\( \DeclareMathOperator{\abs}{abs} \)

Kelvinov most, izvodjenje

potenciometarski razdelnik

(%i1) vb: R2 / (R1 + R2) * E;
\[\mathrm{\tt (\%o1) }\quad \frac{E\cdot \mathit{R2}}{\mathit{R2}+\mathit{R1}}\]

razdelnik na strani nepoznatog otpornika

(%i2) va: v2 + R2 / (R1 + R2) * (v1 - v2);
\[\mathrm{\tt (\%o2) }\quad \frac{\left( \mathit{v1}-\mathit{v2}\right) \cdot \mathit{R2}}{\mathit{R2}+\mathit{R1}}+\mathit{v2}\]
(%i3) va: ratsimp(va);
\[\mathrm{\tt (\%o3) }\quad \frac{\mathit{v2}\cdot \mathit{R1}+\mathit{v1}\cdot \mathit{R2}}{\mathit{R2}+\mathit{R1}}\]

da nadjemo v1 i v2

(%i4) v2: R4 / (R4 + Rz + R3) * E;
\[\mathrm{\tt (\%o4) }\quad \frac{E\cdot \mathit{R4}}{\mathit{R4}+\mathit{R3}+\mathit{Rz}}\]
(%i5) v1: (R4 + Rz) / (R4 + Rz + R3) * E;
\[\mathrm{\tt (\%o5) }\quad \frac{E\cdot \left( \mathit{Rz}+\mathit{R4}\right) }{\mathit{R4}+\mathit{R3}+\mathit{Rz}}\]

da sredimo izraze, prljav posao za wxMaxima

(%i6) va;
\[\mathrm{\tt (\%o6) }\quad \frac{\mathit{v2}\cdot \mathit{R1}+\mathit{v1}\cdot \mathit{R2}}{\mathit{R2}+\mathit{R1}}\]
(%i7) va: ev(va);
\[\mathrm{\tt (\%o7) }\quad \frac{\frac{E\cdot \mathit{R1}\cdot \mathit{R4}}{\mathit{R4}+\mathit{R3}+\mathit{Rz}}+\frac{E\cdot \mathit{R2}\cdot \left( \mathit{Rz}+\mathit{R4}\right) }{\mathit{R4}+\mathit{R3}+\mathit{Rz}}}{\mathit{R2}+\mathit{R1}}\]
(%i8) va: ratsimp(va);
\[\mathrm{\tt (\%o8) }\quad \frac{\mathit{Rz}\cdot E\cdot \mathit{R2}+\left( E\cdot \mathit{R2}+E\cdot \mathit{R1}\right) \cdot \mathit{R4}}{\left( \mathit{R1}+\mathit{R2}\right) \cdot \mathit{R4}+\left( \mathit{R1}+\mathit{R2}\right) \cdot \mathit{R3}+\mathit{Rz}\cdot \mathit{R2}+\mathit{Rz}\cdot \mathit{R1}}\]
(%i9) va: factor(va);
\[\mathrm{\tt (\%o9) }\quad \frac{E\cdot \left( \mathit{Rz}\cdot \mathit{R2}+\mathit{R1}\cdot \mathit{R4}+\mathit{R2}\cdot \mathit{R4}\right) }{\left( \mathit{R2}+\mathit{R1}\right) \cdot \left( \mathit{R4}+\mathit{R3}+\mathit{Rz}\right) }\]

ravnoteza mosta

(%i10) e: va - vb;
\[\mathrm{\tt (\%o10) }\quad \frac{E\cdot \left( \mathit{Rz}\cdot \mathit{R2}+\mathit{R1}\cdot \mathit{R4}+\mathit{R2}\cdot \mathit{R4}\right) }{\left( \mathit{R2}+\mathit{R1}\right) \cdot \left( \mathit{R4}+\mathit{R3}+\mathit{Rz}\right) }-\frac{E\cdot \mathit{R2}}{\mathit{R2}+\mathit{R1}}\]
(%i11) e: ratsimp(e);
\[\mathrm{\tt (\%o11) }\quad \frac{E\cdot \mathit{R1}\cdot \mathit{R4}-E\cdot \mathit{R2}\cdot \mathit{R3}}{\left( \mathit{R1}+\mathit{R2}\right) \cdot \mathit{R4}+\left( \mathit{R1}+\mathit{R2}\right) \cdot \mathit{R3}+\mathit{Rz}\cdot \mathit{R2}+\mathit{Rz}\cdot \mathit{R1}}\]
(%i12) e: factor(e);
\[\mathrm{\tt (\%o12) }\quad \frac{E\cdot \left( \mathit{R1}\cdot \mathit{R4}-\mathit{R2}\cdot \mathit{R3}\right) }{\left( \mathit{R2}+\mathit{R1}\right) \cdot \left( \mathit{R4}+\mathit{R3}+\mathit{Rz}\right) }\]

imenilac me ne zanima, hocu nulu brojioca

(%i13) ne: num(e);
\[\mathrm{\tt (\%o13) }\quad E\cdot \left( \mathit{R1}\cdot \mathit{R4}-\mathit{R2}\cdot \mathit{R3}\right) \]

i da resimo

(%i14) Rx: solve(ne, R4);
\[\mathrm{\tt (\%o14) }\quad [\mathit{R4}=\frac{\mathit{R2}\cdot \mathit{R3}}{\mathit{R1}}]\]

to je sve, lakse nego rukom na tabli!

Created with wxMaxima.