异步电动机

${\displaystyle \omega _{s}={\frac {2{\pi }n_{s}}{60}}={\frac {4{\pi }f_{s}}{p}}}$

电动机輸入等效阻抗

${\displaystyle Z_{m}=R_{s}+jX_{s}+{\frac {({\frac {R_{r}^{'}}{s}}+jX_{r}^{'})(jX_{m})}{{\frac {R_{r}^{'}}{s}}+j(X_{r}^{'}+X_{m})}}}$

定子電流

${\displaystyle I_{s}=V_{s}/Z_{m}=V_{s}/(R_{s}+jX_{s}+{\frac {({\frac {R_{r}^{'}}{s}}+jX_{r}^{'})(jX_{m})}{{\frac {R_{r}^{'}}{s}}+j(X_{r}^{'}+X_{m})}})}$

轉子電流，對應一次側的數值

${\displaystyle I_{r}^{'}={\frac {jX_{m}}{{\frac {R_{r}^{'}}{s}}+j(X_{r}^{'}+X_{m})}}I_{s}}$

根據斯泰因梅茨等效電路，可得

${\displaystyle {\frac {R_{r}^{'}}{s}}={\frac {R_{r}^{'}(1-s)}{s}}+R_{r}^{'}}$

氣隙功率等於電機功率加上轉子銅損

${\displaystyle P_{gap}=P_{em}+P_{r}}$

${\displaystyle P_{r}=3R_{r}^{'}I_{r}^{'2}}$

${\displaystyle P_{gap}={\frac {3R_{r}^{'}I_{r}^{'2}}{s}}}$

${\displaystyle P_{em}={\frac {3R_{r}^{'}I_{r}^{'2}(1-s)}{s}}}$

${\displaystyle P_{em}=P_{gap}(1-s)}$

用轉子角速度表示電機功率輸出

${\displaystyle P_{em}={\frac {3R_{r}^{'}I_{r}^{'2}n_{r}}{sn_{s}}}}$ (watts)

${\displaystyle P_{em}={\frac {3R_{r}^{'}I_{r}^{'2}n_{r}}{746sn_{s}}}}$ (hp)

將 ${\displaystyle T_{em}}$ 用 ft.-lb表示：

${\displaystyle P_{em}={\frac {T_{em}n_{r}}{5252}}}$ (hp)

${\displaystyle T_{em}={\frac {P_{em}}{\omega _{r}}}={\frac {\frac {P_{r}}{s}}{\omega _{s}}}={\frac {3I_{r}^{'2}R_{r}^{'}}{\omega _{s}s}}}$ (newton-meters)

為了將${\displaystyle T_{em}}$用${\displaystyle s}$來表示，IEEE建議${\displaystyle R_{s},X_{s}}$及${\displaystyle X_{m}}$ 需轉換為戴维南等效電路

IEEE建議戴维南等效電路

其中

${\displaystyle V_{TE}={\frac {X_{m}}{\sqrt {R_{s}^{2}+(X_{s}+X_{m})^{2}}}}V_{s}}$

${\displaystyle Z_{TE}=R_{TE}+jX_{TE}={\frac {jX_{m}(R_{s}+jX_{s})}{R_{s}+j(X_{s}+X_{m})}}}$

因為${\displaystyle R_{s}^{2}\gg {(X_{s}+X_{m})^{2}}}$及${\displaystyle X_{s}\ll {X_{m}}}$，令 ${\displaystyle K_{TE}={\frac {X_{m}}{X_{s}+X_{m}}}}$

${\displaystyle V_{TE}\approx {Z_{TE}V_{s}}}$ and ${\displaystyle Z_{TE}\approx {K_{TE}^{2}R_{s}+jX_{s}}}$[50]

${\displaystyle T_{em}={\frac {3V_{TE}^{2}}{(R_{TE}+{\frac {R_{r}^{'}}{s}})^{2}+(X_{TE}+X_{r}^{'})^{2}}}.{\frac {R_{r}^{'}}{s}}.{\frac {1}{\omega _{s}}}(N.m)}$[50]

針對滑差較低時：

Since ${\displaystyle R_{TE}+R_{r}^{'}\gg {R_{TE}+X_{r}^{'}}}$ and ${\displaystyle R_{r}^{'}\gg {R_{TE}}}$

${\displaystyle T_{em}\approx {\frac {1}{\omega _{s}}}.{\frac {3V_{TE}^{2}}{R_{r}^{'}}}.s}$ (N.m)

針對滑差較高時：

Since ${\displaystyle R_{TE}+R_{r}^{'}\ll {R_{TE}+X_{r}^{'}}}$

${\displaystyle T_{em}\approx {\frac {1}{\omega _{s}}}.{\frac {3V_{TE}^{2}}{(X_{s}+X_{r}^{'})^{2}}}.{\frac {R_{r}^{'2}}{s}}}$ (N.m)

最大轉矩或是崩潰轉矩和轉子電阻無關

${\displaystyle T_{max}={\frac {1}{2\omega _{s}}}.{\frac {3V_{TE}^{2}}{R_{TE}+{\sqrt {R_{TE}^{2}+(X_{TE}+X_{r}^{'})^{2}}}}}}$ (N.m)[50]

最大轉矩或是崩潰轉矩的滑差為

${\displaystyle s={\frac {R_{r}^{'}}{\sqrt {R_{TE}^{2}+(X_{TE}+X_{r}^{'})^{2}}}}}$[50]

若用foot-pound的單位

${\displaystyle T_{em}={\frac {21.21I_{r}^{'2}R_{r}^{'}}{n_{r}s}}}$ (ft-lb)

${\displaystyle T_{em}={\frac {7.04P_{gap}}{n_{s}}}}$ (ft-lb)