All transformers have leakage inductance, but the leakage inductance of switching transformers has a particularly significant impact on the performance of switching power supplies. Due to the presence of leakage inductance, a back electromotive force is generated the instant the control switch is turned off, which can easily cause overvoltage breakdown of the switching devices. Leakage inductance can also form an oscillating circuit with the distributed capacitance in the circuit and the distributed capacitance of the transformer coil, causing the circuit to oscillate and radiate electromagnetic energy, resulting in electromagnetic interference. Therefore, analyzing the principle of leakage inductance generation and reducing its generation are also important aspects of switching transformer design.
The leakage inductance between the coils of a switching transformer is caused by the leakage magnetic flux between them; therefore, calculating the leakage magnetic flux between the coils allows us to calculate the value of the leakage inductance. To calculate the leakage magnetic flux between the transformer coils, we first need to know the magnetic field distribution between the two coils. We know that the magnetic field distribution in a helical coil is somewhat similar to the electric field distribution between two plates; that is, the magnetic field strength distribution in a helical coil is basically uniform, and the magnetic field energy is basically concentrated within the helical coil. Furthermore, when calculating the magnetic field strength distribution inside or outside the helical coil, more complex cases can be handled using Maxwell's theorem or the Bisharp-Schwarz theorem, while simpler cases can be handled using Ampere's circuital law or Kirchhoff's laws for magnetic circuits.
Figure 2-30 is a schematic diagram for analyzing and calculating the leakage inductance between the coils of a switching transformer. Below, we will use Figure 2-30 to briefly analyze the principle of leakage inductance between the coils of a switching transformer and perform some simple calculations.
In Figure 2-30, N1 and N2 are the primary and secondary coils of the transformer, respectively, and Tc is the transformer core. r is the radius of the transformer core, and r1 and r2 are the radii of the primary and secondary coils, respectively; d1 is the distance from the primary coil to the core, and d2 is the distance between the primary and secondary coils. For simplicity of analysis and calculation, it is assumed that the number of turns and wire diameter of the primary and secondary coils are equal, and that the current flowing through the coils is concentrated at the center of the wire diameter; therefore, the distance between them is always the center-to-center distance between the two coils, as shown by the dashed line.
Let the cross-sectional area of the iron core be S, S = πr²; the cross-sectional area of the primary coil be S1, S1 = πr²¹; the cross-sectional area of the secondary coil be S2, S2 = πr²²; the cross-sectional area of the gap between the primary coil and the iron core be Sd1, Sd1 = S1 - S; the cross-sectional area of the gap between the secondary coil and the primary coil be Sd2, Sd2 = S2 - S1; the magnetic field strength generated by the current I1 flowing through the primary coil is H1, the magnetic flux generated within area S1 is φ1, and the magnetic flux generated within area Sd2 is φ1'; the magnetic field strength generated by the current I2 flowing through the secondary coil is H2, and the magnetic flux is φ2.
Figure 2.30: The principle of leakage inductance between the coils of a switching transformer
Therefore, the magnetic flux generated by the current I2 flowing through the secondary coil N2 of the transformer can be calculated as follows:
The magnetic flux generated by the current I2 flowing through the secondary coil N2 of the transformer
In equations (2-95) and (2-96), μ0sd2H2=φ2 is the leakage flux of the secondary coil N2 to the primary coil N1 of the transformer; because this part of the flux does not pass through the primary coil N1 of the transformer. The leakage flux can be equivalently represented by an inductor alone, which is called the leakage inductance, denoted as Ls. Similarly, the flux generated by the current I1 flowing through the primary coil N1 of the transformer can also be obtained as:
The magnetic flux generated by the current I1 flowing through the primary coil N1 of the transformer
Magnetic flux calculation formula
In equation (2-96), at first glance, the magnetic flux φ1 generated by the primary coil N1 of the transformer passes entirely through the secondary coil N2 of the transformer, and there should be no leakage flux between them; however, the direction of the magnetic flux φ1 generated by the primary coil in area S1 is exactly opposite to the direction of the magnetic flux φ1 generated in area Sd2; therefore, the magnetic flux φ1 generated by the primary coil N1 of the transformer in area Sd2 is still called the leakage flux of the primary coil N1 of the transformer to the secondary coil N2 of the transformer, and its equivalent inductance is also called leakage inductance.
