A question about the phase of the response function fo the simple 3-loop circuit system

[jiajiasun]

The response function can be written as \frac{\alpha^2 + i \alpha}{1+\alpha^2}. The phase lag is $\phi/2 + tan^{-1}(\alpha)$.

Therefore, if \alpha is very small, say, 1e-4 (for example, in the case of a very poor conductor), then the phase lag is 90 degrees. Conversely, if \alpha is very large, say, 1e+4 (e.g., in the case of a very good conductor), the phase is 180 degrees. And, the phase lag should monotonically increase from 90 to 180 degrees as \alpha increases from 0 to positive infinity.

However, that is not what the following figure on https://em.geosci.xyz/content/maxwell3_fdem/circuitmodel_for_eminduction/understanding_harmonicEMresponse.html shows.

The red curve in the right panel, i.e., the phase values, show values from 0 to 90 degrees. Also, it shows a monotonic decreasing behavior.

Could you please double check the red curve in the right panel?

(Looks like my equations won't show up properly. Sorry about that. I need to figure out how to type equations properly in github)

[sgkang]

HI @jiajiasun,

Here we are plotting phase of Q, not phase lag $\phi/2 + tan^{-1}(\alpha)$.
So, technically there is no problem, but if this still confusing, let me know what's your suggestion is.

[jiajiasun]

Hi, @sgkang

Thanks for your response! But, even for the phase of Q, the red curve in the right panel is still not right.

The phase of Q is defined as $-\phi/2 - tan^{-1}{\alpha}$, as is shown below.

If we talk about the phase of Q, then the phase values should be within [-pi, -pi/2], according to the Q defined above. That is not what the right panel shows.

My suggestion is plotting up the phase lag that ranges from 90 to 180 degrees and increases monotonically, i.e. something similar to the following picture.

This way, what is showing in the figure would be consistent with the following contents that appear on the webpage (https://em.geosci.xyz/content/maxwell3_fdem/circuitmodel_for_eminduction/understanding_harmonicEMresponse.html)