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1962 1969 The Internet is first conceived in the early 60s. Under the leadership of the Department of Defense s Advanced Research Project Agency (ARPA), it grows from a paper architecture into a small network (ARPANET) intended to promote the sharing of supercomputers among researchers in the United States. 1963 1964 1965 Beatles play for the Queen of England. Dr. Strangelove portrays nuclear holocaust, which new networks must survive. The DOD s Advanced Research Project Association begins work on ARPANET. ARPA sponsors research into a cooperative network of time-sharing computers. U.S. Surveyor probe lands safely on moon. First ARPANET papers presented at Association for Computing Machinery Symposium. Delegates at a symposium for the Association for Computing Machinery in Gatlinburg, TN discuss the first plans for the ARPANET. First generation of networking hardware and software designed. ARPANET connects first four universities in the United States. Researchers at four U.S. campuses create the first hosts of the ARPANET, connecting

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 inductance, 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 during the calculation process. Once you have completed the calculation, you can convert the result to any inductance unit.

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In practical electrical circuits, there is almost always some mutual inductance between or among coils when they are wound in a cylindrical shape. The magnetic fields extend significantly outside solenoidal coils, and mutual effects are almost inevitable. The same is true between and among lengths of wire, especially at very-high, ultra-high, and microwave radio frequencies. Sometimes, mutual inductance is all right, and doesn t have a detrimental effect on the behavior of a circuit. But it can be a bad thing.

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Problem 10-3 Suppose there are three inductors, each with a value of 40 H, connected in parallel with no mutual inductance, as shown in Fig. 10-4. What is the net inductance of the combination Let s call the inductances L1 = 40 H, L2 = 40 H, and L3 = 40 H. Use the preceding formula to obtain 1/L = 1/40 + 1/40 + 1/40 = 3/40 = 0.075. Then L = 1/0.075 = 13.333 H. This should be rounded off to 13 H, because the original inductances are specified to only two significant digits. Problem 10-4 Imagine four inductors in parallel, with no mutual inductance and values of L1 = 75.0 mH, L2 = 40.0 mH, L3 = 333 H, and L4 = 7.00 H. What is the net inductance of this combination

188 Inductance Mutual inductance can be minimized by using shielded wires and toroidal inductors. The most common shielded wire is coaxial cable. Toroidal inductors are discussed a little later in this chapter.

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The coefficient of coupling, specified by the letter k, is a number ranging from 0 (no coupling) to 1 (maximum possible coupling). Two coils that are separated by a sheet of solid iron would have essentially k 0; two coils wound on the same form, one right over the other, would have practically k 1.

You can use henrys, millihenrys, or microhenrys as the standard units in this problem. Suppose you decide to use henrys. Then L1 = 0.0750 H, L 2 = 0.0400 H, L 3 = 0.000333 H, and L4 = 7.00 H. Use the preceding formula to obtain 1/L = 13.33 + 25.0 + 3003 + 0.143 = 3041.473. The reciprocal of this is the inductance L = 0.00032879 H = 328.79 H. This should be rounded off to 329 H. This is only a little less than the value of the 333 H inductor alone. If there are several inductors in parallel, and one of them has a value that is much smaller than the values of all the others, then the total inductance is a little smaller than the value of the smallest inductor.

The mutual inductance is specified by the letter M and is expressed in the same units as inductance: henrys, millihenrys, microhenrys, or nanohenrys. The value of M is a function of the values of the inductors, and also of the coefficient of coupling. For two inductors, having values of L1 and L2 (both expressed in the same size units), and with a coefficient of coupling k, the mutual inductance M is found by multiplying the inductance values, taking the square root of the result, and then multiplying by k. Mathematically, M k (L1L2)1/2

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