Inductance Calculations

Frederick W Grover

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486318356
• Topic: Tecnología e ingeniería
• Subtopic: Ciencias aplicadas

This authoritative compilation of formulas and tables simplifies the design of inductors for electrical engineers. It features a single simple formula for virtually every type of inductor, together with tables from which essential numerical factors may be interpolated. Although compiled in the 1940s, before calculators and computers, this book provides fundamental equations that professionals and practitioners can use to produce algorithms for computer programs and spreadsheets.
Starting with a survey of general principles, it explains circuits with straight filaments; parallel elements of equal length; mutual inductance of unequal parallel filaments and filaments inclined at an angle to each other; and inductance of single-layer coils on rectangular winding forms. Additional topics include the mutual inductance of coaxial circular filaments and of coaxial circular coils; self-inductance of circular coils of rectangular cross section; mutual inductance of solenoid and a coaxial circular filament and coaxial single-layer coils; single-layer coils on cylindrical winding forms; and special types of single-layer coil. An esteemed reference, this volume belongs in the library of every electrical engineer.

Chapter 1

GENERAL PRINCIPLES

The electromotive force induced in a circuit A when the current in a circuit B is changed is proportional to the rate of change of the linkages of the flux set up by the current in B with the turns of the circuit A. If the circuits are linked through a core of iron or other magnetic material, nearly all of the flux ϕ, produced by the current, will link with the N turns of circuit A and the induced electromotive force is quite closely

. With magnetic materials, however, it is necessary to know the permeability of the material, which is a function of the magnetizing current and has to be determined by measurement for the current in question. Furthermore, although the knowledge of the permeability permits the reluctance of a complete magnetic circuit of iron to be estimated, the case of a straight magnetic core with the flux lines completed through the air is still further complicated by the difficulty of estimating the reluctance of the air path. It is, therefore, impracticable to do more than to make the roughest of calculations of the flux and therefore of the mutual inductance of circuits coupled by cores of magnetic materials. The treatment of standard apparatus employing complete magnetic circuits of iron, or circuits in which only a short air gap is included, is based on measurements of exciting current and leakage reactance.

With circuits free from iron, the case is different. The magnetic induction at any point due to current in a circuit B is directly proportional to the current i and, although the linkages of flux with the elements of a circuit A will vary, in general, from point to point, the total linkage with the circuit A is capable of being expressed as a constant M times the current. Thus, the electromotive force induced in A may be written as

. The constant M is known as the coefficient of mutual induction or the mutual inductance. If the induced electromotive force is expressed in volts and the current in amperes, then M is expressed in henrys. A mutual inductance of one henry gives rise to an induced electromotive force of one volt, when the inducing current is changing at the rate of one ampere per second. For many simple circuits of only a few turns of wire, a more convenient unit of mutual inductance is the millihenry (mh), which is one thousandth of a henry, or the micro-henry (μh), which is the millionth part of a henry. The latter is especially appropriate for expressing the mutual inductance of straight conductors or small coils of few turns.

The adjective “mutual” emphasizes the fact that if the electromotive force induced in circuit A by a current changing at the rate of one ampere per second in circuit B is equal to e, the same emf e is induced in circuit B when a current is made to change at the rate of one ampere per second in circuit A.

The mutual inductance may also be considered as the number of flux linkages with the circuit A due to unit current in circuit B. In the simple case where B has N₁ turns and circuit A, N₂ turns, the windings being concentrated, it is evident that the magnetic induction at any point due to unit current in B is proportional to N₁ and, therefore, the linkages with each turn of A are proportional to N₁. The total flux linkages with A, due to unit current in B are, consequently, proportional to N₁N₂. If, on the other hand, unit current is set up in coil A, the linkages with each turn of B are proportional to N₂, but there are N₁ turns in B, so that the total number of linkages with B is also proportional to N₁N₂. In general, the magnetic induction is a function of the dimensions of the inducing circuit and the number of linkages with this is a function of the dimensions of the linking circuit.

When the rôles of the two circuits are interchanged, the change in one of these factors is exactly compensated by the change in the other, and the mutual inductance is the same, whichever is the inducing circuit and whichever the circuit in which the electromotive force is induced.

The total electromotive force induced in a circuit at any moment is equal to the algebraic sum of the electromotive forces induced in the various elements of the circuit, opposing electromotive forces being regarded as of opposite signs. If we confine the consideration to frequencies such that the circuit dimensions are negligible with regard to the wave length, the magnetic induction at every point of the field is in phase with the current. In consequence, the induced electromotive forces are at all points in phase.

The total induced emf

may be considered as a summation of the elementary induced emfs around the circuit. This consideration defines what is meant by the mutual inductance of a circuit on a part of another circuit. The partial mutual inductance is the contribution made by the element to the total mutual of the circuit of which it forms a part.

Furthermore, the magnetic flux linked with a circuit element may be considered as the resultant of the fluxes contributed by the separate elements of the inducing circuit. Since, under the quasi-stationary condition assumed, the currents in all the elements are in phase, so are the flux contributions of the separate elements. That is, the mutual inductance of an element of a circuit with the inducing circuit is the algebraic sum of the mutual inductances of the separate elements of the inducing circuit with the circuit element of the second circuit.

Assuming one circuit to be made up of elements A, B, C in series, and the other of elements a, b, c in series, the total mutual inductance is

M = M_Aa + M_Ab + M_Ac + M_Ba + M_Bb + M_Bc + M_Ca + M_Cb + M_Cc,

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