«K12/K13» 5. Hausaufgabe «PDF», «POD»

0.0.1 ↑ 5. Hausaufgabe

0.0.1.1 ↑ Analyse eines verzweigten Stromkreises

Gegeben

R_1$R_1$ bis R_5$R_5$ betragen 10{,}0 \,\Omega$10,0\Omega$, R_6$R_6$ sei variabel.

U = 9{,}00 \,\mathrm{V};$U=9,00\text{V};$

Gesucht

R(R_6)$R\left(R_6\right)$, I(R_6)$I\left(R_6\right)$, \\$$ I_6(R_6)$I_6\left(R_6\right)$, P_6(R_6)$P_6\left(R_6\right)$

Rechnung

R_{4,5} = R_4 + R_5 = 2R_1;$R_{4,5}=R_4+R_5=2R_1;$

R_{3,4,5} = \dfrac{1}{\frac{1}{R_1} + \frac{1}{R_{4,5}}} = \frac{2}{3} R_1;$R_{3,4,5}=\frac{1}{\frac{1}{R_1}+\frac{1}{R_{4,5}}}=\frac{2}{3}{R}_1;$

R_{2,3,4,5,6} = \dfrac{1}{\frac{1}{R_2} + \frac{1}{R_{3,4,5}} + \frac{1}{R_6}} = \dfrac{1}{\frac{1}{R_6} + \frac{5}{2 R_1}};$R_{2,3,4,5,6}=\frac{1}{\frac{1}{R_2}+\frac{1}{R_{3,4,5}}+\frac{1}{R_6}}=\frac{1}{\frac{1}{R_6}+\frac{5}{2R_1}};$

R(R_6) = R = R_1 + R_{2,3,4,5,6} = R_1 + \dfrac{1}{\frac{1}{R_6} + \frac{5}{2 R_1}} = 10{,}0 \,\Omega + \dfrac{1}{\frac{1}{R_6} + 0{,}250 \frac{1}{\,\Omega}};$R\left(R_6\right)=R=R_1+R_{2,3,4,5,6}=R_1+\frac{1}{\frac{1}{R_6}+\frac{5}{2R_1}}=10,0\Omega +\frac{1}{\frac{1}{R_6}+0,250\frac{1}{\Omega }};$

I(R_6) = I = \frac{U}{R} = \dfrac{U}{R_1 + \dfrac{1}{\frac{1}{R_6} + \frac{5}{2 R_1}}} = \dfrac{9{,}00 \,\mathrm{V}}{10{,}0 \,\Omega + \dfrac{1}{\frac{1}{R_6} + 0{,}250 \frac{1}{\,\Omega}}};$I\left(R_6\right)=I=\frac{U}{R}=\frac{U}{R_1+\frac{1}{\frac{1}{R_6}+\frac{5}{2R_1}}}=\frac{9,00\text{V}}{10,0\Omega +\frac{1}{\frac{1}{R_6}+0,250\frac{1}{\Omega }}};$

U_6(R_6) = U_6 = U_2 = \frac{R_{2,3,4,5,6}}{R_1} U_1 = \frac{R_{2,3,4,5,6}}{R_1} R_1 I_1 = R_{2,3,4,5,6} I = \dfrac{I}{\frac{1}{R_6} + \frac{5}{2 R_1}} = \dfrac{U}{\left(R_1 + \dfrac{1}{\frac{1}{R_6} + \frac{5}{2 R_1}}\right) \Biggl(\dfrac{1}{R_6} + \dfrac{5}{2 R_1}\Biggr)} = \dfrac{9{,}00 \,\mathrm{V}}{\left(10{,}0 \,\Omega + \dfrac{1}{\frac{1}{R_6} + 0{,}250 \frac{1}{\,\Omega}}\right) \Biggl(\dfrac{1}{R_6} + 0{,}250 \frac{1}{\,\Omega}\Biggr)};$U_6\left(R_6\right)=U_6=U_2=\frac{R_{2,3,4,5,6}}{R_1}{U}_1=\frac{R_{2,3,4,5,6}}{R_1}{R}_1{I}_1=R_{2,3,4,5,6}I=\frac{I}{\frac{1}{R_6}+\frac{5}{2R_1}}=\frac{U}{\left(R_1+\frac{1}{\frac{1}{R_6}+\frac{5}{2R_1}}\right)\left(\frac{1}{R_6}+\frac{5}{2R_1}\right)}=\frac{9,00\text{V}}{\left(10,0\Omega +\frac{1}{\frac{1}{R_6}+0,250\frac{1}{\Omega }}\right)\left(\frac{1}{R_6}+0,250\frac{1}{\Omega }\right)};$

\renewcommand{\arraystretch}{2.5}\begin{array}{ll}I_6(R_6) = I_6 = \frac{U_6}{R_6} & = \dfrac{U}{\left(R_1 + \dfrac{1}{\frac{1}{R_6} + \frac{5}{2 R_1}}\right) \Biggl(\dfrac{1}{R_6} + \dfrac{5}{2 R_1}\Biggr) R_6} = \\ & = \dfrac{9{,}00 \,\mathrm{V}}{\left(10{,}0 \,\Omega + \dfrac{1}{\frac{1}{R_6} + 0{,}250 \frac{1}{\,\Omega}}\right) \Biggl(\dfrac{1}{R_6} + 0{,}250 \frac{1}{\,\Omega}\Biggr) R_6};\end{array}$\begin{array}{cc}{I}_6\left(R_6\right)=I_6=\frac{U_6}{R_6}\hfill & =\frac{U}{\left(R_1+\frac{1}{\frac{1}{R_6}+\frac{5}{2R_1}}\right)\left(\frac{1}{R_6}+\frac{5}{2R_1}\right)R_6}=\hfill \\ \hfill & =\frac{9,00\text{V}}{\left(10,0\Omega +\frac{1}{\frac{1}{R_6}+0,250\frac{1}{\Omega }}\right)\left(\frac{1}{R_6}+0,250\frac{1}{\Omega }\right)R_6};\hfill \end{array}$

\renewcommand{\arraystretch}{2.5}\begin{array}{ll}P_6(R_6) = P_6 = U_6 I_6 & = \dfrac{U^2}{\left(R_1 + \dfrac{1}{\frac{1}{R_6} + \frac{5}{2 R_1}}\right)^2 \Biggl(\frac{1}{R_6} + \frac{5}{2 R_1}\Biggr)^2 R_6} = \\ & = \dfrac{81{,}0 \,\mathrm{V}^2}{\left(10{,}0 \,\Omega + \dfrac{1}{\frac{1}{R_6} + 0{,}250 \frac{1}{\,\Omega}}\right)^2 \Biggl(\dfrac{1}{R_6} + 0{,}250 \frac{1}{\,\Omega}\Biggr)^2 R_6};\end{array}$\begin{array}{cc}{P}_6\left(R_6\right)=P_6=U_6{I}_6\hfill & =\frac{U^2}{{\left(R_1+\frac{1}{\frac{1}{R_6}+\frac{5}{2R_1}}\right)}^2{\left(\frac{1}{R_6}+\frac{5}{2R_1}\right)}^2{R}_6}=\hfill \\ \hfill & =\frac{81,0{\text{V}}^2}{{\left(10,0\Omega +\frac{1}{\frac{1}{R_6}+0,250\frac{1}{\Omega }}\right)}^2{\left(\frac{1}{R_6}+0,250\frac{1}{\Omega }\right)}^2{R}_6};\hfill \end{array}$

Grenzwertbetrachtungen

\renewcommand{\arraystretch}{1.5} \begin{array}{llcl} {} \lim\limits_{R_6 \to 0 \,\Omega} & R(R_6) & = & 10{,}0 \Omega; \\ {} \lim\limits_{R_6 \to \infty \,\Omega} & R(R_6) & = & 14{,}0 \Omega; \\ {} \lim\limits_{R_6 \to 0 \,\Omega} & I(R_6) & = & 0{,}900 \,\mathrm{A}; \\ {} \lim\limits_{R_6 \to \infty \,\Omega} & I(R_6) & = & 0{,}643 \,\mathrm{A}; \\ {} \lim\limits_{R_6 \to 0 \,\Omega} & I_6(R_6) & = & 0{,}900 \,\mathrm{A}; \\ {} \lim\limits_{R_6 \to \infty \,\Omega} & I_6(R_6) & = & 0 \,\mathrm{A}; \\ {} \lim\limits_{R_6 \to 0 \,\Omega} & P_6(R_6) & = & 0 \,\mathrm{W}; \\ {} \lim\limits_{R_6 \to \infty \,\Omega} & P_6(R_6) & = & 0 \,\mathrm{W}; \end{array}$\begin{array}{cccc}\underset{R_6\to 0\Omega }{lim}\hfill & R\left(R_6\right)\hfill & \hfill =\hfill & 10,0\Omega ;\hfill \\ \underset{R_6\to \infty \Omega }{lim}\hfill & R\left(R_6\right)\hfill & \hfill =\hfill & 14,0\Omega ;\hfill \\ \underset{R_6\to 0\Omega }{lim}\hfill & I\left(R_6\right)\hfill & \hfill =\hfill & 0,900\text{A};\hfill \\ \underset{R_6\to \infty \Omega }{lim}\hfill & I\left(R_6\right)\hfill & \hfill =\hfill & 0,643\text{A};\hfill \\ \underset{R_6\to 0\Omega }{lim}\hfill & I_6\left(R_6\right)\hfill & \hfill =\hfill & 0,900\text{A};\hfill \\ \underset{R_6\to \infty \Omega }{lim}\hfill & I_6\left(R_6\right)\hfill & \hfill =\hfill & 0\text{A};\hfill \\ \underset{R_6\to 0\Omega }{lim}\hfill & P_6\left(R_6\right)\hfill & \hfill =\hfill & 0\text{W};\hfill \\ \underset{R_6\to \infty \Omega }{lim}\hfill & P_6\left(R_6\right)\hfill & \hfill =\hfill & 0\text{W};\hfill \end{array}$

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