The KAT500 will enter a fine-tune mode if you re-initiate a tune cycle right after the first one has finished.  I don't know if the KAT3 will do that, and I can't test it because mine always finds the primary match the first time, but Elecraft would have a lot of incentive to reuse KAT3 algorithms and code in the KAT500.  I believe the Elecraft tuners use some variant of the steepest descent algorithm which can hang up on a less than optimal solution

73,

Fred ["Skip"] K6DGW
Sparks NV DM09dn
Washoe County

Tom Boucher wrote on 12/7/2021 3:59 AM:
Thanks NK7Z, NK7I and K8CN for responses.

Dave and Rick indicated that running the K3 at a higher power level during
the ATU Tune sequence might solve the problem however, as far as I can see,
the power level is fixed at 5 watts during auto tune and cannot be altered.

Mike, there is no problem with the Ant 1 and Ant 2 ports and manual
selection of L & C works OK. The ATU just won't end up with the right
settings for lowest SWR. The problem has not always existed.

73,
Tom G3OLB




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