I am not an electronic engineer. If you feel that I have done anything wrong or made any invalid assumptions please leave a (kind) comment and I'll think about how best to amend the article to fix any mistakes. Alternatively you can find me here where I am an active member.
Now that we all know how unqualified I am to be doing any of this, lets get into things
What am I even doing?
Winding pickups as a hobbyist is a pain in the ass. No way am I winding thousands of turns by hand, only to have it sound like shit and need to start again. The sensible alternative is to build some sort of coil winder using stepper motors and micro-controllers, which has been done very successfully in the past. These are therefore well documented, and consequentially not of much interest to me. The solution I have decided to choose is a painful one, although it should allow for much more precise tuning of parameters than any physical winding setup. Let me introduce you to PCB inductors.
PCBs as inductors
A traditional inductor consists of a coil of wire wrapped around a core. If this sounds like a guitar pickup to you that's because guitar pickups are essentially inductors. With the right mindset and surrounding circuit, any inductor could be used as a guitar pickup.
In recent years, PCB inductors have become more and more widespread and are commonly used in everyday life anytime you use RFID or NFC. If you have a contact less credit card, an ID badge that you can scan wirelessly, a phone that supports NFC, or any number of devices that can communicate over short ranges, they will feature a PCB inductor.
The basic principle of a PCB inductor is that instead of having a number of loops of wire stacked on top of each other, you have a single spiral trace printed. This spiral trace acts no differently to any other spiral of copper, and therefore has inductive properties. I would propose that a stack of spiral traced multi layer PCBs would give enough inductance to allow it to act as a pickup for a conventional electric guitar or bass. The rest of this article will largely be calculations of the parameters of such an inductor, as well as a discussion of the various assumptions made along the way.
Coil properties
Physical dimensions and coil parameters
Let's start by looking at a typical coil on a humbucking pickup. Luckily
Seymour Duncan provide a number of technical drawings on their website. From the drawing, it can be seen that a standard humbucker coil has dimensions of roughly 68x18mm, with a bobbin spacing of 10mm. Standard slugs are 5mm in diameter, and allowing for a 0.5mm border between the slugs and edge gives an internal coil limit of 56x6mm. Since this is a cutout in the center of the main PCB, roughly 5mm of space is available for coils.
But how many coils can fit in 5mm?
To work this out we need to know two things, the minimum trace width and the minimum trace separation. These can be found by checking on the suppliers website, and my supplier of choice quotes 0.09mm as the minimum for both of these. Since we've been fast and loose with the dimensions above, I'm going to assume that each coil needs a total of 0.2mm. 0.09mm for the coil, 0.09mm for the space to the next coil and 0.02mm to contain the sum total of my electronics knowledge. This means we can have a total of 25 coils per PCB layer.
Assuming a pickup height of 16mm, and a PCB thickness of 0.8mm (huh those numbers line up pretty well... ...what a coincidence...) a total of 20 stacked PCBs would be within the realms of possibility. 4 Layer PCBs are commonly available, and would allow for a total of 2000 windings of copper per pickup. This is not quite the 10 000 or so commonly found in regular pickups so some form of active preamp is needed. In order to figure out what sort of preamp, we need to start doing some more advanced maths.
Lets consider the case of a single layer of a single pcb initially, and once we've worked out its properties, we can start stacking them.
Maths level 1 - Resistance and capacitance
Copper, due to the fact that it is decidedly not a superconductor (at least not at ambient conditions) has a small amount of resistance. This is usually negligible but, when you're dealing with long coils of wire, it is large enough that it accounts for the resistance portion of the pickup model discussed in previous posts.
If, like me, you have studied physics to at least A level (if you're the type of person who's overly into freedom you might call this AP physics), you should recall from school that for a given size of conductor, its resistance, length and cross sectional area are related by a property called resistivity. OK, so in order to find the resistance we need to find the total length of wire. If you expect me at this point to pull out a formula for the length of a rectangular spiral of a given pitch then you can head out at this point. I'm a physical chemist dammit and we're making some fucking assumptions here because the maths on the lengths of spirals is stupid.
Now that my opinions on assumptions that simplify things are known, lets start making some. First of all, since the vast majority of our copper is straight traces, lets assume that it's all one long straight rectangle, since the corners contribute far more to the complexity of the calculation than they will to the overall trace length. Secondly, since our trace width is equal to the separation, let's assume that exactly half of our available area is covered in copper. With an outer boundary of 68x18mm, and an inner boundary of 58x8mm. This leaves us with 380mm^2 of copper coated area. Converting the area of one long rectangle of copper to its length is trivial when you already know the width. Per layer of PCB there is an upper bound of 4.25m of copper.
To convert this to resistance, we still need the cross sectional area though. Luckily we already know that the width is 0.09mm. The height will depend on the fab house. The manufacturer I plan on using lists their inner layer thickness as 0.0175mm, and the outer layer thickness as 0.035mm. Since a 4 layer board consists of 2 inner and 2 outer layers, an average thickness of 0.02625 can be used when working out the resistance of a single layer. Using these parameters a total resistance of 30ohm per layer, or 2.4Kohm for a stack of 80 layers can be calculated.
When working out capacitance, we need to first figure out where the capacitance
comes from. Most of you will probably know that when you have two metal plates of area A separated by distance d they have a capacitance that is related to A and d by the following formula
C=kεA/d
where C is capacitance, ε is the permittivity of free space and k is a modifier to account for the permittivity of different inter plate materials.
Since we have many layers of stacked coils of wire, we can assume our PCB can be modeled as shown on the right. Each layer forms a capacitor with the layers above and below it, in parallel to the resistance of the layer. This of course is not strictly true, as it assumes the resistance is all at once rather than distributed evenly as is the case in real life, however this will have little impact on value calculations.
Lets assume that our spirals are stacked directly on top of one another, they aren't, but for a rough calculation of values they're close enough. This means that the plate area is equal to the area of copper we used to work out the resistance, or 380mm^2. The boards are 0.8mm thick, with 4 layers per board, so the layer separation is 0.267mm. The k factor for this material is quoted by the fab house as 4.5. Plugging those into our formula from earlier, we can calculate the inter layer capacitance as 56pF, with a total stacked capacitance of 0.71pF.
It's worth pointing out that there will be some inter loop capacitance within the layers themselves, as the loops consist of flat lengths of copper separated by a distance, but the length of wire between these is short enough that it's effectively shorted, and can safely be ignored.
Inductance
The per pickup inductance was determined empirically to be approximately 10H (see part 4). This is likely to be higher than the true value, but should at least be of the correct order of magnitude.
Comparison to regular coils
My preferred humbucker is the dimarzio super 2. According to this forum post it has an inductance of 5.49H, an overall resistance of 8.4Kohm and a capacitance of 56.6pf unloaded. These parameters give a resonant peak at 2.3kHz when loaded. The stack of coils we have would have a resistance of 4.8Kohm (since the coils are in series we need to add two to each other), 20H inductance and a capacitance of 0.35pf. While these are not very similar, the orders of magnitude are close enough that with some extra active circuitry I firmly believe a stack of PCBs can function as a replacement for traditional guitar pickups
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