omega_{0}^{2}}{\omega_{0}^{2}{{- \omega}^{2} + j2\xi\omega}_{0}\omega} \bullet \frac{\omega_{0}^{2}{{- \omega}^{2} - j2\xi\omega}_{0}\omega}{\omega_{0}^{2}{{- \omega}^{2} - j2\xi\omega}_{0}\omega}$$

$$G\left( \text{jω} \right) = \frac{k\omega_{0}^{2}\left( \omega_{0}^{2}{- \omega}^{2} \right) - jk\omega_{0}^{2}\left( {2\xi\omega}_{0}\omega \right)}{\left( \omega_{0}^{2}{- \omega}^{2} \right)^{2} + \left( {2\xi\omega}_{0}\omega \right)^{2}}$$

$$G\left( \text{jω} \right) = \frac{k\omega_{0}^{2}\left( \omega_{0}^{2}{- \omega}^{2} \right)}{\left( \omega_{0}^{2}{- \omega}^{2} \right)^{2} + \left( {2\xi\omega}_{0}\omega \right)^{2}} - j\frac{k\omega_{0}^{2}\left( {2\xi\omega}_{0}\omega \right)}{\left( \omega_{0}^{2}{- \omega}^{2} \right)^{2} + \left( {2\xi\omega}_{0}\omega \right)^{2}}$$

G(jω) = P + jQ

$$P = \frac{k\omega_{0}^{2}\left( \omega_{0}^{2}{- \omega}^{2} \right)}{\left( \omega_{0}^{2}{- \omega}^{2} \right)^{2} + \left( {2\xi\omega}_{0}\omega \right)^{2}};\ \ \ \ \ \ \ \ \ \ Q = - \frac{k\omega_{0}^{2}\left( {2\xi\omega}_{0}\omega \right)}{\left( \omega_{0}^{2}{- \omega}^{2} \right)^{2} + \left( {2\xi\omega}_{0}\omega \right)^{2}}$$

$$G\left( \text{jω} \right) = \sqrt{P^{2} + Q^{2}}e^{- j\varphi} = Me^{- j\varphi}$$

$$M = \frac{A_{y}}{A_{x}} = \sqrt{P^{2} + Q^{2}}$$

$$M = \sqrt{\frac{k^{2}\left( \omega_{0}^{2}\left( \omega_{0}^{2}{- \omega}^{2} \right) \right)^{2}}{\left( \left( \omega_{0}^{2}{- \omega}^{2} \right)^{2} + \left( {2\xi\omega}_{0}\omega \right)^{2} \right)^{2}} + \frac{k^{2}\left( \omega_{0}^{2}\left( {2\xi\omega}_{0}\omega \right) \right)^{2}}{\left( \left( \omega_{0}^{2}{- \omega}^{2} \right)^{2} + \left( {2\xi\omega}_{0}\omega \right)^{2} \right)^{2}}}$$

$$M = \frac{A_{y}}{A_{x}} = \frac{k\omega_{0}^{2}}{\sqrt{\left( \omega_{0}^{2}{- \omega}^{2} \right)^{2} + \left( {2\xi\omega}_{0}\omega \right)^{2}}}$$

$$\tan\varphi = \frac{Q}{P} = - \frac{\omega\left( T_{1} + T_{2} \right)}{1 - T_{1}T_{2}\omega^{2}}$$

$$\varphi = - arc\tan\left( \frac{{2\xi\omega}_{0}\omega}{\omega_{0}^{2}{- \omega}^{2}} \right)$$

$$G\left( \text{jω} \right) = \frac{k\omega_{0}^{2}}{\sqrt{\left( \omega_{0}^{2}{- \omega}^{2} \right)^{2} + \left( {2\xi\omega}_{0}\omega \right)^{2}}}e^{- j\varphi}$$

6. Integral I

$$G\left( s \right) = \frac{1}{T_{i}s}$$

$$P\left( \omega \right) = 0;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Q\left( \omega \right) = - \frac{1}{T_{i}\omega}$$

$$M\left( \omega \right) = \frac{1}{T_{i}\omega};\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \varphi\left( \omega \right) = - 90$$

7. Integral I real

$$G\left( s \right) = \frac{1}{T_{i}s\left( Ts + 1 \right)}$$

$$G\left( \text{jω} \right) = \frac{1}{\text{jT}_{i}\omega\left( Tj\omega + 1 \right)}$$

$$G\left( \text{jω} \right) = \frac{1}{- T_{i}T\omega^{2} + jT_{i}\omega} \bullet \frac{- T_{i}T\omega^{2} - jT_{i}\omega}{- T_{i}T\omega^{2} - jT_{i}\omega}$$

$$G\left( \text{jω} \right) = \frac{- T_{i}T\omega^{2} - jT_{i}\omega}{T_{i}^{2}{T^{2}\omega}^{4} + T_{i}^{2}\omega^{2}}$$

$$G\left( \text{jω} \right) = \frac{- T_{i}T\omega^{2}}{T_{i}^{2}{T^{2}\omega}^{4} + T_{i}^{2}\omega^{2}} - j\frac{T_{i}\omega}{T_{i}^{2}{T^{2}\omega}^{4} + T_{i}^{2}\omega^{2}}$$

$$P = \frac{- T}{T_{i}\left( {T^{2}\omega}^{2} + 1 \right)};\ \ \ \ \ \ \ \ \ \ Q = - \frac{1}{T_{i}\omega\left( {T^{2}\omega}^{2} + 1 \right)}$$

$$G\left( \text{jω} \right) = \sqrt{P^{2} + Q^{2}}e^{- j\varphi} = Me^{- j\varphi}$$

$$M = \frac{A_{y}}{A_{x}} = \sqrt{P^{2} + Q^{2}}$$

$$M = \sqrt{\frac{T_{i}^{2}{T^{2}\omega}^{4}}{\left( T_{i}^{2}{T^{2}\omega}^{4} + T_{i}^{2}\omega^{2} \right)^{2}} + \frac{T_{i}^{2}\omega^{2}}{\left( T_{i}^{2}{T^{2}\omega}^{4} + T_{i}^{2}\omega^{2} \right)^{2}}}$$

$$M = \sqrt{\frac{T_{i}^{2}{T^{2}\omega}^{4} + T_{i}^{2}\omega^{2}}{\left( T_{i}^{2}{T^{2}\omega}^{4} + T_{i}^{2}\omega^{2} \right)^{2}}}$$

$$M = \sqrt{\frac{1}{T_{i}^{2}{T^{2}\omega}^{4} + T_{i}^{2}\omega^{2}}}$$

$$M = \frac{A_{y}}{A_{x}} = \frac{1}{T_{i}\omega\sqrt{{T^{2}\omega}^{2} + 1}}$$

$$\tan\varphi = \frac{Q}{P} = \frac{- T_{i}\omega}{- T_{i}T\omega^{2}} = \frac{1}{\text{Tω}}$$

$$\varphi = arc\tan\left( \frac{1}{\text{Tω}} \right)$$

$$G\left( \text{jω} \right) = \frac{1}{T_{i}\omega\sqrt{{T^{2}\omega}^{2} + 1}}e^{- j\varphi}$$

8. Differential D

G(s) = T_ds ;

P(ω) = 0;                           Q(ω) = T_dω

M(ω) = T_dω;                           φ(ω) = 90

9. Differential D real

$$G\left( s \right) = \frac{T_{d}s}{Ts + 1}$$

$$G\left( \text{jω} \right) = \frac{T_{d}\text{jω}}{Tj\omega + 1}$$

$$G\left( \text{jω} \right) = \frac{T_{d}\text{jω}}{1 + Tj\omega} \bullet \frac{1 - Tj\omega}{1 - Tj\omega}$$

$$G\left( \text{jω} \right) = \frac{TT_{d}\omega^{2} + T_{d}\text{jω}}{1 + {T^{2}\omega}^{2}}$$

$$G\left( \text{jω} \right) = \frac{TT_{d}\omega^{2}}{1 + {T^{2}\omega}^{2}} + j\frac{T_{d}\omega}{1 + {T^{2}\omega}^{2}}$$

G(jω) = P + jQ

$$P = \frac{TT_{d}\omega^{2}}{1 + {T^{2}\omega}^{2}};\ \ \ \ \ \ \ \ \ \ Q = \frac{T_{d}\omega}{1 + {T^{2}\omega}^{2}}$$

$$G\left( \text{jω} \right) = \sqrt{P^{2} + Q^{2}}e^{- j\varphi} = Me^{- j\varphi}$$

$$M = \frac{A_{y}}{A_{x}} = \sqrt{P^{2} + Q^{2}}$$

$$M = \sqrt{\frac{{T^{2}T_{d}^{2}\omega}^{4}}{\left( 1 + {T^{2}\omega}^{2} \right)^{2}} + \frac{{T_{d}^{2}\omega}^{2}}{\left( 1 + {T^{2}\omega}^{2} \right)^{2}}}$$

$$M = \sqrt{\frac{{T_{d}^{2}\omega}^{2}\left( 1 + {T^{2}\omega}^{2} \right)}{\left( 1 + {T^{2}\omega}^{2} \right)^{2}}}$$

$$M = \sqrt{\frac{{T_{d}^{2}\omega}^{2}}{1 + {T^{2}\omega}^{2}}}$$

$$M = \frac{A_{y}}{A_{x}} = \frac{T_{d}\omega}{\sqrt{1 + {T^{2}\omega}^{2}}}$$

$$\tan\varphi = \frac{Q}{P} = \frac{T_{d}\omega}{TT_{d}\omega^{2}} = \frac{1}{\text{Tω}}$$

$$\varphi = arc\tan\left( \frac{1}{\text{Tω}} \right)$$

$$G\left( \text{jω} \right) = \frac{T_{d}\omega}{\sqrt{1 + {T^{2}\omega}^{2}}}e^{- j\varphi}$$

10. Delay

$$G\left( s \right) = \frac{Y(s)}{X(s)} = e^{- \tau s}$$

G(jω) = e^−τjω

G(jω) = cos(τω) − jsin(τω)

G(jω) = P + jQ

P = cos(τω);           Q = −sin(τω)

$$G\left( \text{jω} \right) = \sqrt{P^{2} + Q^{2}}e^{- j\varphi} = Me^{- j\varphi}$$

$$M = \frac{A_{y}}{A_{x}} = \sqrt{P^{2} + Q^{2}}$$

M = 1

φ = −τω

G(jω) = e^−τjω