Calculation of volt-second capacity and coil turns

In Figure 2-1, when a DC pulse voltage is input to both ends of the primary coil a and b of the transformer, an excitation current flows through the primary coil of the transformer, and the excitation current generates a magnetic flux in the transformer core. At the same time, a back electromotive force is generated at both ends of the primary coil of the transformer; the amplitude of the back electromotive force voltage is equal to the amplitude of the input voltage, but the direction is opposite.

Therefore, according to the law of electromagnetic induction, the magnetic flux in the transformer core The change process is determined by the following formula:

In the above formulas (2-13), (2-14), and (2-15), US is the volt-second capacity of the transformer, US = E×τ, that is, the volt-second capacity is equal to the product of the input pulse voltage amplitude and the pulse width, in volt-seconds, E is the input pulse voltage amplitude, in volts, τ is the pulse width, in seconds; Δ is the flux increment in Maxwell (Mx), Δ = S×ΔB; ΔB is the magnetic flux density increment, ΔB = Bm-Br, in Gauss (Gs); S is the cross-sectional area of ​​the core, in square centimeters; N1 is the number of turns of the transformer primary coil N1 winding, and K is the proportional constant.

The volt-second capacity indicates how high the input voltage and how long the impact of a transformer can withstand. Therefore, the larger the volt-second capacity US of the transformer, the smaller the excitation current flowing through the primary coil of the transformer. The excitation current of a general transformer does not provide power output, and the only exception is the flyback switching power supply. Therefore, in a forward transformer switching power supply or a dual-excitation transformer switching power supply, the smaller the excitation current, the higher the working efficiency of the switching power supply. Under certain transformer volt-second capacity conditions, the higher the input voltage, the shorter the time the transformer can withstand the impact, and conversely, the lower the input voltage, the longer the transformer can withstand the impact; and under certain working voltage conditions, the larger the volt-second capacity of the transformer, the lower the magnetic flux density in the transformer core, and the transformer core is less likely to saturate. The volt-second capacity of the transformer has basically nothing to do with the size and power of the transformer, but only with the magnitude of the change in magnetic flux.

If we slightly transform equation (2-15), we can get the formula for calculating the number of turns of the primary coil of the single-excitation switching power supply transformer:

Formula (2-16) is the formula for calculating the number of turns of the primary coil N1 winding of the single-excitation switching power supply transformer. In the formula, N1 is the minimum number of turns of the primary coil N1 winding of the transformer, S is the magnetic conductive area of ​​the transformer core (unit: square centimeters), Bm is the maximum magnetic flux density of the transformer core (unit: Gauss), Br is the residual magnetic flux density of the transformer core (unit: Gauss), τ is the pulse width, or the width of the power switch conduction time (unit: seconds), and E is the pulse voltage amplitude, that is, the operating voltage amplitude of the switching power supply, in volts.

The exponent 108 in equation (2-16) is exactly equal to the proportionality coefficient K in equations (2-13), (2-14), and (2-15). Therefore, the value of the proportionality coefficient K will be different if different units are used. Here, the CGS unit system is used, that is, the length is centimeters (cm), the magnetic flux density is Gauss (Gs), and the magnetic flux unit is Maxwell (Mx).

It can also be seen from Figures 2-2 and 2-3 that directly using the parameters of Figures 2-2 and 2-3 to design a single-excitation switching power supply transformer is not of much value in practical applications. This is because the maximum magnetic flux density Bm of ordinary transformer core materials is not large, approximately between 3000 and 5000 Gauss, while the residual magnetic flux density Br is generally as high as more than 80% of the maximum magnetic flux density Bm. Therefore, the actual applicable magnetic flux density increment ΔB is generally very small, only about 500 Gauss, and generally does not exceed 1000 Gauss. In order to increase the magnetic flux density increment ΔB, it is generally necessary to leave a certain length of air gap in the transformer core to reduce the value of the residual magnetic flux density Br.

From equations (2-13) and (2-14), we can see that although the magnetization curve is not linear, when the input voltage is a square wave, the magnetic flux generated by the excitation current flowing through the primary coil of the transformer still increases according to a linear law; however, the excitation current flowing through the primary coil of the transformer and the magnetic field strength do not necessarily increase according to a linear law. It is for this reason that a proportional constant K appears in equations (2-13) and (2-14).

That is to say, when we regard the coefficient K in equations (2-13), (2-14), and (2-15) as a proportional constant, it also means that we have regarded the magnetic permeability of the transformer core as a constant. However, due to the nonlinearity of the magnetic permeability of the transformer core and the nonlinearity of the excitation current, the two nonlinear parameters compensate each other, which makes the magnetic flux in the transformer core change according to the linear law. Therefore, when the transformer core is about to approach saturation, the excitation current in the transformer primary coil is very large.

In a single-excitation transformer switching power supply, although the magnetic flux generated by the current flowing through the primary coil of the transformer increases according to the linear law, when the transformer core is demagnetized, the change in magnetic flux does not necessarily decrease according to the linear law. This problem has been basically explained in the content of the first chapter. After the DC pulse voltage passes, the secondary coil of the transformer generates a flyback voltage output. In a pure resistance load, its output voltage is generally a voltage pulse that decreases exponentially. Therefore, the corresponding magnetic flux increment cannot change according to the linear law, but should also change according to the exponential law. However, the latter exponential law is just the result of integrating the former exponential law. This corresponding relationship can also be easily seen from equations (2-13) and (2-14).

It should be pointed out here that in a single-excitation transformer switching power supply, the only thing that has a magnetizing effect on the transformer core is the excitation current flowing through the transformer primary coil. Therefore, the excitation current is also called the magnetizing current. The demagnetizing effect on the transformer core is the back electromotive force generated by the primary and secondary coils of the transformer, as well as the current generated by the back electromotive force, namely: the flyback output voltage and current. The forward output voltage and current have no effect on the magnetization and demagnetization of the transformer core.

Because, although the excitation current will generate a forward voltage, it cannot provide a forward current output, which is equivalent to the situation when the secondary coil of the transformer is in an open circuit; when the secondary coil of the transformer has a forward current output, a current must be added to the primary coil of the transformer accordingly, and this current is increased accordingly on the basis of the original excitation current; the magnetic flux generated by this additional current is exactly equal to the magnetic flux generated by the forward output current in value, but in opposite directions, and the two cancel each other out, that is, they have no effect on magnetization or demagnetization.

Calculation of the volt-second capacity and primary coil turns of a double-excitation switching power supply transformer

In Figure 2-7, for the dual-excitation switching power supply transformer, each time an AC pulse voltage is input, except for the first input pulse whose magnetic flux density changes from 0 to the maximum value Bm, the range of the magnetic flux density changes for the remaining input pulses is from the negative maximum value -Bm to the positive maximum value Bm, or from the positive maximum value Bm to the negative maximum value -Bm, that is: each time an AC pulse voltage is input, the increment ΔB of the magnetic flux density is 2 times the maximum magnetic flux density Bm (2Bm). Therefore, substituting this result into equations (2-13) and (2-14), we can obtain:

Formulas (2-17) and (2-18) are the formulas for calculating the number of turns of the primary coil N1 winding of the dual-excitation switching power supply transformer. In the formula, N1 is the minimum number of turns of the primary coil N1 winding of the transformer, S is the magnetic conductive area of ​​the transformer core (unit: square centimeters), Bm is the maximum magnetic flux density of the transformer core (unit: Gauss), τ is the pulse width, or the width of the power switch conduction time (unit: seconds), E is the amplitude of the pulse voltage, that is, the operating voltage amplitude of the switching power supply, in volts, and F is the operating frequency of the switching power supply, in Hertz.

Similarly, we define the product of the input pulse voltage amplitude E and the pulse width τ in formula (2-17) as the volt-second capacity of the transformer, expressed as US (unit: volt-second), that is: US = E×τ.

It should also be pointed out here that the use of (2-17) and (2-18) to calculate the number of turns of the primary coil N1 winding of the double-excitation switching power supply transformer is conditional, that is, the volt-second capacity Us of the positive and negative half-cycles of the input AC pulse voltage must be equal. If they are not equal, the flux density increment ΔB in (2-17) and (2-18) cannot be expressed by 2Bm, but should be expressed by the difference between the two actual variables Bm and -Bm, that is: ΔB = Bm-(-Bm). Here, it is more appropriate to regard Bm and -Bm as variables.

Comparing equations (2-17) and (2-18) with equation (2-16), it is easy to see that, under the condition that the magnetic conductive area of ​​the transformer core and the input voltage amplitude are completely equal, the range of variation of the magnetic flux density in the iron core of the dual-excitation switching power supply transformer is much larger than that in the iron core of the single-excitation switching power supply transformer; or under the condition that the volt-second capacity is completely equal, the number of turns of the primary coil of the dual-excitation switching power supply transformer is much less than that of the primary coil of the single-excitation switching power supply transformer. Therefore, for dual-excitation switching power supply transformers, it is generally not necessary to leave an air gap in the transformer core.

In equations (2-17) and (2-18), for the core of a high-power dual-excitation switching power supply transformer, the maximum magnetic flux density Bm is generally not to exceed 3000 Gauss. If the Bm value is too high, when the switching device is accidentally triggered and the phase in Figure 2-7 is wrong, it is easy to cause magnetic saturation of the transformer core, causing the switching power supply to be damaged by excessive operating current.

Calculation of the number of turns of the primary coil of various waveform power transformers

Although formula (2-18) is used to calculate the number of turns of the primary coil N1 of the dual-excitation switching power supply transformer, it can also be used to calculate the number of turns of the primary coil of other waveform power supply transformers by only slightly changing or modifying some individual parameters in the formula.

Here, we first derive the formula for calculating the number of turns of the primary coil of the sine wave power transformer. The method is shown in Figure 2-8. First, calculate the half-cycle average value Ua of the sinusoidal voltage, because the half-cycle average value Ua of the sinusoidal voltage is exactly equal to the amplitude E of the square wave voltage. Therefore, just substitute the half-cycle average value of the sinusoidal voltage into formula (2-18) to get the formula for calculating the number of turns of the primary coil of the sine wave power transformer.

However, the half-cycle average value Ua of the sinusoidal voltage is generally rarely used. Therefore, it is also necessary to convert the half-cycle average value Ua of the sinusoidal voltage into the effective value U of the sinusoidal voltage; since the effective value U of the sinusoidal voltage is equal to 1.11 times the half-cycle average value Ua of the sinusoidal voltage, that is: U = 1.11Ua. The calculation formula for the number of turns of the primary coil of the sinusoidal power transformer is obtained as follows:

Formula (2-19) is the formula for calculating the number of turns of the primary coil N1 of the sinusoidal power transformer. In the formula, N1 is the minimum number of turns of the transformer primary coil N1 winding, S is the magnetic conductive area of ​​the transformer core (unit: square centimeters), Bm is the maximum magnetic flux density of the transformer core (unit: Gauss), U is the effective value of the sinusoidal input voltage, in volts, and F is the frequency of the sinusoidal wave, in Hertz. This calculation method is also valid for non-sinusoidal waves. Figure 2-9 is an AC pulse waveform with unequal positive and negative pulse amplitudes and pulse widths. We can also use the method of calculating their positive and negative half-cycle average values ​​Ua and -Ua respectively, and then use the average value Ua to replace the rectangular pulse amplitude E in formula (2-17) or (2-18).

Of course, the condition in Figure 2-9 is that the volt-second capacity of the positive and negative pulses should be equal. If they are not equal, the calculation method of the primary coil turns of the single- and double-excitation switching power supply transformer can be adopted, that is: consider both methods at the same time and make a compromise based on the emphasis.

The half-cycle average value Ua of various waveforms is obtained by the following formula:

In formulas (2-19) and (2-20), Ua and Ua- are respectively the average values ​​of the positive and negative half cycles of various waveforms, Pu(t) and Nu(t) are respectively the positive waveform function (positive half cycle) and negative waveform function (negative half cycle) of various waveforms, and T is the period of the waveform. For most AC voltage waveforms, the absolute values ​​of the average values ​​of the positive and negative half cycles are equal, but the signs are opposite.

By the way, the half-cycle average value here is not the average value of the positive half-cycle or negative half-cycle of the AC voltage with completely symmetrical positive and negative half-cycle waveforms in the general sense. The half-cycle average value here refers to the average value of the positive half-wave voltage or negative half-wave voltage in the entire cycle within half a cycle. As shown in Figure 2-9. In addition, the half-cycle average values ​​Ua and Ua- in equations (2-19) and (2-20) are slightly different from the half-wave average values ​​Upa and Upa- defined in equations (1-73), (1-74), and (1-75) in Chapter 1. The difference between Ua and Ua- and Upa and Upa- is mainly in the denominator.