APPENDIX 8 (continued)

$$
\begin{align*}
\text{substitute, } \int_r^a Il(r \cdot k \cdot i \cdot q) \, dr &= \frac{1}{k \cdot i \cdot q} \cdot I0(a \cdot k \cdot i \cdot q) - \frac{1}{k \cdot i \cdot q} \cdot I0(r \cdot k \cdot i \cdot q) \\
\text{substitute, } \int_r^a K1(r \cdot k \cdot i \cdot q) \, dr &= \frac{-1}{k \cdot i \cdot q} \cdot K0(a \cdot k \cdot i \cdot q) + \frac{1}{k \cdot i \cdot q} \cdot K0(r \cdot k \cdot i \cdot q) \\
\text{zero = zero3} \qquad \qquad \qquad \qquad \qquad \qquad \rightarrow \text{zero = 0!} \\
\text{substitute, } \int_b^a Il(r \cdot k \cdot i \cdot q) \, dr &= \frac{1}{k \cdot i \cdot q} \cdot I0(a \cdot k \cdot i \cdot q) - \frac{1}{k \cdot i \cdot q} \cdot I0(b \cdot k \cdot i \cdot q) \\
\text{substitute, } \int_b^a K1(r \cdot k \cdot i \cdot q) \, dr &= \frac{-1}{k \cdot i \cdot q} \cdot K0(a \cdot k \cdot i \cdot q) + \frac{1}{k \cdot i \cdot q} \cdot K0(b \cdot k \cdot i \cdot q) \\
\text{substitute } j_{DCWS} &= j_{DC} + j_W + j_S \\
\text{substitute, } A1 &= \frac{K1(b \cdot k \cdot i \cdot q)}{I0(a \cdot k \cdot i \cdot q) \cdot K1(b \cdot k \cdot i \cdot q) + I1(b \cdot k \cdot i \cdot q) \cdot K0(a \cdot k \cdot i \cdot q)} \\
\text{substitute, } A2 &= \frac{I1(b \cdot k \cdot i \cdot q)}{I0(a \cdot k \cdot i \cdot q) \cdot K1(b \cdot k \cdot i \cdot q) + I1(b \cdot k \cdot i \cdot q) \cdot K0(a \cdot k \cdot i \cdot q)} \\
\text{simplify}
\end{align*}
$$

Finding the eddy-free solution at: $p := 0 \quad q := \sqrt{2 \cdot p + 1} \quad q = 1$

$$
j_{10}(r) := \frac{j_{DC} \cdot k \cdot q^2 \cdot (Il(b \cdot k \cdot q \cdot i) \cdot K0(k \cdot q \cdot r \cdot i) + K1(b \cdot k \cdot q \cdot i) \cdot I0(k \cdot q \cdot r \cdot i))}{-k \cdot p \cdot I0(a \cdot k \cdot q \cdot i) \cdot K1(b \cdot k \cdot q \cdot i) - k \cdot p \cdot K0(a \cdot k \cdot q \cdot i) \cdot Il(b \cdot k \cdot q \cdot i) + k \cdot p \cdot I0(b \cdot k \cdot q \cdot i) \cdot K1(b \cdot k \cdot q \cdot i) + k \cdot p \cdot K0(b \cdot k \cdot q \cdot i) + k \cdot q^2 \cdot I0(a \cdot k \cdot q \cdot i) \cdot K1(b \cdot k \cdot q \cdot i) + k \cdot q^2 \cdot K0(a \cdot k \cdot q \cdot i) \cdot Il(b \cdot k \cdot q \cdot i) - a \cdot k_5^2 \cdot q \cdot \ln(s) \cdot K1(a \cdot k \cdot q \cdot i) \cdot Il(b \cdot k \cdot q \cdot i) \cdot i + a \cdot k_5^2 \cdot q \cdot \ln(s) \cdot Il(a \cdot k \cdot q \cdot i) \cdot K1(b \cdot k \cdot q \cdot i) \cdot i} *
$$

$$
j_1(r) = j_{10}(r) \text{simplify} \rightarrow j_1(r) = \frac{j_{DC} \cdot k \cdot (Il(b \cdot k \cdot i) \cdot K0(k \cdot r \cdot i) + K1(b \cdot k \cdot i) \cdot I0(k \cdot r \cdot i))}{k \cdot I0(a \cdot k \cdot i) \cdot K1(b \cdot k \cdot i) + k \cdot K0(a \cdot k \cdot i) \cdot Il(b \cdot k \cdot i) - a \cdot k_5^2 \cdot \ln(s) \cdot K1(a \cdot k \cdot i) \cdot Il(b \cdot k \cdot i) \cdot i + a \cdot k_5^2 \cdot \ln(s) \cdot Il(a \cdot k \cdot i) \cdot K1(b \cdot k \cdot i) \cdot i} *
$$

* Long expressions have been edited by moving the terms to the next line and placing them in square brackets.

**Vertical blocks of commands edited by splitting into two parts.
APPENDIX 9: Search for total currents and impedances for cylindrical tubular wire

$$
\text{Eddy-free case} \qquad j_1 (r) = \frac{j_{DC} \cdot k \cdot (I_0 (r \cdot k \cdot i) \cdot K_1 (b \cdot k \cdot i) + K_0 (r \cdot k \cdot i) \cdot I_1 (b \cdot k \cdot i))}{D_{tA} + D_{tB}}
$$

$$
\text{where}
\begin{align*}
D_{tA} &= k \cdot (K_1 (b \cdot k \cdot i) \cdot I_0 (a \cdot k \cdot i) + I_1 (b \cdot k \cdot i) \cdot K_0 (a \cdot k \cdot i)) \\
D_{tB} &= i \cdot a \cdot k_s^2 \ln(s) \cdot (I_1 (a \cdot k \cdot i) \cdot K_1 (b \cdot k \cdot i) - I_1 (b \cdot k \cdot i) \cdot K_1 (a \cdot k \cdot i))
\end{align*}
$$

$$
\begin{align*}
& I_1 = 2 \cdot \pi \cdot \int_b^a j_1 (r) \cdot r \, dr \quad \left. \substack{\text{substitute, } j_1 (r) = \frac{j_{DC} \cdot k \cdot (I_0 (r \cdot k \cdot i) \cdot K_1 (b \cdot k \cdot i) + K_0 (r \cdot k \cdot i) \cdot I_1 (b \cdot k \cdot i))}{D_{tA} + D_{tB}} \\ \text{simplify} \\ \text{factor} \quad \right| \rightarrow \\
& \rightarrow I_1 = \frac{2 \cdot \pi \cdot j_{DC} \cdot k \cdot I_1 (b \cdot k \cdot i) \cdot \int_b^a r \cdot K_0 (k \cdot r \cdot i) \, dr + 2 \cdot \pi \cdot j_{DC} \cdot k \cdot K_1 (b \cdot k \cdot i) \cdot \int_b^a r \cdot I_0 (k \cdot r \cdot i) \, dr}{D_{tA} + D_{tB}} \quad \left. \rule{10em}{0.4pt} \right|^{*}
\end{align*}
$$

$$
\substack{\text{substitute,} \\ \text{substitute,} \\ \text{simplify} \\ \text{factor}} \quad \int_b^a r \cdot I_0 (r \cdot k \cdot i) \, dr = \frac{a}{k \cdot i} \cdot I_1 (a \cdot k \cdot i) - \frac{b}{k \cdot i} \cdot I_1 (b \cdot k \cdot i) \quad \rightarrow
$$

$$
\substack{\text{substitute,} \\ \text{simplify} \\ \text{factor}} \quad \int_b^a r \cdot K_0 (r \cdot k \cdot i) \, dr = \frac{-a}{k \cdot i} \cdot K_1 (a \cdot k \cdot i) + \frac{b}{k \cdot i} \cdot K_1 (b \cdot k \cdot i) \\
\rightarrow I_1 := \frac{2i \cdot \pi \cdot a \cdot j_{DC} \cdot (K_1 (a \cdot k \cdot i) \cdot I_1 (b \cdot k \cdot i) - I_1 (a \cdot k \cdot i) \cdot K_1 (b \cdot k \cdot i))}{D_{tA} + D_{tB}}
$$

$$
Z_1 = \frac{E \cdot l}{I_1} \text{ substitute, } \frac{E}{j_{DC}} = \rho \rightarrow Z_1 = \frac{\rho \cdot l \cdot (D_{tA} + D_{tB}) \cdot i}{2 \cdot \pi \cdot a \cdot (I_1 (a \cdot k \cdot i) \cdot K_1 (b \cdot k \cdot i) - K_1 (a \cdot k \cdot i) \cdot I_1 (b \cdot k \cdot i))}
$$

$$
\text{Eddy case} \qquad j(r) = \frac{j_{DC} \cdot q^2 \cdot k \cdot (I_0 (r \cdot k \cdot q \cdot i) \cdot K_1 (b \cdot k \cdot q \cdot i) + K_0 (r \cdot k \cdot q \cdot i) \cdot I_1 (b \cdot k \cdot q \cdot i))}{D_{tC} + D_{tD} + D_{tE}}
$$

$$
\text{where } D_{tC} = (q^2 - p) \cdot k \cdot (K_1 (b \cdot k \cdot q \cdot i) \cdot I_0 (a \cdot k \cdot q \cdot i) + I_1 (b \cdot k \cdot q \cdot i) \cdot K_0 (a \cdot k \cdot q \cdot i))
$$

$$
D_{tD} = i \cdot a \cdot q \cdot k_s^2 \ln(s) \cdot (I_1 (a \cdot k \cdot q \cdot i) \cdot K_1 (b \cdot k \cdot q \cdot i) - I_1 (b \cdot k \cdot q \cdot i) \cdot K_1 (a \cdot k \cdot q \cdot i))
$$

$$
D_{tE} = p \cdot k \cdot (K_1 (b \cdot k \cdot q \cdot i) \cdot I_0 (b \cdot k \cdot q \cdot i) + I_1 (b \cdot k \cdot q \cdot i) \cdot K_0 (b \cdot k \cdot q \cdot i))
$$

$$
\begin{align*}
& I = 2 \cdot \pi \cdot \int_b^a j(r) \cdot r \, dr \quad \left. \substack{\text{substitute, } j(r) = \frac{j_{DC} \cdot q^2 \cdot k \cdot (I_0 (r \cdot k \cdot q \cdot i) \cdot K_1 (b \cdot k \cdot q \cdot i) + K_0 (r \cdot k \cdot q \cdot i) \cdot I_1 (b \cdot k \cdot q \cdot i))}{D_{tC} + D_{tD} + D_{tE}} \\ \text{simplify} \\ \text{factor} \quad \right| \rightarrow \\
& \rightarrow I = \frac{2 \cdot \pi \cdot j_{DC} \cdot k \cdot q^2 \cdot I_1 (b \cdot k \cdot q \cdot i) \cdot \int_b^a r \cdot K_0 (k \cdot q \cdot r \cdot i) \, dr + 2 \cdot \pi \cdot j_{DC} \cdot k \cdot q^2 \cdot K_1 (b \cdot k \cdot q \cdot i) \cdot \int_b^a r \cdot I_0 (k \cdot q \cdot r \cdot i) \, dr}{D_{tC} + D_{tD} + D_{tE}} \quad \left. \rule{10em}{0.4pt} \right|^{*}
\end{align*}
$$

$$
\substack{\text{substitute,} \\ \text{substitute,} \\ \text{simplify} \\ \text{factor}} \quad \int_b^a r \cdot I_0 (r \cdot k \cdot i \cdot q) \, dr = \frac{a}{k \cdot i \cdot q} \cdot I_1 (a \cdot k \cdot i \cdot q) - \frac{b}{k \cdot i \cdot q} \cdot I_1 (b \cdot k \cdot i \cdot q) \quad \rightarrow
$$

$$
\substack{\text{substitute,} \\ \text{simplify} \\ \text{factor}} \quad \int_b^a r \cdot K_0 (r \cdot k \cdot i \cdot q) \, dr = \frac{-a}{k \cdot i \cdot q} \cdot K_1 (a \cdot k \cdot i \cdot q) + \frac{b}{k \cdot i \cdot q} \cdot K_1 (b \cdot k \cdot i \cdot q) \\
\rightarrow I := - \frac{-2i \cdot \pi \cdot a \cdot j_{DC} \cdot q \cdot K_1 (a \cdot k \cdot q \cdot i) \cdot I_1 (b \cdot k \cdot q \cdot i) + 2i \cdot \pi \cdot a \cdot j_{DC} \cdot q \cdot I_1 (a \cdot k \cdot q \cdot i) \cdot K_1 (b \cdot k \cdot q \cdot i)}{D_{tC} + D_{tD} + D_{tE}}
$$

$$
Z = \frac{E \cdot l}{I} \text{ substitute, } \frac{E}{j_{DC}} = \rho \rightarrow Z = \frac{\rho \cdot l \cdot (D_{tC} + D_{tD} + D_{tE}) \cdot i}{2 \cdot \pi \cdot a \cdot q \cdot (I_1 (a \cdot k \cdot q \cdot i) \cdot K_1 (b \cdot k \cdot q \cdot i) - K_1 (a \cdot k \cdot q \cdot i) \cdot I_1 (b \cdot k \cdot q \cdot i))}
$$

<footnote>* Vertical blocks of commands have been moved to the next line</footnote>
APPENDIX 10: Comparison of graphs for solid and tubular wire for the eddy case with an optimal ratio of skin depth to wall thickness, providing a lower resistance of tubular wire compared to solid wire

$$
\begin{align*}
& f \equiv 1 \cdot 10^7 \quad \omega \equiv 2\pi f \quad E \equiv 0.11508 \quad s \equiv 1, \quad k_b \equiv 0.9820179 \quad 1 \equiv 10^3 \quad \rho \equiv 2.877 \times 10^{-8} \quad k_a \equiv k_b \quad a \equiv 3.338 \times 10^{-3} \\
& \mu_r \equiv 1 \quad \mu \equiv \mu_o + i\mu_r \quad \mu_{sr} \equiv 1 \quad \mu_s \equiv \mu_o + i\mu_{sr} \quad p \equiv 0.5 \quad q \equiv \sqrt{1+2p} \quad n \equiv s \cdot a \quad b \equiv k_b \cdot a \quad b1 \equiv k_a \cdot a \quad S \equiv \pi \cdot (a^2 - b^2) \\
& j_{DC} \equiv \frac{E}{\rho} \quad \delta \equiv \sqrt{\frac{2\rho}{\mu \cdot \omega}} \quad k \equiv \sqrt{\frac{-i \cdot \mu \cdot \omega}{\rho}} \quad k_s \equiv \sqrt{\frac{-i \cdot \mu_s \cdot \omega}{\rho}} \quad R_{DC} \equiv \rho \cdot \frac{1}{S} \quad \frac{\delta}{a-b} = 0.45 \quad R_{DC} = 23.06 \quad R_{DC0} = 0.8219 \\
& S = 1.248 \times 10^{-6} \quad \delta = 2.7 \times 10^{-5} \quad b = 3.278 \times 10^{-3} \quad j_{DC} = 4 \times 10^6 \quad I_{DC} = 4.99 \quad I_{DC0} = 140.02
\end{align*}
$$

$$
|j_{b=0}(b)| = 2.32 \times 10^5, \quad |j_{b=0}(a)| = 5.33 \times 10^6, \quad |I_0| = 1.51, \quad \arg(I_0) = -0.78, \quad \arg(j_{b=0}(b)) = 3.14 \arg(j_{b=0}(a)) = 0
$$

$$
|j_{b>0}(b)| = 4.76 \times 10^5, \quad |j_{b>0}(a)| = 5.49 \times 10^6, \quad |I| = 1.55, \quad \arg(I) = -0.78, \quad \arg(j_{b>0}(b)) = 3.14 \arg(j_{b>0}(a)) = 0
$$

$$
|j_{b>0}(b)| = 4.76 \times 10^5, \quad |j_{b>0}(a)| = 5.49 \times 10^6, \quad |I| = 1.55, \quad \arg(I) = -0.78, \quad \arg(j_{b>0}(b)) = 3.14 \arg(j_{b>0}(a)) = 0
$$

<figure><img src="image_39.png" /></figure>

<figure><img src="image_40.png" /></figure>

<table>
  <tr>
    <td>23.061</td>
  </tr>
  <tr>
    <td>59</td>
  </tr>
  <tr>
    <td>85.079</td>
  </tr>
  <tr>
    <td>104.65</td>
  </tr>
  <tr>
    <td>120.772</td>
  </tr>
  <tr>
    <td>134.93</td>
  </tr>
  <tr>
    <td>147.756</td>
  </tr>
  <tr>
    <td>159.573</td>
  </tr>
  <tr>
    <td>170.582</td>
  </tr>
</table>

<table>
  <tr>
    <td>0.822</td>
  </tr>
  <tr>
    <td>60.413</td>
  </tr>
  <tr>
    <td>85.372</td>
  </tr>
  <tr>
    <td>104.524</td>
  </tr>
  <tr>
    <td>120.67</td>
  </tr>
  <tr>
    <td>134.895</td>
  </tr>
  <tr>
    <td>147.756</td>
  </tr>
  <tr>
    <td>159.582</td>
  </tr>
  <tr>
    <td>170.59</td>
  </tr>
</table>

<table>
  <tr>
    <td>22.238</td>
  </tr>
  <tr>
    <td>-1.413</td>
  </tr>
  <tr>
    <td>-0.293</td>
  </tr>
  <tr>
    <td>0.126</td>
  </tr>
  <tr>
    <td>0.101</td>
  </tr>
  <tr>
    <td>0.035</td>
  </tr>
  <tr>
    <td>7.005·10<sup>-4</sup></td>
  </tr>
  <tr>
    <td>-8.511·10<sup>-3</sup></td>
  </tr>
  <tr>
    <td>-7.512·10<sup>-3</sup></td>
  </tr>
</table>

<table>
  <tr>
    <td>9.398·10<sup>-4</sup></td>
  </tr>
  <tr>
    <td>59.583</td>
  </tr>
  <tr>
    <td>85.7</td>
  </tr>
  <tr>
    <td>104.558</td>
  </tr>
  <tr>
    <td>120.52</td>
  </tr>
  <tr>
    <td>134.706</td>
  </tr>
  <tr>
    <td>147.576</td>
  </tr>
  <tr>
    <td>159.417</td>
  </tr>
  <tr>
    <td>170.433</td>
  </tr>
</table>

<table>
  <tr>
    <td>0.031</td>
  </tr>
  <tr>
    <td>60.258</td>
  </tr>
  <tr>
    <td>85.218</td>
  </tr>
  <tr>
    <td>104.37</td>
  </tr>
  <tr>
    <td>120.516</td>
  </tr>
  <tr>
    <td>134.741</td>
  </tr>
  <tr>
    <td>147.601</td>
  </tr>
  <tr>
    <td>159.428</td>
  </tr>
  <tr>
    <td>170.435</td>
  </tr>
</table>

<table>
  <tr>
    <td>-0.03</td>
  </tr>
  <tr>
    <td>-0.675</td>
  </tr>
  <tr>
    <td>0.483</td>
  </tr>
  <tr>
    <td>0.188</td>
  </tr>
  <tr>
    <td>3.602·10<sup>-3</sup></td>
  </tr>
  <tr>
    <td>-0.035</td>
  </tr>
  <tr>
    <td>-0.026</td>
  </tr>
  <tr>
    <td>-0.011</td>
  </tr>
  <tr>
    <td>-2.05·10<sup>-3</sup></td>
  </tr>
</table>
APPENDIX 11: Comparison of graphs for solid and tubular wire for the eddy-free case with an optimal ratio of skin depth to wall thickness, providing a lower resistance of tubular wire compared to solid wire

<figure><img src="image_41.png" /></figure>
APPENDIX 12: Comparison of eddy case graphs for the inner wire of a coaxial cable and for a conventional wire, graphs for the outer wire-screen

$$
\begin{align*}
& f \equiv 1.5 \cdot 10^5 \qquad \omega \equiv 2\pi f \qquad E \equiv 2 \qquad s \equiv 1.75 \qquad k_b \equiv 0.8 \qquad 1 \equiv 10^3 \rho \equiv 2.877 \times 10^{-8} \qquad a \equiv 5 \times 10^{-3} \qquad p \equiv 0.5 \qquad q \equiv \sqrt{1+2p} \\
& \mu_r \equiv 1 \quad \mu \equiv \mu_0 \cdot \mu_r \quad \mu_{sr} \equiv 1 \quad \mu_s \equiv \mu_0 \cdot \mu_{sr} \quad b \equiv k_b \cdot a \quad k_a \equiv k_b \\
& n \equiv s \cdot a \qquad b1 \equiv k_a \cdot a \qquad S \equiv \pi \cdot (a^2 - b^2) \qquad b = 0.004 \\
& \delta \equiv \sqrt{\frac{2\rho}{\mu \cdot \omega}} \qquad k \equiv \sqrt{\frac{-i \cdot \mu \cdot \omega}{\rho}} \qquad k_s \equiv \sqrt{\frac{-i \cdot \mu_s \cdot \omega}{\rho}} \qquad \frac{\delta}{a-b} = 0.22 \\
& \Delta \equiv \sqrt{2 \cdot a \cdot \min(\delta, a-b) - \min(\delta, a-b)^2 + n^2} - n \qquad t \equiv \sqrt{a^2 - b^2 + n^2} \qquad k_\Phi \equiv \frac{2 \cdot a - \min(\delta, a-b)}{2 \cdot n + \min(\Delta, t-n)} \cdot \frac{\min(\Delta, t-n)}{\min(\delta, a-b)} \\
& k_i \equiv \sqrt{k^2 \cdot \frac{1}{2} \cdot (1 + k_\Phi)} \qquad k_{sc} \equiv \sqrt{k_s^2 \cdot \frac{1}{2}} \qquad n = 0.00875 \qquad t1 = 0.00925 \qquad t1 \equiv \sqrt{a^2 - b1^2 + n^2} \qquad r := b1, b1 + \frac{a-b1}{50} .. a \\
& R_{DC} \equiv \rho \cdot \frac{1}{S} \qquad j_{DC} \equiv \frac{E}{\rho} \qquad u := n, n + \frac{t1 - n}{50} .. t1 \quad \delta = 2.204 \times 10^{-4} \qquad \Delta = 1.223 \times 10^{-4} \qquad k_\Phi = 0.308 \cdot \frac{\Delta}{t-n} = 0.245
\end{align*}
$$

$$
|j(b)| = 1.92 \times 10^4 \quad |j(a)| = 5.29 \times 10^6 \quad |I| = 18.18 \quad \arg(I) = -1.53 \quad \arg(j(b)) = -0.88 \quad \arg(j(a)) = -0.75
$$

$$
|j_i(b)| = 1.03 \times 10^5 |j_i(a)| = 8.36 \times 10^6 |I_i| = 35.43 \arg(I_i) = -1.51 \arg(j_i(b)) = 0.37 \arg(j_i(a)) = -0.73
$$

$$
s = 1.75 \qquad j_{DC} = 6.952 \times 10^7 \qquad I_{DC} = 1.966 \times 10^3 \qquad R_{DC} = 1.02 \qquad R = 4.48232 \qquad R_i = 3.62672
$$

<figure><img src="image_42.png" /></figure>

<figure><img src="image_43.png" /></figure>

$$
|j_o(t)| = 1.03 \times 10^5 |j_o(n)| = 8.36 \times 10^6 |I_o| = 35.43 \arg(I_o) = -1.51 \arg(j_o(t)) = 0.37 \arg(j_o(n)) = -0.73 \\
s = 1.75 \qquad R_o = 3.62672
$$

<figure><img src="image_44.png" /></figure>

<figure><img src="image_45.png" /></figure>

$$
s = 1 \quad |j(b)| = 3.35 \times 10^5 \quad |j(a)| = 9.26 \times 10^7 \quad |I| = 318.04 \arg(I) = -0.78 \quad \arg(j(b)) = -0.12 \arg(j(a)) = 0
$$

$$
|j_i(b)| = 3.83 \times 10^5 |j_i(a)| = 9.26 \times 10^7 |I_i| = 324.72 \arg(I_i) = -0.78 \arg(j_i(b)) = 0.01 \arg(j_i(a)) = 0
$$

$$
b = 0.004 \qquad t = 0.005831 \qquad n = 0.005 \qquad j_{DC} = 6.952 \times 10^7 \qquad I_{DC} = 1.966 \times 10^3 \qquad R_{DC} = 1.02 \qquad R = 4.48232 \qquad R_i = 4.39019
$$

<figure><img src="image_46.png" /></figure>

<figure><img src="image_47.png" /></figure>
APPENDIX 13: Comparison of eddy-free case graphs for the inner wire of a coaxial cable and for a conventional wire, graphs for the outer wire-screen

$$f \equiv 1.5 \cdot 10^5 \quad \omega \equiv 2\pi f \quad E \equiv 2 \quad s \equiv 1.75 \quad k_b \equiv 0.8 \quad 1 \equiv 10^3 \rho \equiv 2.877 \times 10^{-8} \quad a \equiv 5 \times 10^{-3} \quad p \equiv 0 \quad q \equiv \sqrt{1+2p}$$

$$\mu_r \equiv 1 \quad \mu \equiv \mu_0 \cdot \mu_r \quad \mu_{sr} \equiv 1 \quad \mu_s \equiv \mu_0 \cdot \mu_{sr} \quad b \equiv k_b \cdot a \quad k_a \equiv k_b$$

$$n \equiv s \cdot a \quad b1 \equiv k_a \cdot a \quad S \equiv \pi \cdot (a^2 - b^2) \quad b = 0.004$$

$$\delta \equiv \sqrt{\frac{2\rho}{\mu \cdot \omega}} \quad k \equiv \sqrt{\frac{-i \cdot \mu \cdot \omega}{\rho}} \quad k_s \equiv \sqrt{\frac{-i \cdot \mu_s \cdot \omega}{\rho}} \quad \frac{\delta}{a-b} = 0.22$$

$$\Delta \equiv \sqrt{2 \cdot a \cdot \min(\delta, a-b) - \min(\delta, a-b)^2 + n^2} - n \quad t \equiv \sqrt{a^2 - b^2 + n^2} \quad k_{\Phi} \equiv \frac{2 \cdot a - \min(\delta, a-b)}{2 \cdot n + \min(\Delta, t-n)} \cdot \frac{\min(\Delta, t-n)}{\min(\delta, a-b)}$$

$$k_i \equiv \sqrt{k^2 \cdot \frac{1}{2} \cdot (1 + k_{\Phi})} \quad k_{sc} \equiv \sqrt{k_s^2 \cdot \frac{1}{2}} \quad n = 0.00875 \quad t = 0.00925 \quad t1 \equiv \sqrt{a^2 - b1^2 + n^2} \quad r := b1, b1 + \frac{a - b1}{50} \quad a$$

$$R_{DC} \equiv \rho \cdot \frac{1}{S} \quad j_{DC} \equiv \frac{E}{\rho} \quad u := n, n + \frac{t1 - n}{50} \quad t1 \quad \delta = 2.204 \times 10^{-4} \quad \Delta = 1.223 \times 10^{-4} \quad k_{\Phi} = 0.308 \cdot \frac{\Delta}{t - n} = 0.245$$

$$|j(b)| = 8.89 \times 10^4 \quad |j(a)| = 3.76 \times 10^6 \quad |I| = 18.23 \quad \arg(I) = -1.53 \quad \arg(j(b)) = 1 \quad \arg(j(a)) = -0.76$$

$$|j_i(b)| = 3.34 \times 10^5 \quad |j_i(a)| = 5.96 \times 10^6 \quad |I_i| = 35.58 \quad \arg(I_i) = -1.51 \quad \arg(j_i(b)) = 1.89 \quad \arg(j_i(a)) = -0.74$$

$$s = 1.75 \qquad j_{DC} = 6.952 \times 10^7 \quad I_{DC} = 1.966 \times 10^3 \quad R_{DC} = 1.02 \quad R = 4.24729 \quad R_i = 3.45931$$

<figure><img src="image_48.png" /></figure>

<figure><img src="image_49.png" /></figure>

$$|j_o(t)| = 3.34 \times 10^5 \quad |j_o(n)| = 5.96 \times 10^6 \quad |I_o| = 35.58 \quad \arg(I_o) = -1.51 \quad \arg(j_o(t)) = 1.89 \quad \arg(j_o(n)) = -0.74$$

$$s = 1.75 \qquad R_o = 3.45931$$

<figure><img src="image_50.png" /></figure>

<figure><img src="image_51.png" /></figure>

$$s = 1 \quad |j(b)| = 1.64 \times 10^6 \quad |j(a)| = 6.95 \times 10^7 \quad |I| = 336.72 \quad \arg(I) = -0.77 \quad \arg(j(b)) = 1.76 \quad \arg(j(a)) = 0$$

$$|j_i(b)| = 1.81 \times 10^6 \quad |j_i(a)| = 6.95 \times 10^7 \quad |I_i| = 343.84 \quad \arg(I_i) = -0.77 \quad \arg(j_i(b)) = 1.85 \quad \arg(j_i(a)) = 0$$

$$b = 0.004$$

$$t = 0.005831 \quad n = 0.005 \quad j_{DC} = 6.952 \times 10^7 \quad I_{DC} = 1.966 \times 10^3 \quad R_{DC} = 1.02 \quad R = 4.24729 \quad R_i = 4.16066$$

<figure><img src="image_52.png" /></figure>

<figure><img src="image_53.png" /></figure>
APPENDIX 14: Search and verification of solutions for a flat plate

Finding a solution to an equation :

$$
\begin{align*}
j(x) &= j_{DC} + q^2 \cdot k^2 \cdot \int_x^t \int_0^x j(x) \, dx \, dx - k^2 \cdot p \cdot \int_x^t \int_0^x j(x) \, dx \, dx + t_s \cdot k_s^2 \cdot \int_0^t j(x) \, dx \\
&\qquad \text{where: } q^2 = 1 + 2p \\
&\qquad -k^2 \cdot p \cdot \int_0^t \int_0^x j(x) \, dx \, dx = j_W
\end{align*}
$$

$$
t_s \cdot k_s^2 \cdot \int_0^t j(x) \, dx = j_S \quad \text{-- constants; we represent the solution in the form: } \quad j(x) = \frac{j_{DC} + j_W + j_S}{\cos(q \cdot t \cdot k)} \cdot \cos(q \cdot k \cdot x)
$$

Finding the values of constants *j*<sub>W</sub> and *j*<sub>S</sub> :

$$
\begin{align*}
-k^2 \cdot p \cdot \int_0^t \int_0^x j(x) \, dx \, dx &= j_W \\
\left.
\begin{array}{l}
\text{substitute, } j(x) = \frac{j_{DC} + j_W + j_S}{\cos(q \cdot t \cdot k)} \cdot \cos(q \cdot k \cdot x) \rightarrow \\
\quad 2 \cdot p \cdot \left( \frac{\cos(k \cdot q \cdot t)}{2} - \frac{1}{2} \right) \cdot (j_S + j_{DC}) \\
\text{solve, } j_W \\
\text{simplify}
\end{array}
\right\}
\quad \text{Then let: } j_{W1} := \frac{p - p \cdot \cos(k \cdot q \cdot t) + q^2 \cdot \cos(k \cdot q \cdot t)}{p - p \cdot \cos(k \cdot q \cdot t) + q^2 \cdot \cos(k \cdot q \cdot t)}
\end{align*}
$$

$$
t_s \cdot k_s^2 \cdot \int_0^t j(x) \, dx = j_S
\quad
\left|
\begin{array}{@{}l@{\quad}l@{}}
\text{substitute, } j(x) = \frac{j_{DC} + j_W + j_S}{\cos(q \cdot t \cdot k)} \cdot \cos(q \cdot k \cdot x) & \rightarrow -\frac{k_s^2 \cdot t_s \cdot \sin(k \cdot q \cdot t) \cdot (j_W + j_{DC})}{k_s^2 \cdot t_s \cdot \sin(k \cdot q \cdot t) - k \cdot q \cdot \cos(k \cdot q \cdot t)} \\
\text{solve, } j_S \\
\text{simplify} & \\
\text{Then let: } j_{S1} & := -\frac{k_s^2 \cdot t_s \cdot \sin(k \cdot q \cdot t) \cdot (j_W + j_{DC})}{k_s^2 \cdot t_s \cdot \sin(k \cdot q \cdot t) - k \cdot q \cdot \cos(k \cdot q \cdot t)}
\end{array}
\right.
$$

$$
\begin{array}{@{}l@{\quad}c@{\quad}l@{}}
\left( \begin{array}{l} j_S = j_{S1} \\ j_W = j_{W1} \end{array} \right) \left| \begin{array}{l} \text{solve, } j_S, j_W \\ \text{simplify} \end{array} \right. \rightarrow & \left[ \begin{array}{@{}l@{}} \displaystyle \frac{j_{DC} \cdot k_s^2 \cdot q \cdot t_s \cdot \sin(k \cdot q \cdot t)}{k \cdot p - k \cdot p \cdot \cos(k \cdot q \cdot t) + k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - k_s^2 \cdot q \cdot t_s \cdot \sin(k \cdot q \cdot t)} \\ \displaystyle \frac{j_{DC} \cdot k \cdot p \cdot (\cos(k \cdot q \cdot t) - 1)}{k \cdot p - k \cdot p \cdot \cos(k \cdot q \cdot t) + k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - k_s^2 \cdot q \cdot t_s \cdot \sin(k \cdot q \cdot t)} \end{array} \right] & \\
j_S := \frac{j_{DC} \cdot k_s^2 \cdot q \cdot t_s \cdot \sin(k \cdot q \cdot t)}{k \cdot p - k \cdot p \cdot \cos(k \cdot q \cdot t) + k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - k_s^2 \cdot q \cdot t_s \cdot \sin(k \cdot q \cdot t)} & \quad j_{W1} := \frac{j_{DC} \cdot k \cdot p \cdot (\cos(k \cdot q \cdot t) - 1)}{k \cdot p - k \cdot p \cdot \cos(k \cdot q \cdot t) + k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - k_s^2 \cdot q \cdot t_s \cdot \sin(k \cdot q \cdot t)} & \\
\text{Substituting } j_W \text{ and } j_S \text{ into:} & & \\
j(x) = \frac{j_{DC} + j_W + j_S}{\cos(q \cdot t \cdot k)} \cdot \cos(q \cdot k \cdot x) \text{ simplify} \rightarrow j(x) = \frac{j_{DC} \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot x)}{k \cdot p - k \cdot p \cdot \cos(k \cdot q \cdot t) + k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - k_s^2 \cdot q \cdot t_s \cdot \sin(k \cdot q \cdot t)} & &
\end{array}
$$

$$
j(x) = \frac{j_{DC} + j_W + j_S}{\cos(q \cdot t \cdot k)} \cdot \cos(q \cdot k \cdot x) \text{ simplify} \rightarrow j(x) = \frac{j_{DC} \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot x)}{k \cdot p - k \cdot p \cdot \cos(k \cdot q \cdot t) + k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - k_s^2 \cdot q \cdot t_s \cdot \sin(k \cdot q \cdot t)} !
$$

Testing a solution:

$$
j(x) := \frac{j_{DC} \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot x)}{k \cdot p - k \cdot p \cdot \cos(k \cdot q \cdot t) + k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - k_s^2 \cdot q \cdot t_s \cdot \sin(k \cdot q \cdot t)}
$$

$$
-j(x) + j_{DC} + q^2 \cdot k^2 \cdot \int_x^t \int_0^x j(x) \, dx \, dx - k^2 \cdot p \cdot \int_0^t \int_0^x j(x) \, dx \, dx + t_s \cdot k_s^2 \cdot \int_0^t j(x) \, dx \, \text{simplify} \rightarrow 0 !
$$

Finding a solution for the eddy-free equation at: $p := 0 \quad q := \sqrt{2 \cdot p + 1} \quad q = 1$

$$
j_I(x) = \frac{j_{DC} \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot x)}{k \cdot p - k \cdot p \cdot \cos(k \cdot q \cdot t) + k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - k_s^2 \cdot q \cdot t_s \cdot \sin(k \cdot q \cdot t)} \xrightarrow{\text{simplify}} j_I(x) = \frac{j_{DC} \cdot k \cdot \cos(k \cdot q \cdot x)}{k \cdot \cos(k \cdot t) - k_s^2 \cdot t_s \cdot \sin(k \cdot q \cdot t)} !
$$
APPENDIX 15: Comparison of solutions for a flat plate with the existing solution for a semi-infinite space

$$
\begin{array}{l}
f \equiv 1 \cdot 10^3 \qquad \omega \equiv 2\pi f \qquad E \equiv 0.11508 k_a \equiv 0.5 \qquad z \equiv 2.0868406193 \qquad 1 \equiv 10^3 \rho \equiv 2.877 \times 10^{-8} t \equiv 3.338 \times 10^{-3} S \equiv t \cdot z \\
\mu_r \equiv 1 \mu \equiv \mu_o \cdot \mu_r \mu_{sr} \equiv 1\mu_s \equiv \mu_o \cdot \mu_{sr} \quad j_{DC} \equiv \frac{E}{\rho} R_{DC} \equiv \rho \cdot \frac{1}{S} \delta \equiv \sqrt{\frac{2\rho}{\mu \cdot \omega}} k \equiv \sqrt{\frac{-i \cdot \mu \cdot \omega}{\rho}} k_s \equiv \sqrt{\frac{-i \cdot \mu_s \cdot \omega}{\rho}} \frac{\delta}{t} \equiv 0.81 \\
b1 \equiv k_a \cdot t \qquad x \equiv b1, b1 + \frac{t - b1}{50} \quad .. \quad t \qquad p \equiv .5 \qquad \delta = 2.7 \times 10^{-3} \qquad S = 6.966 \times 10^{-3} \quad t_s = 0
\end{array}
$$

$$
\begin{align*}
|j_1 (b1)| &= 2.48 \times 10^6 \quad |j_1 (t)| = 4 \times 10^6 \quad |I_1| = 1.82 \times 10^4 \quad \arg(I_1) = -0.68 \quad \arg(j_1 (b1)) = -1.18 \quad \arg(j_1 (t)) = 0 \\
|j_1 (b1)| &= 1.94 \times 10^6 \quad |j_1 (t)| = 5.42 \times 10^6 \quad |I_1| = 1.61 \times 10^4 \quad \arg(I_1) = -0.69 \quad \arg(j_1 (b1)) = -1.64 \arg(j_1 (t)) = 0.12 \\
|j_{hs} (b1)| &= 1.16 \times 10^6 \quad |j_{hs} (t)| = 4 \times 10^6 \quad |I_{hs}| = 1.51 \times 10^4 \quad \arg(I_{hs}) = -0.49 \quad \arg(j_{hs} (b1)) = -1.24 \arg(j_{hs} (t)) = 0 \\
b1 = 0 \quad j_{DC} = 4 \times 10^6 \quad I_{DC} = 2.79 \times 10^4 \quad R_{DC} = 4.13 \times 10^{-3} \quad R_1 = 4.92 \times 10^{-3} \quad R = 5.51 \times 10^{-3} \quad R_{hs} = 6.74 \times 10^{-3}
\end{align*}
$$

<figure><img src="image_54.png" /></figure>

$$
\begin{array}{l}
f \equiv 1 \cdot 10^7 \qquad \omega \equiv 2\pi f \qquad E \equiv 0.11508 k_a \equiv 0.95 \qquad s \equiv 0 \qquad z \equiv 2.0868406193 \qquad 1 \equiv 10^3 \rho \equiv 2.877 \times 10^{-8} t \equiv 3.338 \times 10^{-3} S \equiv t \cdot z \\
\mu_r \equiv 1 \mu \equiv \mu_o \cdot \mu_r \mu_{sr} \equiv 1\mu_s \equiv \mu_o \cdot \mu_{sr} \quad j_{DC} \equiv \frac{E}{\rho} R_{DC} \equiv \rho \cdot \frac{1}{S} \delta \equiv \sqrt{\frac{2\rho}{\mu \cdot \omega}} k \equiv \sqrt{\frac{-i \cdot \mu \cdot \omega}{\rho}} k_s \equiv \sqrt{\frac{-i \cdot \mu_s \cdot \omega}{\rho}} \\
b1 \equiv k_a \cdot t \qquad x \equiv b1, b1 + \frac{t - b1}{50} \quad .. \quad t \qquad p \equiv .5 \qquad b1 = 3.171 \times 10^{-3} \qquad \delta = 2.7 \times 10^{-5} \qquad S = 6.966 \times 10^{-3} \quad t_s = 0
\end{array}
$$

$$
\begin{align*}
|j_1 (b1)| &= 8.26 \times 10^3 \quad |j_1 (t)| = 4 \times 10^6 \quad |I_1| = 159.34 \quad \arg(I_1) = -0.79 \quad \arg(j_1 (b1)) = 0.1 \quad \arg(j_1 (t)) = 0 \\
|j_1 (b1)| &= 850.71 \quad |j_1 (t)| = 5.33 \times 10^6 \quad |I_1| = 150.23 \quad \arg(I_1) = -0.79 \quad \arg(j_1 (b1)) = -2.46 \arg(j_1 (t)) = 0 \\
|j_{hs} (b1)| &= 8.26 \times 10^3 \quad |j_{hs} (t)| = 4 \times 10^6 \quad |I_{hs}| = 159.34 \quad \arg(I_{hs}) = -0.79 \quad \arg(j_{hs} (b1)) = 0.1 \quad \arg(j_{hs} (t)) = 0 \\
j_{DC} &= 4 \times 10^6 \quad I_{DC} = 2.79 \times 10^4 \quad R_{DC} = 4.13 \times 10^{-3} \quad R_1 = 0.51 \quad R = 0.54 \quad R_{hs} = 0.51
\end{align*}
$$

<figure><img src="image_55.png" /></figure>
APPENDIX 16: Comparison of solution for a flat plate with solution for a thin-walled tubular cylinder

$$ \begin{gathered} \text{s1} \equiv 1, \quad \text{k}_{\text{b1}} \equiv 0.99 \quad \text{k}_{\text{a1}} \equiv \text{k}_{\text{b1}} \quad \text{a} \equiv 3.338 \cdot 10^{-1} \quad \text{n} \equiv \text{s1} \cdot \text{ab} \equiv \text{k}_{\text{b1}} \cdot \text{a} \cdot \text{b11} \equiv \text{k}_{\text{a1}} \cdot \text{a} \quad \text{S1} \equiv \pi \cdot (\text{a}^2 - \text{b}^2) \quad \text{R}_{\text{DC1}} \equiv \rho \cdot \frac{1}{\text{S1}} \\ \delta = 2.7 \times 10^{-3} \quad \text{S1} = 6.966 \times 10^{-3} \quad \text{j}_{\text{DC}} = 4 \times 10^6 \quad \text{I}_{\text{DC1}} = 2.78635 \times 10^4 \quad \text{b} = 0.33 \quad \text{b11} = 0.33 \quad \text{n} = 0.334 \end{gathered} $$

$$ |j_{\text{ws}}(\text{b})| = 2.49 \times 10^6 \quad |j_{\text{ws}}(\text{a})| = 4 \times 10^6 \quad |I_{\text{ws}}| = 1.82 \times 10^4 \arg(I_{\text{ws}}) = -0.68 \quad \arg(j_{\text{ws}}(\text{b})) = -1.1785 \quad \arg(j_{\text{ws}}(\text{a})) = 0 $$

$$ |j_1(\text{b1})| = 2.48 \times 10^6 \quad |j_1(\text{t})| = 4 \times 10^6 \quad |I_1| = 1.82 \times 10^4 \arg(I_1) = -0.68 \quad \arg(j_1(\text{b1})) = -1.1806 \quad \arg(j_1(\text{t})) = 0 $$

$$ |j_{\text{hs}}(\text{b1})| = 1.16 \times 10^6 \quad |j_{\text{hs}}(\text{t})| = 4 \times 10^6 \quad |I_{\text{hs}}| = 1.51 \times 10^4 \arg(I_{\text{hs}}) = -0.49 \quad \arg(j_{\text{hs}}(\text{b1})) = -1.2365 \quad \arg(j_{\text{hs}}(\text{t})) = 0 $$

<figure><img src="image_56.png" /></figure>

<figure><img src="image_57.png" /></figure>

$$ \frac{\delta}{a-b} = 0.809 $$

$$ 2\pi \cdot \frac{a+b}{2} = 2.0868406193 \quad a-b = 3.338 \times 10^{-3} $$

$$ z = 2.0868406193 \quad t = 3.338 \times 10^{-3} $$

$$ \frac{\delta}{t} = 0.809 $$

$$ r \equiv \text{b11}, \text{b11} + \frac{a - \text{b11}}{50} \quad .. \quad a \cdot \frac{(a-b)}{a} = 0.01 \quad f = 1 \times 10^3 $$
APPENDIX 17: Search and verification of solutions for a long solenoid

Finding a solution to an equation :

$$
\begin{align*}
j(x) &= j_{DC} + q^2 \cdot k^2 \cdot \int_{x}^{t} \int_{0}^{x} j(x) \, dx \, dx + \frac{S_s}{P_s} \cdot k_s^2 \cdot \int_{0}^{t} \int_{0}^{x} j(x) \, dx \, dx \\
&\quad - p \cdot k^2 \cdot \int_{0}^{t} \int_{0}^{x} j(x) \, dx \, dx = j_W \\
&\quad \text{We represent the solution in the form: } j(x) = \frac{j_{DC} + j_S + j_W}{\cos(q \cdot k \cdot x)} \cdot \cos(q \cdot k \cdot x) \\
&\quad \text{Finding the constants } j_S, j_W \\
\frac{S_s}{P_s} \cdot k_s^2 \cdot \int_{0}^{t} j(x) \, dx &= j_S \\
&\quad \left|
  \begin{array}{@{}l@{}}
    \text{substitute } j(x) = \frac{j_{DC} + j_S + j_W}{\cos(q \cdot k \cdot x)} \cdot \cos(q \cdot k \cdot x) \\
    \text{solve } j_S \\
    \text{simplify}
  \end{array}
\right.
\quad \rightarrow \quad
j_{S1} := - \frac{S_s \cdot k_s^2 \cdot \sin(k \cdot q \cdot t) \cdot (j_W + j_{DC})}{S_s \cdot k_s^2 \cdot \sin(k \cdot q \cdot t) - P_s \cdot k \cdot q \cdot \cos(k \cdot q \cdot t)} \\
- p \cdot k^2 \cdot \int_{0}^{t} \int_{0}^{x} j(x) \, dx \, dx = j_W \\
&\quad \left|
  \begin{array}{@{}l@{}}
    \text{substitute } j(x) = \frac{j_{DC} + j_S + j_W}{\cos(q \cdot k \cdot t)} \cdot \cos(q \cdot k \cdot x) \\
    \text{solve } j_W \\
    \text{simplify}
  \end{array}
\right.
\quad \rightarrow \quad
j_{W1} := \frac{2 \cdot p \cdot \left( \frac{\cos(q \cdot k \cdot t)}{2} - \frac{1}{2} \right) \cdot (j_S + j_{DC})}{p - p \cdot \cos(k \cdot q \cdot t) + q^2 \cdot \cos(k \cdot q \cdot t)} \\
\left(
  \begin{array}{@{}l@{}}
    j_S = j_{S1} \\
    j_W = j_{W1}
  \end{array}
\right) \quad \rightarrow \quad
j_S := \frac{S_s \cdot j_{DC} \cdot k_s^2 \cdot q \cdot \sin(k \cdot q \cdot t)}{P_s \cdot k \cdot p + P_s \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - S_s \cdot k_s^2 \cdot q \cdot \sin(k \cdot q \cdot t) - P_s \cdot k \cdot p \cdot \cos(k \cdot q \cdot t)} \\
\rightarrow \quad
j_W := - \frac{P_s \cdot j_{DC} \cdot k \cdot p - P_s \cdot j_{DC} \cdot k \cdot p \cdot \cos(k \cdot q \cdot t)}{P_s \cdot k \cdot p + P_s \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - S_s \cdot k_s^2 \cdot q \cdot \sin(k \cdot q \cdot t) - P_s \cdot k \cdot p \cdot \cos(k \cdot q \cdot t)}
\end{align*}
$$

$$
j(x) = \frac{j_{DC} + j_S + j_W}{\cos(q \cdot k \cdot t)} \cdot \cos(q \cdot k \cdot x) \text{simplify} \rightarrow j(x) = \frac{P_s \cdot j_{DC} \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot x)}{P_s \cdot k \cdot p + P_s \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - S_s \cdot k_s^2 \cdot q \cdot \sin(k \cdot q \cdot t) - P_s \cdot k \cdot p \cdot \cos(k \cdot q \cdot t)}
$$

Testing a solution:

$$
j(x) := \frac{j_{DC} \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot x)}{k \cdot p + k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - \frac{S_s}{P_s} \cdot k_s^2 \cdot q \cdot \sin(k \cdot q \cdot t) - k \cdot p \cdot \cos(k \cdot q \cdot t)}
$$

$$
-j(x) + j_{DC} + q^2 \cdot k^2 \cdot \int_{x}^{t} \int_{0}^{x} j(x) \, dx \, dx + \frac{S_s}{P_s} \cdot k_s^2 \cdot \int_{0}^{t} j(x) \, dx - p \cdot k^2 \cdot \int_{0}^{t} \int_{0}^{x} j(x) \, dx \, dx \text{ simplify } \rightarrow 0
$$

$$
E_t(x) = q^2 \cdot k^2 \cdot \rho \cdot \int_x^t \int_0^x j(x) \, dx \, dx \text{simplify} \rightarrow 2 \cdot P_s \cdot \rho \cdot j_{DC} \cdot k \cdot q^2 \cdot \sin\left(\frac{k \cdot q \cdot t}{2}\right)^2 - 2 \cdot P_s \cdot \rho \cdot j_{DC} \cdot k \cdot q^2 \cdot \sin\left(\frac{k \cdot q \cdot x}{2}\right)^2 \\
\rightarrow E_t(x) := \frac{P_s \cdot k \cdot q^2 + 2 \cdot P_s \cdot k \cdot p \cdot \sin\left(\frac{k \cdot q \cdot t}{2}\right)^2 - S_s \cdot k_s^2 \cdot q \cdot \sin(k \cdot q \cdot t) - 2 \cdot P_s \cdot k \cdot q^2 \cdot \sin\left(\frac{k \cdot q \cdot t}{2}\right)^2}{P_s \cdot k \cdot q^2 + 2 \cdot P_s \cdot k \cdot p \cdot \sin\left(\frac{k \cdot q \cdot t}{2}\right)^2 - S_s \cdot k_s^2 \cdot q \cdot \sin(k \cdot q \cdot t) - 2 \cdot P_s \cdot k \cdot q^2 \cdot \sin\left(\frac{k \cdot q \cdot x}{2}\right)^2}
$$

$$
E_{tm} = \frac{1}{t} \int_{0}^{t} E_{t}(x) \, dx \text{ simplify} \rightarrow E_{tm} = \frac{P_{s} \cdot \rho \cdot j_{DC} \cdot q \cdot \sin(k \cdot q \cdot t) - P_{s} \cdot \rho \cdot j_{DC} \cdot k \cdot q^2 \cdot t \cdot \cos(k \cdot q \cdot t)}{t \cdot (P_{s} \cdot k \cdot p + P_{s} \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - S_{s} \cdot k_{s}^2 \cdot q \cdot \sin(k \cdot q \cdot t) - P_{s} \cdot k \cdot p \cdot \cos(k \cdot q \cdot t))}
$$

$$
E_s = \frac{S_s}{P_s} \cdot k_s^2 \cdot \rho \cdot \int_0^t j(x) \, dx \text{ simplify} \rightarrow E_s = \frac{S_s \cdot \rho \cdot j_{DC} \cdot k_s^2 \cdot q \cdot \sin(k \cdot q \cdot t)}{P_s \cdot k \cdot p + P_s \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - S_s \cdot k_s^2 \cdot q \cdot \sin(k \cdot q \cdot t) - P_s \cdot k \cdot p \cdot \cos(k \cdot q \cdot t)}
$$

$$
E_w = -p \cdot k^2 \cdot \rho \cdot \int_0^t \int_0^x j(x) \, dx \, dx \text{simplify} \rightarrow E_w = \frac{P_s \cdot \rho \cdot j_{DC} \cdot k \cdot p \cdot (\cos(k \cdot q \cdot t) - 1)}{P_s \cdot k \cdot p + P_s \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - S_s \cdot k_s^2 \cdot q \cdot \sin(k \cdot q \cdot t) - P_s \cdot k \cdot p \cdot \cos(k \cdot q \cdot t)}
$$

$$
I = \frac{z}{N} \int_{0}^{t} j(x) \, dx \text{ simplify} \rightarrow I = \frac{P_s \cdot j_{DC} \cdot q \cdot z \cdot \sin(k \cdot q \cdot t)}{N \cdot (P_s \cdot k \cdot p + P_s \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - S_s \cdot k_s^2 \cdot q \cdot \sin(k \cdot q \cdot t) - P_s \cdot k \cdot p \cdot \cos(k \cdot q \cdot t))}
$$

For the eddy-free version $p := 0$ $q := \sqrt{1 + 2p}$ $q = 1$

$$
j_1(x) = j_{DC} + k^2 \cdot \int_{x}^{t} \int_{0}^{x} j_1(x) \, dx \, dx + \frac{S_s}{P_s} \cdot k_s^2 \cdot \int_{0}^{t} j_1(x) \, dx
$$

$$
j_1(x) = \frac{j_{DC} \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot x)}{k \cdot p + k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - \frac{S_s}{P_s} \cdot k_s^2 \cdot q \cdot \sin(k \cdot q \cdot t) - k \cdot p \cdot \cos(k \cdot q \cdot t)} \quad \text{simplify} \rightarrow
$$

$$
\rightarrow j_1(x) := \frac{j_{DC} \cdot k \cdot \cos(k \cdot x)}{k \cdot \cos(k \cdot t) - \frac{S_s}{P_s} \cdot k_s^2 \cdot \sin(k \cdot t)} - j_1(x) + j_{DC} + k^2 \cdot \int_x^t \int_0^x j_1(x) \, dx \, dx + \frac{S_s}{P_s} \cdot k_s^2 \cdot \int_0^t j_1(x) \, dx \text{ simplify} \rightarrow 0
$$
APPENDIX 18: Search and verification of solutions for a long solenoid with a series-connected capacitor

$$
I = \frac{z}{N} \int_{0}^{t} j(x) \, dx \quad -U_c = -I \cdot \frac{-i}{\omega \cdot C} \frac{-U_c}{\rho \cdot P_s \cdot N} = \frac{-i}{\rho \cdot P_s \cdot N \cdot \omega \cdot C} \cdot \frac{-z}{N} \int_{0}^{t} j(x) \, dx = -z \cdot k_c^2 \cdot \int_{0}^{t} j(x) \, dx \quad \text{where constant} \\
k_c = \sqrt{\frac{-i}{\rho \cdot P_s \cdot N^2 \cdot \omega \cdot C}}
$$

Finding a solution to an equation :

$$
j(x) = j_{DC} + q^2 \cdot k^2 \cdot \int_{x}^{t} \int_{0}^{x} j(x) \, dx \, dx + \frac{S_s}{P_s} \cdot k_s^2 \cdot \int_{0}^{t} j(x) \, dx - p \cdot k^2 \cdot \int_{0}^{t} \int_{0}^{x} j(x) \, dx \, dx - z \cdot k_c^2 \cdot \int_{0}^{t} j(x) \, dx
$$

where constants

$$
\frac{S_s}{P_s} \cdot k_s^2 \cdot \int_0^t j(x) \, dx = j_S - p \cdot k^2 \cdot \int_0^t \int_0^x j(x) \, dx \, dx = j_W - z \cdot k_c^2 \cdot \int_0^t j(x) \, dx = j_C
$$

We represent the solution in the form:

$$
j(x) = \frac{j_{DC} + j_S + jW + jC}{\cos(q \cdot t \cdot k)} \cdot \cos(q \cdot k \cdot x)
$$

Finding the values of constants $j_S$, $j_W$ and $j_C$:

$$
\frac{S_s}{P_s} \cdot k_s^2 \cdot \int_0^t j(x) \, dx = j_S
\begin{array}{l}
\text{substitute, } j(x) = \frac{j_{DC} + j_S + jW + jC}{\cos(q \cdot t \cdot k)} \cdot \cos(q \cdot k \cdot x) \\
\text{solve } j_S \\
\text{simplify} \quad \rightarrow \quad j_{S1} := - \frac{S_s \cdot k_s^2 \cdot \sin(k \cdot q \cdot t) \cdot (j_C + jW + j_{DC})}{S_s \cdot k_s^2 \cdot \sin(k \cdot q \cdot t) - P_s \cdot k \cdot q \cdot \cos(k \cdot q \cdot t)}
\end{array}
$$
$$
-p \cdot k^2 \cdot \int_0^t \int_0^x j(x) \, dx \, dx = j_W
\begin{array}{l}
\text{substitute, } j(x) = \frac{j_{DC} + j_S + jW + jC}{\cos(q \cdot t \cdot k)} \cdot \cos(q \cdot k \cdot x) \\
\text{solve } j_W \\
\text{simplify} \quad \rightarrow \quad j_{W1} := \frac{2 \cdot p \cdot \left( \frac{\cos(k \cdot q \cdot t)}{2} - \frac{1}{2} \right) \cdot (j_C + j_S + j_{DC})}{p - p \cdot \cos(k \cdot q \cdot t) + q^2 \cdot \cos(k \cdot q \cdot t)}
\end{array}
$$
$$
-z \cdot k_c^2 \cdot \int_0^t j(x) \, dx = j_C
\begin{array}{l}
\text{substitute, } j(x) = \frac{j_{DC} + j_S + jW + jC}{\cos(q \cdot t \cdot k)} \cdot \cos(q \cdot k \cdot x) \\
\text{solve } j_C \\
\text{simplify} \quad \rightarrow \quad j_{C1} := - \frac{k_c^2 \cdot z \cdot \sin(k \cdot q \cdot t) \cdot (j_S + jW + j_{DC})}{z \cdot \sin(k \cdot q \cdot t) \cdot k_c^2 + k \cdot q \cdot \cos(k \cdot q \cdot t)}
\end{array}
$$
$$
\begin{pmatrix} j_S = j_{S1} \\ j_W = j_{W1} \\ j_C = j_{C1} \end{pmatrix} \quad \begin{array}{l} \text{solve } j_S, j_W, j_C \rightarrow \\ \text{simplify} \quad \rightarrow \quad j_S := \frac{S_s \cdot j_{DC} \cdot k_s^2 \cdot q \cdot \sin(k \cdot q \cdot t)}{P_s \cdot k \cdot p + P_s \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - S_s \cdot k_s^2 \cdot q \cdot \sin(k \cdot q \cdot t) - P_s \cdot k \cdot p \cdot \cos(k \cdot q \cdot t) + P_s \cdot k_c^2 \cdot q \cdot z \cdot \sin(k \cdot q \cdot t)} \\ \rightarrow \quad j_W := \frac{P_s \cdot j_{DC} \cdot k \cdot p \cdot (\cos(k \cdot q \cdot t) - 1)}{P_s \cdot k \cdot p + P_s \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - S_s \cdot k_s^2 \cdot q \cdot \sin(k \cdot q \cdot t) - P_s \cdot k \cdot p \cdot \cos(k \cdot q \cdot t) + P_s \cdot k_c^2 \cdot q \cdot z \cdot \sin(k \cdot q \cdot t)} \\ \rightarrow \quad j_C := - \frac{P_s \cdot j_{DC} \cdot k_c^2 \cdot q \cdot z \cdot \sin(k \cdot q \cdot t)}{P_s \cdot k \cdot p + P_s \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - S_s \cdot k_s^2 \cdot q \cdot \sin(k \cdot q \cdot t) - P_s \cdot k \cdot p \cdot \cos(k \cdot q \cdot t) + P_s \cdot k_c^2 \cdot q \cdot z \cdot \sin(k \cdot q \cdot t)} \end{array}
$$

Substituting $j_S$, $j_W$ and $j_C$ into:

$$
j(x) = \frac{j_{DC} + j_S + jW + jC}{\cos(q \cdot t \cdot k)} \cdot \cos(q \cdot k \cdot x) \text{simplify} \rightarrow \frac{j_{DC} \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot x)}{k \cdot p + k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - \frac{S_S}{P_S} \cdot k_S^2 \cdot q \cdot \sin(k \cdot q \cdot t) - k \cdot p \cdot \cos(k \cdot q \cdot t) + k_C^2 \cdot q \cdot z \cdot \sin(k \cdot q \cdot t)}!
$$
APPENDIX 18 (continued)

Testing a solution:

$$j(x) := \frac{j_{DC} \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot x)}{k \cdot p + k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - \frac{S_s}{P_s} \cdot k_s^2 \cdot q \cdot \sin(k \cdot q \cdot t) - k \cdot p \cdot \cos(k \cdot q \cdot t) + k_c^2 \cdot q \cdot z \cdot \sin(k \cdot q \cdot t)}$$

$-j(x)+j_{DC}+q^2 \cdot k^2 \cdot \int_x^t \int_0^x j(x) dx \, dx + \frac{S_s}{P_s} \cdot k_s^2 \cdot \int_0^t j(x) dx - p \cdot k^2 \cdot \int_x^t \int_0^x j(x) dx \, dx - z \cdot k_c^2 \cdot \int_0^t j(x) dx \text{ simplify } \to 0!$

$$E_t(x) = q^2 \cdot k^2 \cdot \rho \cdot \int_x^t \int_0^x j(x) \, dx \, dx \text{ simplify} \to \\ \qquad \to E_t(x) := \frac{2 \cdot P_s \cdot \rho \cdot j_{DC} \cdot k \cdot q^2 \cdot \sin\left(\frac{k \cdot q \cdot t}{2}\right)^2 - 2 \cdot P_s \cdot \rho \cdot j_{DC} \cdot k \cdot q^2 \cdot \sin\left(\frac{k \cdot q \cdot x}{2}\right)^2}{P_s \cdot k \cdot q^2 + 2 \cdot P_s \cdot k \cdot p \cdot \sin\left(\frac{k \cdot q \cdot t}{2}\right)^2 - S_s \cdot k_s^2 \cdot q \cdot \sin(k \cdot q \cdot t) - 2 \cdot P_s \cdot k \cdot q^2 \cdot \sin\left(\frac{k \cdot q \cdot t}{2}\right)^2 + P_s \cdot k_c^2 \cdot q \cdot z \cdot \sin(k \cdot q \cdot t)}$$

$$E_{tm} = \frac{1}{t} \int_{0}^{t} E_{t}(x) \, dx \text{ simplify} \to \\ \qquad \to E_{tm} = \frac{P_{s} \cdot \rho \cdot j_{DC} \cdot q \cdot \sin(k \cdot q \cdot t) - P_{s} \cdot \rho \cdot j_{DC} \cdot k \cdot q^2 \cdot t \cdot \cos(k \cdot q \cdot t)}{t \cdot \left( P_{s} \cdot k \cdot p + P_{s} \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - S_{s} \cdot k_{s}^2 \cdot q \cdot \sin(k \cdot q \cdot t) - P_{s} \cdot k \cdot p \cdot \cos(k \cdot q \cdot t) + P_{s} \cdot k_{c}^2 \cdot q \cdot z \cdot \sin(k \cdot q \cdot t) \right)}$$

$$E_s = \frac{S_s}{P_s} \cdot k_s^2 \cdot \rho \cdot \int_0^t j(x) \, dx \text{ simplify} \to \\ \qquad \to E_s = \frac{S_s \cdot \rho \cdot j_{DC} \cdot k_s^2 \cdot q \cdot \sin(k \cdot q \cdot t)}{P_s \cdot k \cdot p + P_s \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - S_s \cdot k_s^2 \cdot q \cdot \sin(k \cdot q \cdot t) - P_s \cdot k \cdot p \cdot \cos(k \cdot q \cdot t) + P_s \cdot k_c^2 \cdot q \cdot z \cdot \sin(k \cdot q \cdot t)}$$

$$E_w = -p \cdot k^2 \cdot \rho \cdot \int_0^t \int_0^x j(x) \, dx \, dx \text{ simplify} \to \\ \qquad \to E_w = \frac{P_s \cdot \rho \cdot j_{DC} \cdot k \cdot p \cdot (\cos(k \cdot q \cdot t) - 1)}{P_s \cdot k \cdot p + P_s \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - S_s \cdot k_s^2 \cdot q \cdot \sin(k \cdot q \cdot t) - P_s \cdot k \cdot p \cdot \cos(k \cdot q \cdot t) + P_s \cdot k_c^2 \cdot q \cdot z \cdot \sin(k \cdot q \cdot t)}$$

$$E_c = -z \cdot k_c^2 \cdot \rho \cdot \int_0^t j(x) \, dx \text{ simplify} \to \\ \qquad \to E_c = \frac{P_s \cdot \rho \cdot j_{DC} \cdot k_c^2 \cdot q \cdot z \cdot \sin(k \cdot q \cdot t)}{P_s \cdot k \cdot p + P_s \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - S_s \cdot k_s^2 \cdot q \cdot \sin(k \cdot q \cdot t) - P_s \cdot k \cdot p \cdot \cos(k \cdot q \cdot t) + P_s \cdot k_c^2 \cdot q \cdot z \cdot \sin(k \cdot q \cdot t)}$$

$$I = \frac{z}{N} \int_{0}^{t} j(x) \, dx \text{ simplify} \to \\ \qquad \to I = \frac{P_s \cdot j_{DC} \cdot q \cdot z \cdot \sin(k \cdot q \cdot t)}{N \cdot \left( P_s \cdot k \cdot p + P_s \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - S_s \cdot k_s^2 \cdot q \cdot \sin(k \cdot q \cdot t) - P_s \cdot k \cdot p \cdot \cos(k \cdot q \cdot t) + P_s \cdot k_c^2 \cdot q \cdot z \cdot \sin(k \cdot q \cdot t) \right)}$$

For the eddy-free version $p := 0$ $q := \sqrt{1+2p}$ $q = 1$

$$j_1(x) = j_{DC} + k^2 \cdot \int_x^t \int_0^x j_1(x) \, dx \, dx + \frac{S_s}{P_s} \cdot k_s^2 \cdot \int_0^t \int_0^x j_1(x) \, dx - z \cdot k_c^2 \cdot \int_0^t \int_0^x j_1(x) \, dx$$

$$j_1(x) = \frac{j_{DC} \cdot k \cdot q^2 \cdot \cos(k \cdot q \cdot x)}{k \cdot p + k \cdot q^2 \cdot \cos(k \cdot q \cdot t) - \frac{S_s}{P_s} \cdot k_s^2 \cdot q \cdot \sin(k \cdot q \cdot t) - k \cdot p \cdot \cos(k \cdot q \cdot t) + k_c^2 \cdot q \cdot z \cdot \sin(k \cdot q \cdot t)} \text{ simplify } \to$$

$$\to j_1(x) := \frac{j_{DC} \cdot k \cdot \cos(k \cdot x)}{z \cdot \sin(k \cdot t) \cdot k_c^2 - \frac{S_s}{P_s} \cdot \sin(k \cdot t) \cdot k_s^2 + k \cdot \cos(k \cdot t)}$$

$-j_1(x) + j_{DC} + k^2 \cdot \int_x^t \int_0^x j_1(x) \, dx \, dx + \frac{S_s}{P_s} \cdot k_s^2 \cdot \int_0^t \int_0^x j_1(x) \, dx - z \cdot k_c^2 \cdot \int_0^t \int_0^x j_1(x) \, dx \text{ simplify } \to 0!$
APPENDIX 19: Transition from the balance equation of specific voltages to the equation for general quantities

Going from an equation for specific quantities to an equation for general quantities

$$j(r) = j_{DC} + q^2 \cdot k^2 \cdot \int_{r}^{a} \frac{1}{r} \cdot \int_{b}^{r} j(r) \cdot r \, dr \cdot dr + k_s^2 \cdot \ln(s) \cdot \int_{b}^{a} j(r) \cdot r \, dr - p \cdot k^2 \cdot \int_{b}^{a} \frac{1}{r} \cdot \int_{b}^{r} j(r) \cdot r \, dr \cdot dr, \text{ where: } q^2 = 1 + 2p$$
solution for b=0

$$j(r) = \frac{j_{DC} \cdot k \cdot q^2 \cdot J0(k \cdot q \cdot r)}{(q^2 - p^2) \cdot k \cdot J0(a \cdot k \cdot q) - a \cdot k_s^2 \cdot q \cdot \ln(s) \cdot J1(a \cdot k \cdot q) + k \cdot p} \quad I = 2 \cdot \pi \cdot \int_{0}^{a} j(r) \cdot r \, dr \cdot \int_{0}^{a} J0(r \cdot k \cdot q) \cdot r \, dr = \frac{a}{k \cdot q} \cdot J1(a \cdot k \cdot q)$$

Then the balance of specific voltages and EMF

$$j(r) \cdot \rho = E + E_{ab}(r) + E_s + E_w \quad \int_{0}^{r} J0(r \cdot k) \cdot r \, dr = \frac{r}{k \cdot q} \cdot J1(r \cdot k \cdot q)$$

Let's write down what the product will be equal to

$$I \cdot R_{DC} = \rho \cdot \frac{l}{S} \cdot 2 \cdot \pi \cdot \int_{0}^{a} j(r) \cdot r \, dr \quad \substack{\text{substitute } j(r) = \frac{E + E_{ab}(r) + E_s + E_w}{\rho} \\ \text{substitute } S = \pi \cdot a^2 \\ \text{expand} \\ \rightarrow I \cdot R_{DC} = E \cdot l + E_s \cdot l + E_w \cdot l + \frac{2 \cdot l \cdot \int_{0}^{a} r \cdot E_{ab}(r) \, dr}{a^2}}$$

$$\text{where: } R_{DC} = \rho \cdot \frac{l}{S} \quad S = \pi \cdot a^2 \quad E := j_{DC} \cdot \rho \quad \text{and the values found in Appendix 1 are:}$$

$$\begin{align*}
& j(r) := \frac{j_{DC} \cdot k \cdot q^2 \cdot J0(k \cdot q \cdot r)}{k \cdot p + k \cdot q^2 \cdot J0(a \cdot k \cdot q) - k \cdot p \cdot J0(a \cdot k \cdot q) - a \cdot k_s^2 \cdot q \cdot \ln(s) \cdot J1(a \cdot k \cdot q)} & I := \frac{2 \cdot \pi \cdot a \cdot j_{DC} \cdot q \cdot J1(a \cdot k \cdot q)}{\left[ k \cdot p + k \cdot q^2 \cdot J0(a \cdot k \cdot q) - k \cdot p \cdot J0(a \cdot k \cdot q) - a \cdot k_s^2 \cdot q \cdot \ln(s) \cdot J1(a \cdot k \cdot q) \right]^*} \\
& E_{ab}(r) := \frac{\rho \cdot j_{DC} \cdot k \cdot q^2 \cdot (J0(a \cdot k \cdot q) - J0(k \cdot q \cdot r))}{k \cdot p + k \cdot q^2 \cdot J0(a \cdot k \cdot q) - k \cdot p \cdot J0(a \cdot k \cdot q) - a \cdot k_s^2 \cdot q \cdot \ln(s) \cdot J1(a \cdot k \cdot q)} & E_s := \frac{a \cdot \rho \cdot j_{DC} \cdot k_s^2 \cdot q \cdot \ln(s) \cdot J1(a \cdot k \cdot q)}{\left[ k \cdot p + k \cdot q^2 \cdot J0(a \cdot k \cdot q) - k \cdot p \cdot J0(a \cdot k \cdot q) - a \cdot k_s^2 \cdot q \cdot \ln(s) \cdot J1(a \cdot k \cdot q) \right]^*} \\
& E_w := \frac{\rho \cdot j_{DC} \cdot k \cdot p \cdot (J0(a \cdot k \cdot q) - 1)}{k \cdot p + k \cdot q^2 \cdot J0(a \cdot k \cdot q) - k \cdot p \cdot J0(a \cdot k \cdot q) - a \cdot k_s^2 \cdot q \cdot \ln(s) \cdot J1(a \cdot k \cdot q)} & -j(r) \cdot \rho + E + E_{ab}(r) + E_s + E_w \quad \text{simplify} \rightarrow 0
\end{align*}$$

$$\text{We introduce the notation } E_{am} = \frac{2 \cdot \int_{0}^{a} r \cdot E_{ab}(r) \, dr}{a^2} \quad \substack{\text{simplify} \\ \text{substitute, } \int_{0}^{a} J0(r \cdot k \cdot q) \cdot r \, dr = \frac{a}{k \cdot q} \cdot J1(a \cdot k \cdot q)} \rightarrow$$

$$\rightarrow E_{am} := \left[ \frac{2 \cdot \rho \cdot j_{DC} \cdot q \cdot J1(a \cdot k \cdot q) - a \cdot \rho \cdot j_{DC} \cdot k \cdot q^2 \cdot J0(a \cdot k \cdot q)}{a \cdot k \cdot p - a \cdot k \cdot p \cdot J0(a \cdot k \cdot q) + a \cdot k \cdot q^2 \cdot J0(a \cdot k \cdot q) - a^2 \cdot k_s^2 \cdot q \cdot \ln(s) \cdot J1(a \cdot k \cdot q)} \right]_* \quad \substack{R_{DC} := \rho \cdot \frac{l}{\pi \cdot a^2} \\ -I \cdot R_{DC} + E \cdot l + E_{am} \cdot l + E_s \cdot l + E_w \cdot l \quad \text{simplify} \rightarrow 0}$$

Separating the specific EMF of self-induction and mutual induction.

$$E_m := \frac{E_{am}}{q^2} \quad E_{em} := E_{am} + E_w - \frac{E_{am}}{q^2}$$

$$\text{we have a balance: } -I \cdot R_{DC} + E \cdot l + E_m \cdot l + E_s \cdot l + E_{em} \cdot l \quad \text{simplify} \rightarrow 0$$

$$\begin{align*}
E_m &\rightarrow \frac{2 \cdot \rho \cdot j_{DC} \cdot q \cdot J1(a \cdot k \cdot q) - a \cdot \rho \cdot j_{DC} \cdot k \cdot q^2 \cdot J0(a \cdot k \cdot q)}{q^2 \cdot \left( a \cdot k \cdot p - a \cdot k \cdot p \cdot J0(a \cdot k \cdot q) + a \cdot k \cdot q^2 \cdot J0(a \cdot k \cdot q) - a^2 \cdot k_s^2 \cdot q \cdot \ln(s) \cdot J1(a \cdot k \cdot q) \right)} \\
E_{em} \text{ factor} &\rightarrow \frac{\rho \cdot j_{DC} \cdot \left( 2 \cdot J1(a \cdot k \cdot q) - 2 \cdot q^2 \cdot J1(a \cdot k \cdot q) - a \cdot k \cdot q \cdot J0(a \cdot k \cdot q) + a \cdot k \cdot p \cdot q + a \cdot k \cdot q^3 \cdot J0(a \cdot k \cdot q) - a \cdot k \cdot p \cdot q \cdot J0(a \cdot k \cdot q) \right)}{a \cdot q \cdot \left( k \cdot p \cdot J0(a \cdot k \cdot q) - k \cdot q^2 \cdot J0(a \cdot k \cdot q) - k \cdot p + a \cdot k_s^2 \cdot q \cdot \ln(s) \cdot J1(a \cdot k \cdot q) \right)}
\end{align*}$$

$$\text{For the eddy-free version } p := 0 \quad q := \sqrt{1 + 2p} \quad q = 1$$

$$\begin{align*}
E_a(r) &= - \frac{\rho \cdot j_{DC} \cdot k \cdot q^2 \cdot (J0(a \cdot k \cdot q) - J0(k \cdot q \cdot r))}{k \cdot p + k \cdot q^2 \cdot J0(a \cdot k \cdot q) - k \cdot p \cdot J0(a \cdot k \cdot q) - a \cdot k_s^2 \cdot q \cdot \ln(s) \cdot J1(a \cdot k \cdot q)} & \text{simplify} & \rightarrow & \frac{\rho \cdot j_{DC} \cdot k \cdot (J0(a \cdot k) - J0(k \cdot r))}{k \cdot J0(a \cdot k) - a \cdot k_s^2 \cdot \ln(s) \cdot J1(a \cdot k)} \\
& & & & \rightarrow E_a(r) = - \frac{a \cdot \rho \cdot j_{DC} \cdot k_s^2 \cdot q \cdot \ln(s) \cdot J1(a \cdot k \cdot q)}{k \cdot p + k \cdot q^2 \cdot J0(a \cdot k \cdot q) - k \cdot p \cdot J0(a \cdot k \cdot q) - a \cdot k_s^2 \cdot q \cdot \ln(s) \cdot J1(a \cdot k \cdot q)} & \text{simplify} & \rightarrow & \frac{a \cdot \rho \cdot j_{DC} \cdot k_s^2 \cdot q \cdot \ln(s) \cdot J1(a \cdot k \cdot q)}{k \cdot p + k \cdot q^2 \cdot J0(a \cdot k \cdot q) - k \cdot p \cdot J0(a \cdot k \cdot q) - a \cdot k_s^2 \cdot q \cdot \ln(s) \cdot J1(a \cdot k \cdot q)}
\end{align*}$$

$$\begin{align*}
E_{s-} &= \frac{2 \cdot \rho \cdot j_{DC} \cdot q \cdot J1(a \cdot k \cdot q) - a \cdot \rho \cdot j_{DC} \cdot k \cdot q^2 \cdot J0(a \cdot k \cdot q)}{a \cdot k \cdot p - a \cdot k \cdot p \cdot J0(a \cdot k \cdot q) + a \cdot k \cdot q^2 \cdot J0(a \cdot k \cdot q) - a^2 \cdot k_s^2 \cdot q \cdot \ln(s) \cdot J1(a \cdot k \cdot q)} & \text{simplify} & \rightarrow & \frac{2 \cdot \rho \cdot j_{DC} \cdot J1(a \cdot k) - a \cdot \rho \cdot j_{DC} \cdot k \cdot J0(a \cdot k)}{a \cdot k \cdot J0(a \cdot k) - a^2 \cdot k_s^2 \cdot \ln(s) \cdot J1(a \cdot k)} \\
E_{am-} &= \frac{2 \cdot \rho \cdot j_{DC} \cdot q \cdot J1(a \cdot k \cdot q) - a \cdot \rho \cdot j_{DC} \cdot k \cdot q^2 \cdot J0(a \cdot k \cdot q)}{a \cdot k \cdot p - a \cdot k \cdot p \cdot J0(a \cdot k \cdot q) + a \cdot k \cdot q^2 \cdot J0(a \cdot k \cdot q) - a^2 \cdot k_s^2 \cdot q \cdot \ln(s) \cdot J1(a \cdot k \cdot q)} & \text{simplify} & \rightarrow & \frac{2 \cdot \rho \cdot j_{DC} \cdot J1(a \cdot k) - a \cdot \rho \cdot j_{DC} \cdot k \cdot J0(a \cdot k)}{a \cdot k \cdot J0(a \cdot k) - a^2 \cdot k_s^2 \cdot \ln(s) \cdot J1(a \cdot k)}
\end{align*}$$

$$E_a(r) := \frac{\rho \cdot j_{DC} \cdot k \cdot (J0(a \cdot k) - J0(k \cdot r))}{k \cdot J0(a \cdot k) - a \cdot k_s^2 \cdot \ln(s) \cdot J1(a \cdot k)} \quad \rightarrow \quad E_{am-} = \frac{2 \cdot \rho \cdot j_{DC} \cdot J1(a \cdot k) - a \cdot \rho \cdot j_{DC} \cdot k \cdot J0(a \cdot k)}{a \cdot k \cdot J0(a \cdot k) - a^2 \cdot k_s^2 \cdot \ln(s) \cdot J1(a \cdot k)}$$

$$E_w_- := 0 \quad E_{am-} := \frac{2 \cdot \rho \cdot j_{DC} \cdot J1(a \cdot k) - a \cdot \rho \cdot j_{DC} \cdot k \cdot J0(a \cdot k)}{a \cdot k \cdot J0(a \cdot k) - a^2 \cdot k_s^2 \cdot \ln(s) \cdot J1(a \cdot k)} \quad E_{s-} := \frac{a \cdot \rho \cdot j_{DC} \cdot k_s^2 \cdot \ln(s) \cdot J1(a \cdot k)}{k \cdot J0(a \cdot k) - a \cdot k_s^2 \cdot \ln(s) \cdot J1(a \cdot k)}$$

$$\begin{align*}
I_1 &= \frac{2 \cdot \pi \cdot a \cdot j_{DC} \cdot q \cdot J1(a \cdot k \cdot q)}{k \cdot p + k \cdot q^2 \cdot J0(a \cdot k \cdot q) - k \cdot p \cdot J0(a \cdot k \cdot q) - a \cdot k_s^2 \cdot q \cdot \ln(s) \cdot J1(a \cdot k \cdot q)} & \text{simplify} & \rightarrow & \frac{2 \cdot \pi \cdot a \cdot j_{DC} \cdot J1(a \cdot k)}{k \cdot J0(a \cdot k) - a \cdot k_s^2 \cdot \ln(s) \cdot J1(a \cdot k)} \\
& & & & \rightarrow I_1 := \frac{2 \cdot \pi \cdot a \cdot j_{DC} \cdot J1(a \cdot k)}{k \cdot J0(a \cdot k) - a \cdot k_s^2 \cdot \ln(s) \cdot J1(a \cdot k)} & -I_1 \cdot R_{DC} + E \cdot l + E_{am-} \cdot l + E_s \cdot l + E_w \cdot l \quad \text{simplify} \rightarrow 0
\end{align*}$$

<footnote>*Long expressions have been edited by moving the terms to the next line and placing them in square brackets.</footnote>
APPENDIX 20: Comparison of vector diagrams of electromotive forces and voltages of the pre-
sented and existing solutions for a solid cylinder for the eddy-free case

$$
\mathrm{El}_x := \begin{pmatrix} 0 \\ \mathrm{Re}(\mathrm{E} \cdot \mathrm{l}) \end{pmatrix}, \quad \mathrm{El}_y := \begin{pmatrix} 0 \\ \mathrm{Im}(\mathrm{E} \cdot \mathrm{l}) \end{pmatrix}, \quad \mathrm{IR}_{DCx} := \begin{pmatrix} 0 \\ \mathrm{Re}(\mathrm{I}_1 \cdot \mathrm{R}_{DC}) \end{pmatrix}, \quad \mathrm{IR}_{DCy} := \begin{pmatrix} 0 \\ \mathrm{Im}(\mathrm{I}_1 \cdot \mathrm{R}_{DC}) \end{pmatrix}
$$

$$
\mathrm{IR}_y := \begin{pmatrix} 0 \\ \mathrm{Im}(\mathrm{I}_1 \cdot \mathrm{R}_1) \end{pmatrix}, \quad \mathrm{IR}_x := \begin{pmatrix} 0 \\ \mathrm{Re}(\mathrm{I}_1 \cdot \mathrm{R}_1) \end{pmatrix}, \quad \mathrm{iIX}_x := \begin{pmatrix} \mathrm{Re}(\mathrm{E} \cdot \mathrm{l}) \\ \mathrm{Re}(\mathrm{E} \cdot \mathrm{l}) + \mathrm{Re}(-\mathrm{i} \cdot \mathrm{I}_1 \cdot \mathrm{X}_1) \end{pmatrix}
$$

$$
\mathrm{iIX}_y := \begin{pmatrix} \mathrm{Im}(\mathrm{E} \cdot \mathrm{l}) \\ \mathrm{Im}(\mathrm{E} \cdot \mathrm{l}) + \mathrm{Im}(-\mathrm{i} \cdot \mathrm{I}_1 \cdot \mathrm{X}_1) \end{pmatrix}, \quad \mathrm{El}_{sx} := \begin{pmatrix} \mathrm{Re}(\mathrm{E} \cdot \mathrm{l}) \\ \mathrm{Re}(\mathrm{E} \cdot \mathrm{l}) + \mathrm{Re}(\mathrm{E}_s \cdot \mathrm{l}) \end{pmatrix}, \quad \mathrm{El}_{sy} := \begin{pmatrix} \mathrm{Im}(\mathrm{E} \cdot \mathrm{l}) \\ \mathrm{Im}(\mathrm{E} \cdot \mathrm{l}) + \mathrm{Im}(\mathrm{E}_s \cdot \mathrm{l}) \end{pmatrix}
$$

$$
\mathrm{El}_{amx} := \begin{pmatrix} \mathrm{Re}(\mathrm{E} \cdot \mathrm{l}) + \mathrm{Re}(\mathrm{E}_s \cdot \mathrm{l}) \\ \mathrm{Re}(\mathrm{E} \cdot \mathrm{l}) + \mathrm{Re}(\mathrm{E}_s \cdot \mathrm{l}) + \mathrm{Re}(\mathrm{E}_{am} \cdot \mathrm{l}) \end{pmatrix}, \quad \mathrm{El}_{amy} := \begin{pmatrix} \mathrm{Im}(\mathrm{E} \cdot \mathrm{l}) + \mathrm{Im}(\mathrm{E}_s \cdot \mathrm{l}) \\ \mathrm{Im}(\mathrm{E} \cdot \mathrm{l}) + \mathrm{Im}(\mathrm{E}_s \cdot \mathrm{l}) + \mathrm{Im}(\mathrm{E}_{am} \cdot \mathrm{l}) \end{pmatrix}
$$

$f. = 1 \times 10^5$

$$
s = 1.01 - E \cdot l + I_1 \cdot R_{DC} - E_s \cdot l - E_{am} \cdot l = 0
$$

$$
-E \cdot l + I_1 \cdot R_1 + i \cdot I_1 \cdot X_1 = 0
$$

<figure><img src="image_58.png" /></figure>

<figure><img src="image_59.png" /></figure>

$f. = 1 \times 10^6$

$$
s = 1.01 - E \cdot l + I_1 \cdot R_{DC} - E_s \cdot l - E_{am} \cdot l = 0
$$

$$
-E \cdot l + I_1 \cdot R_1 + i \cdot I_1 \cdot X_1 = 0
$$

<figure><img src="image_60.png" /></figure>

<figure><img src="image_61.png" /></figure>
APPENDIX 21: Vector diagrams of specific electromotive forces and voltages for various distances and comparison of power diagrams of the presented and existing solutions for a solid cylinder for the eddy-free version

<figure><img src="image_62.png" /></figure>

$$
\begin{align*}
\mathrm{IEI}_X &:= \begin{pmatrix} 0 \\ \mathrm{Re}(\bar{I}_1 \cdot \mathrm{I} \cdot \mathrm{E} \cdot \mathrm{l}) \end{pmatrix} \quad \mathrm{IEI}_Y &:= \begin{pmatrix} 0 \\ \mathrm{Im}(\bar{I}_1 \cdot \mathrm{I} \cdot \mathrm{E} \cdot \mathrm{l}) \end{pmatrix} \quad \mathrm{IIR}_X &:= \begin{pmatrix} 0 \\ \mathrm{Re}(\bar{I}_1 \cdot \mathrm{I}_1 \cdot \mathrm{R}_1) \end{pmatrix} \quad \mathrm{IIR}_Y &:= \begin{pmatrix} 0 \\ \mathrm{Im}(\bar{I}_1 \cdot \mathrm{I}_1 \cdot \mathrm{R}_1) \end{pmatrix} \\
\mathrm{iIIX}_X &:= \begin{pmatrix} \mathrm{Re}(\bar{I}_1 \cdot \mathrm{E} \cdot \mathrm{l}) \\ \mathrm{Re}(\bar{I}_1 \cdot \mathrm{E} \cdot \mathrm{l}) + \mathrm{Re}(-\mathrm{i} \cdot \bar{I}_1 \cdot \mathrm{I}_1 \cdot \mathrm{X}_1) \end{pmatrix} \quad \mathrm{iIIX}_Y &:= \begin{pmatrix} \mathrm{Im}(\bar{I}_1 \cdot \mathrm{E} \cdot \mathrm{l}) \\ \mathrm{Im}(\bar{I}_1 \cdot \mathrm{E} \cdot \mathrm{l}) + \mathrm{Im}(-\mathrm{i} \cdot \bar{I}_1 \cdot \mathrm{I}_1 \cdot \mathrm{X}_1) \end{pmatrix} \\
\mathrm{IEI}_{\mathrm{amx}} &:= \begin{pmatrix} \mathrm{Re}(\bar{I}_1 \cdot \mathrm{E} \cdot \mathrm{l}) + \mathrm{Re}(\bar{I}_1 \cdot \mathrm{E}_s \cdot \mathrm{l}) \\ \mathrm{Re}(\bar{I}_1 \cdot \mathrm{E} \cdot \mathrm{l}) + \mathrm{Re}(\bar{I}_1 \cdot \mathrm{E}_s \cdot \mathrm{l}) + \mathrm{Re}(\bar{I}_1 \cdot \mathrm{E}_{\mathrm{am}} \cdot \mathrm{l}) \end{pmatrix} \quad \mathrm{IEI}_{\mathrm{sx}} &:= \begin{pmatrix} \mathrm{Re}(\bar{I}_1 \cdot \mathrm{E} \cdot \mathrm{l}) \\ \mathrm{Re}(\bar{I}_1 \cdot \mathrm{E} \cdot \mathrm{l}) + \mathrm{Re}(\bar{I}_1 \cdot \mathrm{E}_s \cdot \mathrm{l}) \end{pmatrix} \\
\mathrm{IEI}_{\mathrm{amy}} &:= \begin{pmatrix} \mathrm{Im}(\bar{I}_1 \cdot \mathrm{E} \cdot \mathrm{l}) + \mathrm{Im}(\bar{I}_1 \cdot \mathrm{E}_s \cdot \mathrm{l}) \\ \mathrm{Im}(\bar{I}_1 \cdot \mathrm{E} \cdot \mathrm{l}) + \mathrm{Im}(\bar{I}_1 \cdot \mathrm{E}_{\mathrm{am}} \cdot \mathrm{l}) + \mathrm{Im}(\bar{I}_1 \cdot \mathrm{E}_{\mathrm{am}} \cdot \mathrm{l}) \end{pmatrix} \quad \mathrm{IEI}_{\mathrm{sy}} &:= \begin{pmatrix} \mathrm{Im}(\bar{I}_1 \cdot \mathrm{E} \cdot \mathrm{l}) \\ \mathrm{Im}(\bar{I}_1 \cdot \mathrm{E} \cdot \mathrm{l}) + \mathrm{Im}(\bar{I}_1 \cdot \mathrm{E}_s \cdot \mathrm{l}) \end{pmatrix} \\
\mathrm{IIR}_{\mathrm{DCx}} &:= \begin{pmatrix} 0 \\ \mathrm{Re}(\bar{I}_1 \cdot \bar{I}_1 \cdot \mathrm{R}_{\mathrm{DC}}) \end{pmatrix} \quad \mathrm{IIR}_{\mathrm{DCy}} &:= \begin{pmatrix} 0 \\ \mathrm{Im}(\bar{I}_1 \cdot \bar{I}_1 \cdot \mathrm{R}_{\mathrm{DC}}) \end{pmatrix}
\end{align*}
$$

<figure><img src="image_63.png" /></figure>
APPENDIX 22: Power calculation and graphs for a solid cylinder for eddy-free and eddy variants

<figure><img src="image_64.png" /></figure>
APPENDIX 23: Power calculation and graphs for a solid cylinder with a dedicated part of the eddy specific electromotive force

<figure><img src="image_65.png" /></figure>

<figure><img src="image_66.png" /></figure>

$f = 1 \times 10^6$

$s = 1.01$

<figure><img src="image_67.png" /></figure>

$c = 3.259 \times 10^{-3}$
APPENDIX 24: Power calculation and graphs for a long solenoid

$t = 0.2 \times 10^{-3}$ $D_e = 20 \times 10^{-3}$ $D_i = 12 \times 10^{-3}$ $H_s = 12 \times 10^{-3}$ $N = 1$ $\rho = 1.72 \times 10^{-8}$ $p = 0.5$ $q = \sqrt{1+2p}$ $f = 3.05826 \times 10^5$ $\omega = 2\pi f E = 469$ U $\therefore E \cdot 1k_a = 0$ $W_s = 0.5 \cdot (D_e - D_i) P_s = 2 \cdot (W_s + H_s)$ $D_s = 0.5 \cdot (D_e + D_i)$ $\mu_r = 1$ $\mu = \mu_0 \cdot \mu_r$ $\mu_{sr} = 100$ $\mu_s = \mu_0 \cdot \mu_{sr}$ $z = \pi \cdot D_s$ $S_s = W_s \cdot H_s$ $1 = P_s \cdot N$ $b1 = k_a \cdot t$ $S = 1.005 \times 10^{-5}$

$k_s = \sqrt{\frac{-i \cdot \mu_s \cdot \omega}{\rho}} \quad k = \sqrt{\frac{-i \cdot \mu \cdot \omega}{\rho}} \quad \delta = \sqrt{\frac{2\rho}{\mu \cdot \omega}} \quad S = t \cdot \frac{z}{N}$
$R_{DC} = \rho \cdot \frac{1}{S} j_{DC} = \frac{E}{\rho} x = b1, b1 + \frac{t - b1}{50} \quad t \quad P_s = 0.032$
$\delta = 0.000119 \qquad j_{DC} = 2.73 \times 10^{10} \qquad I_{DC} = 2.74 \times 10^5 \qquad R_{DC} = 5.47 \times 10^{-5} \qquad R = 9.56 \times 10^{-5}$
$z = 0.05$

$|j(b1)| = 4.06 \times 10^6$ $|j(t)| = 2.17 \times 10^7$ $|I| = 65.06$ $\arg(j(b1)) = -1.57$ $\arg(j(t)) = 3.14 \arg(j(t)) = -0.77$

<figure><img src="image_68.png" /></figure>

<figure><img src="image_69.png" /></figure>

$c = 1.32910829607 \times 10^{-4}$ $\text{Re}(\overline{j(c)} \cdot E) = 0$ $I_{0c} = -21.686 - 21.657i$ $I_{ct} = 21.713 - 43.404i$ $I = 0.027 - 65.06i$

$\overline{I \cdot R_{DC}} = 0.23171 \overline{I \cdot E} \cdot 1 = 0.405 + 976.427i$ $\overline{I \cdot E_{tm}} \cdot 1 = -0.308 - 0.559i$ $\overline{I \cdot E_s} \cdot 1 = -976.04i$ $\overline{I \cdot E_w} \cdot 1 = 0.1351 + 0.1761i$

$Q_{0t} = 0.539747$ $P_{0t} = 0.405 + 976.427i$ $T_{0t} = -0.559i$ $S_{0t} = -976.04i$ $W_{0t} = 0.1351 + 0.1761i$

$Q_{0c} = 0.11177$ $P_{0c} = -325.47 + 325.02i$ $T_{0c} = 0.119 - 0.366i$ $S_{0c} = 325.48 - 324.76i$ $W_{0c} = -0.0138 + 0.1036i$

$Q_{ct} = 0.42798$ $P_{ct} = 325.88 + 651.4i$ $T_{ct} = -0.119 - 0.193i$ $S_{ct} = -325.48 - 651.28i$ $W_{ct} = 0.1489 + 0.0724i$

<figure><img src="image_70.png" /></figure>

<figure><img src="image_71.png" /></figure>

<figure><img src="image_72.png" /></figure>

<figure><img src="image_73.png" /></figure>
APPENDIX 25: Power calculation and graphs for a long solenoid for a large frequency

$t = 0.2 \times 10^{-3}$ $D_e = 20 \times 10^{-3}$ $D_i = 12 \times 10^{-3}$ $H_s = 12 \times 10^{-3}$ $N = 1$ $\rho = 1.72 \times 10^{-8}$ $p = 0.5$ $q = \sqrt{1+2p}$ $f = 3.05826 \times 10^6$ $\omega = 2\pi f E = 469$ $U := E \cdot l k_a = 0$ $W_s = 0.5 \cdot (D_e - D_i) P_s = 2 \cdot (W_s + H_s)$ $D_s = 0.5 \cdot (D_e + D_i)$ $\mu_r = 1$ $\mu = \mu_0 \cdot \mu_r$ $\mu_{sr} = 100$ $\mu_s = \mu_0 \cdot \mu_{sr}$ $z = \pi \cdot D_s$ $S_s = W_s \cdot H_s$ $l = P_s \cdot N$ $b1 = k_a \cdot t$ $S = 1.005 \times 10^{-5}$

$k_s = \sqrt{\frac{-i \cdot \mu_s \cdot \omega}{\rho}} \quad k = \sqrt{\frac{-i \cdot \mu \cdot \omega}{\rho}} \quad \delta = \sqrt{\frac{2\rho}{\mu \cdot \omega}} \quad S = t \cdot \frac{z}{N} \quad R_{DC} = \rho \cdot \frac{1}{S} \cdot j_{DC} = \frac{E}{\rho} \cdot x = b1, b1 + \frac{t - b1}{50} \quad t \cdot P_s = 0.032$
$\delta = 3.774393 \times 10^{-5} = 2.73 \times 10^{10}$ $I_{DC} = 2.74 \times 10^5$ $R_{DC} = 5.47 \times 10^{-5}$ $R = 3.08 \times 10^{-4}$ $S_s = 4.8 \times 10^{-5}$

$|j(b1)| = 7.64 \times 10^3$ $|j(t)| = 6.86 \times 10^6$ $|I| = 6.51$ $\arg(I) = -1.57$ $\arg(j(b1)) = -2$ $\arg(j(t)) = -0.79$

<figure><img src="image_74.png" /></figure>

<figure><img src="image_75.png" /></figure>

$c = 1.79034988099 \times 10^{-4}$ $\text{Re}(\overline{j(c)} \cdot E) = 0$ $I_{0c} = -2.098 - 2.098i$ $I_{ct} = 2.099 - 4.41i$ $I = 0.001 - 6.508i$

$\overline{I \cdot R_{DC}} = 0.00231 \cdot E \cdot I = 0.013 + 97.668i$

$\overline{I \cdot E_{tm}} \cdot I = -0.015 - 0.017i$ $\overline{I \cdot E_s} \cdot I = -97.65i$ $\overline{I \cdot E_w} \cdot I = 0.0043 + 0.0043i$

$Q_{0t} = 0.017375$ $P_{0t} = 0.013 + 97.668i$

$T_{0t} = -0.017i$ $S_{0t} = -97.65i$ $W_{0t} = 0.0043 + 0.0043i$

$Q_{0c} = 0.00361$ $P_{0c} = -31.48 + 31.48i$

$T_{0c} = 0.004 - 0.011i$ $S_{0c} = 31.48 - 31.48i$

$W_{0c} = 0.0028i$

$Q_{ct} = 0.01376$ $P_{ct} = 31.5 + 66.18i$

$T_{ct} = -0.004 - 0.006i$ $S_{ct} = -31.48 - 66.18i$

$W_{ct} = 0.0043 + 0.0015i$

<figure><img src="image_76.png" /></figure>

<figure><img src="image_77.png" /></figure>

<figure><img src="image_78.png" /></figure>
APPENDIX 26: Power calculation and graphs for a long solenoid with a capacitor

$$
\begin{align*}
t &= 0.2 \times 10^{-3} & D_e &= 20 \times 10^{-3} & D_i &= 12 \times 10^{-3} & H_s &= 12 \times 10^{-3} & N &= 1 & \rho &= 1.72 \times 10^{-8} & p &= 0.5 & q &= \sqrt{1+2p} \\
f &= 3.05826 \times 10^{5} & \omega &= 2\pi f & E &= 0.194357 & k_a &= 0 & W_s &= 0.5 \cdot (D_e - D_i) P_s &= 2 \cdot (W_s + H_s) & D_s &= 0.5 \cdot (D_e + D_i) \\
\mu_r &= 1 & \mu &= \mu_0 \cdot \mu_r & \mu_{sr} &= 100 & \mu_s &= \mu_0 \cdot \mu_{sr} & z &= \pi \cdot D_s & S_s &= W_s \cdot H_s & 1 &= P_s \cdot N & b1 &= k_a \cdot t & S &= 1.005 \times 10^{-5}
\end{align*}
$$

$$
\begin{align*}
k_s &= \sqrt{\frac{-i \cdot \mu_s \cdot \omega}{\rho}} && k = \sqrt{\frac{-i \cdot \mu \cdot \omega}{\rho}} && \delta = \sqrt{\frac{2\rho}{\mu \cdot \omega}} && S = t \cdot \frac{z}{N} && R_{DC} = \rho \cdot \frac{1}{S} j_{DC} = \frac{E}{\rho} x = b1, b1 + \frac{t-b1}{50} \quad \therefore t && P_s = 0.032 \\
\delta &= 0.0001194 && j_{DC} = 1.13 \times 10^7 && I_{DC} = 113.6 && R_{DC} = 5.47 \times 10^{-5} && R = 9.56 \times 10^{-5} && z = 0.05
\end{align*}
$$

$$
|j(b1)| = 4.06 \times 10^6 \quad |j(t)| = 2.17 \times 10^7 \quad |I| = 65.06 \quad \arg(I) = 0 \quad \arg(j(b1)) = -1.58 \quad \arg(j(t)) = 0.8
$$

<figure><img src="image_79.png" /></figure>

<figure><img src="image_80.png" /></figure>

$$
C = 2.256004023447185 \times 10^{-6} \qquad k_c = \sqrt{\frac{-i}{\rho \cdot P_s \cdot N^2 \cdot \omega \cdot C}}
$$

$$
E_c = 469i \quad E_s + E_w + E_{tm} = -0.08 - 469i
$$

$$
c = 1.32910829607 \cdot 10^{-4} \quad \mathrm{Re}(\overline{j(c)} \cdot \overline{E_c}) = -1.8 \times 10^6 \quad I_{0c} = 21.648 - 21.696i \quad I_{ct} = 43.413 + 21.696i \quad I = 65.061
$$

$$
I \cdot I \cdot R_{DC} = 0.231 \cdot E \cdot I = 0.4
$$

$$
I \cdot E_{\text{tm}} \cdot I = -0.31 - 0.56 I \cdot E_s \cdot I = -976.06i \quad I \cdot E_w \cdot I = 0.14 + 0.18i \quad I \cdot E_c \cdot I = 976.44i
$$

$$
Q_{0t} = 0.54 \quad P_{0t} = 0.4
$$

$$
T_{0t} = -0.56i \quad S_{0t} = -976.06i \quad W_{0t} = 0.14 + 0.18i \quad C_{0t} = 976.44i
$$

$$
C_{0c} = -325.61 + \\ \qquad + 324.89i
$$

$$
Q_{0c} = 0.1 P_{0c} = 0.13 + 0.13i \quad T_{0c} = 0.12 - 0.37i \quad S_{0c} = 325.48 - 324.76i \quad W_{0c} = -0.01 + 0.1i
$$

$$
Q_{ct} = 0.43 \quad P_{ct} = 0.27 - 0.13i \quad T_{ct} = -0.12 - 0.19i \quad S_{ct} = -325.48 - 651.2 \quad W_{ct} = 0.15 + 0.07i \quad C_{ct} = 325.61 + \\ \qquad + 651.55i
$$

<figure><img src="image_81.png" /></figure>

<figure><img src="image_82.png" /></figure>

<figure><img src="image_83.png" /></figure>

<figure><img src="image_84.png" /></figure>
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[4] M. S. Raven, Experimental measurements of the skin effect and internal inductance at low frequencies, Acta Technica 60 (2015), pp. 51–69, Institute of Thermomechanics AS CR, v.v.i.

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[6] H. Lamb, "XIII. On electrical motions in a spherical conductor", Philosophical Transactions of the Royal Society of London. Vol. 174: pp. 519–549 (1883), doi:10.1098/rstl.1883.0013. S2CID 111283238.

[7] F. E. Terman, Radio engineers handbook, McGraw-Hill Book Company, Inc., New York and London (1943).