## 16

4.06(P„/Vdc)

1.62Vdc +

2.3.9 Forward converter magnetics

2.3.9.1 First-quadrant operation only. The transformer core in the forward converter operates in the first quadrant of the hysteresis loop only. This can be seen in Fig. 2.10. When Q1 is on, the dot end of T1 is positive with respect to the no-dot end and the core is driven, say, in a positive direction on the hysteresis loop and the magnetizing current ramps up linearly in the magnetizing inductance.

When Q1 turns off, stored current in the magnetizing inductance reverses the polarity of voltages on all windings. The dot end of Nr goes negative until it is caught one diode drop below ground by catch diode D1. Now the magnetizing current which really is stored in the magnetic core continues to flow. It simply transfers from Np, where it had ramped upward during the Q1 on time, into Nr. It flows out of the no-dot end of Nr into the positive end of the supply voltage Vdc, out of the negative end of Vdc, through the anode, then through the cathode of D1, and back into Nr.

During the Q! off time, as the dot end of Nr is positive with respect to its no-dot end, the magnetizing current Id ramps linearly downward as can be seen in Fig. 2.10. When it has ramped down to zero (at the end of area A2 in Fig. 2.10), there is no longer any stored energy in the magnetizing inductance and nothing to hold the dot end of Nr below the D1 cathode. The voltage at the dot end of Nr starts rising toward Vdc, and that at the no-dot end of Np (Q1 collector) starts falling from 2Vdc back down toward V,k.

Thus operation on the hysteresis loop is centered about half the: peak magnetizing current (VdcTon/2Lm). Nothing ever reverses the di-: rection of the magnetizing current—it simply builds up linearly to a peak and relaxes back down linearly to zero.

This first-quadrant operation has some favorable and some unfavorable consequences. First, compared to a push-pull circuit, it halves the available output power from a given core. This can be seen from Faraday's iaw (Eq. 1.17), which fixes the number of turns on the primary.

Solving Faraday's law for the number of primary turns, Np = E dtj Ae dB x 10 ~8. H'dB in the forward converter is limited to an excursion from zero to some Bmax instead of from -Bmax to +Bmax as in a push-pull topology, the number of primary turns for the forward converter will be twice that in each half primary for a push-pull operating from the same Vdc. Although the push-pull has two half primaries, each of which must support the same volt-second product as the forward converter, the push-pull offers two power pulses per period as compared to one for the forward converter. The end result is that a core used in a forward converter offers only half the available output power it is capable of in a push-pull configuration.

But the push-pull core at twice the output power will run somewhat warmer as its flux excursion is twice that of the forward converter. Since core losses are proportional to the area of the hysteresis loop traversed, the push-pull core losses Eire twice that of the forward converter.

Yet total copper losses in both half primaries of a push-pull are no greater than that of the forward converter of half the output power. For the rms current in each push-pull half primary is equal to that in the forward converter of half the output power. But since the number of turns in each push-pull half primary is half that of the forward converter of half the output power, each push-pull half primary has half the resistance of the forward converter. Thus, total copper losses of a forward converter is then equal to the total losses of the two half primaries in a push-pull of twice the output power.

2.3.9.2 Core gapping in a forward converter. In Fig. 2.3, the hysteresis loop of a ferrite core with no air gap is shown. It is seen that at zero magnetizing force (0 Oe) there is a residual magnetic flux density of about ± 1000 G. This residual flux is referred to as remanence.

In a forward converter, if the core started at 0 Oe and hence at 1000 G, the maximum flux change dB possible before the core is driven up into the curved part of the hysteresis loop is about 1000 G. It is desirable to stay off the curved part of the hysteresis loop, and hence the forward converter core with no air gap is restricted to a maximum dB of 1000 G. As shown above, the number of primary turns is inversely proportional to dB. Such a relatively small dB requires a relatively large number of primary turns. A larger number of primary turns requires smaller wire size and hence decreases the current and power available from the transformer.

By introducing an air gap in the core, the hysteresis loop is tilted as shown in Fig. 2.5 and magnetic remanence is reduced significantly.

The hysteresis loop tilts over and still crosses the zero flux density at the same point on the H axis (referred to as coercive force. Coercive force for ferrites is seen to be about 0.2 Oe in Fig. 2.3. An air gap of about 2 to 4 mils will reduce remanence to about 200 G for most cores used at a 200- to 500-W output power level. With remanence of 200 G, the permissible dB before the core enters the curved part of the hysteresis loop is now about 1800 G and fewer turns are permissible.

However, there is a penalty paid in introducing an air gap. Figure 2.5 shows the slope of the hysteresis loop tilted over. The slope is dBI dH or the core permeability, which has been decreased by adding the gap. Decreasing permeability has decreased the magnetizing inductance and increased the magnetizing current (7m = VdcTon/Lm). Magnetizing current contributes no output power to the load; it simply moves the core across the hysteresis loop and is wasteful if it exceeds 10 percent of the primary load current.

2.3.9.3 Magnetizing inductance with gapped core. Magnetizing inductance with a gapped core can be calculated as follows. Voltage across the magnetizing inductance is Lm dlm/dt and from Faraday's law:

where Lm = magnetizing inductance, H Np = number of primary turns Ae = core area, cm2 dB = core flux change, G dlm = change in magnetizing current, A

Now a fundamental law in magnetics is Ampere's law:

This states that if a line is drawn encircling a number of ampere turns NI, the dot product H • dl along that line is equal to 0.4ttM. If the line is taken through the core parallel to the magnetic flux lines and across the gap, since H is uniform at a level Hi within the core and uniform at a value Ha across the gap, then where Hi = magnetic field intensity in iron (ferrite), Oe li = length of iron path, cm Ha = magnetic field intensity in air gap, Oe la = length of the air gap, cm Im = magnetizing current, A

However, H: = BJu, where B, is the magnetic flux density in iron and u is the iron permeability; Ha = Ba as the permeability of air is 1; and Ba = B^(flux density in iron = flux density in air) if there is no fringing flux around the air gap. Then Eq. 2.36 can be written as

Then dB/dIm = 0.4ttA7(7u + IJu), and substituting this into Eq. 2.35

Thus, introducing an air gap of length la to a core of iron path length li reduces the magnetizing inductance in the ratio of

It is instructive to consider a specific example. Take an international standard core such as the Ferroxcube 783E608-3C8. It has a magnetic path length of 9.7 cm and an effective permeability of 2300. Then if a 4-mil (= 0.0102-cm) gap were introduced into the magnetic-path, from Eq. 2.39

9.7/2300

■^m (with gap) Q Q202 + 9 7/2300 (with0Llt W = 0.29Lln (w;thout gap)

A useful way of looking at a gapped core is to examine the denominator in Eq. 2.38. In most cases, u is so high that the term IJu is small compared to the air gap la and the inductance is determined primarily by the length of the air gap.

### 2.3.10 Power transformer design relations

2.3.10.1 Core selection. As discussed in Sec. 2.2.9.1 on core selection for a push-pull transformer, the amount of power available from a core for a forward converter transformer is related to the same parameters—peak flux density, core iron and bobbin areas, frequency, and coil current density in circular mils per rms ampere.

In Chap. 7, an equation will be derived giving the amount of available output power as a function of these parameters. This equation will be converted to a chart which permits selection of core size and operating frequency at a glance.

For the present, it is assumed that a core has been selected and its iron and bobbin area are known.

2.3.10.2 Primary turns calculation. The number of primary turns is calculated from Faraday's law as given in Eq. 2.7. Recall (Sec. 2.3.9.2) that in the forward converter, with a gapped core, flux moves from about 200 G to some higher value 7?max. As in the push-pull topology, this peak value will be set at 1600 G for ferrites even at low frequencies where core losses are not a limiting factor.

This, as discussed in Sec. 2.2.9.4, is to avoid the problem of a much larger and dangerous flux swing resulting from rapid changes in DC input voltage or load current. Such rapid changes are not immediately corrected for because the limited error-amplifier bandwidth does not permit a rapid correction in the power transistor on time.

During the error-amplifier delay, the peak flux density can, for a number of cycles, go up to as high as 50 percent above the calculated for normal steady-state operation. This can be tolerated if the normal peak flux density in the absence of a line or load transient is set to a low value of 1600 G. As discussed above, the excursion from approximately zero to 1600 G will take place in 80 percent of a half period to ensure that the core can be reset before the start of the next period (Fig. 2.126).

Thus the number of primary turns is set by Faraday's law at

where Vdc = minimum DC input, V T = operating period, s Ae = iron area, cm2 dB = 1600 G

2.3.10.3 Secondary turns calculation. Secondary turns are calculated from Eqs. 2.25 to 2.27. In those relations, all values except the secondary turns are specified or already calculated. Thus (Fig. 2.10):

minimum DC input, V

maximum on time, s ( = 0.8772) numbers of main and slave turns number of primary turns rectifier forward drop

Usually the main output is a high-current 5-V one for which a Schottky diode of 0.5 V is used. The slaves usually have higher output voltages which require the use of diodes with higher reverse-voltage

x on

ratings. Such fast-recovery higher-reverse-voltage diodes have forward drops of 1.0 V over a large range of currents.

2.3.10.4 Primary rms current and wire size selection. Primary equivalent flat-topped current is given by Eq. 2.28. That current flows a maximum of 80 percent of a half period out of each full period, and hence its maximum duty cycle is 0.4. Recalling that the rms value of a flat-topped pulse of amplitude 7 is 7_ = /„VtI it, the rms primary current is

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