This paper addresses the problem of the stability and the performance analysis of N-nodes Cartesian networks of self-sampled all digital phase-locked loops. It can be demonstrated that under certain conditions (such as filter coefficients value) a global and a local synchronization can be obtained. Our approach to find the optimal conditions consists in analyzing an corresponding linear average system of the Cartesian network rather than constructing a piecewise-linear system which is extremely difficult to analysis. The constructed corresponding system takes into account the non-linearity of the network and especially the self-sampling property. It is then analyzed by linear performance criteria such as modulus margin to guarantee a robust stability of the Cartesian network. The reliability of our approach is proved by transient simulations in networks of different sizes.