This paper considers the problem of optimum design of the impedance matching, lumped-parameter, wide-band transformer model with all transmission zeros at infinity. The spiral-tape-winding transformer is compared with the wire-winding transformer with an electrostatic screen, and it is shown that the former has no advantage over the latter except (perhaps) at quite low impedance levels. The low-pass-filter model of the transformer is considered in more detail than is customary, and it is shown that the HF performance of any lumped-parameter wideband transformer may be extended in proportion to the number of branches selected for the ladder network model of the transformer; building out the transformer-coupling network to include additional inductors and capacitors thus extends the frequency response. For a given network model and a given low-end cutoff frequency, it is shown that the high-end cutoff frequency is a function of the peripheral distance around the winding, the mean impedance level of the source and load, and the turns ratio. From this relation it is shown that the most favorable turns ratio is unity and that the most favorable impedance levels are low ones.