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}

Step-by-Step

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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}

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