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Komplexe Widerstände

Definitionen

$U$ : $[U] = V$ : Spannung in Volt

$\boxed{U(t) = U_0 e^{j(wt + \phi)}}$ : komplexe Spannung

$I$ : $[I] = A$ : Strom in Ampere

$\boxed{I(t) = I_0 e^{j(wt + \psi)}}$ : komplexer Strom

$\boxed{Z = Z_0 e^{j\phi} = R + jX}$ : komplexer Widerstand

Hilfsmittel: Komplexe Algebra

www.openhardsoftware.de : Komplexe Algebra

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als PDF-Dokument : 2304031800_ComplexResistance.pdf

Das Ohmsche Gesetz

Ohmsches Gesetz reell : $\boxed{U = R I}$

Spannung reell
$\boxed{U = U(t)}$ : $[U] = V$

Strom reell
$\boxed{I = I(t)}$ : $[I] = A = \dfrac{C}{s}$

Ohmscher Widerstand reell
$\boxed{R = \dfrac{U}{I}}$ : $[R] = \dfrac{V}{A} = \Omega$

Ohmsches Gesetz komplex : $\boxed{U = Z I}$

Spannung komplex
$\boxed{U = U(t) = U_{0} \exp[j(\omega t + \phi)]}$ : $[U] = V$

Strom komplex
$\boxed{I = I(t) = I_{0} \exp[j(\omega t + \psi)]}$ : Strom, $[I] = A$

Ohmscher Widerstand komplex
$\boxed{Z = \dfrac{U}{I} = \dfrac{U_{0} \exp(j \omega t + j\phi)}{I_{0} \exp(j \omega t + j\psi)} = \dfrac{U_0}{I_0}\exp(j(\phi - \psi))}$ : $[Z] = \dfrac{V}{A} = \Omega$

Beispiel: Kondensator

Ohmsches Gesetz : $\boxed{U_c = X_c I_c}$

Komplexer Widerstand Kondensator : $\boxed{Z_c = -j\dfrac{1}{\omega C} = \dfrac{1}{j \omega C}}$

Nebenrechnung Dimension :
$[Z_c] = \dfrac{V}{A} = [-j\dfrac{1}{\omega C}] = [-j\dfrac{U}{2 \pi f Q}]= \dfrac{V s}{A s} = \Omega$ {mit : $C = \dfrac{Q}{U}$ }

$U_c = \dfrac{I_c}{j \omega C} = -j\dfrac{1}{\omega C} I_c = Z_c I_c$

$Z_c = -j\dfrac{1}{\omega C}$ , $Re(Z_c) = 0$ , $Im(Z_c) = -\dfrac{1}{\omega C}$

$\vert Z_c \vert = \sqrt{Re(Z_c)^2 + Im(Z_c)^2} = \sqrt{0^2 + \dfrac{1}{\omega^2 C^2}} = \dfrac{1}{\omega C}$

$\phi_c = \arctan(\dfrac{Im(Z_c)}{Re(Z_c)}) = -\arctan(\dfrac{1}{0 \cdot \omega C}) = -\dfrac{\pi}{2}$

$U_c = Z_c I_c$ $\Rightarrow U_c = \vert Z_c \vert I_c \exp(-j\dfrac{\pi}{2})= $

$U_0 \exp(jwt + j\phi_u) = \vert Z_c \vert I_0 \exp(jwt + j\phi_i + j\phi_c) = \vert Z_c \vert I_0 \exp(jwt -j\dfrac{\pi}{2})$

$U_0 \exp(j\phi_u)= \vert Z_c \vert I_0 \exp(j\phi_i - j\dfrac{\pi}{2}) = \vert Z_c \vert I_0 (-j) \exp(j\phi_i)$

Frequenzabhängiger Widerstand eines Kondensators:
$\boxed{U_0 = \vert Z_c \vert I_0 = \dfrac{1}{\omega C} I_0}$

$\exp(j\phi_u) = -j \exp(j\phi_i) = \exp(j\phi_i - j \dfrac{\pi}{2})$

$\exp(j(\phi_u - \phi_i)) = \exp(-j \dfrac{\pi}{2})$

Phasenverschiebung zwischen Spannung und Strom am Kondensator:
$\boxed{\phi_u - \phi_i = -\dfrac{\pi}{2}}$

Beispiel: Tiefpass

$\boxed{\dfrac{U_a}{U_e} = \dfrac{Z_a}{Z_e + Z_a} := T_{lp}}$

Sei $Z_a = \dfrac{1}{j \omega C}$ und $Z_e = R$

$\Rightarrow T = \dfrac{\dfrac{1}{j \omega C}}{R + \dfrac{1}{j \omega C}} = \dfrac{1}{1 + j \omega R C} = \dfrac{1 - j \omega R C}{(1 + j \omega R C)(1 - j \omega R C)}$

Übertragungsfunktion :
$\boxed{T_{lp} = \dfrac{U_a}{U_e} = \dfrac{1}{1 + \omega^2 R^2 C^2} - j \dfrac{\omega R C}{1 + \omega^2 R^2 C^2}}$

$\vert T_{lp} \vert = \sqrt{Re(T_{lp})^2 + Im(T_{lp})^2} = \sqrt{\dfrac{1}{(1 + \omega^2 R^2 C^2)^2} + \dfrac{\omega^2 R^2 C^2}{(1 + \omega^2 R^2 C^2)^2}} = \sqrt{\dfrac{1 + \omega^2 R^2 C^2}{(1 + \omega^2 R^2 C^2)^2}}$

Amplitudengang der Übertragungsfunktion Tlp:
$\boxed{\vert T_{lp} \vert = \dfrac{1}{\sqrt{1 + \omega^2 R^2 C^2}}}$

$\phi_T = \arctan(\dfrac{Im(T_{lp})}{Re(T_{lp})}) = \arctan\bigg(-\dfrac{\dfrac{\omega R C}{1 + \omega^2 R^2 C^2}}{\dfrac{1}{1 + \omega^2 R^2 C^2}}\bigg)$

Phasengang der Übertragungsfunktion Tlp:
$\boxed{\phi_T = -\arctan(\omega R C )}$

Polardarstellung der Übertragungsfunktion Tlp:
$\boxed{T_{lp} = \dfrac{U_a}{U_e} = \dfrac{1}{\sqrt{1 + \omega^2 R^2 C^2}} \exp[-j \arctan(\omega R C)]}$

Beispiel: Hochpass

$\boxed{\dfrac{U_a}{U_e} = \dfrac{Z_a}{Z_e + Z_a} := T_{hp}}$

Sei $Z_a = R$ und $Z_e = \dfrac{1}{j \omega C}$

$\Rightarrow T_{hp} = \dfrac{R}{\dfrac{1}{j \omega C} + R} = \dfrac{j \omega R C}{1 + j\omega R C} = \dfrac{j \omega R C(1 - j\omega R C)}{(1 + j\omega R C)(1 - j\omega R C)}$

$\boxed{T_{hp} = \dfrac{U_a}{U_e} = \dfrac{\omega^2 R^2 C^2}{1 + \omega^2 R^2 C^2} + j \dfrac{\omega R C}{1 + \omega^2 R^2 C^2} }$

$\vert T_{hp} \vert = \sqrt{Re(T_{hp})^2 + Im(T_{hp})^2} = \sqrt{\dfrac{\omega^4 R^4 C^4}{(1 + \omega^2 R^2 C^2)^2} + \dfrac{\omega^2 R^2 C^2}{(1 + \omega^2 R^2 C^2)^2}}$

$\vert T_{hp} \vert = \sqrt{\dfrac{\omega^2 R^2 C^2(1 + \omega^2 R^2 C^2)}{(1 + \omega^2 R^2 C^2)^2}} = \dfrac{\omega R C}{1 + \omega^2 R^2 C^2} \sqrt{1 + \omega^2 R^2 C^2}$

Amplitudengang der Übertragungsfunktion Thp:
$\boxed{\vert T_{hp} \vert = \dfrac{\omega R C }{\sqrt{1 + \omega^2 R^2 C^2}}}$

$\phi_T = \arctan(\dfrac{Im(T_{hp})}{Re(T_{hp})}) = \arctan\bigg(\dfrac{\dfrac{\omega R C}{1 + \omega^2 R^2 C^2}}{\dfrac{\omega^2 R^2 C^2}{1 + \omega^2 R^2 C^2}} \bigg) = \arctan(\dfrac{1}{\omega R C})$

Phasengang der Übertragungsfunktion Thp:
$\boxed{\phi_T = \arctan(\dfrac{1}{\omega R C})}$

Polardarstellung der Übertragungsfunktion Thp:
$\boxed{T_{hp} = \dfrac{U_a}{U_e} = \dfrac{\omega R C }{\sqrt{1 + \omega^2 R^2 C^2}} \exp\big[j \arctan(\dfrac{1}{\omega R C})\big]}$

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