In this paper, we propose a Convolutional Neural Network (CNN) based speaker recognition model for extracting robust speaker embeddings. The embedding can be extracted efficiently with linear activation in the embedding layer. To understand how the speaker recognition model operates with text-independent input, we modify the structure to extract frame-level speaker embeddings from each hidden layer. We feed utterances from the TIMIT dataset to the trained network and use several proxy tasks to study the networks ability to represent speech input and differentiate voice identity. We found that the networks are better at discriminating broad phonetic classes than individual phonemes. In particular, frame-level embeddings that belong to the same phonetic classes are similar (based on cosine distance) for the same speaker. The frame level representation also allows us to analyze the networks at the frame level, and has the potential for other analyses to improve speaker recognition.