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I tried to put together a quote for a 4-layer PCB with controlled impedance (Displayport, so 100ohm differential +/- 10%). I used the online quote tool on Advanced Circuits (4pcb.com). The quote couldn't be generated automatically, and when I called they said is because I had checked the "controlled impedance" box but I still had "FR-4" (default) as the dielectric. They said the quote can only be generated if I select 370HR as the dielectric or some other specific one (I think the other was 185HR). Why is that the case? What is special about these "HR" dielectrics that make them better suited for controlled impedance?

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370HR is a specific laminate product made by Isola Group. It has a well-known Dk of about 4.04.

"FR-4" is a generic term for a wide variety of products from numerous manufacturers. They have Dk values typically in the range of 4.0 - 5.0.

When you specify "FR-4" you're basically telling the shop to use whatever material they have on hand. This reduces the price of your board. But it also means that on one run you might get a Dk of 4.2 and on another run you might get 4.8, because they happened to have different "FR-4" products in stock that week. That means that if you tune the trace width for 50 ohm impedance control for the first run, the impedance will be different on the second run (i.e. the impedance is not well controlled at all).

If you want impedance control, you should specify a specific laminate with a known Dk value.

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