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For high temperature superconducting (HTS) motor field coils two types of quench initiation are of practical importance. The first type will be referred to as current induced quench (CIQ). It typically occurs during coil or assembled rotor testing, when the current is increased in small increments until the quench is detected. The current at which the quench starts is referred to as the quench current Iq -. The objective of these experiments is to establish actual limits of operational conditions. The second type will be referred to as temperature induced quench (TIQ) and it is caused by failure or malfunctioning of the cooling system. The examples include vacuum deterioration, cryocooler failure etc. Numerous CIQ experiments -,  have showed a certain pattern of quench event, including a well defined Iq, slow near-linear increase with respect to time of coil voltage and temperature with subsequent transition to fast non-linear voltage and temperature growth. TIQ experiments  showed the same behavior when coil cooling system was shut down while the current was maintained constant. The presented here work is a computational 2-D and 3-D Finite Element Analysis (FEA) study of CIQ and TIQ in HTS coils. The effects of magnetic field, non-homogeneity of tape properties, current sharing and amount of stabilizer are studied. It is shown that the formation and propagation of the normal zone is preceded by the process that we chose to call pre-quench instability. The pre-quench instability starts when the current exceeds Iq and demonstrates experimentally observed features mentioned above. It is shown that with an insufficient amount of stabilizer for second generation (2-G) HTS tapes the pre-quench instability may be practically undetectable. The comparison with some of the experiments ,  is presented. We intensively used the concepts developed in articles -. Many (but not all) our conclusions are in agreement - with -.