Capacitors, Magnetic Circuits, and Transformers is a free introductory textbook on the physics of capacitors, coils, and transformers. See the editorial for more information....

Core Size and Stability of Reactors

Reactors that, for example, are used for controlling current in arc furnaces or a-c welders carry no appreciable d-c component of current under steady-state conditions and no d-c magnetic saturation will result. Nevertheless, if such a reactor is to be stable, the inductance for a given tap setting should remain nearly constant at a given frequency for normal fluctuations in the applied voltage. This means that if there are no air gaps in the core of the reactor, the operation must be such that the flux density is confined to the linear portion below the knee of the magnetization curve.

Figure 5-12 shows the manner in which variations in the voltage applied to a given number of turns in an iron-core reactor affect the inductance. If N is the number of turns in the reactor winding, A the cross-sectional area of the core at right angles to the flux path, and l the length of the flux path, the maximum instantaneous flux linkage is

 [5-56]

where Bm is the maximum instantaneous flux density and the induced voltage is

 [5-57]

also, the maximum value of the exciting current is

 [5-58]

where Hm is the magnetic field intensity required to produce the maximum flux density Bm in the core.

When hysteresis and eddy-current effects are neglected, the vertical axis of the BH curve of the iron can be provided with two scales in addition to the B scale. These are for the maximum flux linkage λm and the rms voltage E, and are obtained simply by multiplying the 5-scale by the area turns to give λ, and by 4.44f times the area turns to give E, as shown in Eqs. 5-56 and 5-57. The horizontal axis can likewise be provided with a scale to give the maximum value Im of the exciting current by multiplying the Hm scale by the ratio of length to turns as indicated in Eq. 5-58.

 Figure 5-12, Magnetization curve of iron-core reactor.

In Fig. 5-12, ab represents the normal voltage when projected on the voltage scale of the vertical axis. When ab is projected on the flux linkage scale of the vertical axis, it represents the normal value of the maximum instantaneous flux linkage. The current required to produce this value of flux linkage is represented by ob projected on the current scale of Fig. 5-12. Although we are dealing with a nonlinear magnetic circuit, for the present purpose let the inductance be defined as the flux linkage per ampere. Then the normal inductance is given by

 [5-59]

Suppose, because of line fluctuations or some other reason, the voltage rises from its normal value ab to that of a'b'. The inductance for this increased voltage is now less than normal, as shown by

 [5-60]

If, on the other hand, the voltage drops to a value of a"b" the inductance will be greater than normal, rising to a value of

 [5-61]

This shows that, when there are no air gaps and the iron is operated in the saturated region of the magnetization curve, relatively small variations in the voltage can lead to sizeable changes in the inductance of the reactor. This seems to suggest that the operation should be confined to the region below the knee of the magnetization curve where the characteristic is nearly linear. An alternative to this might be operation in region of high magnetic saturation to keep the inductance fairly constant. This, however, means high flux densities and corresponding large core losses, which are objectionable because of heat dissipation and poor efficiency. However, operation in the unsaturated region requires a core of excessive mass, as is shown in the following. If the resistance of the winding is neglected, the applied voltage and the induced emf are equal. Then, if E is the rated rms value of the voltage and I the rated rms value of the current, we have

 [5-62]

where L is the inductance of the reactor and ω is 2πf radians per sec. Equation 5-62 neglects the effects of the nonlinearity of the iron. These effects, however, are small enough in the unsaturated region so that they may be neglected in this discussion.

The value of inductance alone does not determine the size of the reactor. The size is also a function of current and frequency. In order to take these into account, multiply both sides of Eq. 5-62 by I2 to give

 [5-63]

which expresses the volt ampere rating of the reactor for a given frequency. The magnetic energy stored in the iron at any instant is proportional to the square of the instantaneous current and is a maximum when the current reaches its maximum. Hence

 [5-64]

However, the maximum energy stored in the field on the basis of Eq. 3-72 is expressed by

 [5-65]

where Vol is the volume of the core.

A comparison of Eqs. 5-63, 5-64, and 5-65 shows that

 [5-66]

If Bm and Hm are in webers per square inch and ampere turns per inch respectively, the volume in cubic inches per volt ampere is given by

 [5-67]

When Bm is expressed in lines per sq in. and Hm in ampere turns per in. according to the Mixed English System of Units, the volume in cubic inches per volt-ampere is expressed by

 [5-68]

 Example 5-2: (a) Determine the volume of a core comprised of U.S.S. Annealed Electrical Sheet Steel (see Fig. 3-12) for a reactor rated at 90 v, 200 amp, and and 60 cps. The maximum flux density Bm is to be 80,000 lines per sq in. Neglect the effects of air gaps introduced by the joints in the core. (b) What is the weight of the core if its density is 0.276 Ib per cu in. ? Solution: A flux density of Bm = 80 kilolines per sq in. requires a magnetic field intensity Hm of about 10 amp turns per in. for annealed sheet steel according to Fig. 3-13.

The values computed in Example 5-2 show the size of the core to be excessive when the operation is restricted to the region below the knee of the magnetization curve. Actually, a value of 80 kilolines per sq in. is well in the knee of the curve and a lower value of flux density, perhaps 60 kilolines or less, is required to insure stability.

The excessive amount of iron is due to the high value of the relative permeability in the unsaturated region. Since Bm = μrμ0Hm, Eq. 5-68 can be expressed in terms of B2m as

 [5-69]

where μr is the relative permeability of the core material and μ0 is the permeability of free space having the value of 3.19 in the Mixed English System of Units, Hence, Eq. 5-69 can be reduced to

 [5-70]

where Bm is expressed in maxwells or lines per square inch.

Equation 5-70 shows that, for a given value of maximum flux density Bm, the volume of the core is directly proportional to the relative permeability μr This relationship suggests the use of air, or some material with magnetic characteristics similar to those of air, instead of iron for the core. Air alone or other nonmagnetic material will not do, as it is extremely difficult to arrange the exciting winding in a manner such as to confine the flux path to a definite volume of air. This can be achieved by means of a toroid, an arrangement that is impractical for most reactors. For that reason, as well as for structural considerations, both iron and air are used, the main function of the iron being to impart the proper size and configuration to the one or more air gaps that may be required in the core and to accommodate the exciting winding. The air gap gives the magnetic circuit a more linear characteristic, thus reducing the variation of inductance with voltage fluctuation.

The value of μr is very nearly unity for air. If the air gap or air gaps are so proportioned in relation to the iron that the reluctance of the iron is negligible compared with that of the air, and if the dimensions of the one or more gaps are such that the effect of fringing can be neglected, then expression for the volume of air gap per volt ampere is obtained by substituting μr = 1.00 in Eq. 5-70. This yields

 [5-71]

 Example 5-3: Determine the volume of air to be used in the core of a reactor of the same rating as the one in Example 5-1 if the maximum flux density is to be 80 kilolines per sq in. Neglect the reluctance of the iron, leakage, and fringing. Solution: The reactor has the following rating When these values are substituted in Eq. 5-71, the volume of the air is found to be

Although Eq. 5-71 is useful for determining the volume of the air in the core, it gives no indication as to the configuration, i.e., length and area, of the one or more air gaps. The cross sections of reactor cores are generally rectangular and the air gaps are usually between plane parallel surfaces of iron. So once the volume is known, the length and cross-sectional area can be determined from the number of turns in the exciting winding, the current, the voltage, and the maximum flux density as follows. On the basis of sinusoidal voltage and current we have

 [5-72]

and

 [5-73]

also

 [5-74]

from which

 [5-75]

In the Mixed English System, the area in square inches is

 [5-76]

and the length in inches is

 [5-77]

Equations 5-76 and 5-77 show that in a reactor, of a given frequency rating, volt ampere rating, and flux density Bm, the cross-sectional area of the core is proportional to the volts per turn and the length of air gap is proportional to the number of turns. The volume of the winding is a function of the number of turns; the volume of the core is a function of the volts per turn. The coordination of these two volumes is usually dictated by considerations of economical size and construction of the reactor.

Last Update: 2011-08-01