Power Loss in an Inductor
POWER LOSS IN AN INDUCTOR
Since an inductor (coil) consists of a number of turns of wire, and since all wire has some resistance, every inductor has a certain amount of resistance. Normally this resistance is small. It is usually neglected in solving various types of ac circuit problems because the reactance of the inductor (the opposition to alternating current, which will be discussed later) is so much greater than the resistance that the resistance has a negligible effect on the current.
However, since some inductors are designed to carry
relatively large amounts of current, considerable power can be
dissipated in the inductor even though the amount of resistance in the
inductor is small. This power is wasted power and is called
The copper loss of an inductor can be calculated by multiplying the
square of the current in the inductor by the resistance
In addition to copper loss, an iron-core coil (inductor) has two iron losses. These are called HYSTERESIS LOSS and EDDY-CURRENT LOSS. Hysteresis loss is due to power that is consumed in reversing the magnetic field of the inductor core each time the direction of current in the inductor changes.
Eddy-current loss is due to heating of the core by circulating currents that are induced in the iron core by the magnetic field around the turns of the coil. These currents are called eddy currents and circulate within the iron core only.
All these losses dissipate power in the form of heat. Since this power cannot be returned to the electrical circuit, it is lost power.
Whenever two coils are located so that the flux from one coil links with the turns of the other coil, a change of flux in one coil causes an emf to be induced in the other coil. This allows the energy from one coil to be transferred or coupled to the other coil. The two coils are said to be coupled or linked by the property of MUTUAL INDUCTANCE (M). The amount of mutual inductance depends on the relative positions of the two coils. This is shown in figure (12). If the coils are separated a considerable distance, the amount of flux common to both coils is small and the mutual inductance is low. Conversely, if the coils are close together so that nearly all the flux of one coil links the turns of the other, the mutual inductance is high. The mutual inductance can be increased greatly by mounting the coils on a common iron core.
Two coils are placed close together as shown in figure (13). Coil 1 is connected to a battery through switch S, and coil 2 is connected to an ammeter (A). When switch S is closed as in figure (13-A), the current that flows in coil 1 sets up a magnetic field that links with coil 2, causing an induced voltage in coil 2 and a momentary deflection of the ammeter. When the current in coil 1 reaches a steady value, the ammeter returns to zero. If switch S is now opened as in figure (13-B), the ammeter (A) deflects momentarily in the opposite direction, indicating a momentary flow of current in the opposite direction in coil 2. This current in coil 2 is produced by the collapsing magnetic field of coil 1.
FACTORS AFFECTING MUTUAL INDUCTANCE
The mutual inductance of two adjacent coils is dependent
upon the physical dimensions of the two coils, the number of turns in
each coil, the distance between the two coils, the relative positions of
the axes of the two coils, and the permeability of the cores.
The COEFFICIENT OF COUPLING
between two coils is equal to the ratio of the flux cutting one coil to the flux originated in the other coil. If the two coils are so positioned with respect to each other so that all of the flux of one coil cuts all of the turns of the other, the coils are said to have a unity coefficient of coupling. It is never exactly equal to unity (1), but it approaches this value in certain types of coupling devices. If all of the flux produced by one coil cuts only half the turns of the other coil, the coefficient of coupling is 0.5. The coefficient of coupling is designated by the letter K.
The mutual inductance between two coils, L1 and L2, is expressed in terms of the inductance of each coil and the coefficient of coupling K. As a formula:
One 10-H coil and one 20-H coil are connected in series and are physically close enough to each other so that their coefficient of coupling is 0.5. What is the mutual inductance between the coils?
SERIES INDUCTORS WITHOUT MAGNETIC COUPLING
When inductors are well shielded or are located far enough apart from one another, the effect of mutual inductance is negligible. If there is no mutual inductance (magnetic coupling) and the inductors are connected in series, the total inductance is equal to the sum of the individual inductances. As a formula:
where LT is the total inductance; L 1, L2, L3 are the inductances of L1, L2, L3; and Ln means that any number (n) of inductors may be used. The inductances of inductors in series are added together like the resistances of resistors in series.
SERIES INDUCTORS WITH MAGNETIC COUPLING
When two inductors in series are so arranged that the field of one links the other, the combined inductance is determined as follows:
The plus sign is used with M when the magnetic fields of the two inductors are aiding each other, as shown in figure (14).The minus sign is used with M when the magnetic field of the two inductors oppose each other, as shown in figure 2-15. The factor 2M accounts for the influence of L1 on L2 and L2 on L1.
Figure (14). - Series inductors with aiding fields.
Figure (15). - Series inductors with opposing fields.
A 10-H coil is connected in series with a 5-H coil so the fields aid each other. Their mutual inductance is 7 H. What is the combined inductance of the coils?
PARALLEL INDUCTORS WITHOUT COUPLING
The total inductance (LT) of inductors in parallel is calculated in the same manner that the total resistance of resistors in parallel is calculated, provided the coefficient of coupling between the coils is zero. Expressed mathematically: