The manual analysis and design of microwave circuits are generally tedious and error prone. The Smith chart provides a very useful graphical tool to these problems. A great deal of knowledge can be acquired from a Smith chart e.g. standing wave ratio, single and double stub tunings and much more. Although Smith charts are valuable and contain significant amount of information, inaccurate observations can lead to erroneous results and frustration. In this work, an artificial neural network (ANN) model of the Smith chart is achieved. In this model, the two bilinear transformations between the rectangular Z(Y)-plane and the reflection coefficient Gamma-plane are employed in both directions for the training data. In the current work, the feed forward Multilayer Perceptron (MLP) type of neural network is utilized with the two hidden layers, five inputs and two outputs. Input impedance variations along the transmission line are given as a typical example for the utilization of the Neural Smith chart.