[Lisa~]

You should never feel planks in your low back, ever, not during execution or afterward in DOMS. It's a sign that technique was off. Even though you stayed "straight as a board" you were not in correct pelvic tilt from the get-go. You were in anterior pelvic tilt. If you know what I mean, I can probably stop there. Let me know if I need to explain further.

Set up in a posterior pelvic tilt for the plank and maintain that position throughout. You should feel your rectus abdominis first, then you'll find your obliques, then finally your serratus (under your arm pits). If you ever feel anything in your back, then you've changed your body position and you need to either correct it or, if you can't correct it, terminate the set.