## Inductance of Single-Layer Solenoids

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The approximate value of the low-frequency inductance of a single-layer solenoid is*

* Equations and Fig. 1 are derived from equations and tables in Bureau of Standards Circular No. C74.

*L *= *Fn2d*

where,

*L *= inductance in microhenrys,

*F *= form factor, a function of the ratio *dli *(value ofF may be read from Fig. 1),

*n *= number of turns,

*d *diameter of coil (inches) between centers of conductors,

length of coil (inches) = *n *times the distance between centers of adjacent turns.

The equation is based on the assumption of a uniform current sheet, but the correction due to the use of spaced round wires is usually negligible for practical purposes. For higher frequencies, skin effect alters the inductance slightly. This effect is not readily calculated, but is often negligibly small. However, it must be borne in mind that the equation gives approximately the true value of inductance. In contrast, the apparent value is affected by the shunting effect of the distributed capacitance of the coil.

*Example: *Required, a coil of 100 microhenrys inductance, wound on a form 2 inches in diameter by 2 inches winding length. The *dli *= 1.00, and *F *= 0.0173 in Fig. 1.

*n *= *(LIFd)112 *

= [100/(0.0173 x 2)11/2

= 54 turns

Reference to Table 1 will assist in choosing a desirable size of wire, allowing for a suitable spacing between turns according to the application of the coil. A slight correction may then be made for the increased diameter (diameter of form, plus two times radius of wire), if this small correction seems justified.

**Approximate Equation **For single-layer solenoids of the proportions normally used in radio work, the inductance in microhenrys is given to an accuracy of about 1 percent by the formula

*L *= *n2[r2/(9r + 101)] *

* *where

*r d12*and the other quantities are as defined for the previous inductance formula.

**General Remarks **In the use of various charts, tables, and calculators for designing inductors, the following relationships are useful in extending the range of the devices. They apply to coils of any type or design.

(A) If all dimensions are held constant, inductance is proportional to

*n2.*

(B) If the proportions of the coil remain unchanged, then for a given number of turns the inductance is proportional to the dimensions of the coil. A coil with all dimensions

*m*times those of a given coil (having the same number of turns) has

*m*times the inductance of the given coil. That is, inductance has the dimensions of length.

**Decrease of Solenoid Inductance by Shielding* **When a solenoid is enclosed in a cylindrical shield, the inductance is reduced by a factor given in Fig. 2. This effect has been evaluated by considering the shield

to be a short-circuited single-turn secondary. The curves in Fig. 2 are reasonably accurate provided the clearance between each end of the coil winding and the corresponding end of the shield is at least equal to the radius of the coil. For square shield cans, take the equivalent shield diameter (for Fig. 2) as being 1.2 times the width of one side of the square.

*Example: *Let the coil winding length be 1.5 inches and its diameter 0.75 inch, while the shield diameter is 1.25 inches. What is the reduction of inductance due to the shield? The proportion are

(winding length)/(winding diameter) = 2.0

(winding diameter)/(shield diameter) = 0.6

Referring to Fig.2, the actual inductance in the shield is 72 percent of the inductance of the coil in free space.

**Q of Unshielded Solenoid **Fig. 3 can be used to obtain the unloaded Q of an unshielded solenoid.

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