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If possible, what are calculations and precautions involved in making a simple transformer without core? The only purpose of this transformer is to trigger an opto-coupler for zero-crossing detection of mains supply.

schematic

simulate this circuit – Schematic created using CircuitLab

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    \$\begingroup\$ Why do you want to avoid a core? For a mains-connected transformer a core will be necessary for a practical design. \$\endgroup\$ – John D May 1 at 14:52
  • \$\begingroup\$ @john-d: space (Lining Box) and cost. \$\endgroup\$ – M.Youcef May 1 at 15:01
  • \$\begingroup\$ This transformer would be big as your house. \$\endgroup\$ – Marko Buršič May 1 at 15:13
  • \$\begingroup\$ Any mains connected transformer will need to have a primary inductance of a henry or so so, if air-cored, think about how many turns are needed. \$\endgroup\$ – Andy aka May 1 at 15:16
  • \$\begingroup\$ @M.Youcef Does your output need to be isolated? There are other ways to get a zero cross signal. \$\endgroup\$ – Aaron May 1 at 15:19
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Since you said that it didn't need to be isolated, maybe something like this would work:

enter image description here

Output looks like this: enter image description here

If you are on 120Vac, then change the 2.2k to ~1.1k.
Again, beware that the left hand side is not isolated from the mains.

Source: https://circuitdigest.com/electronic-circuits/zero-crossing-detector-circuit-diagram

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