why is mutual inductance a complex number in my ADS Momentum?

Hello all,

I have been thinking about this for a long time and am quite confused now. Need some help or brainstorming...

I was simulating two square loop coils (antennas) in Agilent ADS 2011 layout editor using Momentum. ADS gives me S-parameters, then I convert them to Z parameters. Then I can get my mutual inductance (M) between these two antennas by M=Z_12/(jW) or Z_21/(jw).

The confusing part is: why is my M a complex number? The imagnarity part is about 1e4 times smaller than the real part. But why is it even there? Even I am not using a substrate and the coils are in air, i.e. the relative permittivity is a real number, hence no substrate loss, no loss in the air. The material for my metal trace is perfect conductor (conductivity=5.8e7 siemens/m, hence the resistivity is almost zero). There should be no loss at all to cause a complex M, right?

Please give me some idea.. thanks!

I am thinking that the mutual inductance relates to the phase of the voltage induced in the second coil, i.e. if its leads the current by 90 degrees its inductive, if its 89.9 degrees its what you have got. With all those straight sided coils- anything is possible?
Frank

Interesting. Thank you.

The subtle phase difference is indeed what happened. Why will do you think the sharp turn cause this?

Thinking about the circular magnetic field around a conductor. In a normal coil, all the circles are mutually supporting. With a square shaped "coil", there is a definite problem with the corners. the magnetic flux will mutually repel inside the corners, leading to a peculiar magnetic field within the centre of the core. Like wise on the outside of the corners the same field is distributed around the radius of the corner, so at some flux density might actually be inside the physical corner of the "coil". Could this have something to do with it?
Another thought is the very limited length to number of turns . i.e. the magnetic axis is not in line with the physical axis. If you had some flux detector and moved it around the coil, the flux would go from + at one end to - at the other end, so if there was an orthogonal component of the field, the flux detector would register some 90 degree component when it was at half way round the coil. But we are not using a flux detector, but another similar coil, so any orthogonal voltage induced in the receiving coil will result in a short circuit current - or a current that is phase with the original current in the transmit coil, i.e. a "real" part to the mutual inductance?
One way to explore this would be to use a receiving coil made in a circular form, but with the same inductance and see if the mutual inductance changes.
Frank ?(over 40 years since I went to College!)

M is only real for low frequency coupled inductors where the is negligible time delay for flux changes in the primary to affect the secondary. Any accurate representation of coupled coils (particularly at high frequencies and where they are separated) will show a phase delay to the secondary. In a frequency domain circuit theory representation this will be a complex value of M.

Thanks so much, Chuckey.

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Hi, fred23, thank you. It makes lots of sense to me. Could you please provide some good reference such as book or paper for me to read to get a more details on what's going on? I do need a precise circuit model to represent coupled coils. All I read so far are the transformer models from which I do not see any circuit components resulting in a complex M.

Thanks!

Presumed this is a pure AC magnetic problem, you get already complex coupling parameters due to conductor losses and eddy currents. In electromagnetic problems, coupling to free space (radiation losses) add.

Hi, FvM, thanks a lot.

However, even when I changed the metal trace to perfect conductor, i.e. no eddy current loss and ohmic loss. The M is still a complex number. I don't think in free space will make M complex because there is no loss (there is attenuation or path loss but I don't think it will make M complex), right? Any thought, please?

To understand the nature of complex inductance, we would need to see the setup details. But generally speaking, in an electromagnetic simulation of coupled inductors in free space, there will be of course radiation "losses".

I was also thinking of losses first. However, chuckeys explanation of complex k factor due to the distributed nature of the inductors makes perfect sense to me. The physical size will lead to some small phase offset between the inductors -> complex k.