Filter Inductor Design

In Wave Filter Principles and Limitations of Wave Filters, it was pointed out that inductors for wave filters must have Q great enough to provide low attenuation in the pass band. In design, attention must be given as much to Q as to inductance.

Low-loss core material is essential for high Q. Nickel-iron alloys are widely used; the lamination thickness depends on frequency. At frequencies up to 400 cycles, 0.014-in.-thick laminations are used, and at frequencies higher than 400 cycles, 0.005 in. thick. This is an approximate practical guide.

Fig. 140. Core loss in laminations 0.014 and 0.005 in. thick.

Figure 140 shows how loss varies with thickness, frequency, and flux density. At frequencies higher than 1,000 cycles, flux density must be quite small for low core loss. In the majority of audio applications, low flux density conditions prevail. Under such conditions, core loss is largely eddy-current loss and may be treated as a linear resistance.

Core gaps are used in filter reactors to obtain better Q. For any core, inductance per turn, and frequency, there is a maximum value of Q. The reason for this is that the a-c resistance is composed of at least two elements: the winding resistance and the equivalent core loss. In previous chapters the core loss has been regarded as an equivalent resistance across a winding. But it can also be regarded as an equivalent resistance in series with the winding.

Fig. 141. Shunt and series equivalent core-loss resistance.

Figure 141 shows this equivalence, which may be stated:

For values of Q > 5,

[84]

where

R_sh = equivalent shunt resistance
R_ser = equivalent series resistance
X = winding reactance = 2πfL.

The equivalence depends upon frequency. The formula for large Q may then be changed to

[85]

or Q is proportional to shunt resistance, the winding resistance being neglected. Thus Q can be increased by lowering L, and L is lowered by increasing core gap, but there are limits on the increase of Q that can be obtained in this way.

First, the winding resistance is not negligible. With small gaps, maximum Q is obtained when winding resistance and equivalent series core-loss resistance are equal. For a given air gap there is a certain frequency f_m at which this maximum Q holds. At higher and lower frequencies, the manner in which Q falls below the maximum is found as follows: Let R_c be the coil winding resistance. Then

If for R_ser we substitute the value obtained from equation 84, we have, approximately,

[86]

Equation 86 therefore gives the relation of Q to frequency. When it is plotted on log-log coordinates with frequency as the independent variable,⁽¹⁾ it is symmetrical about the frequency f_m for which Q is a maximum. If the core gap is changed, frequency f_m changes.

Fig. 142. Frequency variation of Q for an iron-core coil with air gaps.

Figure 142 shows how the Q of a small inductor varies with frequency for several values of air gap in the core. All these curves have the same shape, a fact which suggests the use of a template for interpolating such curves.

Another phenomenon that limits Q is the flux fringing at the core gap, the influence of which on inductance was discussed in Chapter 3. As the air gap increases, the flux across it fringes more and more, like that shown in Fig. 143, and L ceases to be inversely proportional to the gap.

Fig. 143. Magnetic flux fringe at core gap.

Some of the fringing flux strikes the core perpendicular to the laminations and sets up eddy currents which cause additional loss. Accurate prediction of gap loss depends on the amount of fringing flux. For laminated cores it can be estimated from

[87]

where

G = a constant (17 · 10 ^-10 for silicon steel)
d = lamination tongue width in inches
μ = permeability
t_e = lamination tks. in inches
f = frequency in cycles
B_m = peak core induction in gauss
l_g = gap length in inches.

In Power Supply Frequency, it was shown that under certain conditions maximum transformer rating for a given size is obtained when core and winding losses are equal. The same would be true for inductors with zero core gap. Similarly it may be shown that, if the core gap is large enough to cause appreciable gap loss, maximum Q is obtained with core, winding, and gap losses equal. In a given design, if this triple equality does not result in the required Q, size must be increased. Losses may be compared by finding either the equivalent series resistances or the equivalent shunt resistances.

(1)	See "How Good Is an Iron-Cored Coil?" by P. K. McElroy and R. F. Field, General Radio Experimenter, XVI (March, 1942). This article also discusses choke design from the standpoint of similitude.