Optimizing Target Responses

[Carl-NC]

Part 1: Step Response

Several recent threads have touched on aspects of target response dynamics so I thought a more in-depth look at the physics is in order. I will mostly look at PI behavior but concepts easily apply to VLF, multifrequency, etc. Several key things to keep in mind are:

• The transmitted magnetic field has the same waveform as the coil current.
• The coil voltage is roughly the derivative of the coil current. "Roughly" because the coil has some resistance that makes it a non-ideal inductor.
• The transmitted magnetic field induces an electromotive force (EMF) in a target. The EMF is -dPhi/dt. Because both the coil voltage and the target EMF are derivatives of the magnetic field, they both have (roughly) the same waveform (but opposite polarity).
• The target's EMF produces eddy currents. The relationship between the EMF and the eddy currents is close to that of an RL circuit.
• The eddy currents create their own magnetic field, which is in opposition to the transmitted magnetic field.

Here is an ideal TX current pulse:



It has zero rise & fall times and a peak of 1 amp. The coil voltage required to produce this would consist of infinite voltage spikes at the transitions. We can't produce this pulse but we can produce one with finite rise/fall times:



Now the drive voltage is more modest (but still maybe 100V). This current produces a magnetic field with the same looking waveform and whose field strength depends on the number of coil turns and the distance and position from the coil. We won't worry about the field strength, it's all relative.

The magnetic field induces an EMF in a target which now has this waveform:



That is, the rising and falling edges of the magnetic field produce EMF pulses whose durations match the rise/fall times and whose amplitudes are proportional to the rise/fall slopes. The faster you make those pulse edges, the narrower the EMF, and the higher its voltage. The same conditions also apply to the coil drive voltage.

In the target, the EMF drives the eddy currents with an RL response. Eddy currents are generated during the TX field slews, and they decay when the TX field is constant: either zero or 1 amp, it doesn't matter. The TX pulse above is imagined to have a 100us width with fairly slow (say, 10us) transition slews. A target with a tau of 20us (T20) will have an eddy response as follows:



Again, the EMF pulse drives the eddy current; when the EMF suddenly stops, the eddy current then decays back to zero. Note that this happens even as the TX coil still has 1 amp of current flowing in it, but since dPhi/dt is zero there is no induced EMF. If we are lucky, then the eddies have died out to zero before the TX current turns off (point "a") so that the turn-off eddy drive (If) is about the same as the turn-on eddy drive (Ir).



For a 100us target (T100) things are similar but slower. The slow response means that the driven eddy current achieves a lower amplitude and then proceeds to decay at a much slower rate. At point "a" the turn-on eddies have not decayed to zero and will directly subtract from the turn-off driven eddies, resulting in a weaker target signal after TX turn-off.

We can put these two targets on a common plot and expand it a bit to see how they compare:



During TX turn-on T20 has a much higher driven eddy current but decays quickly (a). T100 reaches a lower eddy drive (b) but is slow to decay out before the end of the TX pulse (c). The residual on-time eddy of T100 (c) reduces its turn-off eddy drive. During TX turn-off T20 clearly has a stronger initial eddy current than T20 (d) and is therefore easier to detect. But T20 also decays more quickly and at some point (e) T20 drops below T100, after which T100 is easier to detect than T20. This answers the question of why a US nickel has a stronger response than a US quarter, at least when you sample before (e). Even though this exercise used a square wave TX which we normally don't use in PI, this effect is the same for standard PI.

It is important to note that the only time eddy currents are increasing in the target is when there is an EMF drive, and this only happens during a changing magnetic field. When the field is flat (even if the coil current is 1 amp) there is zero EMF drive. This is why I find the term "fully charging the target" misleading. In normal PI, during the TX-on pulse, we want the eddies to fully decay to zero before the pulse turns off. It is then the turn-off slew that "charges the target," but never fully. The only way to fully charge a target is with a ramp transmitter, and that will be the next topic of this thread.

[dbanner]

Very informative.

Looking forward to more.

[dbanner]

So by ramping the voltage, the turn on Eddy current is mitigated in the target.??? Clearing the way for a faster decay to zero before the collapsing field Eddy currents are generated??? Fascinating.

[Elliot]

Carl, thank you for your insightful response to many questions around this topic.
I am beginning to realise that selecting the "best" pulse frequency is more critical than I originally thought - very much dependant upon the desired targets' Tau. A multi-frequency pi machine clearly may have some advantages over single frequency (within limits of course).
I have started developing a spreadsheet to better understand/optimise the interplay between the following design factors:

- Pulse frequency
- Pulse width
- Target sample width
- Coil sweep distance and time
- Pules per swing
- Coil width vs effective coil pick-up width
- Target size
- Positive samples per coil swing
- Pre-amp output voltage
- Integrator component values
- Integrator current
- Integrator capacitor charge and discharge rates
- Net integrator output voltage after X integrations

Each of your posts helps me to piece things together better, and I can't wait for your next chapter!

Thanks for the great work!

[waltr]

Well explained Carl. Thanks.

Elliot,
I read that it is the TX Pulse length (allow target eddy current to decay to zero before TX pulse off) that should match the target Tau. Then optimize the Sample times for target Tau.
Pulse rate (frequency) needs matching to integrator design not target Tau.

For a PI it is Multi-Pulse widths (like SD2000) that has advantage for targets of different Tau (not frequency).
For VLF it is Frequency of the waveform.

Carl will correct me if this is wrong.

[Carl-NC]

So by ramping the voltage, the turn on Eddy current is mitigated in the target.??? Clearing the way for a faster decay to zero before the collapsing field Eddy currents are generated??? Fascinating.

Not really. Ideally you would want to step the current 'on' and then wait for the eddies to decay to zero, then step it 'off'. Slower ramps produce less eddy drive, but you still have to wait for it to decay. Things will become more clear as I present more cases.

[Carl-NC]

Well explained Carl. Thanks.

Elliot,
I read that it is the TX Pulse length (allow target eddy current to decay to zero before TX pulse off) that should match the target Tau. Then optimize the Sample times for target Tau.
Pulse rate (frequency) needs matching to integrator design not target Tau.

For a PI it is Multi-Pulse widths (like SD2000) that has advantage for targets of different Tau (not frequency).
For VLF it is Frequency of the waveform.

Carl will correct me if this is wrong.

Pretty much correct.

[Elliot]

Well explained Carl. Thanks.

Elliot,
I read that it is the TX Pulse length (allow target eddy current to decay to zero before TX pulse off) that should match the target Tau. Then optimize the Sample times for target Tau.
Pulse rate (frequency) needs matching to integrator design not target Tau.

For a PI it is Multi-Pulse widths (like SD2000) that has advantage for targets of different Tau (not frequency).
For VLF it is Frequency of the waveform.

Carl will correct me if this is wrong.

Yes, I agree that multi-pulse widths would certainly offer an advantage, but I was trying to also explain that to maximise the number of pulses being sampled, you can squeeze in more by having a multi-frequency arrangement too. In essence, you'd have multi pulse width and multi frequency machine. The multi frequency part enables you to optimise the accumulated integrated voltage to be higher, due to more samples being taken for a given time period.

[Carl-NC]

Yes, I agree that multi-pulse widths would certainly offer an advantage, but I was trying to also explain that to maximise the number of pulses being sampled, you can squeeze in more by having a multi-frequency arrangement too. In essence, you'd have multi pulse width and multi frequency machine. The multi frequency part enables you to optimise the accumulated integrated voltage to be higher, due to more samples being taken for a given time period.

Depends on how the processing is done. Let's say you have a 100us pulse every millisecond. That's 1kHz. Now you add a 20us pulse in between each 100us pulse. Normally you would process those in independent RX channels, each running at 1kHz. So you are still effectively at 1kHz as the two processing channels are independent of each other. OTOH, you might also blend them right into the first demod and then have an effective 2kHz.

[waltr]

Yes, I agree that multi-pulse widths would certainly offer an advantage, but I was trying to also explain that to maximise the number of pulses being sampled, you can squeeze in more by having a multi-frequency arrangement too. In essence, you'd have multi pulse width and multi frequency machine. The multi frequency part enables you to optimise the accumulated integrated voltage to be higher, due to more samples being taken for a given time period.

Depends on how the processing is done. Let's say you have a 100us pulse every millisecond. That's 1kHz. Now you add a 20us pulse in between each 100us pulse. Normally you would process those in independent RX channels, each running at 1kHz. So you are still effectively at 1kHz as the two processing channels are independent of each other. OTOH, you might also blend them right into the first demod and then have an effective 2kHz.

Like this?

[Carl-NC]

Part 2: Ramp Response

In Part 1 the TX waveform was a fairly idealized pulsed current with equally fast rise & fall times. I call this "constant-current PI." Achieving a fast turn-off is easy, we do it all the time in std PI. But a fast turn-on is difficult, it requires a very high voltage; in fact, exactly as high as the flyback voltage you get at turn-off. That is, 100V-600V. It is possible to do that either by brute force or by a clever self-regenerating kick-start circuit. I believe the latter was covered by Deemon in a thread somewhere.

If we're lazy, then instead of a high-voltage kick-start to get that fast rising edge, we simply apply a low voltage and let the coil charge as it will. If the coil is ideal (no R) then a DC voltage applied produces a linear ramping current:



So we've applied a low voltage for, say, 100us and then suddenly turned it off. This is a bit closer to the standard PI most people are working with. The EMF induced in a target now looks like this:



The slow-ramping magnetic field induces a low EMF (-Vr), and the sudden turn-off induces a much higher EMF (+Vf). Because the turn-on magnetic field is a perfect linear ramp, the EMF is a perfect constant level. Since the target can be modeled as an RL circuit, this is akin to applying a stepped DC voltage to an RL circuit. From Circuits 101, the current will exponentially rise to a peak value and stay there. The target EMF for a 20us target looks like this:



Now during the turn-on period the eddy current exponentially increases to a maximum value and would remain at that value is we could sustain the ramping magnetic field. In this case, you could say that the target will reach "saturation" at 5*tau. However, this effect is undesirable. As with the square pulse, the turn-on residual eddy current subtracts from the turn-off eddy drive current to weaken the target response. That is to say, we don't want to saturate the target with turn-on eddy currents, we want them to be zero.

While the fast target can mostly overcome the turn-on eddy offset, a slow target has more trouble:



In this case a 100us target wastes about 2/3rds of its flyback drive just overcoming the turn-on eddy offset. You might be tempted to suggest that we just use a shorter turn-on ramp, so that we cut the turn-on eddy losses earlier in the exponential. But that also means the magnetic field peak is proportionately cut, and the flyback EMF drive as well. It's a losing proposition. You're better off going the other way: let the turn-on eddies reach their exponential peak before turning off the TX. This may be what people refer to when they talk about letting the target fully saturate. Problem is, this isn't what really happens in a real design. That will be Part 3.

While on the subject of ramps, they are also used in VLF designs, namely multifrequency. You can drive a coil with a square wave to produce a triangle wave current, which continuously ramps up & down. The result is a square wave EMF in the target, which then creates a periodic exponential eddy current:



In the receive coil this looks like exponential spikes:



You can either demodulate this with time domain methods much like a PI, or with frequency domain methods like a normal VLF. Although this is really a subject for a separate thread, I thought I would include it as another example of how all this stuff works.

[Carl-NC]

Part 3: Exponential Response

Part 1 showed the responses to a stepped TX current. We don't normally do PI that way. Part 2 showed the responses to a ramped TX current. That's a little closer to standard PI. Now we'll look at what actually happens in std PI.

When we turn on the coil the current starts at zero and charges exponentially to a maximum current with a time constant of tau(on) = L/Rs, where L is the coil inductance and Rs is the total series resistance. If tau(on) is very large compared to the actual turn-on pulse width then the current begins to approximate the ramped current discussed in Part 2. If tau(on) is small compared to the actual turn-on pulse width then the current will completely flat-top before turn-off. Usually tau(on) is around 100-200us and the pulse width is maybe 100us, so it's somewhere in between the ramp case and flat-topping.

The TX coil current, induced target EMF, and target eddy waveforms look like this:



The exponential turn-on current is (a) which induces an exponentially decreasing stepped EMF (b) in the target. The target EMF has the exact same tau as the coil's tau(on) regardless of the target's own tau (tau(tgt)). The EMF, in turn, produces an eddy current (c) which has an odd waveform that is the difference of two exponentials and very much depends on the relationship between tau(on) and tau(tgt). The curvature of (c) as shown might represent a medium conductive target like a copper coin. Other target taus will have different curvatures. In all cases one thing is consistent: as the target EMF exponentially decays toward zero, the target eddies will also decay toward zero. Whether the eddies die out before the end of the TX pulse depends on the taus and the TX-on time.

At point (d) the TX turns off and the current ceases in a non-zero amount of time. The results are as before: an EMF step (e) is induced, which cause the eddy current to abruptly rise (f) and then decay back to zero (g).

The important thing to understand from all this is that, for the purpose of detecting the target, it is the driven eddy current (f) that creates the decaying eddy current (g) that we actually detect. The eddy currents at (c) play no role except to slightly hinder sensitivity. If you want to say we're "charging the target" (and I don't really care for that term) then it is (f) where the charging occurs. The turn-on response (c) is simply an artifact that is necessary to get the TX current up to a peak value for our turn-off event. Ideally, we would always stretch the TX pulse out to ensure all target eddies (c) have completely died out before turn-off but this is impractical for 2 reasons: 1) For high conductors it means the TX pulse width may have to be very large, maybe 300-500us, which severely eats into the timing budget; and 2) leaving the TX turned on for an extended time wastes a lot of power.

More later...

[Elliot]

Thank you Carl! This makes complete sense to me. There is so much false/misleading information buzzing around, that I hope this key information can help clear things up once and for all.

On a more mundane matter, is it possible to extend the amount of time in between having to keep re-logging in? It seems to last only about 10 minutes or so (at least it feels that short).

[Qiaozhi]

Thank you Carl! This makes complete sense to me. There is so much false/misleading information buzzing around, that I hope this key information can help clear things up once and for all.

On a more mundane matter, is it possible to extend the amount of time in between having to keep re-logging in? It seems to last only about 10 minutes or so (at least it feels that short).

There is no time-out associated with the Geotech forum.
When you log in, click on the "Remember Me" button.

[Carl-NC]

Part 4: Optimizing the Turn-Off

Now we see that the energizing "kick" that gives us the target response occurs at the pulse turn-off event. Specifically, it is the dI/dt of the pulse turn off that determines the amount of induced EMF, which in turn determines the strength of the eddy currents. If we want a stronger target response then we can either increase the coil current at turn-off or decrease the turn-off time. Or both. Increasing the coil current is a no-brainer; you can reduce the amount of parasitic series resistance or increase the drive voltage. Decreasing the resistance might mean a heavier coil wire gauge, a better MOSFET, or a low-ESR tank cap. Reducing turn-off time is a bit more difficult; maybe a faster MOSFET, faster MOSFET driver, or lower coil parasitic C. Note that changing the coil turns can reduce C and R but impacts other aspects of the design; it is worthy of its own discussion.

You might be tempted to reduce the turn-off time to as close to zero as possible but there is a point of diminishing return. Here are 3 turn-off slew rates:



All 3 cases have the same peak current (1A), and the turn-off time is progressively halved for the middle and right cases (t -> t/2 -> t/4). This progressively doubles the slew rate which progressively doubles the induced EMF (Vf -> 2Vf -> 4Vf). If the EMF doubles, then the maximum peak eddy will also double (Ip -> 2Ip -> 4Ip); this is shown by the heavy dashed lines. However, the time to reach that doubled peak current is halved, so the gain in actual peak eddy current is marginal. The bottom curves are simply the eddy curves expanded a bit so you can better see the improvement in eddy drive vs slew rate.

The amount of eddy drive depends on tau(tgt). If you imagine the first case (t) with a much faster tau(tgt) then you can imagine that the t/2 and t/4 cases provide a much higher drive improvement. You can calculate hard numbers for all this. Given a fixed peak coil current, the turn-off slew efficiency vs tau(tgt) is:

= 63.2%
/2 = 78.7%
/3 = 85.0%
/4 = 88.5%
/5 = 90.6%
/10 = 95.2%
/20 = 97.5%
/50 = 99.0%
/∞ = 100%

In another thread Joe/bbsailor suggested you want the turn-off time to be at least 5x faster than the target tau you want to detect. That's pretty fair, as anything better than that is hard fightin' for minimal gain.