Now, Ethereum can handle only 8-12 transactions per second by using PoW Consensus. I know that Kovan and Rinkeby is using PoA Consensus. PoA is lightweight than PoW. So how many transactions per second can Kovan or Rinkeby Network handle?
Both Kovan and Rinkeby have a gas limit of ~7 million, which is about 330 regular ETH transfers per block (21,000 gas each). Kovan has a ~4 second blocktime and Rinkeby has a ~15 second blocktime, so the tx/s for Kovan is ~80 and for Rinkeby is ~20. These are just based on the max gas/block. Since they are both PoA, they might impose their own artificial limits though.
Note that the Ethereum mainnet actually has the capability of ~20 tx/s if all the tx were regular ETH transfers.