Theory

Terms and Formulas to Know Link to heading

  • Ohms Law Link to heading

  • Maxwell’s Equation Link to heading

    • F : Force.
    • n : Number of turns.
    • I : Current.
    • μ0 : permeability of air.
    • A: Area
    • g : the gap that is separating the electromagnet and the object.

The reasons why this equations are important is that we can see what variables we need to target in order to control the direction of the force for our linear motor. Although we can technically change the number of turns in a magnet, the area and the gap, these are extremely hard to do dynamically for speed. This means that the only real feasible way to control the strength of an electromagnet is to change the current. This is generally speaking, how most of the industry controls magnet based movement for things such linear motors (like the one we will build), stepper motors and even magnetic door locks.

The challenges for us: Link to heading

  • control the electromagnet with enough fidelity in the current to drive the motor accurately
  • create a board that will be able to control multiple electromagnet system (with reverse current drive)
  • create a construction method that is both cheap and accessible
  • pair linear motor with encoder or known-location calculations
  • make all designs parametric (or at least scalable) to increase usability and applicability (as well as increase availability for those under more-constrained budgets)
  • validate the motor applicability with a paired application design

Design Link to heading

Magnetic Alignment Design Link to heading

Stationary Representation Link to heading

Here, we know that if we have an electromagnet next to each other with opposite polarity, there will be no lateral movement. This is due to the fact that each of these magnets is directly and vertically (with respect to orientation of the image presented) on top of each other that this magnet will have no movement. Additionally, we know that each of these magnets

Spacing Electromagnets Link to heading

Now, lets extend the rail and then create some spacing between the electro magnets

Note, the important thing here is not actually the size of the electromagnet but rather the size of the magnets on your rail. As long as your electromagnets can be fit between 1.5x the size of your magnet base, you should be able to control the force that you have. Mathematically, we can simplify this to aggregate the force of the electromagnet at the center of the magnet. Note, this is not generally the case but as long as we create a constant sinusoidal force curve, the residual lateral forces should be cancelled out

Here, we can now see the relative forces between the magnets. By controlling the amplitude of the attracting and opposing forces, we can create movement. Here we want to near-zero out electromagnet #3 (we will explain the near-zero part out later as this is mostly to conserve the amperage fluctuations). This is because if the magnet was fully powered, block wouldn’t move. It would want to be stabilized on the force concentration of block #3. For Block #2, we want to maximize the strength. This is when the block should be the highest as it is currently doing all of the driving laterally. It is both creating and leveraging the opposing and attractive forces. Finally for electromagnet #1, we also want to zero the force out. This is due to it’s instability at the height of opposition between its corresponding vertical magnet.

1/2 Step Down Link to heading

Here, going in reverse numerical order, we can see why block 3 needs to be zeroed out. This is due to continual motion will cause it to repel when it moves towards its next half step. Instead with the advent of electromagnets, we can instead, reverse the current flow. As seen in Maxwell’s Law, this will reverse the direction of the force, or in the event of the magnetic field, it will reverse the polarity. Here, block #2 has reached its optimal state, and now block #1 is driving the entire motor.

Reversing Polarity - Electromagnets Link to heading

As we can see in the image above, electromagnet #3 is actually pushing against movement. Here is where we can introduce the technique of reversing current direction. As Maxwell’s Equation suggests, negative current will cause a negative force (or one in the opposite direction). This is important because in our electromagnet example, now electromagnets 1 and 3 can both drive direction. In the below setup, we would want electromagnet #1 and #3 to be peak amplitude (different directions); while electromagnet #2 would be nearly zeroed out.

This concept of zero force should be thought of as zero current. In the time-scale of objects, this should mean that it would be in the process of reversing current. In the example above, the next frame in sequence (if we use a motion-picture analogy) would have the North pole of electromagnet #2 be below (closer to the rigid-linear rail).

If we continue this but now increase our half step so electromagnet #3 is aligned with its next opposite pole, we get this:

This picture depicts the fact that electromagnet #2 is the only one that drives when #1 and #3 are near-zero current.

https://www.youtube.com/watch?v=0McH4fIHtuc

Calculating Values Link to heading

In the above diagrams, there are a few things to note. Because we are not perfectly aligned with the magnets, we will have magnets on opposite ends. If you notice from our above diagrams, two of the magnets would be at opposite amplitudes regarding their sinusoidal current values. The third one would be at one-step out of phase; this would be determined by half wavelength in sinusoidal terms or the distance between the center of the magnets in on the RIGID LINEAR RAIL.

Here it is very important that we distinguish the geometric constraints of the device are determined by the size of the magnets on the linear rail and NOT the electromagnets.

Our goal at this stage is to determine distance, velocity and acceleration calculations. This would let us determine a very accurate guess of known position of the motor with respect to the axis. For our machine, the goal would be to test the motor. We can then back-calculate the position, velocity and acceleration based off a known distance (the length of the rail movement). However, this is only as good as the accuracy of our measurements, in this case both time and distance. Instead, we can calculate it a few ways both having the known mass and estimating the distance between the center of the electromagnet and the specific magnet it is attracted to on the rail. This would let us calculate the acceleration once we know the current we are supplying (using Newton’s second law - F= m * a ).

The problem with this method is that we want to use the motor to move an object or tool around so our mass is dependent on our

Non-Optimized Design Disclaimer Link to heading

There are a few aspects we can talk about when looking at optimizing our design. Firstly, is the number of electromagnets. This design can be more optimized with coils (as mentioned earlier). However, we can also optimize this with 4 electro magnets rather than 3. In modern create-your-own-coil/custom, electromagnetically-driven linear motors, these can be optimized in manufacturing by utilizing a 3-phase coil. This decision in most modern motors seems to be regarding indexing electrical and manufacturing performance. This uses significantly less overall parts, electrical components and has very little risk of backflow as the loads are almost always well balanced. However, in our case, there is less “smooth” power strokes because two electromagnets are perfectly opposed to one another. This is due to a few reasons:

  • Our electromagnets are much bigger than the size of our magnets in our linear motor system. Getting powerful, small electro magnets requires tighter windings, more robust wiring, and better casing materials (both thinner and more rugged). All of this generally adds to cost.
  • Secondly, because two of our electro magnets are opposed to one another, this means that the third one (mostly the middle one), is forced to oppose it. Note, we can overcome this by reducing the current flow and strength of the paired magnets. However, this is not as simple as adding a multiple as Maxwell’s Law states it has an exponential relationship.
  • Finally, we don’t want to add a fourth linear electromagnet because of bulk/weight/size. A fourth electromagnet would cause our head to be extremely large (more so than it already is). This is less efficient to move, less applicable and while drastically reducing the range of motion of our motor.

Design reconsideration Link to heading

After the third or fourth design iteration using three magnets, we were able to find smaller electromagnets. This means that we will be using a FOUR ELECTROMAGNET SYSTEM NOW

Control Theory Link to heading

In order to get the correct control, we need to understand how to time the current changes both in amplitude and direction. This means that we first have to understand at what amplitudes that this requires. In order to move the linear motor, based on the changing of force vectors, we want to nearly mirror a sinusoidal motion.

Below is a generic sinusoidal graph.

The red dots represent where the amplitude of the current (and therefore the EM force with respect to Maxwells Equation) should be. We notice a few things with this projection. Firstly, the odd and even electromagnets should be directly opposite but equal amplitude. If we think about our spacing (which is 1.5x magnet width) this means that if the first EM is directly over a north pole, the third EM will be directly over a south pole because it is three magnets away (S, N then S).

However, the relationship should be sinusoidal as the electromagnetic force is an i^2 component. The below graph is what happens when the graph is not

<