Does the husband owe an expiation after taking back his wife following the second divorce?

So long as the second divorce occurred, and you took back your wife before her 'iddah expired, she has returned to your bond, and there is no need for expiation or the like. Note that the 'iddah of a divorced woman ends with three menstrual cycles, or with childbirth, or with three months for one who does not menstruate.