68 lines
1.5 KiB
Markdown
68 lines
1.5 KiB
Markdown
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![[TD_FGE_2025.pdf]]
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# Exercise 1.
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a.
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$\frac{1}{R_{eq}}=\sum{\frac{1}{R_i}} = \frac{1}{R_1}+\frac{1}{R_2}$
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$R_1=R_2 \implies R_{eq}=\frac{R^2}{2R}$
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b.
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$\frac{1}{R_{eq}}=\sum{\frac{1}{R_i}} = \frac{1}{R_1}+\frac{1}{R_2}$
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$\implies R_{eq}=\frac{R_1R_2}{R_1+R_2}$
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c.
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$R_{eq}=0$
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d.
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$\frac{1}{R_{eq}}=\sum{\frac{1}{R_i}} = \frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}$
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$=\frac{R_1R_2R_3}{R_1+R_2+R_3}, R_1=R_2=R_3 \implies R_{eq}=\frac{R^3}{3R}$
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$R_1=R_2=R3, R_{eq}=\frac{R}{3}$
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e.
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En série, $R_{eq}=R_1+R_2$
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f.
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$\frac{1}{R_{eq}}=\frac{1}{2R}+\frac{1}{R}+\frac{1}{2R}=\frac{5}{2R}$
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$R_{eq}=\frac{2R}{5}$
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g.
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$R_{brancheDroite}=2R$
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$\frac{1}{R_{rectangle}}=\frac{1}{2R}+\frac{1}{2R}=\frac{2}{2R}=\frac{1}{R}$
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$\implies R_{rectangle}=R$
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$R_{eq}=R+R=2R$
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h.
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$R_{eq}=R$ -> Court circuit entre les autres résistances.
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e.
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$R_{eq}=800+\frac{1}{\frac{1}{3k\Omega}+\frac{1}{2k\Omega}}=2k\Omega$
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# Exercise 2.
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Loi des noeuds en $A$:
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$i_1+i_2=i_3$
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Loi des mailles en $m_1$ et $m_2$:
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$E_1=U_{R_1}+U_{R_3}$
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$E_2=U_{R_2}+U_{R_3}$
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Loi d'Ohm:
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$$
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\begin{equation}
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\begin{cases}
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E_1=i_1R_1+i_3R_3\\
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E_2=i_2R_2+i_3R_3
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\end{cases}
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\end{equation}
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$$
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$$
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\begin{equation}
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\begin{cases}
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i_3=i_1+i_2 \\
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i_1=\frac{E_1-i_2R_3}{R_1+R_3} \\
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E_2=i_2R_2+i_1R_3+(\frac{E_1-i_2R_3}{R_1+R_3})R_3
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\end{cases}
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\end{equation}
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$$
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$E_2=\frac{E_1R_3}{R_1+R_3}+i_2(R_2+R_3-\frac{R_3^2}{R_1+R_3})$
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$\implies i_2=(E_2-\frac{E_1R_3}{R_1+R_3})\times \frac{R_1+R_3}{R_2(R_1+R_3)+R_3(R_1+R_3)-R_3^2}$
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$i_2=\frac{E_2(R_1R_3)-E_1R_3}{R_1R_2+R_1R_3+R_2R_3}$
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