In this report we present a model for phased array feed (PAF) and compare the model predictions with measurements. A theory for loss-less PAF is presented first. To develop the theory we ask the question -- what is the best $T_{sys}/eta_{ap}$ that can be achieved when a PAF is used on a telescope to observe a source at an angle $theta_s, phi_s$ from the boresight direction ? We show that a characteristic matrix for the {em system} (i.e. PAF+telescope+receiver) can be constructed starting from the signal-to-noise ratio of the observations and the best $T_{sys}/eta_{ap}$ can be obtained from the maximum eigenvalue of the characteristic matrix. For constructing the characteristic matrix, we derive the open-circuit voltage at the output of the antenna elements in the PAF due to (a) radiation from source, (b) radiation from ground (spillover), (c) radiation from sky background and (d) noise due to the receiver. The characteristic matrix is then obtained from the correlation matrices of these voltages. We then describe a modeling program developed to implement the theory presented here. Finally the model predictions are compared with results from test observations made toward Virgo A with a prototype PAF (Kite array) on the GBT (Roshi et al. 2015).
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