Switching Power Supply Principle and Design (Series 62)

When the maximum value Hm of the magnetic field strength on the surface of the core or core piece is higher than the average value Ha of the magnetic field strength, the difference is:

The ratio of this value to the magnetic field intensity increment ∆H is equal to: μaδ2/12ρcτ, which characterizes the influence of eddy currents and is proportional to the average magnetic permeability μa and the square of the iron core thickness δ, and inversely proportional to the resistivity ρc of the iron core material and the pulse width τ.

According to formula (2-62), the magnetic field on the surface of the iron core or iron core piece consists of two parts:

(1) The average magnetic field, which grows linearly with time and is induced by a fixed electromotive force in the coil;

(2) The constant part, which does not change with time, is formed by the demagnetization field generated by the compensating eddy currents.

Corresponding to the two parts of the magnetic field on the surface of the iron chip, we can regard them as being generated by and two parts of the current respectively. According to Ampere's circuit law: the line integral of the magnetic field intensity vector along any closed path is equal to the algebraic sum of the currents passing through the area enclosed by the closed path. And Kirchhoff's law of magnetic circuit: in the magnetic field loop, the algebraic sum of the magnetic flux potential NI (N is the number of coil turns, I is the current intensity) in any winding direction is always equal to the algebraic sum of the magnetic voltage drop Hili (Hi is the magnetic field intensity, li is the average length of the magnetic field intensity Hi in the magnetic circuit). That is:

Hm=N*i/l =N(iμ+ib)/l (2-64)

In formula (2-64), l is the average length of the magnetic circuit; i =iμ +ib, iμ is the excitation current in the transformer coil; ib is the current flowing through the transformer coil due to the influence of eddy current.
According to formula (2-62) and formula (2-7), we can get:

Figure 2-20-a is the equivalent circuit diagram of the magnetization process at the input end of the switching transformer when it is affected by eddy current, drawn based on equations (2-67) and (2-68).

In Figure 2-20-a, Rb is the eddy current loss equivalent resistance, and N is the transformer primary coil. It can be seen that due to the influence of eddy current loss, when the transformer core is magnetized, it is equivalent to an eddy current loss equivalent resistance Rb connected in parallel with the transformer primary coil N.

Figure 2-20-b more vividly equates the eddy current loss to a transformer secondary coil N2 that provides energy output to the loss resistor Rb2. The current flowing through the transformer secondary coil N2 can generate current ib1 in the transformer primary coil N1 through electromagnetic induction.

According to formula (2-66) and Figure 2-20, the eddy current loss of the transformer can be obtained as:

In the formula (2-69), Sl=Vc is the volume of the transformer core, S is the area of ​​the transformer core, l is the average length of the magnetic circuit, δ is the thickness of the core, N is the number of turns of the transformer primary coil, ρc is the resistivity of the core, τ is the pulse width, and ∆B is the increment of magnetic flux density.

From this, we can see that the eddy current loss of the transformer core is proportional to the increment of magnetic induction and the volume of the core, proportional to the square of the thickness of the iron core, and inversely proportional to the resistivity and the square of the pulse width.

It is worth noting that the property representing the area S in the above formulas can represent the cross-sectional area of ​​a certain iron core or the total area of ​​the transformer core. When S is the total area of ​​the transformer core, the above result is equivalent to the algebraic sum of the eddy current losses of many individual iron cores. Similarly, the δ representing the thickness of the iron core in the above formulas can represent the thickness of a certain iron core or the total thickness of the transformer core, because the value of the thickness δ of the iron core is arbitrary.

However, when the total area of ​​the transformer core is equal, the eddy current loss of the transformer core composed of one iron core or multiple iron cores of the same thickness is different. For example, when the total area of ​​the transformer core is equal, the eddy current loss of the transformer core composed of one iron core is 4 times the eddy current loss of the transformer core composed of two iron cores; if the ratio of the number of iron cores of the two is 3 times, then the ratio of eddy current loss is 9 times. It can be seen that the eddy current loss decreases according to n2, where n is the number of transformer core chips.

In practice, using formula (2-69) to calculate the eddy current loss of the switching transformer still has certain limitations, because the mutual influence of the eddy current magnetic field between the two iron core pieces is not considered in the derivation of formula (2-69). In principle, the degree of mutual influence of the eddy current magnetic field between the iron core piece in the middle of the transformer core and the iron core piece at the edge is different; and it is impossible for the iron core pieces to be completely insulated from each other.

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In addition, the core materials used in most switching transformers are basically ferrite magnetic materials. These transformer cores made of ferrite are made according to the ceramic production process. The ferromagnetic mixed material is first stamped into shape and then sintered at high temperature. Therefore, it is a whole, or it is divided into two parts for easy installation.

If the shape of the ferrite transformer core is regarded as a cylinder, then the Maxwell one-dimensional equations (2-50) and (2-51) can be regarded as the electromagnetic field energy propagating and radiating from the center of the cylinder to the surroundings; in this way, the cylindrical transformer core is equivalent to a combination of multiple cylinders with different inner and outer diameters and thickness variables of δ. Alternatively, the entire ferrite transformer core can be regarded as a single cylinder with a thickness of d/2, where d is the diameter of the cylinder.

Figure 2-21 is a schematic diagram for calculating the magnetic field distribution of a certain cross section in the iron core of a ferrite cylindrical transformer. The dotted line in the figure indicates that the alternating magnetic field induces eddy currents inside the transformer core. We use the same method to integrate equation (2-58) representing the magnetic field distribution from (2-59) to find the average value, and then find the integral constant c2, that is, we can find the magnetic field distribution in the cylindrical iron core:

In formula (2-70), ∆H is the increment of magnetic field strength in the transformer core, d is the diameter of the cylindrical core, μa is the average magnetic permeability of the transformer core, ρc is the resistivity of the core, and τ is the pulse width.

The above formula (2-70) represents the magnetic field distribution diagram of the cylindrical iron core section along the x-axis direction. In fact, the magnetic field distribution is symmetrical about the center in the xy plane of the entire iron core section. In this way, the distribution function H(x,y) surface of the magnetic field intensity in the cylindrical transformer iron core in the xy plane is equivalent to the new figure obtained by rotating the function curve of Figure 2-19-a around the center of the circle.

Figure 2-22-a and Figure 2-22-b are the surface graphs of the function H(x,y) of the horizontal distribution of the magnetic field strength in the cylindrical transformer core and the curve graphs of the function H(t) of the time distribution.
Based on the above analysis, we can use the same method to calculate the eddy current loss of the cylindrical transformer core:

In formula (2-71), is the volume of the transformer core, S is the area of ​​the transformer core, l is the average length of the magnetic circuit, d is the diameter of the cylindrical core, ρc is the resistivity of the core, τ is the pulse width, and ∆B is the flux density increment.

From this, we can also draw the conclusion about the cylindrical transformer core: the eddy current loss of the cylindrical transformer core is proportional to the magnetic induction increment and the volume of the core, proportional to the square of the core diameter, and inversely proportional to the resistivity and the square of the pulse width.
Alternatively, the eddy current loss of the cylindrical transformer core is proportional to the magnetic induction increment and the fourth power of the core diameter, and inversely proportional to the resistivity and the square of the pulse width.

There is no essential difference between equation (2-71) and equation (2-69) in principle, so the equivalent circuit of Figure 2-20 is also valid for equation (2-71).

Although the above analysis of the working principle of eddy current does not seem very complicated, it is very difficult to accurately calculate the energy of eddy current loss. This is because it is difficult to accurately measure the loss resistance of the transformer core, especially because the core materials used in most switching transformers are basically ferrite magnetic materials; these ferrite transformer cores are made of a mixture of various ferromagnetic metal materials and non-metallic materials, and then the ferromagnetic mixed materials are stamped into shape according to the ceramic production process, and finally sintered at high temperature.

Since ferrites are metal oxides, most metal oxides have the common properties of semiconductor materials, that is, the resistivity changes with temperature, and the rate of change is very large. Thermistors are manufactured based on these properties. Every time the temperature doubles, the resistivity will drop (or increase) several times or even hundreds of times. Most thermistor materials are also metal oxides, so ferrites also have the properties of thermistors.

Although the resistivity of the ferrite transformer core is very high at room temperature, it will drop rapidly when the temperature rises; it is equivalent to the Rb eddy current equivalent resistance in Figure 2-20-a becoming smaller, and the current flowing through Rb increases; when the temperature rises to a certain limit, the effective inductance of the transformer primary coil drops to almost 0, which is equivalent to the magnetic permeability also dropping to 0, or the primary and secondary coils of the transformer are short-circuited. The temperature at this time is called the Curie temperature, represented by Tc. Therefore, the resistivity and magnetic permeability of ferrite are unstable, and the operating temperature of the ferrite switch transformer cannot be very high, generally not exceeding 120 degrees.

Figure 2-23 is a graph showing the change in initial magnetic permeability μi of the H5C4 series magnetic core of Japan TDK Corporation's high magnetic permeability material as a function of temperature.

By the way, the initial magnetic permeability μi in Figure 2-23 is generally obtained by testing with a magnetic ring as a sample. The frequency of the test signal is generally low, only 10kHz, and the maximum magnetic permeability is generally selected as the result during the test; therefore, the magnetic permeability of the switching transformer core in actual applications is not that high.