Analysis of Eddy Current Losses in Switching Power Supply Transformers

　　The eddy current loss of the switching power supply transformer accounts for a large proportion of the total loss of the switching power supply. How to reduce the eddy current loss of the switching power supply transformer is an important part of the design of the switching power supply transformer or the switching power supply. The principle of eddy current loss in transformer production is relatively simple. In addition to being a good magnetic conductive material, the transformer core is also a conductor. When the alternating magnetic lines of force pass through the conductor, an induced electromotive force will be generated in the conductor. Under the action of the induced electromotive force, a loop current will be generated in the conductor to make the conductor heat up. This phenomenon of alternating magnetic lines of force passing through the conductor and generating induced electromotive force and loop current in the conductor is called eddy current, because the loop current it generates is not output as energy, but is lost in its own conductor.

　　There are differences in the methods for calculating the eddy current loss of a single-excitation switching power supply transformer and the eddy current loss of a dual-excitation switching power supply transformer. However, the method used to calculate the eddy current loss of a single-excitation switching power supply transformer can be used to calculate the eddy current loss of a dual-excitation switching power supply transformer with only a slight change. For example, the bipolar input voltage of the dual-excitation switching power supply transformer is regarded as two unipolar input voltages with different polarities, so that the eddy current loss of the dual-excitation switching power supply transformer can be calculated. Therefore, the following only analyzes the eddy current loss calculation of the single-excitation switching power supply transformer in detail.

　　When a DC pulse voltage is applied to both ends of the primary coil of the transformer, an excitation current flows through the primary coil of the transformer, and generates a magnetic field intensity H and a magnetic flux density B in the transformer core, both of which are determined by the following formula:

　　B =ΔB*t/τ +B(0) (2-44)

　　H = ΔH*t/ΔH + H(0) (2-45)

　　In the above formula, ΔB and ΔH are the increment of magnetic flux density and magnetic field intensity respectively, τ is the DC pulse width, B(0) and H(0) are the magnetic flux density B and magnetic field intensity H at t = 0 respectively.

　　In order to reduce eddy current loss, the traditional transformer core is generally designed to be composed of many thin iron sheets, referred to as iron sheets, which are overlapped and insulated from each other. Figure 2-18 shows a transformer core or an iron sheet in a transformer core. We can regard these iron sheets as being composed of a large number of "coils" (as shown by the dotted lines in the figure) tightly combined together; when the alternating magnetic lines of force pass vertically through these "coils", induced electromotive force and induced current will be generated in these "coils". Since these "coils" have resistance, these "coils" will lose electromagnetic energy.

　　During the DC pulse, the mechanism of eddy current is basically the same as that of forward voltage output. The direction of the magnetic field generated by eddy current is exactly opposite to the direction of the magnetic field generated by the excitation current. The demagnetization force is strongest at the center of the iron core and zero at the edge. Therefore, the magnetic flux density distribution in the iron core is uneven, that is, the magnetic field intensity is the largest in the outermost layer and the smallest in the center. If the eddy current demagnetization effect is very strong, the maximum value of the magnetic flux density may far exceed its average value, which is determined by the amplitude and width of the known pulse.

　　The magnetic field distribution along the cross section of the iron core can be obtained using Maxwell's differential equation:

　　In the above formula, μa is the average magnetic permeability of the transformer core, ρc is the resistivity of the core, and the negative sign indicates that the direction of the magnetic field generated by the eddy current is opposite to the direction of the magnetic field generated by the excitation current. rot E and rot Hx represent the curl of the electric field and magnetic field, that is, the intensity of the eddy electric field and eddy magnetic field. Hx, Hy, Hz are the three components of the magnetic field intensity H; Bx, By, Bz are the three components of the magnetic induction intensity B; Ex, Ey, Ez are the three components of the electric field intensity H.

　　Since the hysteresis loop area of ​​the single-excitation switching power supply transformer core is very small, its magnetization curve can basically be regarded as a straight line, and the magnetic permeability μ can also be regarded as a constant; therefore, the average magnetic permeability is used here to replace the widely used magnetic permeability.

　　As can be seen from Figure 2-18, the magnetic field strength is composed of H = Hz: and Hx = Hy = 0; for the electric field strength, its direction parallel to the Y axis is E = Ey, Ex = Ez = 0. Therefore, the above two equations can be rewritten as:

　　Differentiate equation (2-53) and then substitute it into equation (2-52) to obtain the one-dimensional distribution equation of the magnetic field intensity:

　　Since the voltage applied to both ends of the primary coil of the transformer is a DC pulse square wave, under steady-state conditions, the growth of the magnetic field intensity or magnetic flux density generated by the excitation current should be linearly related to time, that is:

　　When x = 0, it is located exactly at the center of the iron core, where the magnetic field intensity is the smallest, that is, the derivative value at this point is equal to 0, thus obtaining the integral constant c1 = 0.

　　Integrate (2-57) again and we get:

　　Since the average value Ha of the cross-sectional magnetic field strength in the transformer iron core must be equal to the value required by electromagnetic induction at any time, that is, it meets the requirements of formula (2-45), the average value of formula (2-58) is calculated corresponding to Figure 2-18:

　　Substituting (2-60) into (2-58), the magnetic field intensity in the iron core under steady-state conditions can be obtained as:

　　Figure 2-19-a and Figure 2-19-b are graphs of the magnetic field intensity in the iron core piece as a function of the horizontal distribution H(x) and the time distribution H(t), respectively, given by equation (2-61).

　　As can be seen from Figure 2-19-a, due to the reverse magnetization effect produced by eddy currents, the magnetic field intensity is lowest at the center of the iron core or iron chip, while the magnetic field intensity is highest at the edge.

　　In Figure 2-19-b, the part that increases linearly with time is the magnetic field generated by the excitation current of the primary coil of the transformer; Hb is the magnetic field generated by the current provided by the primary coil of the transformer in order to compensate for the demagnetizing field caused by eddy current.

　　As can be seen from Figure 2-19-b, the effect of eddy current loss on the magnetic field strength (average value) in the transformer core is basically the same as the effect of the magnetic field generated by the current in the secondary coil on the magnetic field of the transformer core when the transformer is in forward output. It is worth noting that if the same method is used to analyze the y-axis direction, the same result can be obtained.

　　It can be seen from Figure 2-19-a that when x =δ/2, the maximum value of the magnetic field intensity on the surface of the iron core is: