Optimal thin absorber of electromagnetic energy

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The forum discussion focuses on deriving Equation (10) from Equations (8) and (9) in the context of microwave absorption. The user, Tom, seeks clarification on whether k_2 in Equation (8) is a complex number. The consensus is that the complex part of k_2 can be neglected due to its small magnitude, leading to a real value in Equation (10). The derivation involves expanding the impedance formula Z_{1-2}(d) and considering the properties of good-conducting metals.

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Leo2024
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Hi, I am a material engineer and have a question about a formula derivation relative to microwave absorption. I really cannot figure it out after days of trying. This should be simple for a specialist.

In this attached paper, how could one derive Eq(10) based on Eq (8) and (9)? Is k_2 in Eq (8) a complex number?
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Source: https://www.researchgate.net/profil...of-the-permeable-base-transistor.pdf#page=168
 
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I read eq. (9), and its surrounding text, as saying that the complex part is so small that that it can be ignored - leading to eq. (10) being real (with the complex part negligible).

Caveat: Not an expert in the field, just my interpretation of the text and equations.

Cheers,
Tom
 
Leo2024 said:
In this attached paper, how could one derive Eq(10) based on Eq (8) and (9)? Is k_2 in Eq (8) a complex number?
You can rewrite eq.(8) of the paper in the form:$$Z_{1-2}\left(d\right)=\eta_{0}\left(\frac{\eta_{\mathrm{met}}}{\eta_{0}}\right)\left(\frac{1-j\left(\frac{\eta_{\mathrm{met}}}{\eta_{0}}\right)\tan\left(k_{2}d\right)}{\left(\frac{\eta_{\mathrm{met}}}{\eta_{0}}\right)-j\tan\left(k_{2}d\right)}\right)\tag{1}$$You already know from eq.(9) that ##\left|k_{2}d\right|\ll1##, but it's also true that ##\left|\eta_{\mathrm{met}}/\eta_{0}\right|\ll1## because the impedance of a good-conducting metal is very small. So we can expand (1) to lowest order in the two small quantities to get:$$Z_{1-2}\left(d\right)\approx\frac{\eta_{0}\left(\frac{\eta_{\mathrm{met}}}{\eta_{0}}\right)}{\left(\frac{\eta_{\mathrm{met}}}{\eta_{0}}\right)-jk_{2}d}=\frac{\eta_{0}\,\eta_{\mathrm{met}}}{\eta_{\mathrm{met}}-j\eta_{0}\,k_{2}d}\tag{2}$$Now rewrite eq.(6) and the definition of impedance, in terms of the metal's conductivity ##\sigma_{\mathrm{met}}## and skin-depth ##\delta_{s}\equiv\sqrt{\frac{2}{\omega\mu\sigma_{\mathrm{met}}}}##:$$k_{2}=\frac{1+j}{\delta_{s}},\;\eta_{\mathrm{met}}\equiv\frac{\omega\mu}{k_{2}}=\frac{1-j}{\sigma_{\mathrm{met}}\,\delta_{s}}\tag{3a,b}$$Finally, insert this into (2) to yield:$$Z_{1-2}\left(d\right)\approx\frac{\eta_{0}}{1+\eta_{0}\,\sigma_{\mathrm{met}}d}=\frac{R_{S}\,\eta_{0}}{R_{S}+\eta_{0}}\tag{4}$$where ##R_{S}\equiv1/\left(\sigma_{\mathrm{met}}d\right)##.

(Edited to include absolute-value signs in the inequalities.)
 
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