Calculate Diameter of Quad Coil Array for Speed of Light Rotation

[Fluxation]

TL;DR Summary

Calculate Diameter of Quad Coil Array for Speed of Light Rotation of Generated EM Field

I would like to know how to calculate, for any specified frequency in Hz, the required diameter of a 90 degree phased, quadrature coil array such that its generated EM field achieves rotation at the speed of light.

Could someone please provide an example of the calculation using a specific frequency?

 

[hutchphd]

Summary:: Calculate Diameter of Quad Coil Array for Speed of Light Rotation of Generated EM Field

rotation at the speed of light.

You need to be aware that this phrase has no meaning. Rotation is not a speed.

 

[Ibix]

I can't really make sense of this question. Googling "quadrature coil array" leads me to pages about generating circularly polarised radiation in MRI machines, but nothing there rotates in any sense that could be compared to the speed of light.

Can you explain what it is you expect to rotate?

 

[Keith_McClary]

At low frequency and small radius this is equivalent to a rotating magnet, so the field will appear to rotate. But at larger radii the point of max B (e.g.) is moving faster and can reach the speed of light for large enough frequency and radius. (The boundary between near field and far field?)

Is this what they are asking, not the size of the coils but the radius from the center?

 

[Fluxation]

That is a "yes" to Keith's question.

To take a specific example, if the signal to the quadrature coils is 150MHz, what diameter, or radius, is required to reach the speed of light?

I would please like to see the actual calculation so I can apply it to any frequency of interest.

 

[Ibix]

But at larger radii the point of max B (e.g.) is moving faster and can reach the speed of light for large enough frequency and radius.

But aren't quadrature coils just supposed to produce a circularly polarised field? Is there also both a radial and an angular variation in the field strength? Because if not I don't see how you're going to get anything like a velocity out of this.

 

[Fluxation]

I see your point, and realize the wording of my original question was open to misinterpretation. What I should have stated is "stepped" coils, not quadrature. My apologies for the misstep.

Can anyone please answer the question regarding the calculation? I have rephrased it below.

I would like to know how to calculate, for any specified frequency of complete rotation, the required diameter of a stepped four coil array such that its generated EM field rotates at the speed of light.

As an aside, a circularly polarized field can transmit angular momentum. This has been described by Richard Beth in relation to light.

 

[hutchphd]

Perhaps you should just describe what you are trying to do. Again your question makes no sense to me. Changing on word really doesn't help. How about a drawing? Otherwise this will not be useful.

 

[Baluncore]

What I should have stated is "stepped" coils, not quadrature.

What equipment uses "stepped" coils.
We need a good example of what you plan to do.

 

[mfb]

speed of light / (150 MHz) = 2 meters

The answer will depend on the geometry but that's the general scale.

Note that nothing moves faster than light here. It's like sweeping a laser beam across the Moon: You can easily do that in less than the 10 milliseconds light would need to travel from one side to the other on the Moon. But there is nothing traveling along the spot your laser does on the Moon, so nothing moves faster than light.

 

[Keith_McClary]

"stepped" coils

I don't know what you mean, and google does not help.

 

[alan123hk]

Summary:: Calculate Diameter of Quad Coil Array for Speed of Light Rotation of Generated EM Field

I would like to know how to calculate, for any specified frequency in Hz, the required diameter of a 90 degree phased, quadrature coil array such that its generated EM field achieves rotation at the speed of light.

I believe you are studying the things of http://mriquestions.com/lp-vs-cp-quadrature.html and want to know how to calculate the diameter of the coils to generate a fixed frequency circularly polarized electromagnetic wave.

Please note that this circularly polarized electromagnetic wave has the angular frequency of the polarization vector. Its propagation speed is the speed of light, but it does not produce anything that rotates at the speed of light.

As for the diameter of the coil, I think that in general, the diameter needs to be much smaller than the wavelength, so that the generated radiation pattern can be simpler, but when everything is completely symmetrical and the current phase difference is 90 degrees, even if the coil diameter is relatively large, I believe maybe this circular polarization or elliptical polarization may still exist, but the radiation pattern in three-dimensional space will become very complicated.

 

[Ibix]

Please note that this circularly polarized electromagnetic wave has the angular frequency of the polarization vector. Its propagation speed is the speed of light, but it does not produce anything that rotates at the speed of light.

I'd state this last a little stronger: it does not produce anything that rotates in a way that can be compared to any speed at all. I suspect that you are correct and @Fluxation is talking about circularly polarised radiation, in which case the question is unanswerable. The electric and magnetic field vectors at a point rotate, but since they don't have spatial extent this can't be translated into a speed.

 

[Fluxation]

"Stepper" motors are used in hard drives and positioning systems. I will try to explain more clearly.

Imagine four solenoid coils evenly spaced upon the circumference of a certain diameter. Each is energized in sequence until a full rotation is achieved. The rate in Hz at which the energizing current moves from one coil to the next describes a conceptual "speed", or rate of propagation along the circumference.

Question: If the diameter of the coil array is 2 meters, how frequently does each successive coil need to be energized to achieve upon the circumference one complete rotation at the speed of light? 100 million times a second, 200 million times a second, etc. By "rotation" I mean as if circumscribed, not an actual field rotation.

That is as simply as I can put it. Can someone provide an example of the calculation used to derive this figure?

 

[hutchphd]

Yes I can. But why should I? Please motivate me.

 

[Baluncore]

Can someone provide an example of the calculation used to derive this figure?

Yes.
A 2 metre diameter circle has a circumference of 2π.
The speed of light is 299792458 m/sec, on a good day.
The frequency would need to be 299792458 / 2π = 47.7134516 MHz.
With those dimensions the 4 loops would make an interesting radio antenna.
A "thing" that travels at or above the speed of light is probably a "phase velocity".

 

[Ibix]

Can someone provide an example of the calculation used to derive this figure?

Baluncore's calculation is the correct answer to the question you asked. However, as hutchphd implies, it's not clear what the calculation has to do with anything. For example, given the specified symmetry a circumference of $4\sqrt 2\approx 5.66\mathrm{m}$ would be at least as justifiable as a circle. And you'd need to be careful about coil design to be able to energise and de-energise them at the required rate, even in principle.

Can you explain why you want to know this?

 

[Delta2]

I think that the frequency has nothing to with the diameter of the quad (or more) array.

Given n coils, evenly spaced (every 360/n degrees) around a circle, if we want the field to do one complete rotation every $\frac{1}{c}$ seconds then the frequency of the energizing current must be $c$ (in Hz) and its duty cycle $1/n$.

P.S don't be confused because the units of 1/c is not seconds. What I say I believe it makes sense.

 

[Ibix]

I think that the frequency has nothing to with the diameter of the quad (or more) array.

But OP is not specifying a frequency. OP is specifying the "speed" at the rim of the device, and this means that the required frequency does depend on the dimensions of the device.

What I say I believe it makes sense.

It can't - as you note, the units are incorrect. $2\pi r/c$ is the fundamental frequency required if the device is of radius $r$ and we assume that the circular perimeter is at all relevant.

 

[Delta2]

But OP is not specifying a frequency. OP is specifying the "speed" at the rim of the device, and this means that the required frequency does depend on the dimensions of the device.

Well the way OP states is a bit vague or confusing, at least in post #14. Judging from what I read at #1 I thought that he meant that the B-field from the coils to rotate at 300.000.000 times per second.

 

[Baluncore]

The first thing to note is that the 4 coils are driven as two separate pairs, each pair making one Helmholtz coil. One pair is driven with a sine wave, the other pair with the cosine wave. The inductance of the coils can be tuned with parallel capacitance to resonance at the required frequency.

The magnetic field at the centre of the device is a rotating vector, that rotates once per cycle.
The phase velocity, v, of that vector is a function of frequency, f; and radius, r.
The circumference of a circle is; 2·π·r ;
The period of rotation, T = 1 / f ;
The phase velocity, v = 2·π·r / T ;
v = 2·π·r·f ;

Note: The separation of the coils may be independent of the radius at which the phase velocity is being measured. The problem will be to maintain the phase shift of the current flowing in the two Helmholtz coils. The distance between the coils, and the diameter of the coils, may be a significant part of a wavelength.

 

[alan123hk]

OP mentioned stepper motors and the concept of speed to describe current pulses moving from one coil to another in #14. The current pulse that excite the internal coil of the stepper motor is controlled by the controller. This speed cannot be compared with the speed of light, because the tangential speed of the edge of the stepper motor's rotor cannot be close to the speed of light.

So if there is no meaningful thing moving at the speed of light, is it necessary to use the concept of speed to describe it?

I'm indeed a bit confused, maybe I still haven't understood the meaning of OP until now.

 

[Baluncore]

So is it necessary to use the concept of speed to describe it?

Maybe it is not necessary, but it is possible. If you place two sensor loops, at the specified radius, you can measure the relative phase of the wave passing those sensors. Knowing the sensor separation, you can calculate a phase velocity, in m/s, at that radius.

The unusual thing about this model is that it is radial.
A wave, propagating at a group velocity c, in a rectangular waveguide, can have a phase velocity varying from c to ∞, dependent on the angle of the wavefront to the boundary wall.

 

[Fluxation]

I like the term "phase velocity", but I believe there could be more.

In trying to clarify, I may have confused. The example in my OP was a four phase array.

The second was a square wave stepper. This may be easier to visualize. There would be a near field associated with each coil moving through them in sequence. I contend this to be a real "rotation" in the sense of a circularly progressing field intensity, with a virtual reference to center.

The intent of my OP was how to calculate the activation frequency such that a single progression through all four coils occurs at the speed of light. The duty cycle of each coil would be 1/4 of that.

To better understand Baluncore's equation, I tried an inverse proposition. What circumference array would be required such that a 10KHz rotational speed would progress at the speed of light? 10,000 / 300,000,000 = 3.33e-5 or 0.0000333m. Does that look correct?

 

[jbriggs444]

To better understand Baluncore's equation, I tried an inverse proposition. What circumference array would be required such that a 10KHz rotational speed would progress at the speed of light? 10,000 / 300,000,000 = 3.33e-5 or 0.0000333m. Does that look correct?

That looks inverted. If you kept track of units, I expect that would become obvious...

10,000 rotations per second divided by ( 300,000,000 meters per second ) gives something with units of rotations per meter. You want something with units of meters per rotation.

A bigger size should mean lower frequency to get the same speed. 10 kHz is "slow". So you would need a radius that is "big". 33 microns is not "big". That was the red flag that sent me back to double-check your calculation.

 

[Baluncore]

The second was a square wave stepper. This may be easier to visualize. There would be a near field associated with each coil moving through them in sequence. I contend this to be a real "rotation" in the sense of a circularly progressing field intensity, with a virtual reference to center.

The square wave currents would be impossible to generate, and could not be tuned with a capacitor. Also, with rectangular currents the wave would move in four steps, not in a smooth circle.
If you do not use Helmholtz coils in quadrature, you will have trouble with transit times to different points on the circumference from four different coils in different positions. Don't forget that wavelength is significant and important in the length of coil connections.

To better understand Baluncore's equation, I tried an inverse proposition. What circumference array would be required such that a 10KHz rotational speed would progress at the speed of light? 10,000 / 300,000,000 = 3.33e-5 or 0.0000333m. Does that look correct?

No.
v = 2·π·r·f
∴ r = v / ( 2·π·f )

There is a simple ratio from the earlier problem.
For 1 m radius, f = 47713452. Hz.
1 m * 47713452. Hz / 10 kHz = 4771.3452 m

 

[Fluxation]

Thanks to all for pointing out my mistake. I see that on a practical level I would need to stay within certain parameters of frequency and coil size.

According to Baluncore's calculation above, if one were to use 27MHz off-the-shelf CB equipment, that would enable a manageable 1.767m array.

I am now curious about what the performance characteristics of this would be as a radio antenna.

 

[Baluncore]

I am now curious about what the performance characteristics of this would be as a radio antenna.

It would transmit a vertically polarised signal towards the horizon.
It would transmit a circularly polarised signal towards the zenith.
In between, it would transmit an elliptically polarised signal.
There are many more suitable, simpler and efficient antennas for HF CB radio.

 

[Fluxation]

You previously mentioned quad phase Helmholtz coils as the best approach. Were you referring to a set of bipolar sine and cosine waves? These could be easily generated with a dual channel RF generator.

However, the hangup there might be with standard CB radio amps; coupling and perhaps not bipolar output.

Since you seem to have practical expertise in this area, how would you (in a few words) handle this on a hardware level? We could pretend you might want to for some reason.

 

[Baluncore]

@Fluxation
You seem to be attributing something special to a phase velocity of c that is not evident in reality.

The phase velocity is zero at the centre and increases in proportion to the radius at which you make the phase velocity measurement. At twice the computed radius the phase velocity will be 2c. That is a feature of the rotating field, and not of the coil separation.

You do not need big separated coils spaced about 1.75 metres apart for 27 MHz. You could use a set of coils that are very much smaller, they would generate the same far field and the same phase velocity. The phase velocity would still be c at the computed radius no matter what the coil spacing.

You could also position the coils further out than r, but when they get too far apart you will get directional effects because the four individual coils will become an antenna array. You will also have trouble driving big coils spaced by wavelengths because to maintain quadrature phase you will need to calibrate the length of the transmission lines that interconnect the coils.

 

[Fluxation]

Thank you. I see what you are saying. But I think you may be overlooking the effects of a contained field. For example in the case of a cubical array made of square coils. For sinusoidal bipolar signals, there is a reciprocally moving field gradient between coil faces.

 

[Baluncore]

For example in the case of a cubical array made of square coils.

The sides of the coils would share the same physical space, they would be inductively coupled, yet would have quadrature currents.

 

[tech99]

The sides of the coils would share the same physical space, they would be inductively coupled, yet would have quadrature currents.

Just to mention the radiation mechanism from this structure. Irrespective of any rotating fields, radiation occurs due to the acceleration of the electrons in the wire. As the electrons are accelerated in the circular path of the wire, radiation is maximum in the plane of the coils. In this direction there is only plane polarisation, being the simple addition of the radiation fields from each coil. If the coils are spaced by a significant part of a wavelength and driven with a phase difference then we will see a directional pattern but still in the plane of the coils and plane polarised.