Some small transformers for low-power applications are constructed with air between the two coils. Such transformers are inefficient because the percentage of the flux from the first coil that links the second coil is small. The voltage induced in the second coil is determined as follows.
where N is the number of turns in the coil, dφ/dt is the time rate of change of flux linking the coil, and J is the flux in lines. At a time when the applied voltage to the coil is E and the flux linking the coils is J lines, the instantaneous voltage of the supply is:
The maximum value of φ is given by:
Number of turns in the primary winding
Number of turns in the secondary winding
Assuming BMAX primary = BMAX secondary
The ability of iron or steel to carry magnetic flux is much greater than air. This ability to carry flux is called permeability. Modern electrical steels have permeabilities in the order of 1500 compared with 1.0 for air. This means that the ability of a steel core to carry magnetic flux is 1500 times that of air. Steel cores were used in power transformers when alternating current circuits for distribution of electrical energy were first introduced.
B = μo H
where μo is the permeability of free space = 4 π x 10–7 Wb A–1 m–1.
Replacing B by φ / A and H by (I N) / d, where
φ = core flux in lines
N = number of turns in the coil
I = maximum current in amperes
A = core cross-section area
d = mean length of the coil in meters
A = area of the core in square meters
Then, the equation for the flux in the steel core is:
where μr = relative permeability of steel ≈ 1500.
Since the permeability of the steel is very high compared with air, all of the flux can be considered as flowing in the steel and is essentially of equal magnitude in all parts of the core. The equation for the flux in the core can be written as follows:
E = applied alternating voltage
f = frequency in hertz
N = number of turns in the winding
μ = core permeability. This is basically the increase in
inductance which is obtained when the inductor is
wound on a core instead of just air. A µof 2000, for
instance, will increase inductance by 2000:1.
l = magnetic path length. In a simple toroid this is the
average circumference of the core (see sketch).
A = cross-sectional area of the core (see sketch).
g = thickness of air gap (if any) used to increase the energy storage capability of a core (see sketch).
B = magnetic flux density in the core. If B rises too high, the core will "saturate", allowing µand therefore L, to drop drastically.
N = number of turns in the winding.
I = instantaneous winding current.
VC = volume of actual core material.
In most converter applications, the required inductance is determined by constraints such as maximum output power, ripple requirements, input voltage and transient response. I is determined by load current. For purposes of this discussion, therefore, it is assumed that L and I are known quantities, and the quantities to be determined are N, A, l, VC and g.
Inductance is determined by core permeability, path length, cross sectional area and number of turns:
A properly selected inductor must provide the right value of L without exceeding the maximum limit on flux density, (BM). In other words, the core must not "saturate" under conditions of peak winding current (IP). By combining the formulas for inductance and flux density, it can be shown that core volume (VC) required is a direct function of the energy to be stored by the inductor:
In any given application, the value of IP can be determined
from maximum load current and duty cycle. Formulas for
maximum IP are provided in the individual sections on
In many cases, the maximum load current is much less than the LT1070 is capable of providing. A core designed to handle only full load current may saturate under overload or short-circuit conditions. The cycle-by-cycle current limiting of the LT1070 protects the regulator against damage even with saturated cores. This considerably improves the reliability of converters using the LT1070 and eases the design complexity.
Although core volume is the main criterion for selecting a given core, the volume still consists of two variables, A and l. For minimum overall size of the inductor it is generally best to increase A as much as possible at the expense of l, thereby minimizing the number of turns required to obtain the desired inductance. This process can be taken only so far before the "window" in the core becomes too small to accommodate the windings.
The energy storage capability of a core can be increased by "gapping" the core. A significant portion of the total energy is stored in the air gap. The drawback of a gapped core is that the effective permeability drops, requiring many more turns to achieve the required inductance. More turns require a larger winding window. The overall size of the inductor, however, can be considerably less with a properly gapped core, especially with high permeability core material. The formula for inductance with a gapped core is:
Inductance drops by the factor:
Increase in energy storage is equal to the decrease in permeability.
There are several practical limits on the amount which gap
size may be increased. First, large gaps require many more
turns to achieve the same inductance. This requires smaller
diameter wire which increases copper losses from I2R
heating. Secondly, with large gaps the effective gap size is
considerably less than the actual gap because of fringing
fields around the gap.
When using commercially available cores, data sheet information on length, A and μ is usually given in effective values.The theoretical value of µ, for instance, is the bulk value for the core material. The effective value for a single piece core may approach the bulk value, but with 2-piece cores, the tiny air spaces left in the mating surfaces can reduce the effective permeability by as much as 2:1. This may sound unreasonably pessimistic, but a core with bulk μ = 3000 and length = 1.5", will lose half its permeability for g = 0.0005". Data sheets for gapped cores list effective values of µfor each gap size to make calculations simple. They may also list a parameter, "inductance per (turn)2" for each gap to further simplify inductance calculations.
There are two types of core material which are effectively self-gapped: iron powder and permalloy. These materials distribute the gap evenly throughout the core, allowing gapless core to be constructed with much higher energy storage capability. The permeability of this material is much reduced, but if the winding window will accommodate the extra turns, the current handling capability of the inductor will be much higher for the same inductance compared to a high-μ formulation.
Iron powder cores are cheaper than ferrite and can be custom tailored quickly, but high core loss limits their application to low AC flux density applications such as inductors. A significant advantage of iron powder is that it saturates very "softly," preventing catastrophic total loss of inductance for large overcurrent conditions. Note that commercially available powdered iron inductors are typically "optimized" so that core losses and winding (I2R) losses are the same order of magnitude. Core loss is dependent on peak-to-peak ripple current which depends on the voltage-time product applied to the inductor. The inductors are therefore specified for a maximum DC current and a maximum (volt * microsecond) product to limit heating. For applications which require highest possible efficiency, consider using oversized cores or permalloy, which is more expensive, but has much lower core loss. Consult with inductor manufacturers about trading off DC current for ripple current, or vice versa.
The simplest way to select an inductor is to find an off-theshelf unit that meets the minimum inductance and current requirements. This may not be cost effective, however, if the standard types are not fairly close to your requirements. The next best approach is to have the unit custom wound by one of the many companies in the business. They will select the best core and winding combination for your particular application. A third approach is to scan the literature for standard core types which you can custom wind to meet your particular requirement. This is a quick way to get a prototype up and running. It can also be very cost effective for some production situations. At the end of this application note is a list of core and inductor/transformer manufacturers
The procedure for selecting a do-it-yourself core starts with defining the values of peak winding current and inductance. If the LT1070 is to be used at or near full output power, peak winding current will approach 5A, so a conservative value of 5A should be used for core calculations. If external current limiting is used or if output power levels are lower, peak winding currents can be calculated from the equations supplied in the discussions of each topology. Likewise, inductance values are calculated from specific equations in these sections. Actual values for L generally fall into the range of 50μH to 1000μH, with 200μH to 500μH being most typical
For ferrite cores, the next step is to calculate the core volume required to prevent saturation:
Powdered iron cores, because of their high core loss and ability to operate at very high DC flux densities, generally have a different design procedure based on temperature rise due to core loss and winding loss. AC flux densities generally need to be kept below 400 gauss. This leads to a volume formula based on AC flux density:
To reduce core size, inductance (L) must be increased. This seems backwards according to the formula, but ∆I is inversely proportional to L, so the (∆I)2 term drops rapidly as L is increased, reducing required core volume. The penalty is increased wire (copper) loss due to the increased turns required.
After a tentative core is selected based on volume, a check
must be done to see if the power losses from the winding(s)
and the core itself are within the allowed limits.
The first step is to calculate the number of turns required:
To calculate wire size, the usable winding window area (Aw) must be ascertained from the core dimensions. Many data sheets list this parameter directly. The usable window area must allow for bobbin thickness and clearances. Total copper area is only about 60% of window area due to air gaps around the wire. We can now express the required wire gauge in terms of N and Aw.
The next step is to determine the number of winding layers. This is determined by bobbin length, or toroid inside circumference:
Enter the given values
The reason for calculating layers is that the AC copper losses are very dependent on the number of layers in a winding. To calculate AC losses, a table is used (Figure 48) which requires a factor K:
D = wire diameter or foil thickness
For foil conductors, FP is 1. For round wires it is equal to: