We analyze a class of distributed quantized consensus algorithms for arbitrary networks. In the initial setting, each node in the network has an integer value. Nodes exchange their current estimate of the mean value in the network, and then update their estimate by communicating with their neighbors in a limited capacity channel in an asynchronous clock setting. Eventually, all nodes reach consensus with quantized precision. We start the analysis with a special case of a distributed binary voting algorithm, then proceed to the expected convergence time for the general quantized consensus algorithm proposed by Kashyap et al. We use the theory of electric networks, random walks, and couplings of Markov chains to derive an O(N3 log N) upper bound for the expected convergence time on an arbitrary graph of size N, improving on the state of art bound of O(N4 log N) for binary consensus and O(N 5) for quantized consensus algorithms. Our result is not dependent on the graph topology. Simulations are performed to validate the analysis.