Hai dây dẫn có hệ số nhiệt điện trở \({{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\text{,}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\); ở 00C có điện trở R01, R02. Tìm hệ số nhiệt điện trở chung của hai dây khi chúng mắc:

-Điện trở của hai dây dẫn ở nhiệt độ t: R₁ = R₀₁[1+\){{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\)(t₁ – t₀₁)]; R₂ = R₀₂[1+\({{\text{ }\!\!\alpha\!\!\text{ }}_{2}}\)(t₂ – t₀₂)].

R₁ = R₀₁(1+\({{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\)t); R₂ = R₀₂(1+\[{{\text{ }\!\!\alpha\!\!\text{ }}_{2}}\]t).

với:      t₀₁ = t₀₂ = 0; \({{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\)t, \){{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\)t <<1.

-Gọi R₀ là điện trở chung của hai dây dẫn ở 0⁰C; \(\text{ }\!\!\alpha\!\!\text{ }\) là hệ số nhiệt điện trở chung của hai dây dẫn. Điện trở chung của hai dây dẫn ở nhiệt độ t là:

            R = R₀(1 + \(\text{ }\!\!\alpha\!\!\text{ }\)t)                                       (1)

a)Khi mắc nối tiếp:

            R = R₁ + R₂ = R₀₁(1+\({{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\)t) + R₀₂(1+\){{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\)t)

=>        R = (R₀₁ + R₀₂) + (R₀₁\({{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\) + R02\){{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\))t

=>        R = (R₀₁ + R₀₂)\(\left[ \text{1}+\frac{{{\text{R}}_{\text{01}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}+{{\text{R}}_{\text{02}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}}\text{t} \right]\)              (2)

-Từ (1) và (2) suy ra: \(\text{ }\!\!\alpha\!\!\text{ }=\frac{{{\text{R}}_{\text{01}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}+{{\text{R}}_{\text{02}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}}\).

b)Khi mắc song song:

            R = \(\frac{{{\text{R}}_{\text{1}}}{{\text{R}}_{\text{2}}}}{{{\text{R}}_{\text{1}}}+{{\text{R}}_{\text{2}}}}=\frac{{{\text{R}}_{\text{01}}}\text{(1}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\text{t)}{{\text{R}}_{\text{02}}}\text{(1}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\text{t)}}{{{\text{R}}_{\text{01}}}\text{(1}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\text{t)}+{{\text{R}}_{\text{02}}}\text{(1}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\text{t)}}\)

=>        R = \(\frac{{{\text{R}}_{\text{01}}}{{\text{R}}_{\text{02}}}\text{(1}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\text{t)(1}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\text{t)}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}+{{\text{R}}_{\text{01}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\text{t}+{{\text{R}}_{\text{02}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\text{t)}}\] = \[\frac{{{\text{R}}_{\text{01}}}{{\text{R}}_{\text{02}}}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}}\text{.}\frac{\text{(1}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\text{t)(1}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\text{t)}}{\text{1}+\frac{{{\text{R}}_{\text{01}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}+{{\text{R}}_{\text{02}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}}\text{t}}\)

-Với \({{\text{ }\!\!\varepsilon\!\!\text{ }}_{\text{1}}}\text{,}{{\text{ }\!\!\varepsilon\!\!\text{ }}_{\text{2}}}<<\text{1}\), ta có các công thức gần đúng:

\(\text{(1}+{{\text{ }\!\!\varepsilon\!\!\text{ }}_{\text{1}}}\text{)(1}+{{\text{ }\!\!\varepsilon\!\!\text{ }}_{\text{2}}}\text{)}\approx \text{1}+{{\text{ }\!\!\varepsilon\!\!\text{ }}_{\text{1}}}+{{\text{ }\!\!\varepsilon\!\!\text{ }}_{\text{2}}}\text{; }\frac{\text{1}+{{\text{ }\!\!\varepsilon\!\!\text{ }}_{\text{1}}}}{\text{1}+{{\text{ }\!\!\varepsilon\!\!\text{ }}_{\text{2}}}}\approx \text{1}+{{\text{ }\!\!\varepsilon\!\!\text{ }}_{\text{1}}}-{{\text{ }\!\!\varepsilon\!\!\text{ }}_{\text{2}}}\)

nên      \(\text{(1}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\text{t)(1}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\text{t)}\approx \text{1}+\text{(}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\text{)t}\)

=>        \(\frac{\text{1}+\text{(}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\text{)t}}{\text{1}+\frac{{{\text{R}}_{\text{01}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}+{{\text{R}}_{\text{02}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}}\text{t}}\approx \text{1}+\text{(}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}-\frac{{{\text{R}}_{\text{01}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}+{{\text{R}}_{\text{02}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}}\text{)t}\) \)\approx \) 1 + \)\frac{{{\text{R}}_{\text{01}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}+{{\text{R}}_{\text{02}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}}\text{t}\)

=>        R= \(\frac{{{\text{R}}_{\text{01}}}{{\text{R}}_{\text{02}}}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}}\left[ \text{1}+\frac{{{\text{R}}_{\text{01}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}+{{\text{R}}_{\text{02}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}}\text{t} \right]\)          (3)

-Từ (1) và (3) suy ra: \(\text{ }\!\!\alpha\!\!\text{ }=\frac{{{\text{R}}_{\text{01}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}+{{\text{R}}_{\text{02}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}}\).

Vậy: Hệ số nhiệt điện trở chung của hai dây khi chúng mắc nối tiếp là \(\text{ }\!\!\alpha\!\!\text{ }=\frac{{{\text{R}}_{\text{01}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}+{{\text{R}}_{\text{02}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}}\); khi chúng mắc song song là \)\text{ }\!\!\alpha\!\!\text{ }=\frac{{{\text{R}}_{\text{01}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}+{{\text{R}}_{\text{02}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}}\).