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Sigmoid curves are also common in statistics as cumulative distribution functions (which go from 0 to 1), such as the integrals of the logistic density, the normal density, and Student's t probability density functions. The logistic sigmoid function is invertible, and its inverse is the logit function.
The standard logistic function is the logistic function with parameters =, =, =, which yields = + = + = / / + /.In practice, due to the nature of the exponential function, it is often sufficient to compute the standard logistic function for over a small range of real numbers, such as a range contained in [−6, +6], as it quickly converges very close to its saturation values of 0 and 1.
The generalized logistic function or curve is an extension of the logistic or sigmoid functions. Originally developed for growth modelling, it allows for more flexible S-shaped curves. The function is sometimes named Richards's curve after F. J. Richards, who proposed the general form for the family of models in 1959.
The curve represents a boundary of all operating points in the MW/MVAr plane; it is typically drawn with the real power on the horizontal axis, and, for the synchronous generator, resembles a letter D in shape, thus another name for the same curve, D-curve. In some sources the axes are switched, and the curve gets a dome-shaped appearance.
Synchronous impedance curve (short-circuit characteristic curve), showing armature current as function of field current. The curve is obtained by rotating the generator at the rated RPM with the output terminals shorted and the output current going to 100% of the rated for the device (higher values are typically not tested to avoid overheating ...
The curve is typically plotted alongside the synchronous impedance curve. [ 1 ] At the low field, the permeable iron in the magnetic circuit of the generator is not saturated , therefore the reluctance almost entirely depends on the fixed contribution of the air gap , so the part of the curve that starts at the point of origin is a linear "air ...
The Johnson's S U-distribution is a four-parameter family of probability distributions first investigated by N. L. Johnson in 1949. [ 1 ] [ 2 ] Johnson proposed it as a transformation of the normal distribution : [ 1 ]
Compensated demand curve; Duck curve; Engel curve; Hubbert curve; Indifference curve; J curve; Kuznets curve; Laffer curve; Lorenz curve; Phillips curve; Supply curve. Aggregate supply curve; Backward bending supply curve of labor