Assume the relations derived in this section involving ac and dc resistance require D 6 δ (F). Derive an expression for the minimum frequency at which the formulas are valid for copper wire at 25°C. Construct a graph of the minimum frequency versus AWG wire size for 0 ≤ AWG ≤ 22. Compare your

Question

Assume the relations derived in this section involving ac and dc resistance require [latex] D \geq 6 \delta(f) [/latex] . Derive an expression for the minimum frequency at which the formulas are valid for copper wire at [latex] 25^{\circ} C [/latex] . Construct a graph of the minimum frequency versus A W G wire size for [latex]0 \leq AWG \leq 22[/latex]. Compare your results with Fig. 2.20.

Accepted Answer

From (2.42), the skin depth of copper at [latex] 25^{\circ} C [/latex] and frequency f is given by

[latex] \begin{aligned} \delta(f) &=d \sqrt{\frac{ ho /(1 \mu \Omega cm )}{f /(1 MHz )}}; \\ d &=\sqrt{\frac{1 \mu \Omega cm }{(\pi)\left(4 \pi 10^{-7} Hm ^{-1} ight)(1 MHZ )}}=50.33 \mu m \end{aligned}[/latex] (2.42)

[latex]\begin{aligned} \delta(f) &=50.33 \sqrt{\frac{1.71}{f /(1 MHz )}} \mu m \\ &=\frac{65.82}{\sqrt{f /(1 MHz )}} \mu m . \end{aligned} [/latex]

If we require [latex] D \geq 6 \delta(f) [/latex] , then

[latex] \begin{aligned} \delta(f) &=\frac{65.82}{\sqrt{f /(1 MHz )}} \mu m \leq \frac{D}{6} \\ & \Rightarrow \frac{f}{1 MHz } \geq\left(\frac{395 \mu m }{D} ight)^{2} \end{aligned} [/latex]

From (2.41), the diameter of an AWGdesignated wire is given by

[latex]D_{A W G}=8.251 e^{-(0.1159) A W G} mm[/latex] (2.41)

[latex] =8251 e^{-(0.1159) A W G} \mu m [/latex]

and so we require

[latex] \begin{gathered} \frac{f}{1 MHz } \geq\left(\frac{395}{8251 e^{-(0.1159) A W G}} ight)^{2} \\ \Rightarrow f \geq 2.292 e^{(0.2318) A W G} kHz \end{gathered} [/latex]

Figure 2.22 shows a graph of the minimum frequency given above versus AWG wire size. Computed values below are in good agreement with the frequencies at which the lines in Fig. 2.20 intersect the line [latex] R_{a c} / R_{d c}=1.8 [/latex]

A W G	6	10	14	18	22
Minimum Frequency ( kHz)	9.21	23.3	58.9	149	376

Question 2.14: Assume the relations derived in this section involving ac an...

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