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I'm designing an electronic product that has a circular housing. There needs to be a PCB inside this form. In order to reduce costs as much as possible for high volume production (5K-10K), what shape PCB would cost the least?

A circle PCB works, but so does an octagon and hexagon. I'd like to know if any of these shapes lead to a lower cost due to how production is done.

Panelization and depanelization is not trivial,

When a board is panelized for production, it looks something like (Picture from surfacemountprocess.com)

.

The intent is to maximize panel usage and minimize the number of cuts (tool touches) needed to separate the board. Ultimately what determines the cost of the board will be how many can fit on a panel, if you require extra tongues and spacing for clean depanelizing, you may pay more.

Also you fit fewer round PCB's on a rectangular panel than square PCB's

Ultimately a circular PCB panel will be something like this, with the red parts being the mouse-bite or routing tongues.

There are roughly 3 or 4 approaches that can be used for de-panelization

1. V Groove approach, only good for straight lines, and rectangular PCB's. A narrow groove is cut into both sides of the board along your outline . Images are taken from Murata MLCC Datasheet https://search.murata.co.jp/Ceramy/image/img/A01X/G101/ENG/GRM1882C2A102JA01-01A.pdf

This V Groove can be broken in a few ways

1.a. Hand breaking which is also known "Snapping"

1.b. Cutting Wheel (Paper cutter per @ScottSeidman)

1.c. A router can also be used (alternate to b)

2. "Mouse-bite" approach. Perforation is placed along the board outline, A router is used to remove all the extra board material, then the board is snapped along the perforation, usually by hand. The edge can be cleaned up with router or by sanding

Both of these approaches put flex the board, and still require a small connecting piece with the board.

3. 100% Routing, requires special jigs to hold the PCB, but the entire circle is routed out of the design

Generally a fabricator will charge extra for router use, but this is such a normal process now-a-days that it may be already in the cost. However total routing (without mouse bites or a small v groove edge), is a large premium. The tradeoff there is no board flex and cleanest edge.

I would echo @ScottSeidman suggestion to approach your fabricator for input on pricing and suggestions.

Most of the panelized circular (or similar) PCBs I've seen at Asian factories use a straightforward X-Y array with relatively generous spacing between the circles.

Usually you want to maintain the panelization intact through pick and place assembly and perhaps beyond that to testing and so on, so the board has to be connected well enough it doesn't come apart, and should have panel fiducials and tooling strips compatible with the requirements of your assembly line or assembly house.

You could probably use an X-Y array with a combination of routed and V-groove to get the best of both worlds and avoid those mouse bites entirely.

Anyway, without doing much work, my thought is probably an elongated hex with flattened sides (so really an octagon) in an X-Y array (not packed) with V-groove and routed roughly triangular "holes" is optimal by a small margin. To a first approximation, cost is based on the rectangular area that will totally enclose your PCB if they are panelizing.

V-grooves add a bit of cost but can reduce area by the 'kerf' of the routing tool. Vendors will want to use a relatively large routing tool. Also V-grooves usually are required to be completely across the panel, vertically or horizontally, so they're not suitable for hexagonal packing. It might slightly complicate setup for pick and place if they're not on an X-Y array too, but probably not very significant.

Depanelizing tools are like specialized pizza cutters that open the V-grooves. They work in straight lines. I suspect, but don't know for a fact, that the circle would add expense.

I also suspect that the more operations, the higher the cost. You want to give the depanelizer a flat edge to work with at all times, so the operator doesn't have to hunt around for the right position.

You might also think of whether there's a clever way to panelize your board to make all the cuts horizontal or vertical.

Other than that, I recommend working with your fabricator. First, start by getting a quote for what you like. It might not be as expensive as you think. If you don't like it, work with them to minimize cost.