Problem 3 - A 500-kV, 300-km, 60-Hz three-phase overhead transmission line, assumed to be lossless, has a series inductance of 0.36mH/km per phase and a shunt capacitance of 0.0115 uF/km per phase. If the receiving-end load is 800 MW at 500-kV and 0.9 power factor lagging, determine (a) The surge impedanceZc(N) (b) The SIL (MW) (c) The line-line sending end voltage (kV) (d) The transmission efficiency of the line (e) The line's steady-state stability limit (MW) (f) The transmission line angle (power angle) d

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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?