waveforms in targets

[Anonymous]

This essay pertains primarily to nonferrous metal targets.
Also note that in most cases, the engineer is concerned with the shape of the time-response curve, independently of its magnitude. This is because a metal detector is usually designed to have optimum response for particular classes of targets, which might be either close to the searchcoil or at a considerable distance. Therefore in the following procedures no attempt is made to calculate actual currents, only relative currents.
Part I: VOLT-TIME PRODUCT
Basically, a metal target is a one-turn inductor with series resistance. A ring-shaped metal target behaves pretty much like an L-R circuit with a time response described by a simple exponential. A piece of metal with more complex shape behaves electrically like an assembly of inductors with series resistors, exhibiting a time response which is a sum of exponentials.
When a metal target is exposed to a sudden change in magnetic field, the voltage produced by this change sets current to flowing in the target. For a change that takes place over a time interval much less than the time-constant of the target, the current induced in the target is proportional to the voltage and to the duration of the voltage. This is the same equation as that for current flow in an inductor-- the current is proportional to the applied voltage and the length of time that voltage is applied.
When designing the "flyback pulse", this means that for targets having a time-constant at least several times the duration of flyback, you can trade off flyback voltage and flyback duration. What matters to the target is the product of those two numbers. To look at another way, the analytical treatment can be simplified to a current step function (having units of delta amperes) and a volt-time unit impulse (having units of volt-seconds).
Part II: ANALYSIS BY SUPERPOSITION
There has been a lot of discussion recently about what happens during the so-called "transmit pulse". This refers to the time preceding flyback, when a transistor switch is turned on to apply voltage to the transmitter coil, causing current to build up in it. In most cases the current waveform during the "transmit pulse" is triangular or exponential, making it difficult to figure out what corresponding current is flowing in the target.
There are several ways to solve the mystery: PSPICE simulation, hifalutin engineering calculus, constructing very carefully an induction balance to 'scope the target waveform. There is another way which is tedious, but fairly straightforward. That way is to break up the process into discrete time-intervals, and then solve and sum them using the principle of linear superposition.
Although this method is tedious if done by hand, it is ideally suited to computation using a computer spreadsheet program.
STEP 1. You have to know what the transmit current waveform is. Usually you'll know that through circuit design calculations or by oscilloscope measurements.
STEP 2. You have to know what the unit impulse function of the target is. Usually you won't know that in detail. For a particular target you may have published information on its dominant time-constant, or you may have data from your own prior measurements. Or, you may just pick a time-constant, in an attempt to answer the question "what would happen if we had a target with this time-constant?"
STEP 3. Chop the graph of transmit current vs. time into small time-slot intervals. Intervals about one tenth the target time-constant or one-tenth the transmit time (whichever is smaller) will usually give sufficient accuracy.
STEP 4. For each interval, perform the following calculations.
1. Determine the change in current over the interval. Call this number the volt-time unit impulse for that interval.
2. Plot the time response of the target to that unit impulse, scaling its height in proportion to the magnitude of the unit impulse. The shape of the curve is the same in each case: all that changes is its height.
3. Chop up the time response graph into the same interval units that the intervals themselves are chopped up into.
STEP 5.
When the unit impulse time response has been calculated for each time interval, add them all up, each part in its corresponding time-slot.
For instance, suppose that the transmit time is given at 100 microseconds, the target time-constant is given at 200 microseconds (a high-conductivity target), and for purposes of analysis you broke it down into ten 10-microsecond time slots, labelled (naturally) 1 through 10.
The graph for the target response to the impulse of time-slot 1 extends for 10 time-slots. That gives you 10 numbers, one for each time-slot.
The graph for the target response to the impulse of time-slot 2 extends for 9 time-slots. Those nine numbers are added to those from the first graph, each in its corresponding time-slot.
You do the same up to graph #10, which only has one value, which gets summed into time-slot #10.
Now, plot the sums. The result is the magnitude of the current with respect to time, in response to the transmit current.
A MODIFICATION TO THE METHOD.
The procedure described above allows you to plot the current in the target during the transmit period. However, you can extend it as long as you like by adding additional time-slots. During flyback you have to calculate volt-time unit impulse, the same as during the transmit time. After flyback, the coil current is zero, and the volt-time products are also zero, so no additional graphs have to be generated for time-slots after the flyback.
Remember that if the change in current is negative, the volt-time product is also negative. When summing the graphs, you have to keep track of the arithmetic signs, subtracting when necessary.
If you're concerned about the effects of increasing or decreasing the pulse repetition rate, then you can extend the whole process through two or more cycles.
--Dave Johnson
PS: sorry, I ain't no mathemetician. If you're a math whiz and think the above description stinks, please rewrite it more clearly and correctly and by all means post it on this forum. Thanks.

 

[Anonymous]

Dave
Here are your plots for 3 different drive conditions.
This is for a simple non-ferrous target whose time constant is one horizontal division on the graph. There are 10 time increments per division.
The drive circuit is a simple RL with a constant drive voltage during the on time. At turn off the voltage is allowed to rise to whatever value is required to collapse the field in one time increment. The on time is equal to 3 target time constants.
The 3 cases are:
1. The drive RL time constant is 10 times the target time constant.
2. The drive RL time constant is 1 times the target time constant.
3. The drive RL time constant is 0.1 times the target time constant.
In each case the drive voltage is set so the current will reach 1 at turn off.
The flyback pulse is not shown on the target signal. This is not realistic, but this is just a simulation of ideal conditions.
Robert

 

[Anonymous]

YES! Thank you, Robert! Your "pictures" say it much better than anything I have been able to say with words.
The simplifying assumption of making flyback pulsewidth one time-slice was a nice touch for this purpose.
-Dave J.

 

[Anonymous]

Hi Robert / Dave
To me it seems crystal clear that their most be a lot of power to save by using the right curveform for feeding the coil.
For instance Thomas Breuer's idea using a constant current instead of a constant voltage to feed the coil.
What about making a simple D/A by using two, three or more mosfet in parallel each having a resistor in series with the drain of different sizes.
Lets say we have a PI coil and we wand to feed this with a 100 uS pulse, the coil has a DC resistance of 4 ohm.
We will use three mosfet (T1..T3) that all three has the source (S) connected to ground, and two of the transistors Drain (D) is connected through a series resistor to the coil. The third one has its drain directly connected to the coil.
T1 is having 10 ohm's as drain resistor
T2 is having 5 ohm's as drain resistor
T3 is having 0 ohm's as drain resistor
Lets say ssytem is running on 12 volt and total pulse is 100uS.
If we then first switch T1 for 50uS we will have:
12/(4+10)=0,86 amps
Next we switch on T2 too for the next 30uS and we will have:
12/(4+(5//10)=1,64 amps
Finally we switch the direct connecting T3 on for the last 20uS and know we have:
12/3= 4 amps
We could even by binary input split this to a more fine grained scala but even three steps will do better than the ordinary "one".
What this will give us is a hughe saving in power, a long pulse for the bigger target and a final short strong pulse for the small nuggets (or gold ring).
Of course the time constant for LR or mosfet on resistance and the inductance most be balanced so the final pulse lenght is capable of fully charge the coil.
By fully charging the coil with current just before we final the full pulse will also create a good strong flyback pulse.
I think that both Eric and Dave are both right when claiming the flyback pulse matter or does not matter, its all related to type of taget object.
From my point of view the reflected pulse or absorbed energy from the target is a complex process where both the current send to the target and the reflected pulse from the coil is part of the final result.
For instance I have created a circuit like the one Eric did for us back in time, and depending on the inductance, the coil's capacity and the gain of the amplifier will have different capabilities in the terms of the flyback pulse. You can easy create an amplifier that saturated at a very long part of the reflected pulse so you will never se a gold target ...SCARRY URGH.
For instance I can make the circuit work so only the very first part of the decaying flyback pulse is changing when a thin copper coin is target and only iron or a very big non magnetic target is having biggest influence on the middle to final part of the reflected pulse.
For most cases it depend on the gain you put in the frontend which part of the decaing curve you are able to study the falling voltage of. This again of couse is under influence of the TC of the PI coil as well as TC of the target. But also voltage supply of the front end circuit will make a difference, the input protection diodes and the rise/fall time of the used amplifier.... Not an easy task to predict.
If the target is small low conductive and fast decaying its more the reflected pulse that count, good for small gold nugget.
If the target is a big heavy chunk of metal its more the stored energy that influence because the saturated amplifier is first seeing the signal a long time after the high voltage in the coil is dead. Like Eric's Car detector example.
Thanks to all of you and a Happy New MD year, especially Thanks to Eric for having this wonderfull page for us.
Mark

 

[Anonymous]

Robert, I wonder if you could be persuaded to calculate and publish a set of curves identical to the last set, except with a target time-constant longer than the "transmit pulse", say 6 divisions (twice the "transmit pulse" duration).
Thanks.
-Dave J.

 

[Anonymous]

Dave
The target time constant is now 2 divisions and the on time is one division. So the target TC is twice the on time. I kept the shape of the coil currents the same as before, but it is compressed horizontally. Vertical scaling is the same. The signals are weaker now.
Robert

 

[Anonymous]

Thanks, Robert.
What these waveforms show, is that for high-condutivity targets (targets with a time-constant greater than the transmit on-time):
1. The current flowing in the target as a result of volt-time product during the so-called "transmit pulse" severely impacts the signal available to the receiver after the flyback pulse.
2. For a given total transmit on-time, it is advantageous to get the transmit current rise (i.e. the volt-time partial products) done as early in the "transmit pulse" as possible.
The data do, however, allow for a different interpretation, specifically:
For this particular ratio of target time-constant to on-time, the available receiver signal is roughly proportional to the area under the transmit current-time curve. .....I think that in general, this approximation would hold true for longer time-constants but not for shorter ones, but haven't thought it through carefully.
--Dave J.

 

[Anonymous]

Hi Robert,
I am curious as to the spice program you are using to provide the simulations.
Reg

 

[Anonymous]

Reg
Actually I am not using any circuit simulation software at all. It is all just math. So you could say I am just making it all up.
The Mathcad sheet that plots one of the cases is below. Most of it could easily be duplicated in a spread sheet. It is only the last step that would be difficult.
This is an explanation of the variables used:
tmax - the number of time steps (really tmax+1 because I start from 0).
ton - the on time
ttc - the target time constant
tc - the coil time constant
v - the supply voltage needed to get a peak current of 1.
r(t) - a function, the impulse response of the target
t - a vector of the time values. In a spreadsheet this could be a column of cells, if needed at all.
i - a vector of the current levels. In a spreadsheet this would be a column of cells.
s - a vector of the target signal levels. This is the one that would be harder to do in a spreadsheet. Each of these is a sum of products of the change in current times the impulse response of the target. This is the superposition step. I do not know if you can do a sum of products in one cell. I would hate to have to type in a 60 x 60 array of cells to solve the problem in a spreadsheet. The calculation of s is also a discrete approximation of convolution.
Then you just plot i and s.
Robert

 

[Anonymous]

To model an exponential-decay time-constant impulse response in a spreadsheet is easy. (I'm guessing this is how Robert H. actually did it.)
The width of the time-slot increment has to be several times smaller than the time-constant being modelled if you want reasonably accurate results.
First, divide the time-slot increment width by the time-constant. For instance, if the increment is 10 microseconds and the TC = 70 microseconds, the result is .143. Subtract that from unity. In this example, (1.000 - .143) = .857.
In each row where a target impulse response is being calculated, there will be a series of columns going from left to right, which will be filled with zeroes up till the impulse being calculated. If the calculations being done are all relative, the number in that column will normally be unity, 1.000.
In each such row, starting from the leftmost column, multiply the number in that column by (for instance) .857 to get the number in the next column to the right. The only exception is the column corresponding to the impulse, which is defined as unity.
MULTIPLE TC TARGETS
So far in this discussion, we've assumed that targets can be accurately represented by a single TC. For ring-shaped nonferrous objects this is true, and for many applications, the inaccuracy of this assumption about other types of targets does not present any difficulty-- still good enough for the crude purpose at hand.
When designing discriminators, the single-TC model will usually not be good enough. There are a variety of ways to describe a more complicated target. If you are using the spreadsheet method and like its computational straightforwardness and simplicity (no calculus, not even the direct computation of an exponential), you can calculate several TC impulse responses for each impulse, and sum them up in the usual fashion when the partial values have all been calculated.
One serious difficulty in the multiple-TC approach is that you need to know which TC's will represent the target in question. You'll usually need an experimental test setup of some type to get the answer. I believe that in most cases, having TC's in a 1:3:9:etc. progression will give satisfactory results, but you might find that tighter ratios are needed.
DETERMINING TC'S FROM AN EXPERIMENTAL TEST SETUP.
If you are using a conventional PI for measuring TC's of multi-TC targets, chances are that you are able to scope only what happens during the receiver on-time. The waveform you see there was not produced by a single impulse corresponding to flyback, so you can't just plot the slopes and extract the time-constants.
But, what you can do is model the system in a spreadsheet, and by trial and error find the TC's which produce the response you see on the 'scope. As a starting point, you can play around with two TC's as follows:
1. Let the first (long) TC have a time-constant equal to the pulse repetition interval.
2. Let the second (short) TC be equal to 3 times the duration of the flyback pulse. Let the coefficient of its impulse be unity. (NOTE: the time-slot intervals will have to be half the flyback pulse time or smaller, in order to get a satisfactory simulation.)
3. Starting with a coefficient of unity and working down, play around with the coefficient of the first TC impulse until you get the best fit.
This approach presupposes that you're trying to model medium to high-conductivity or iron targets. It will probably be accurate enough for doing the initial design of PI's which demodulate two or three delays. NOTE: if your 'scope shows less than a 70% decay from the beginning of receiver on-time to just prior to the beginning of the next "transmit pulse", you won't be able to model the target with the simple approach outlined above, and to model it you'll have to stretch the simulation through two or three full pulse cycles before the target response stabilizes.
-------
If anyone tries this, I hope they'll post their results here and offer suggestions on how to improve the method.
--Dave J.

 

[Anonymous]

Hi Robert,
Thanks for the information.
Been a long time since I have seen many of the equations that have surfaced lately. Just for fun, try to do the same problem without mathcad, excel, home computer, or even a scientific calculator. When I went to college none of those things existed. Now I remember why I didn't like some of the classes.
Old timers like me may remember those days when math problems were done with a slide rule and look up tables for the exponential functions.
I admire your knowledge of the subject from a mathematical standpoint. For me it is too much work for my one remaining braincell. Besides my curiousity is one primarily coming from a user standpoint with some technical knowledge on the subject.
As for simulation of what might happen, I will have to stick with a relatively inexpensive spice programs to see end results. I've gotten too old to try my nerves trying to look up the necessary equations and then try to remember how to do them.
I truly admire all of you who can do this.
Reg

 

[Anonymous]

Dave
For a small number of time increments you can do a spreadsheet solution the way you describe. But as you increase the number of time steps the array grows in two dimensions. I am too lazy to work with that many cells.
I almost have a solution that only grows in one dimension and can be extended just by duplicating cells and not editing them. I have created columns for time, current, delta current, impulse response, and signal. And I found a function sumproduct() that gives a sum of products. The problem is the expression deltaI(x)*r(t-x). The minus sign means that you have to index backwards through one of the arrays, and the spreadsheet does not seem to allow that. I can flip the r array end for end and then sequence through them in the same direction, but then the cell range has to be calculated for r because instead of using the range (0 to t) you have to use (end-t to end). I cannot find spreadsheet functions that will allow me to calculate a cell range. I can make it work if I type in a unique range for each cell of s, but even that is too much work for me.
Robert

 

[Anonymous]

Reg
I still have my slide rules, but they don't get much use any more.
Robert

 

[Anonymous]

OK, I found the function I needed to compute a range for sumproduct(). So now I have an Excel spreadsheet that will plot the target signal for a simple RL drive circuit. If anyone wants to play with it you can download it from this site.
Robert

 

[Anonymous]

Hi Robert,
Maybe you made it up but it sure looks good to me.
Really shows what Eric has spoke about for years, the time and waveform shape that will cause the best signal back, is the one which the currents have died down in the target before turn off of the transmit current.
Pictures really do show alot. I have used Mathcad since the old dos version 2.0 on to now 7.0. Don't have the new 2000 or whatever, but even 2.0 was cool. The jump from the slide rule to the calculator is like the jump from the calculator to Mathcad.
Anyway, the waveforms look like you got something right, all makes sense to me.
Thanks for sharing, also enjoyed reading your vlf work.
JC

 

[Anonymous]

JC writes "also enjoyed reading your VLF work".
I presume that refers to published work. Robert (or JC), can you refer us to that work?
--Dave J.
PS. Thanks, Robert, for "running with the ball" on that eddy current computation method and publishing the results here.

 

[Anonymous]

http://www.thunting.com/cgi-bin/geotech/forum/view.cgi?action=index&forum=tech
http://home.cdsnet.net/~roberth/
Robert posted 35 parts to IB metal detector on the geotech forum.
Then there is the stuff on his web page.
Interesting stuff, thoughts, and comments.
Sure he can tell you better than I about all this.
JC

 

[Anonymous]

I added a brief description of the variables used in the spreadsheet to the download site. I also updated the spreadsheet to add a second target so that a short time constant target and a long time constant target can be compared on the same graph.
Robert