A
Anonymous
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In Dave Emery's thread, "Question for Dave Johnson", I said that for a VLF detector the current in the target is given by It = j w M Ic / (Rt + j w Lt).
Where:
It - is the target current
w - is the frequency
M - is the mutual inductance between the coil and the target
Ic - is the coil current
Rt - is the target resistance
Lt - is the target inductance
j - is the square root of -1
Rc - is the coil resistance
Lc - is the coil inductance
I want to show where that comes from.
Below is a diagram of a simple metal detector system. The left side is the detector and the right side is the target. This diagram is too simple for a practical detector but the equations that come out of it are useful for understanding metal detectors. Also this shows the target as a single inductor and resistor. Actually the target would more accurately be represented as distributed inductors and resistors. For a PI detector, where the drive signal has a wide bandwidth, it might be necessary to model the target as several LR circuits especially for high conductivity targets. But for a single frequency VLF machine a single LR is good enough.
I can write loop equations for both sides of this circuit.
On the left : Ic Rc + I'c Lc + I't M = E
Where:
I' is the derivative of the current = dI/dt.
Ic Rc is the voltage across the coil resistance due to the coil current.
I'c Lc is the voltage across the coil inductance due to the coil current.
I't M is the voltage across the coil inductance due to the target current.
E is the drive voltage.
On the right: It Rt + I't Lt + I'c M = 0
Where:
It Rt is the voltage across the target resistance due to the target current.
I't Lt is the voltage across the target inductance due to the target current.
I'c M is the voltage across the target inductance due to the coil current.
These are differential equations which in general are difficult to solve. But there is a special case in which they can be solved easily. That is when the current is of the form e ^ct where c is a complex number. In that case I' is just c I. Substituting c I for I' changes these into ordinary equations that can be easily solved.
So far, all of this applies to both simple PI machines and single frequency VLF machines. But now it is time to separate the two kinds of machines. The current of e^ct has a transient part and a steady state part. The complex number c is of the form -1/tc + jw where 1/tc is the time constant of the transient part and w is the frequency of the steady state part.
In a simple PI machine the drive voltage is constant during the on time and is also constant (zero) during the off time. If you solve separately for the on time and the off time the frequency of the drive voltage is 0 during those times so the w part of c drops out and the current is of the form e^-t/tc, an exponential decay. The PI detector is only interested in the transient current.
In the VLF machine we do not care about the transient part and just care about the steady state part of the current. So we can ignore 1/tc, and the current is of the form e^jwt which is a sine wave. So in a VLF machine I' is j w I. Substituting that into the original left and right equations we get:
left: Ic Rc + j w Ic Lc + j w It M = E
right: It Rt + j w It Lt + j w Ic M = 0
Solving the right equation for It we get:
It = - j w Ic M / (Rt + j w Lt)
Which is the equation I gave for target current. I left the minus sign out of the other thread because I did not think it was relevant to the discussion.
I can go farther if I solve the left equation for Ic. In a VLF machine the j w It M term is very small compared to j w Ic Lc so I can ignore it when calculating Ic.
Ic = E / (Rc + j w Lc)
If I substitute this for Ic in the solution of It we get
It = - j w E M / ((Rc + j w Lc) (Rt + j w Lt))
Notice that the coil current does not appear in this solution, the target current just depends on the drive voltage and frequency and the characteristics of the coil and target.
It is interesting that if the drive voltage is kept fixed and the coil inductance is kept fixed and only the frequency changes, then the target current approaches 0 as the frequency approaches 0, and the target current also approaches 0 as the frequency approaches infinity. The peak current occurs somewhere in-between. In fact the peak current occurs between w = Rc/Lc and w = Rt/Lt. And it looks to me like the peak will be highest if Rc/Lc = Rt/Lt.
Getting back to Carl's question about how you would go about finding the optimum operating frequency. It looks to me like if you were looking for just one target it would be best to use a coil R/L equal to the target R/L and a frequency w = R/L.
This is all for the simple circuit I described and ignoring skin effect.
Robert
Where:
It - is the target current
w - is the frequency
M - is the mutual inductance between the coil and the target
Ic - is the coil current
Rt - is the target resistance
Lt - is the target inductance
j - is the square root of -1
Rc - is the coil resistance
Lc - is the coil inductance
I want to show where that comes from.
Below is a diagram of a simple metal detector system. The left side is the detector and the right side is the target. This diagram is too simple for a practical detector but the equations that come out of it are useful for understanding metal detectors. Also this shows the target as a single inductor and resistor. Actually the target would more accurately be represented as distributed inductors and resistors. For a PI detector, where the drive signal has a wide bandwidth, it might be necessary to model the target as several LR circuits especially for high conductivity targets. But for a single frequency VLF machine a single LR is good enough.
I can write loop equations for both sides of this circuit.
On the left : Ic Rc + I'c Lc + I't M = E
Where:
I' is the derivative of the current = dI/dt.
Ic Rc is the voltage across the coil resistance due to the coil current.
I'c Lc is the voltage across the coil inductance due to the coil current.
I't M is the voltage across the coil inductance due to the target current.
E is the drive voltage.
On the right: It Rt + I't Lt + I'c M = 0
Where:
It Rt is the voltage across the target resistance due to the target current.
I't Lt is the voltage across the target inductance due to the target current.
I'c M is the voltage across the target inductance due to the coil current.
These are differential equations which in general are difficult to solve. But there is a special case in which they can be solved easily. That is when the current is of the form e ^ct where c is a complex number. In that case I' is just c I. Substituting c I for I' changes these into ordinary equations that can be easily solved.
So far, all of this applies to both simple PI machines and single frequency VLF machines. But now it is time to separate the two kinds of machines. The current of e^ct has a transient part and a steady state part. The complex number c is of the form -1/tc + jw where 1/tc is the time constant of the transient part and w is the frequency of the steady state part.
In a simple PI machine the drive voltage is constant during the on time and is also constant (zero) during the off time. If you solve separately for the on time and the off time the frequency of the drive voltage is 0 during those times so the w part of c drops out and the current is of the form e^-t/tc, an exponential decay. The PI detector is only interested in the transient current.
In the VLF machine we do not care about the transient part and just care about the steady state part of the current. So we can ignore 1/tc, and the current is of the form e^jwt which is a sine wave. So in a VLF machine I' is j w I. Substituting that into the original left and right equations we get:
left: Ic Rc + j w Ic Lc + j w It M = E
right: It Rt + j w It Lt + j w Ic M = 0
Solving the right equation for It we get:
It = - j w Ic M / (Rt + j w Lt)
Which is the equation I gave for target current. I left the minus sign out of the other thread because I did not think it was relevant to the discussion.
I can go farther if I solve the left equation for Ic. In a VLF machine the j w It M term is very small compared to j w Ic Lc so I can ignore it when calculating Ic.
Ic = E / (Rc + j w Lc)
If I substitute this for Ic in the solution of It we get
It = - j w E M / ((Rc + j w Lc) (Rt + j w Lt))
Notice that the coil current does not appear in this solution, the target current just depends on the drive voltage and frequency and the characteristics of the coil and target.
It is interesting that if the drive voltage is kept fixed and the coil inductance is kept fixed and only the frequency changes, then the target current approaches 0 as the frequency approaches 0, and the target current also approaches 0 as the frequency approaches infinity. The peak current occurs somewhere in-between. In fact the peak current occurs between w = Rc/Lc and w = Rt/Lt. And it looks to me like the peak will be highest if Rc/Lc = Rt/Lt.
Getting back to Carl's question about how you would go about finding the optimum operating frequency. It looks to me like if you were looking for just one target it would be best to use a coil R/L equal to the target R/L and a frequency w = R/L.
This is all for the simple circuit I described and ignoring skin effect.
Robert