Some of the ways that you can get free PlayStation Network cards are to join websites or use apps that offer them as payment for completing certain tasks. These tasks include shopping on their site, using the app, surfing the web, watching videos and taking surveys or quizzes.
A number of sites offer PSN cards as rewards for participating in offers on the site. Some of the websites that offer legitimate ways to obtain PlayStation cards are Points2Shop, Rewards1, Prize Rebel and SwagBucks. Whaff Rewards is an Android app that offers money toward prizes such as PlayStation cards for using and maintaining the app.
All of theses sites contain offers you can complete to obtain currency to redeem for PlayStation cards. In addition, Points2Shop and Swagbucks have exercises such as watching videos, surfing the web or taking surveys that can earn you site currency you can use to obtain a PSN card.