# Comparing PRAM and Circuit Complexity, $NC^i$

I wondered about the following quote from NC (Wikipedia):

$$NC^i$$ is the class of decision problems decidable by uniform boolean circuits with a polynomial number of gates of at most two inputs and depth $$O(\log^i n)$$, or the class of decision problems solvable in time $$O(\log^i n)$$ on a parallel computer with a polynomial number of processors.

Are these classes actually the same?

Looking at the proofsketch of Lemma 2.4.2 in Limits to Parallel Computation, we have a logarithmic depth overhead when converting from the PRAM model to a circuit. The reason seems to be a uniform cost measure for the PRAMs operations. Hence, I would expect $$NC^{i + 1}$$ to be the class of decision problems solvable in time $$O(\log^i n)$$ (uniform cost measure) on a parallel computer with a polynomial number of processors.

If the classes are different, does the additional requirement of a logarithmic cost measure fix this?