# Magnetic Flux Propagation?

Hey all,

I'm working on a project now where I am using a solenoid driven at a 10Hz AC current to create a magnetic field on a solid piece of low carbon iron, and then using a smaller solenoid to capture the magnetic flux coming from the iron.

By measuring the phase of the resulting waves from the drive coil and the sense coil I notice that the phase difference between the two coils increases as I move the sense coil away from the drive coil. This tells me that there is a measurable delay between the transmission of the magnetic flux into the iron, and when that flux is picked up by the sense coil. It also seems to depend on the distance from the drive coil.

Can anyone give me a reason that this might be happening or some theory behind this? I was under the impression that magnetic flux moved at the speed of light, but this might have to do with the magnetic domains aligning as H changes.

Thanks!

Simon Bridge
Homework Helper
Changes in the magnetic flux travel at the speed of light.

vanhees71
Gold Member
I'm a bit puzzled how you define the velocity or speed of magnetic flux, which is
$$\Phi_{\vec{B}}[A,t]=\int_A \mathrm{d}^2 \vec{F} \cdot \vec{B}(t,\vec{x}),$$
where $A$ is an arbitrary (maybe even timedependent!) two-dimensional oriented area in $\mathbb{R}^3$.

Concerning the original question, I'd think the phase-shift difference is rather an effect of the change in mutual inductance rather than a retardation effect. I'd also think that at such small frequencies retardation effects should be negligible at all. Note that the wave length is $\lambda=c/f \approx 30 \cdot 10^6 \text{m}$, and the solenoids and distance between them are far smaller than that.

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Philip Wood
Gold Member
I find this puzzling. You talk about measuring […] 'waves' from the two coils. Are you actually applying the p.d.s across the coils to your measuring instruments? And how are you measuring the phase difference?

Simon Bridge
Homework Helper
vanhees said:
I'm a bit puzzled how you define the velocity or speed of magnetic flux, ...
How who defines the speed of... ?
One may define the speed of propagation of a flux change by timing it over a known distance - same as any speed - but you know that!

I think ronanrice is being imprecise in post #1.

Concerning the original question, I'd think the phase-shift difference is rather an effect of the change in mutual inductance rather than a retardation effect.
That's what I'm thinking - the delay should not be measurable. I think we need clarification though. There appear to be a number of misunderstandings here.

marcusl
Gold Member
Please describe your experiment precisely. How are you measuring phase? Are both coils around the iron core when you translate them or is one around the core and the other below it in air? What are the dimensions? It is hard to offer assistance with incomplete information.

Please describe your experiment precisely. How are you measuring phase? Are both coils around the iron core when you translate them or is one around the core and the other below it in air? What are the dimensions? It is hard to offer assistance with incomplete information.

Sure, I have one 8" solenoid with 300 turns being driven by an audio amplifier at 10Hz sine wave. I also have a "sense" coil which is 2" in diameter with 20k turns. These are laying flat on a half inch thick piece of low-carbon iron.

The sense coil signal goes into a voltage follower to convert the signal to low impedance so it can be read by a DAQ. The phase is calculated in LabVIEW as the difference between the phase of the signal used to drive the drive coil, and the signal recieved by the sense coil. As the sense coil is moved away from the drive coil, I see a measurable change in the phase difference between the two coils.

I measured the phase at different distances from the drive coil and saw that the change was largely linear. I calculated that the speed that this magnetic flux is "moving" is ~155 m/s. I am trying to investgate why this is happening.

Philip Wood
Gold Member
What a weird set up! The magnetic coupling via the flat sheet will, I think, be rather weak. Much more usual to have the coils coaxial, and put a rod of soft (low carbon) iron along the common axis, going through both coils.

You'd expect a phase difference of about pi/2 between the two voltages, because I'm guessing that the primary circuit is resistance-dominated, so the current will be roughly in phase with the voltage, so the flux will be, too. in the secondary (sensing) circuit, the induced voltage will be proportional to the rate of change of flux, and will therefore be pi/2 out of phase with the flux, and so roughly pi/2 out of phase with the primary voltage.

But none of this explains why the phase difference should vary with tightness of coupling (ie. with how much of the flux generated by the primary goes through thew secondary). Possibly some non-linearity associated with the iron.

What a weird set up! The magnetic coupling via the flat sheet will, I think, be rather weak. Much more usual to have the coils coaxial, and put a rod of soft (low carbon) iron along the common axis, going through both coils.

You'd expect a phase difference of about pi/2 between the two voltages, because I'm guessing that the primary circuit is resistance-dominated, so the current will be roughly in phase with the voltage, so the flux will be, too. in the secondary (sensing) circuit, the induced voltage will be proportional to the rate of change of flux, and will therefore be pi/2 out of phase with the flux, and so roughly pi/2 out of phase with the primary voltage.

But none of this explains why the phase difference should vary with tightness of coupling (ie. with how much of the flux generated by the primary goes through thew secondary). Possibly some non-linearity associated with the iron.

You are right, the phase difference when the two coils are concentric is 270 degrees and as I move the sense coil away from the drive coil the phase difference increases. At a 6 inch separation I get a phase difference reading of 274.74 degrees.

marcusl
Gold Member
The diagram didn't come through so it's still not clear what your geometry is. I can say, however, that you've chosen an experiment that is dramatically more complex and subtle than you think. The magnetic behavior of your experiment is dominated by eddy currents. It is a property of eddy currents that the phase of the current varies with position, so it is believable that you are seeing a position-dependent phase shift. Note that this is not a speed-of-light issue.

There is nothing simple about eddy current calculations. The most common calculation (because it is relatively simple) contains complex variables and Bessel as well as special ber and bei functions. The propagation of energy laterally along a sample is even more complicated, and I won't venture to guess what it looks like.

As for your measurement apparatus, electrostatic and inductive coupling from drive to sense coil, although small, are probably significant compared to the size of the magnetically-induced signal. You can measure this as a baseline by removing the core. Thus the observed behavior is probably a combination of very complicated magnetic behavior with experimental artifact.

marcusl
Gold Member
The small size (5 degrees) of variation with position indicates that inductive coupling dominates your measurement, as I said. To reiterate Phillip Wood's comments, there is a -90 degree relative phase between primary current and secondary voltage induced through mutual inductive coupling--this is known as Faraday's law.

The diagram didn't come through so it's still not clear what your geometry is. I can say, however, that you've chosen an experiment that is dramatically more complex and subtle than you think. The magnetic behavior of your experiment is dominated by eddy currents. It is a property of eddy currents that the phase of the current varies with position, so it is believable that you are seeing a position-dependent phase shift. Note that this is not a speed-of-light issue.

There is nothing simple about eddy current calculations. The most common calculation (because it is relatively simple) contains complex variables and Bessel as well as special ber and bei functions. The propagation of energy laterally along a sample is even more complicated, and I won't venture to guess what it looks like.

As for your measurement apparatus, electrostatic and inductive coupling from drive to sense coil, although small, are probably significant compared to the size of the magnetically-induced signal. You can measure this as a baseline by removing the core. Thus the observed behavior is probably a combination of very complicated magnetic behavior with experimental artifact.

Could you explain why inductive coupling and eddy currents would give rise to this phase change? Not the -90 degree phase change, but the change in phase change with position.

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marcusl
Gold Member
I doubt that your seeing much from the core, due to the small penetration (skin depth). The coupling is probably mainly inductive, giving you a 90 degree phase shift. Parasitic electric coupling, from the wires in one coil directly to those in the other, is also present though much smaller than the inductive coupling. This coupling occurs with no phase shift, so it has the effect of pulling the phase slightly away from 90. The variations you see probably arise from position-dependent changes in the relative contributions of these two coupling mechanisms.

Try turning your sense coil over. The inductive part of signal will change sign of phase (from -90 to +90). If the electric coupling also reverses, you'll see the phase go from -85 to +85 degrees. Depending on your setup and grounding, however, the sense of electric coupling might not reverse in which case you seel the phase go from -85 to +95.

I doubt that your seeing much from the core, due to the small penetration (skin depth). The coupling is probably mainly inductive, giving you a 90 degree phase shift. Parasitic electric coupling, from the wires in one coil directly to those in the other, is also present though much smaller than the inductive coupling. This coupling occurs with no phase shift, so it has the effect of pulling the phase slightly away from 90. The variations you see probably arise from position-dependent changes in the relative contributions of these two coupling mechanisms.

Try turning your sense coil over. The inductive part of signal will change sign of phase (from -90 to +90). If the electric coupling also reverses, you'll see the phase go from -85 to +85 degrees. Depending on your setup and grounding, however, the sense of electric coupling might not reverse in which case you seel the phase go from -85 to +95.

I ran this test, and here are the results I got.

The phase flipped when I turned the coil as expected. But I'm not sure how to interpret these results. My theory is that this is caused by the alignment of the magnetic domains within the iron. As one domain is effected by the magnetic field due to the drive coil, the adjacent domain is effected by this alignment and aligns itself in the same direction, with some delay. I think what I'm seeing is the speed that these adjacent magnetic domains are aligning. I'm trying to find some science that supports this theory however.

Also consider this possibility:

The sense coil might have flux linkage with two ( or more ) sources where the sources have phase difference. The other sources might be the power supply transformer, the florescent light inductance, etc. The resultant flux could then vary in phase as the distance with the coil changes.

Philip Wood
Gold Member
As Marcusi pointed out (in post 10) eddy currents in the metal sheet will play a big part. I think it more than likely that they are responsible for the phase shift with distance that you report.

Thanks Philip. Post#14 also discusses other possibilities.

vanhees71
Gold Member
How who defines the speed of... ?
One may define the speed of propagation of a flux change by timing it over a known distance - same as any speed - but you know that!

I think ronanrice is being imprecise in post #1.

That's what I'm thinking - the delay should not be measurable. I think we need clarification though. There appear to be a number of misunderstandings here.

Perhaps it's my English tricking me here. For me "magnetic flux" (German: magnetischer Fluß) is not a field quantity but only dependent on time. So how can it have a "speed"? A field can have a speed in the sense of the one or other wave-propagation speed quantity (phase velocity, group velocity, front velocity, etc.). So for the magnetic field (or induction) $\vec{B}(t,\vec{x})$ some kind of wave speed can be considered, but not for the magnetic flux, which is a functional of the magnetic field given in my previous posting.

vanhees71
Gold Member
Ok, let's see if I can help with some math :-).

The whole thing is nothing than a somewhat unusual kind of transformer. You can savely use the quasistationary approximation to understand what's going on. In the following I use an subscript 1 for the "source" circuit and subscript 2 for the "sense" circuit. Then Kirchhoff's rules read
$$(R_1 + \mathrm{i} \omega L_1)I_1 - \mathrm{i} \omega M_{12} I_2=U_1,$$
$$-\mathrm{i} \omega M_{12} I_2 + (R_2+\mathrm{i} \omega) I_2=0.$$
After some algebra (I used Mathematica for it, to be honest ;-)) you get for the current in the secondary ("sense") coil
$$I_2=\frac{\mathrm{i} M_{12} U_1}{R_1 R_2+(M_{12}^2-L_1 L_2) \omega^2 + \mathrm{i} (R_1 L_2 + R_2 L_1) \omega},$$
from which you can get the phase shift between $U_1$ and $I_2$ in the usual way. Writing $I_2=Z U_1$, it's given by
$$\mathrm{arg} \; Z=\mathrm{sign} \; (\mathrm{Im} \; Z) \arccos \left (\frac{\mathrm{Re} \; Z}{|Z|} \right ).$$

It's a much more complicated issue, how to calculate the various quantitities, particularly the mutual inductance, $M_{12}$. The other quantities you can also measure easily on the single coils.

vanhees71,

I doubt your argument. The sense coil is supposed to be connected to a high-impedance oscilloscope, so R2 is large while L and M12 are very small. At 10 Hz, the denominator of Z can be safely approximated by R1R2+iωR2L1. This makes the phase lag independent of the distance.