Strange SONNET Lite result

[Catalyst]

Hi all,

I have created a 4 turn inductor spiral (attached). I simulated it from 0.1 MHz to 20 MHz. The Inductance values are pretty close to experimental, however over the frequency sweep the inductance value seems to drop rapidly from 0.1 MHz to about 1 MHz then starts to slowly increase again. I having a problem understanding why there is a sharp fall in the inductance value at the lower frequencies.

Can anyone help to explain this? (See the attached graph)

Much appreciated

Cat

 

[maxwellian]

Would you call a 7 nH drop in a 500 nH inductor a sharp dropoff? Your inductance scale goes from 507 nH to 514 nH. That doesn't seem like much of a difference on the whole. Thats about a 1.4% change.

--Max

 

[Catalyst]

Hi,

Yes I understand its not a sharp drop and in the greater scheme of things its not jaw dropping but it would be nice to know the reason for it (physically). I just thought the inductance always rose with frequency up to its self resonant frequency.

Any ideas? anyone?

Cat

 

[maxwellian]

I wonder if it has something to do with how the current tends to be more or less uniformly distributed across a transmission line cross-section at low frequencies, but as the frequencies increase, the current tends to flow more on the edges of the transmission line. It seems to me that this would have at least a second-order effect on the overall inductance that is observed from the device.

Have you looked at current density in Sonnet Lite? I think you would probably have to choose a fairly high resolution for sampling the current. I don't know if that would run in Sonnet Lite or not.

--Max

 

[Catalyst]

Well, i've just tried looking at the current density, I can see it but not in a good enough resolution really. there is a variation but can't see a pattern, nothing conclusive.

Max, I agree that maybe it could be the skin effect (assuming thats what you meant) but even then , why does the inductance decrease then increase, it doesn't make sense to me

 

[maxwellian]

catalyst:

Well, skin depth might be part of the answer, but I also wonder if it doesn't have something to do also with how the current density is distributed across the width of the transmission line.

Did you look at the current density generated by Sonnet Lite at the different frequencies? If you notice the current density at 0.1 MHz (100 kHz), you see (even with the low resolution of 2 cells across each trace width) that the current is more or less evenly distributed across the entire line width, except at the corners.

Then, look at the current density at one of the higher frequencies (like 2.5 or 5 MHz). You can see how the current is starting to exhibit the "current crowding" effect, where more current seems to flow on one edge of the winding trace than another. It appears to me that most of the current spends most of its time on an inside edge (see the first 3 windings or so).

Now, considering these two effects, think about the "average" path length taken by the total current at each frequency. At the lower frequency (0.1 MHz), the "average distance" traveled by the current through the inductor seems to be a little bit longer than the "average distance" traveled by the current at a higher frequency. I might postulate that the apparent longer distance traveled by the current at the lower frequency could look like a larger inductance if you view the behavior from the port.

The current crowding effect is interesting. How quickly it happens (with respect to rising frequency) for an open transmission line usually depends on the loss of the metal, skin depth loss, and to some degree the self inductance per unit length of the transmission line itself. There are a few interesting references out there that explain why this happens in microstrip. Here is one from the Sonnet Software web site:

https://www.sonnetsoftware.com/support/downloads/techdocs/MTTmagMetalLoss1.zip

Other references they cite on this effect are:

A. R. Djordjevic, and T. K. Sarkar, “Closed form formulas for frequency-dependent resistance and inductance per unit length of microstrip and strip transmission lines,” IEEE Tran. Microwave Theory Tech., vol. MTT-42, No. 2, Feb. 1994, pp. 241-248.

F. Schnieder, and W. Heinrich, “Model of thin-film microstrip line for circuit design,” IEEE Tran. Microwave Theory Tech., vol. MTT-49, No. 1, Jan. 2001, pp. 104-110.

In essence, this is what they say:

At really low frequencies, the skin dept is very deep, much deeper than the metal thickness. At these frequencies, the matel appears to be electrically very thin. The edge singularity (as they call it) doesn't show up and loss is constant with frequency. Pure resistive loss effects dominate where the current goes, and we know from basic theory that the electrons will spread out as much as possible to minimize loss. Therefore, the current is nearly uniform across the transmission line.

When the frequency goes up to where your metal thickness is about 2 skin depths thick or less, then what happens is that the inductive reactance per unit length of the transmission line becomes comparable to, or greater than the resistance per unit length. As the reactance begins to dominate, the electrons begin to move to the edges of the transmission line. Apparently the forces that cause this are greater than the desire to stay uniformly distributed.

The interesting thing about spiral inductors is that they frequency have winding-to-winding interactions. The interaction between multiple turns apparently influences the field behavior so that the current is "crowded" to one side of the trace or the other, thereby changing the effective path length for the current.

This crowding effect is also something that I think has a major effect on inductor Q. If you think of the metal as a resistor, you can see that forcing more current through a smaller part of the metal would effectively raise the equivalent series resistance of the inductor. If you could find a smart way to counteract the way that current wants to flow on one edge of the inductor trace or the other, you would probably unlock a secret to raising the Q of your inductor.

I hope that some of this is useful or helpful.

--Max

 

[rautio]

Very nice discussion. One point in favor of skin effect for the low frequency increase in inductance is that with skin effect, the surface reactance equals the surface resistance. This is in sharp contrast to an inductor. For an ideal inductor, the reactance decreases linearly with decreasing frequency (X = 2*pi*f*L). Skin effect resistance decreases with the square root of decreasing frequency. Skin effect reactance thus also decreases with the square root of decreasing frequency.

Let's say we are working at a very low frequency where the reactance from the inductor is small compared to the reactance from the skin effect. Now, we look at the total reactance and divide by 2*pi*f to get the inductance, our "inductance" is now dominated by skin effect inductance and we will see it increasing with 1/sqrt(f) as f decreases. If you have skin effect all the way to zero frequency, then inductance at zero frequency is infinite. When the frequency is low enough that you no longer have skin effect, then you will see the inductance go to zero.

Bottom line, if you do not see inductance increasing at low frequency for a spiral inductor, either you have an unusual inductor, or there is something wrong with your EM analysis.

 

[Catalyst]

Thanks you guys, I'm enjoying this topic.

So now I'm slightly confused. Rautio are you saying that the model I am currently working on is wrong as the inductance is actually decreasing at first instead of increasing all the way from 0.1MHz upto 20 MHz?

 

[rautio]

Hi Catalyst -- I do not know the details of the model you are working on so I can not say if it is right or wrong. At very very low frequency, the reactance will be zero. At a little higher frequencythe reactance will be due the skin effect . The skin effect reactance (in Ohms) will be equal to the skin effect resistance (in Ohms). To get the inductance, divide the reactance by 2*pi*f. At higher frequencies, the inductor reactance becomes large enough to take over and now things act as you expect.

Fortunately, this is of little practical consequence most of the time. The inductance can indeed be unexpectedly large at low frequency, but at low frequency a large inductance does not make much difference. In some cases, it can make a difference, so it is worth investigating and understanding what is happening.

 

[Catalyst]

Hi Rautio, the sonnet files (model and graph) are attached in the original post. If you have the time please have a quick look to make sure the model is set up right I can't see anything rong.

Maybe some of this effect we are seeing is due to the crossover bridge as well?

Added after 50 minutes:

Here is something interesting. I am attaching the variation of inductance with frequency for the same spiral with different number of turns (2,4,6 turns):

The spiral with 2 turns just exhibits an overall decrease upto 20 MHz (sharp fall then a gradual fall)

The spiral with 4 turns has its inductance starting to increase at around 5 MHz

The spiral with 6 turns has its inductance swing at around 4 MHz.

I'm a bit stumped here guys. Maybe this model is not correct

 

[Catalyst]

Hi guys,

Rautio you said "with skin effect, the surface reactance equals the surface resistance". Is it possible to have some sources for that as I find that very interesting.

So, I can obtain the skin effects own inductance value by simply dividing the skin effect resistance value by (2*pi*freq) ?

Heres an example (6 Tun Spiral) :
The 6 Turn spiral is made out of copper metal and is 35 microns thick.
Its DC inductance expected to be 750nH (which is near to real measurements)

At 0.5 Mhz
Intrinsic Reactance (for 750nH) = 2.3 Ohms
Skin Effect Reactance/Resistance = 0.254 Ohms

So even at this low frequency the intrinsic inductance / reactance is dominating by far. Yet the inductance is STILL decreasing at 0.5 MHz (see attached graph for 6 turn spiral in my previous post)

Any ideas? or am I misunderstanding this concept Sad

Please help Sad


Sorry for the Q's


Cat

 

[rautio]

Hi Catalyst -- I looked at the 2 turn case. I noticed that your very first data point has lower inductance than the second data point. I think you will find that your metal is around one or two skin depths thick at the second data point. The skin effect inductance appears to have maximum effect at your second data point. For even lower frequencies, you will find the inductance goes to the value of the inductor, because skin effect inductance is gone.

As for skin effect reactance equals skin effect resistance, this result is in all of the EM text books. Read the section where they derive the skin effect by looking at a plane wave incident on an infinite conducting plane. The skin effect surface resistance must be equal to skin effect surface inductance.

Yes, if you know the skin effect surface resistance in Ohms, just divide the Ohms by 2*pi*f and you have the skin effect surface inductance. That is almost certainly what you see in your plots at low frequency.

To check this idea, change your metal to resistor type metal. Resistor metal has no skin effect model included. You should see a flat inductance at low frequency.

If any EM analysis does not include this low frequency increase in inductance, the EM analysis is wrong.

...Played with your numbers a bit. 0.254 Ohms of skin effect at 0.5MHz equals about 80 nH. Looks like you have about 8 nH extra inductance. If you are well into the skin effect region, all the resistance is skin effect. Perhaps at 0.5 MHz, you are partially skin effect and partially DC resistance. Looking at your 6 turn result, the first data point is almost the same inductance as the second data point, suggesting you are very close to the maximum skin effect inductance.

 

[Catalyst]

The metal thickness for the model is 35 um of Copper. The skin depth of Copper at 14 Mhz is around 17um (roughly 2 times the thickness of the metal). Therefore all frequencies below 14 MHz the metal track is much thicker than twice the skin depth. So this doesn't make sense for what you said about the 2 Turn spiral.

Plus the skin depth of copper at 0.5 Mhz is 93um so surely skin depth /skin effect should not be coming into play yet. I'm confused , lol

Regarding the 8nH increase. I'm not sure how you got that. The intrinsic L is 750nH and the skin effect L is 80nH .......where do you get an extra 8nH from?


Cat

 

[rautio]

Hi Catalyst -- I think the wait for my reply might be worth it, even though I do not have a complete understanding of the strange low frequency inductance behavior.

I discovered that you have also been talking with our support, so I got one of your inductors from them to explore. Seems that Max's ideas above, were pretty close. (Max, if you would like a job in our support group, please send me a resume! )

Full description is too long to post here, so I have written it up in a pdf attached. Hope it helps you in your work!

As for the 8 nH I talked about above, that is how much higher the low frequency inductance peak is above the inductance at 2-10 MHz or so. That is how much extra inductive reactance must be explained, and it is well under 0.1 Ohm, so very tiny, but still numerically significant.