## Q. 6.AE.5

Estimate the lifetime reductions of induction machines, transformers, and universal machines for the single- and three-phase voltage spectra of Table E6.5.1 and their associated lifetime reduction for an activation energy of E=1.1 eV. The ambient temperature is $T_{amb}$=23 °C, the rated temperature is $T_2$=85 °C, and the rated lifetime of $t_2$=40 years can be assumed.

Table E6.5.1 Possible Voltage Spectra with High-Harmonic Penetration
$\begin{array}{l|l|l} \boldsymbol{h} & \left(\boldsymbol{V}_h / \boldsymbol{V}_{60 \mathrm{~Hz}}\right)_{1 \Phi}(\%) & \left(\boldsymbol{V}_h / \boldsymbol{V}_{60 \mathrm{~Hz}}\right)_{3 \Phi}(\%) \\ \hline 1 & 100 & 100 \\ 2 & 2.5 & 0.5 \\ 3 & 5.71 & 1.0 \\ 4 & 1.6 & 0.5 \\ 5 & 1.25 & 7.0 \\ 6 & 0.88 & 0.2 \\ 7 & 1.25 & 5.0 \\ 8 & 0.62 & 0.2 \\ 9 & 0.96 & 0.3 \\ 10 & 0.66 & 0.1 \\ 11 & 0.30 & 2.5 \\ 12 & 0.18 & 0.1 \\ 13 & 0.57 & 2.0 \\ 14 & 0.10 & 0.05 \\ 15 & 0.10 & 0.1 \\ 16 & 0.13 & 0.05 \\ 17 & 0.23 & 1.5 \\ 18 & 0.22 & 0.01 \\ 19 & 1.03 & 1.0 \\ & \text { All higher harmonics }<0.2 \% & \\ \hline \end{array}$

## Verified Solution

Calculation of weighted harmonic factor for single-phase spectrum of Table E6.5.1 based on the values for $k_{avg}=0.85\ and\ \ell_{avg}=1.4$ for single-phase induction motors:

$\frac{1}{2^{0.85}} (2.5)^{1.4}=2.000, \frac{1}{3^{0.85}} (5.71)^{1.4}=4.506\\\frac{1}{4^{0.85}} (1.6)^{1.4}=0.594,… \frac{1}{17^{0.85}} (0.23)^{1.4}=0.0115\\ \frac{1}{18^{0.85}} (0.22)^{1.4}=0.0103, \frac{1}{19^{0.85}} (1.03)^{1.4}=0.0853$

Summing all contributions results in the weighted harmonic-voltage factor for single phase induction motors $\sum_{h=2}^{hmax } \frac{1}{h^k}\left(\frac{V_{\mathrm{pk}}}{V_{p 1}}\right)^{\ell} \approx 8.5$.

Calculation of weighted harmonic factor for three-phase spectrum of Table E6.5.1 using the values for $k_{avg}=0.95\ and\ \ell_{avg}=1.6$ for three-phase induction motors:

\begin{aligned} & \frac{1}{2^{0.95}}(0.5)^{1.6}=0.1707, \frac{1}{3^{0.95}}(1.0)^{1.6}=0.3522 \\ & \frac{1}{4^{0.95}}(0.5)^{1.6}=0.0884, \ldots, \frac{1}{17^{0.95}}(1.5)^{1.6}=0.1297 \\ & \frac{1}{18^{0.95}}(0.01)^{1.6}=0.00004, \frac{1}{19^{0.95}}(1.0)^{1.6}=0.0609 \end{aligned}

Summing all contributions results in the weighted harmonic-voltage factor for three-phase induction motors $\sum_{h=2}^{hmax } \frac{1}{h^k}\left(\frac{V_{p k}}{V_{p l}}\right)^{\ell} \approx 8.6$.

Calculation of weighted harmonic factor for single-phase spectrum of Table E6.5.1 based on the values for $k_{avg}=0.90\ and\ \ell_{avg}=1.75$ for single-phase transformers:

\begin{aligned} & \frac{1}{2^{0.9}}(2.5)^{1.75}=2.664, \frac{1}{3^{0.9}}(5.71)^{1.75}=7.847, \\ & \frac{1}{4^{0.9}}(1.6)^{1.75}=0.6536, \ldots, \frac{1}{17^{0.9}}(0.23)^{1.75}=0.00597, \\ & \frac{1}{18^{0.9}}(0.22)^{1.75}=0.00524, \frac{1}{19^{0.9}}(1.03)^{1.75}=0.744 . \end{aligned}

Summing all contributions results in the weighted harmonic-voltage factor for single-phase transformers $\sum_{h=2}^{hmax} \frac{1}{h^k}\left(\frac{V_{p_h}}{V_{F^{\prime}}}\right)^{\ell} \approx 12.3$.

Calculation of weighted harmonic factor for three-phase spectrum of Table E6.5.1 based on the values for $k_{avg}=0.90\ and\ \ell_{avg}=1.75$ for three-phase transformers:

\begin{aligned} & \frac{1}{2^{0.9}}(0.5)^{1.75}=0.1593, \frac{1}{3^{0.9}}(1)^{1.75}=0.37203 \\ & \frac{1}{4^{0.9}}(0.5)^{1.75}=0.0854, \ldots ., \frac{1}{17^{0.9}}(1.5)^{1.75}=0.15876 \\ & \frac{1}{18^{0.9}}(0.01)^{1.75}=0.0000234, \frac{1}{19^{0.9}}(1)^{1.75}=0.07065 \end{aligned}

Summing all above contributions results in the weighted harmonic-voltage factor for three-phase transformers $\sum_{h=2}^{hmax} \frac{1}{h^k}\left(\frac{V_{ph}}{V_{F^{\prime}}}\right)^{\ell} \approx 11.8$.

Calculation of weighted harmonic factor for single-phase spectrum of Table E6.5.1 based on the values for $k_{avg}=1.0\ and\ \ell_{avg}=2.0$ for universal motors:

\begin{aligned} & \frac{1}{2}(2.5)^2=3.125, \frac{1}{3}(5.71)^2=10.868 \\ & \frac{1}{4}(1.6)^2=0.640, \ldots ., \frac{1}{17}(0.23)^2=0.00311 \\ & \frac{1}{18}(0.22)^2=0.0027, \frac{1}{19}(1.03)^2=0.0558 \end{aligned}

Summing all contributions results in the weighted harmonic-voltage factor for universal motor $\sum_{h=2}^{h \max } \frac{1}{h^k}\left(\frac{V_{p h}}{V_{p 1}}\right)^{\ell} \approx 15.6$

Based on Fig. 6.14, $T_2$=85 °C, $T_{amb}$=23 °C, E=1.1 eV, and rated lifetime of $t_2$=40 years the above harmonic factors result in the additional temperature rises and lifetime reductions of Table E6.5.2.

Table E6.5.2 Additional Temperature Rise and Associated Lifetime Reduction of Induction Motors, Transformers, and Universal Motors Due to the Harmonic Spectra of Table E6.5.1
$\begin{array}{l|l|l|l|l|l} & {\begin{array}{l} \text { Single-phase } \\ \text { induction } \\ \text { motors } \end{array}} & {\begin{array}{l} \text { Three-phase } \\ \text { induction } \\ \text { motors } \end{array}} & {\begin{array}{l} \text { Single-phase } \\ \text { transformers } \end{array}} & {\begin{array}{l} \text { Three-phase } \\ \text { transformers } \end{array}} & {\begin{array}{l} \text { Universal } \\ \text { motors } \end{array}} \\ \hline \Delta T_h(\%) & 9.2 & 4.5 & 1.9 & 1.8 & 2.4 \\ \Delta T_h\left({ }^{\circ} \mathrm{C}\right) & 5.7 & 2.8 & 1.2 & 1.1 & 1.5 \\ \text { Lifetime }& 43 & 24 & 11 & 10 & 14 \\ \begin{array}{l} \text { reduction (\%) } \end{array} \\ \hline \end{array}$