Problem 3 A 3-phase, 50 HP, 415Vrms (line-to-line), 50HZ, 4-pole Y-connected induction machine has the following equivalent circuit parameters: Rs = 0.15625ohm Rr'= 0.185ohm Xm = 17.75ohm XI1= 0.7825ohm X12'= 0.7825ohm Rc = 148.75ohm If the machine is operating as a generator at the rated conditions (50 HP mechanical power and 415V terminal voltage), find, a) The slip. the slip in % is: b) Input torque. the input torque in N.m c) the stator current amplitude. the current in A is d) The rotor speed in rpm the rotor speed in rpm is

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Gradient of a Scalar Field

Consider a room or a place where the temperature is denoted by a scalar field T, such that at each point (x, y, z), the temperature is given as T(x, y, z). It is independent of time, in which the temperature rises most quickly. The gradient of T will show the direction at that point by moving away from (x, y, z). How far will the rise in temperature in that direction be determined by the gradient magnitude? Consider a surface that has a height above sea level at point (x, y) which is given by H(x, y). At that point, pointing in the direction of the steepest slope or grade is the gradient of H. The magnitude of the gradient of the vector gives the steepness of the slope at that point. How a scalar field changes in another direction can be measured by the gradient by taking a dot product rather than just the direction of the greatest change?