kircchof 2 loops

[dudung]

      --> I1     I2 -->
  a     R1    b     R2    c
  +---/\/\/---+---/\/\/---+
  |           |           |
  |       I3  <           |
 ___ +     |  > R5       ___ +
  -  E1    v  <           -  E2
  |           <           |
  |           |           |
  +---/\/\/---+---/\/\/---+
  d     R3    e     R4    f

b & e:
$$\tag{1}
I_1 - I_2 - I_3 = 0
$$

a-b-e-d-a:
$$\tag{3}
\varepsilon_1 - I_1 R_1 - I_3 R_5 - I_1 R_3 = 0
$$

c-b-e-f-c:
$$\tag{4}
\varepsilon_2 + I_2 R_2 - I_3 R_5 + I_2 R_4 = 0
$$

a-b-c-f-e-d-a | (3) - (4):
$$\tag{5}
\begin{array}{rcl}
\varepsilon_1 - I_1 R_1 - I_2 R_2 &&\newline

• \varepsilon_2 - I_2 R_4 - I_1 R_3 & = & 0
  \end{array}
  $$

(1) → (2):
$$\tag{6}
\begin{array}{rcl}
\varepsilon_1 \newline

• (R_1+R_3+R_5) I_1 \newline

• R_5 I_2 & = & 0
  \end{array}
  $$

(1) → (3):
$$\tag{7}
\begin{array}{rcl}
\varepsilon_2 \newline

• R_5 I_1 \newline

• (R_2+R_4+R_5) I_2 & = & 0
  \end{array}
  $$

$R_5$ (6) - $(R_1+R_3+R_5)$ (7):
$$\tag{8}
\begin{array}{c}
R_5 \varepsilon_1 - (R_1+R_3+R_5) \varepsilon_2 \newline
-(R_1+R_3+R_5)(R_2+R_4+R_5) I_2 \newline
+R_5^2 I_2 = 0
\end{array}
$$

$$\tag{9}
\begin{array}{c}
R_5 \varepsilon_1 - (R_1+R_3+R_5) \varepsilon_2 \newline
-[(R_1+R_3+R_4+R_4)R_5 \newline

• (R_1+R_3)(R_2+R_4)] I_2 = 0
  \end{array}
  $$

$$\tag{10} I_2 = \frac{ R_5 \varepsilon_1 - (R_1+R_3+R_5) \varepsilon_2 }{ (R_1+R_3+R_4+R_4)R_5 + (R_1+R_3)(R_2+R_4) } $$