PI: Effects of pulse shape and transmit time.

[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]

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]

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

[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]

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.