PI: Effects of pulse shape and transmit time.

[Teleno]

In spite of the above, early samples can still be cleared of the ground component via interpolation.

For each period T1 and T2 we need to take two samples:

Early sample m0 at t = t0. (m0 = s0 + g0; "s": target, "g": ground)
GB sample m1 at t1.

From the GB samples at T1 and T2 we can calculate the ground signal g1 at t1 as in the previous posts. Since we know the ground decay law, we can interpolate to approximate the ground signal at any point in time.

The theoretical ground signal at zero time is:

Now we can calculate g0:



and remove this g0 from the early sample to obtain the target signal: s0 = m0 - g0;

Zero spot of my method compared with the delayed sample method.


Let's assume a Tx pulse is 50us long (T1) and the target is sampled at 8us (t0).



My method:

We add a second measurement period T2 = 400us (long pulse).

8us after the short pulse T0 the target signal s0 is:

which varies depending on tau.

and 8us after the long pulse T1 the target signal s1 is:


applying the ground elimination formula we get:



The attenuation of the ground-eliminated signal relative to the original signal is:





Delayed sample method:


We define t1 as the time when the ground signal has decayed to 50% of the value at t0. This is obtained by solving t i the following equation:



which gives t1 = 14.4 us.

applying the delayed sample we get:



Where K = 1/0.5 = 2, now the attenuation is the following:




Comparison:

Blue line: my method with T1 = 50us, T2 = 400 us, t0 = 8us and k= 1.14
Red line: delayed sample method with t1 = 14.4 us and k=2.




My method has a much larger bandwith, it seems.

[Teleno]

In spite of the above, early samples can still be cleared of the ground component via interpolation.

For each period T1 and T2 we need to take two samples:

Early sample m0 at t = t0. (m0 = s0 + g0; "s": target, "g": ground)
GB sample m1 at t1.

From the GB samples at T1 and T2 we can calculate the ground signal g1 at t1 as in the previous posts. Since we know the ground decay law, we can interpolate to approximate the ground signal at any point in time.

The theoretical ground signal at zero time is:

Now we can calculate g0:



and remove this g0 from the early sample to obtain the target signal: s0 = m0 - g0;

[Teleno]

Actually, you can sample much earlier than 14us if you change the TX pulse widths.

For example, let's say we want to sample at t=7us with k=1.1, and =50us. This is based on the conclusion that t must be at least 14% of .

In this case:

Good observation.

If you keep T2 = 400us then you can sample at t = 6us.

For a given pair { T1, k } there's a lower limit for the sampling time achievable when which is:

In your example, for {T1=50, k=1.1} the earliest t is 5us. The longer T2 the better the approximation. For {T2=800, T1=50, k=1.1}; t = 5.4us

[Qiaozhi]

Excellent eplanation, 100% correct. Now the caveats.

If the delay to sampling is too short then k is very close to 1. In these conditions the system:




becomes unsolvable, because kg and g are the same. Target and ground cannot be separated. This scheme requires a minimum delay in order to be effective, which degrades sensitivity, but then again this is always the case with ground balance.

Let's take, for example, two pulses T1 = 100us and T2= 400us. We want k to be at least 1.1 (10% away from 1 in order to solve the above system of equations). The minimum value of t (sampling delay) can be calculated from this formula: . Thus we have: , which gives t =~ 14us

Actually, you can sample much earlier than 14us if you change the TX pulse widths.

For example, let's say we want to sample at t=7us with k=1.1, and =50us. This is based on the conclusion that t must be at least 14% of .

In this case:

[Teleno]

For those who are wondering what's being discussed here, you can easily understand it with the minimum of mathematics, as follows.

In a standard ground-balancing PI (like White's TDI) two samples are taken that are fairly close together. First you have the main sample, closely followed by a second ground sample (maybe 10us to 15us later). This technique is very similar to the method for Earth Field cancellation, in as much as the Earth Field exists in both samples, so can easily be removed by subtraction. However, with the TDI method, the target signal in the second sample is lower than the first sample, and needs additional gain when compared to the first sample. By adjusting the gain on the second sample, it is possible to cancel targets (specifically ground) with a particular decay constant (or ). One benefit of this approach is that it can be used as a simple form of discrimination based on conductivity. The result is that the audio tone will rise for low conductivity targets, and will lower for high conductivity. In this way, small iron targets (such as nails) can be identified. But, one unfortunate side effect is that it creates a hole in the target response. In other words, any targets that happen to match the selected will also be rejected.

What Teleno is proposing here is subtly different. In this case two pulses are transmitted of different widths. The first pulse is narrow, and the second is wider. The proviso is that both pulse widths are sufficiently wide that they are capable of saturating the target. Or (in other words) the target is much smaller than the time constant of the first TX pulse. The result is that the sample taken after the first TX pulse will return the same amplitude signal as the sample taken after the second [larger] pulse. This is because the target has a finite size, whereas the ground appears to the coil as effectively infinite. Hence the ground signal will be different for the two samples. i.e. larger for the sample following the second [larger] TX pulse.

Consider this formula:



where:
s is the target signal
is the signal from the sample following the first [narrow] TX pulse, which contains both target signal and ground.
is the signal from the sample following the second [wide] TX pulse, which also contains both target signal and ground.
k is a user-adjustable gain to eliminate the ground signal

Here's how it works in simple terms, using "easy" numbers:

If we take . This contains the target signal (let's say 5 units) + a contribution from the ground (let's say 1 unit). So the total for is 6 units.
Next we take . This also contains the target signal (5 units) + a larger contribution from the ground due to the wider TX pulse (let's say 3 units). Therefore the total for is 8 units.
As you can readily see, the ground signal in the second sample is 3x the ground signal in the second sample. So we need to set k (the gain) to 3 if we want to eliminate ground.

Now we're ready to remove the ground signal:



which is the target signal minus the ground signal, but without the hole in the target response that you get with the TDI.

Excellent eplanation, 100% correct. Now the caveats.

If the delay to sampling is too short then k is very close to 1. In these conditions the system:




becomes unsolvable, because kg and g are the same. Target and ground cannot be separated. This scheme requires a minimum delay in order to be effective, which degrades sensitivity, but then again this is always the case with ground balance.

Let's take, for example, two pulses T1 = 100us and T2= 400us. We want k to be at least 1.1 (10% away from 1 in order to solve the above system of equations). The minimum value of t (sampling delay) can be calculated from this formula: . Thus we have: , which gives t =~ 14us

[Qiaozhi]

Sure, it's here: http://www.geotech1.com/forums/showt...063#post197063



The formula comes from this paper.

above is the "on" period, that I noted as T. From the paper you'll see that the origin t = 0 is the end of the flyback transient.

Derivation: k represents the ratio between the ground responses to the "wide" and the "narrow" pulses, whose formulas are and respectively.

Ah, yes! ... I remember that paper now.
We discussed it just over one year ago. How time flies.

[Qiaozhi]

For those who are wondering what's being discussed here, you can easily understand it with the minimum of mathematics, as follows.

In a standard ground-balancing PI (like White's TDI) two samples are taken that are fairly close together. First you have the main sample, closely followed by a second ground sample (maybe 10us to 15us later). This technique is very similar to the method for Earth Field cancellation, in as much as the Earth Field exists in both samples, so can easily be removed by subtraction. However, with the TDI method, the target signal in the second sample is lower than the first sample, and needs additional gain when compared to the first sample. By adjusting the gain on the second sample, it is possible to cancel targets (specifically ground) with a particular decay constant (or ). One benefit of this approach is that it can be used as a simple form of discrimination based on conductivity. The result is that the audio tone will rise for low conductivity targets, and will lower for high conductivity. In this way, small iron targets (such as nails) can be identified. But, one unfortunate side effect is that it creates a hole in the target response. In other words, any targets that happen to match the selected will also be rejected.

What Teleno is proposing here is subtly different. In this case two pulses are transmitted of different widths. The first pulse is narrow, and the second is wider. The proviso is that both pulse widths are sufficiently wide that they are capable of saturating the target. Or (in other words) the target is much smaller than the time constant of the first TX pulse. The result is that the sample taken after the first TX pulse will return the same amplitude signal as the sample taken after the second [larger] pulse. This is because the target has a finite size, whereas the ground appears to the coil as effectively infinite. Hence the ground signal will be different for the two samples. i.e. larger for the sample following the second [larger] TX pulse.

Consider this formula:



where:
s is the target signal
is the signal from the sample following the first [narrow] TX pulse, which contains both target signal and ground.
is the signal from the sample following the second [wide] TX pulse, which also contains both target signal and ground.
k is a user-adjustable gain to eliminate the ground signal

Here's how it works in simple terms, using "easy" numbers:

If we take . This contains the target signal (let's say 5 units) + a contribution from the ground (let's say 1 unit). So the total for is 6 units.
Next we take . This also contains the target signal (5 units) + a larger contribution from the ground due to the wider TX pulse (let's say 3 units). Therefore the total for is 8 units.
As you can readily see, the ground signal in the second sample is 3x the ground signal in the second sample. So we need to set k (the gain) to 3 if we want to eliminate ground.

Now we're ready to remove the ground signal:



which is the target signal minus the ground signal, but without the hole in the target response that you get with the TDI.

[Teleno]

Can you explain the derivation of: ?

;

Sure, it's here: http://www.geotech1.com/forums/showt...063#post197063



The formula comes from this paper.

above is the "on" period, that I noted as T. From the paper you'll see that the origin t = 0 is the end of the flyback transient.

Derivation: k represents the ratio between the ground responses to the "wide" and the "narrow" pulses, whose formulas are and respectively.

[Qiaozhi]

Correct. Thank you.

Actually is the ground signal rather than .

can be simplified as .

In practice the k for ground balance would be simply calculated as when s = 0 (no target) and g > 0 (hot ground).

Actually, the ground signal is: due to the the negative sign on , but that's not important. It's the target signal (s) that you want to extract.

Can you explain the derivation of: ?

;

[Teleno]

...
and therefore:

which is the same result.

Hope this doesn't screw up what looks like a potentially elegant solution.

Correct. Thank you.

Actually is the ground signal rather than .

can be simplified as .

In practice the k for ground balance would be simply calculated as when s = 0 (no target) and g > 0 (hot ground).

[Qiaozhi]

Then we can eliminate the ground signal by solving the linear system:

where and are the measurements taken after the short and long pulses respectively.

This gives:

It appears a sequence of two such pulses could be used to cancel ground for all taus up to the shorter pulse n a simple way.

I think your solution to the simultaneous equations is slightly incorrect:




Solving by elimination gives:

and not

I also double-checked using determinants:







Resulting in:



and therefore:

which is the same result.

Hope this doesn't screw up what looks like a potentially elegant solution.

[Teleno]

I'm retaking this thread to share a quick thought on pulse width and ground elimination.

Let's say a PI has two different transmit periods and with . The target's tau is such that .

The target signal is for both and (because tau is small).

The ground signals are different: for and for .

because , therefore causes more ground response.

is calculated directly as:

;

where t is the moment of measurement counted from the end of flyback.
(formula from a previous post in this thread).

Then we can eliminate the ground signal by solving the linear system:

where and are the measurements taken after the short and long pulses respectively.

This gives:

It appears a sequence of two such pulses could be used to cancel ground for all taus up to the shorter pulse n a simple way.

[CAS]

I had already been playing with injecting a high voltage pulse to speed rise time and have managed to get a constant 1.8A coil current at about 12uS after turn on. If I use a lower current (1A) that time can be reduced significantly (to about 2uS) but I dont want to start reducing the energy put into the coil. This means I can have a TX pulse down around the 12uS but surely there will always be some bit of rise time just due the the coil/component effect.

[Teleno]

The fastest rise time will always be limited by the capacitance/resistance/inductance combination of the coil and fet. Other than picking the best spec components and having a super fast coil, I dont see any way of over coming this.

It is overcome by a very high voltage, as high as the transient's peak at cut-off.

By using a CC circuit, the max voltage will be reduced as compared to no CC. This means that there will not be as much energy put into the coil in a CC circuit.

If you turn the coil on using the high voltage you reach the top current instantly rather than havoing to eait for the ramp to crawl up as in standard. The energy stored in the coil is the same: E = (L x I^2)/2, but it is stored faster.

In the first post of this thread I posted a graph explaining the difference.

Now what happens in the target is this:

- The rapid chage at turn-on induces the typical exponential decaying voltage in the target.

- If the pulse-width is longer than the target's tau it will have decayed completely by the time of turn-off. The exponential decay induced by turn-off (of opposite polarity) starts fronm zero ásn is fully available after turn-off.

- If the pulse-width is longer than the target's tau, by the time of turn-off the target is still decaying. This residual voltage is subtracted form the decay amplitude generated at turn-off (remember, they're opposite polarity). The target's signal is weaker.

In short: targets having tau's longer than the pulse duration get damped, targets with shorter tau's pass through.

[CAS]

Thanks for the answer. While I have a good understanding of electronics theory and practice, how it all relates to MD theory is new to me.

Now as I understand it, max coil current will always be relative to the combination of resistance in the TX circuit. The fastest rise time will always be limited by the capacitance/resistance/inductance combination of the coil and fet. Other than picking the best spec components and having a super fast coil, I dont see any way of over coming this.
By using a CC circuit, the max voltage will be reduced as compared to no CC. This means that there will not be as much energy put into the coil in a CC circuit.

Will this result in a reduction of detecting depth or because, of the benefits of CC, sensitivity of the receive circuit can be increased ?