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.
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.
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:
M = k(L1L2)1/2
where the 1⁄ 2 power represents the square root. The value of Mthus obtained will be in the same size unit as the values of the inductance you input to the equation.
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 in opposing phase, the net inductance is decreased relative to the condition of zero mutual inductance.
When two inductors are connected in series and there is reinforcing mutual inductance between them, the total inductance L is given by the following formula:
L = L1 + L2 + 2M
where L1 and L2 are the inductances, and M is the mutual inductance. All inductances must be expressed in the same size units. When two inductors are connected in series and the mutual inductance is opposing, the total inductance L is given by this formula:
L = L1 + L2 − 2M
where, again, L1 and L2 are the values of the individual inductors. It is possible for mutual inductance to increase the total series inductance of a pair of coils by as much as a factor of 2, if the coupling is total and if the flux reinforces. Conversely, it is possible for the inductances of two coils to completely cancel each other. If two equal-valued inductors are connected in series so their fluxes oppose (or buck each other) and k = 1, the result is theoretically zero inductance.
Suppose two coils, having inductances of 30 μH and 50 μH, are connected in series so that their fields reinforce, as shown in following figure. Suppose that the coefficient of coupling is 0.500. What is the total inductance of the combination?
First, calculate M from k. According to the formula for this, given previously, M = 0.500(50 × 30)1/2 = 19.4 μH. Then figure the total inductance. It is equal to L = L1 + L2 + 2M = 30 + 50 + 38.8 = 118.8 μH, rounded to 120 μH because only two significant digits are justified.
Imagine two coils with inductances of L1 = 835 μH and L2 = 2.44 mH. Suppose they are connected in series so that their coefficient of coupling is 0.922, acting so that the coils oppose each other, as shown in above figure. What is the net inductance of the pair?
First, calculate M from k. The coil inductances are specified in different units. Let’s use microhenrys for our calculations, so L2 = 2440 μH. Then M = 0.922(835 × 2440)1/2 = 1316 μH. Then figure the total inductance. It is L = L1 + L2 − 2M = 835 + 2440 − 2632 = 643 μH.