Inductors in Series and Parallel by Stan Gibilisco

The Unit of Inductance

When a battery is connected across a wire-coil inductor (or any kind of inductor), it takes a while for the current flow to establish itself throughout the inductor. The current changes at a rate that depends on the inductance: the greater the inductance, the slower the rate of change of current for a given battery voltage.

The unit of inductance is an expression of the ratio between the rate of current change and the voltage across an inductor. An inductance of one henry, abbreviated H, represents a potential difference of one volt across an inductor within which the current is increasing or decreasing at one ampere per second.

The henry is an extremely large unit of inductance. Rarely will you see an inductor anywhere near this large, although some power-supply filter chokes have inductances up to several henrys. Usually, inductances are expressed in millihenrys (mH), microhenrys (uH), or even in nanohenrys (nH). You should know your prefix multipliers fairly well by now, but in case you've forgotten;

1 mH = 0.001 H = 10-3 H,
1 µH = 0.001 mH = 0.000001 H = 10-6 H,
and 1 nH = 0.001 uH = 10 -9 H.

Very small coils, with few turns of wire, produce small inductances, in which the current changes quickly and the voltages are small. Huge coils with ferromagnetic cores, and having many turns of wire, have large inductances, in which the current changes slowly and the voltages are large.

Inductors in Series

As long as the magnetic fields around inductors do not interact, inductances in series add like resistances in series. The total value is the sum of the individual values. It's important to be sure that you are using the same size units for all the inductors when you add their values.

Problem 10-1

Three 40-uH inductors are connected in series, and there is no interaction, or mutual inductances, among them (Fig. 10-3). What is the total inductance?

You can just add up the values. Call the inductances of the individual components L1, L2, and L3, and the total inductance L. Then L = L1 + L2 + L3 = 40 + 40 + 40 + 120 uH.

Inductors in Parallel

If there is no mutual inductance among two or more parallel-connected inductors, their values add up like the values of resistors in parallel. Suppose you have inductances L1, L2, L3, ..., Ln all connected in parallel. Then you can find the reciprocal of the total inductance, 1/L, using the following formula:

1/L = 1/L1 + 1/L2 + 1/L3 + ... + 1/Ln

The total inductance, L, is found by taking the reciprocal of the number you get for 1/L.

Again, as with inductances in series, it's important to remember that all the units have to agree. Don't mix microhenrys with millihenrys, or henrys with nanohenrys. The units you use for the individual component values will be the units you get for the final answer.

Problem 10-3

Suppose there are three inductors, each with a value of 40 uH, connected in parallel with no mutual inductance, as shown in Fig. 10-4. What is the net inductance of the set?

Call the inductances L1 = 40 uH, L2 = 40 uH, and L3 = 40 uH. Use the formula above to obtain 1/L = 1/40 + 1/40 + 1/40 = 3/40 = 0.075.Then L = 1/0.075 = 13.333 uH. This should be rounded off to 13 uH, because the original inductances are specified to only two significant digits.

Stan Gibilisco is one of McGraw-Hill's most prolific and popular authors, specializing in electronics and science topics. His clear, reader-friendly writing style makes his science books accessible to a wide audience, and his background in research makes him an ideal editor for professional references and course materials. He is the author of The Encyclopedia of Electronics; The McGraw-Hill Encyclopedia of Personal Computing; and several titles in the popular Demystified library of home-schooling and self-teaching books. His published works have won numerous awards. The Encyclopedia of Electronics was chosen a "Best Reference Book of the 1980s" by the American Library Association, which also named his McGraw-Hill Encyclopedia of Personal Computing a "Best Reference of 1996." Stan Gibilisco maintains a Web site at www.sciencewriter.net.

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