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In real-world circuits, there is almost always some mutual inductance between or among solenoidal coils. The magnetic fields extend significantly outside such coils, and mutual effects are difficult to avoid or eliminate. The same is true between and among lengths of wire, especially at high ac frequencies. Sometimes, mutual inductance has no detrimental effect, but in some situations it is not wanted. Mutual inductance can be minimized by using shielded wires and toroidal inductors. The most common shielded wire is coaxial cable. Toroidal inductors are discussed later in this chapter.

Mutual inductance can operate either to increase the inductance of a pair of series connected inductors, or to decrease it. This is because the magnetic fields might reinforce each other, or they might act against each other. When two inductors are connected in series, and there is reinforcing mutual inductance between them, the total inductance L is given in the formula: L L1 L2 2M

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Coefficient of Coupling The coefficient of coupling, symbolized k, is an expression of the extent to which two inductors interact. It is specified as a number ranging from 0 (no interaction) to 1 (the maximum possible interaction). Two coils separated by a sheet of solid iron, or by a great distance, have a coefficient of coupling of zero (k = 0); two coils wound on the same form, one right over the other, have the maximum possible coefficient of coupling (k = 1). Sometimes, the coefficient of coupling is multiplied by 100 and expressed as a percentage from 0 to 100 percent. Mutual Inductance The mutual inductance between two inductors is symbolized 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. In the case of two inductors having values of L1 and L2 (both expressed in the same size units), and with a coefficient of coupling equal to 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:

5

where L1 and L2 are the values of the individual inductors, and M is the mutual inductance. All inductances must be expressed in the same size units.

M = k(L1L 2 )1/2 where the 1 2 power represents the square root. The value of M thus obtained will be in the same size unit as the values of the inductance you input to the equation.

Problem 10-5

Radio design played a role in improving spectrum utilization. The first systems, such as the one in St. Louis, used a scheme called nontrunked radio. In nontrunked radio systems, the available spectrum was divided into channels and groups of users were assigned to those channels. The advantage of this technique is that non-trunked radios were relatively inexpensive. The downside was that certain channels could become severely overloaded, while others remained virtually unused. The invention of trunked radio relieved this problem immensely. Trunked radios were frequency agile, meaning that all radios could access all channels. When a user placed a call, the radio unit would search for an available channel and use it. With the arrival of this capability, blocking became a non-issue. The downside, of course, was that frequency-agile radios, because of their more complex circuitry, were significantly more expensive than their nontrunked predecessors. Most of this work was conducted during the turbulent 1960s and led to Bell Labs introduction of Improved Mobile Telephone Service (IMTS). IMTS used two 30-kHz, narrowband FM channels for each conversation and provided full-duplex talk paths, direct dialing, and automatic trunking. Introduced commercially in 1965, IMTS is considered to be the predecessor of modern cellular telephony.

Effects of Mutual Inductance Mutual inductance can either increase or decrease the net inductance of a pair of series-connected coils, compared with the condition of zero mutual inductance. The magnetic fields around the coils either reinforce each other or oppose each other, depending on the phase relationship of the ac applied to them. If the two ac waves (and thus the magnetic fields they produce) are in phase, the inductance is increased compared with the condition of zero mutual inductance. If the two waves are

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