Permeability And Frequency

One of the more significant improvements has to do with the method of securing empirically the information needed for a particular lamination structure. This needed information consists of the maximum storage factor Q of a full coil wound on the lamination structure, the frequency at which it occurs, and the law of variation of inductance with center-leg air gap.

Theory

If reference be made to page 9 in the Appendix portion of the March, 1942, article, it will be seen that for a given structure Dc varies inversely while De varies directly as t he product fix. Hence, if frequency is held constant, dissipation factors Dc (copper loss) and De (eddy-current) are functions solely of effective permeability /j.. Thus,

8 ir*fSAa 3Pi109 (27)

(Measurements are postulated at very small B, where hysteresis loss is negligible.) Consequently, the same shape

1P. k. McElroy and R. F. Field, "How Good is an Iron-Cored Coil?" General Radio Experimenter, March, 1942.

of Q curve will be obtained at constant frequency and varying permeability as with constant permeability and varying frequency.

The Q of the coil will vary from a low value with the core interleaved up through a maximum value at some intermediate air gap and down again to a low value at a large air gap.

Technique

It will be necessary to choose an appropriate frequency at which to make the measurements, in order to be sure that the peak occurs somewhere near the geometric center of the inductance range covered. While experience will usually provide a guide to the selection of the measuring frequency, a more reliable one is provided by the expression for fm and a few measurements on various types of cores.

The frequency at which the Q of a coil is a maximum is given by the expression

These frequencies are made lower as the size of the core is increased, as the laminations are made thicker, as the permeability of the core is increased, or as the resistivity of the core is decreased. Both of the last two reasons, for instance, operate to make an A-metal core require a lower measuring frequency than a silicon-steel one.

If the frequency chosen were that corresponding to Qm for an interleaved core, the measured points would all fall at and on one side (the left) of the maximum, while for the frequency corresponding to no iron (air core) the distribution of points would be on the other side of the maximum. A convenient frequency is the geometric mean of the two values of f™ corresponding to Qm for the interleaved core and Qm for the air core. This frequency corresponds to a m which is the geometric mean between that of the interleaved material and the effective value for the air core.2

2This value is greater than unity for at least two reasons:

(1) The turns are not concentrated in a single layer, but fill the whole window, thus yielding a larger inductance; and (2) the flux path external to the center leg ia a much greater fraction of the total.

For example, measurements of a number of coils with cores of 4% silicon steel indicate that the ratio of inductance, and hence equivalent permeability, between air core and full-interleaved core is about 65. For an initial permeability of the core material of 470, the equivalent air core permeability is then about 7. Using the geometric mean of 60 in Equation 23 gives the optimum frequency for taking data for the humped Q curve. For A-metal, the permeability values are 2500 and 7, giving a geometric mean of about 130.

The following table indicates the optimum measuring frequencies for gathering data for humped curves as functions of the lamination size and material (in the order of size). The dimensions of the four standard GR lamination shapes here discussed appear in Figure 1. Lamination 4% Silicon Steel A-Metal 746 1805 cps 782 cps

345 823 439

485 612 327

These figures can be compared with the notes following shortly below and with the distribution of plotted points on the curves in Figures 2 to 11.

In the examples illustrated by those figures, various combinations of lamination shape and material appear. It should be noted, in connection with all the curves mentioned here, that a point representing measurements on an air-core coil (ferro-magnetic material completely removed) generally lies right on the curve which goes through the points representing conditions where there is some ferromagnetic material in the circuit. These points are identified on the plots by an adjacent letter "A." A letter "B" on each plot will be placed adjacent to the point representing a butt joint in the center leg (interleaved outside legs), and a letter "I" beside the point representing the completely interleaved condition.

Some notes correlating the symmetry of the humped curves with the measuring frequency are given in the captions. Unless otherwise stated, fair symmetry was achieved.

Plotting the Curves

The plots are all made with values of Q as ordinates, and values of inductance as abscissae. Since the only information to be extracted from the humped curves is the maximum Q reached, there is no

Figure 1. Dimensions of core laminations used in measurements.

necessity to reduce the data so that plotting can be done against the effective permeability to which the inductance is, by definition, strictly proportional.

Each of Figures 2 to 8 will be seen to carry two additional curves. The second curve, which slopes downward from left to right, is obtained (from the original data) by plotting measured center-leg air gap in mils against measured inductance (at the initial permea-ability level)3. These curves would have a constant downward 45° slope (inductance inverse with air gap), were it not for the effects of fringing, of the reluctance of the magnetic material, and of the equivalent reluctance of a butt joint.

The third curves, sloping upward to the right, are derived from the second ones. For them, the ordinates are the same as for the second ones, namely, measured air gap. The abscissae, however, are frequencies in cycles according to the scale at the top of the figure. These frequencies are the frequencies at which the maximum Q occurs for various air gaps. This curve can be used to decide what air gap will give the best compromise between stability of inductance against voltage changes and high Q at the desired frequency. When the air gap has been decided upon, the frequency at which the maximum Q will occur can be read from the curve.

3No points for air-core, interleaved, or butt joints are included, since it ia impossible to assign specific values of air gap to these conditions on the log-log chart.

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