Selection of air gap in the iron core of switching power supply transformer

Switching power supply Selection of transformer core air gap

As mentioned above, the input voltage of the single-shot switching power supply transformer is a unipolar voltage pulse. When the pulse amplitude and width exceed the volt-second capacity of the transformer, the transformer core will be magnetically saturated. The simplest way to prevent the switching transformer core from being magnetically saturated is to leave an air gap in the transformer core or use a reverse magnetic field.

When there is an air gap in the transformer core, since the magnetic permeability of air is only a few thousandths of the magnetic permeability of the core, the magnetomotive force almost all falls on the air gap; therefore, the average magnetic permeability of the transformer core with an air gap will be greatly reduced; not only will the residual magnetic flux density be reduced, but the maximum magnetic flux density Bm can reach the saturation magnetic flux density Bs; thereby increasing the magnetic flux increment, and the transformer core is no longer prone to magnetic saturation. As shown in Figure 2-24, this is the working principle diagram and magnetization curve diagram of the transformer core with an air gap.

In Figure 2-24-a, assuming that l1 is the air gap length and the total length of the transformer core magnetic circuit is lc, the magnetic flux potential of the magnetic circuit is:

△Hlc=△B(l1-lc)/μc +△Bl1/μ0

In the above formula, μc is the magnetic permeability of the transformer core; μ0 is the magnetic permeability of air, which is approximately equal to 1; lc is the total length of the transformer core magnetic circuit; l1 is the length of the air gap; △H is the increment of magnetic field intensity; △B is the increment of magnetic flux density.

Since lc >> l1, μ0 ≈ 1, so (lc - l1) ≈ lc, so the above formula can be simplified to:

In the above formula, μa is the average magnetic permeability of the air-gap core, μc is the magnetic permeability of the transformer core, l1 is the length of the air gap, and lc is the total length of the transformer core magnetic circuit.

In formula (2-72), since μc is not a constant, we cannot use the derivative method to treat l1 as a variable to find the maximum value of μa; in addition, finding the maximum value of μa is not our main purpose; our wish is to require that the average magnetic permeability μa can also reach the maximum under the condition of the maximum magnetic flux density increment △B.

Let's look at Figure 2-24-b again. In Figure 2-24-b, the dotted line represents the hysteresis loop when the transformer core has no air gap, and the solid line represents the hysteresis loop when the transformer core has an air gap, where the magnetization curve oa is the basic magnetization curve of the core with an air gap. The basic magnetization curve here is not exactly the same as the initial magnetization curve. The basic magnetization curve here is equivalent to the geometric mean of the magnetization curve, so as to be used to analyze the relationship between the magnetic field intensity increment △H and the magnetic induction density increment △B.

[page]Obviously, there is a set of corresponding hysteresis loops for each value of the air gap length; however, no matter how large the air gap length is, the maximum magnetic flux density Bm of the core can only reach the Bs value corresponding to the magnetic saturation of the core, and it will not continue to increase with the increase of the air gap length l1; and the residual magnetic flux density Br of the core will not drop significantly due to the increase of the air gap length l1. Therefore, l1 should have an optimal value, which should take into account both the maximum magnetic flux density increment △B and the conditions for the average magnetic permeability μa to reach the maximum.

In order to find the best value, we can continuously draw tangents along the basic magnetization curve oa, such as the tangent ob in the figure; the tangent value tgβ of the angle β between the tangent and the H-axis is the magnetic permeability of this point; when the tangent point of the tangent is located at one-half of the maximum magnetic flux density increment △B, the tangent value tgβ of this point can be considered to be equal to the average magnetic permeability μa; from this we can see that the average magnetic permeability μa is always less than or equal to the tangent value tgβ.

If we define the flux density increment △B and magnetic field intensity increment △H corresponding to the maximum tangent value tgβ as the best working point of the core, then the corresponding optimal value of l1 can be obtained through the tangent ob. It can be proved that the tangent ob passing through the origin is the tangent with the largest tangent value, because the basic magnetization curve in reality does not exist. The basic magnetization curve is equivalent to the geometric mean of the magnetization curve, which is an exponential curve that changes according to the charging law of the capacitor (please refer to the content of the chapter "2-1-1-9. Switching power supply transformer core hysteresis loop measurement"); in addition, the defined optimal working point is the working point corresponding to the minimum value of the air gap length l1.

It can be seen from Figure 2-24-b and formula (2-72) that when μcl1/lc>>1, the average magnetic permeability μa of the gapped iron core is basically inversely proportional to the length of the air gap l1; therefore, the value of μcl1/lc is exactly the tangent value tgα of the angle α between the tangent ob and the B axis in Figure 2-24-b; △H represents μcl1, and △B represents lc. The multiplication of μc and l1 just normalizes the units of the two orthogonal lines H and B, otherwise the angle between them is meaningless.

It can be seen from Figure 2-24-b that when tgα≈1/2, l1 is the optimal value, which is actually the minimum value of l1; because the average magnetic permeability μa will decrease as l1 increases. Therefore, the optimal value (or minimum value) of l1 is obtained by the following formula:

l1/lc≈2/μc （2-73）

Substituting the result of formula (2-73) into formula (2-72), it can be obtained that when l1 is the optimal value, the average magnetic permeability μa of the iron core with air gap is exactly equal to one third of the magnetic permeability of the iron core without air gap.

It is particularly pointed out here that the result given by equation (2-73) is the condition for finding the maximum value of the average permeability μa of the air-gap iron core under the condition of initially satisfying the requirements of the magnetic flux density increment; of course, the smaller the air gap length, the greater the average permeability μa. However, in actual work, the value of μa should be smaller than this value, because a certain margin should be reserved for the air gap length, and the working point of the transformer core cannot always work at the edge of the optimal value; therefore, the maximum magnetic flux density increment △B and the maximum magnetic field intensity increment △H of the transformer core in actual work will exceed the range of the conditions given by equation (2-73); therefore, the air gap length l1 obtained by equation (2-73) is also the lowest limit value.

For example, when the permeability of the core without an air gap is μυ=1000, the ratio is l1/lc=2•10-3. If the total length of the transformer core magnetic circuit is lc=120mm, the minimum air gap length l1 of the core should be equal to 0.24mm. In practical applications, l1=0.5mm can be taken, which is twice the minimum air gap length. At this time, the average permeability μa is only 1/5 of the core permeability μc, that is, μa=200.

Another simplest way to prevent magnetic saturation in the core of a switching transformer is to use a reverse magnetic field, install a permanent magnet in the core of the transformer, or add a reverse DC to the primary and secondary coils of the transformer, and this DC generally needs to be isolated by a choke inductor, or powered by a constant current source. Since adding a reverse DC to the primary and secondary coils of the transformer will reduce the efficiency of the switching power supply and increase the cost, most switching power supplies currently do not use this method; it is only used in some occasions that require a relatively large dynamic range of magnetization, a particularly large output power, and do not need to consider the cost.

By the way, the air gap length of the core of the forward switching transformer is different from that of the flyback switching transformer; the air gap length of the core of the forward switching transformer is completely to meet the requirements of the maximum magnetic flux density increment, while the air gap length of the core of the flyback switching transformer, in addition to meeting the requirements of the maximum magnetic flux density increment, must also meet the requirements of the minimum inductance. Generally, the air gap length of the core of the flyback switching transformer is larger than that of the core of the forward switching transformer.