As discussed in Section 2.2.9.1 on core selection for a push-pull trans- former, the amount of power available from a core for a forward con- verter transformer is related to the same parameters—peak flux den- sity, core iron and window areas, frequency, and coil current density in circular mils per rms ampere.
In Chapter 7, an equation will be derived giving the amount of available output power as a function of these parameters. This equa- tion will be converted to a chart that permits selection of core size and operating frequency at a glance.
For the present, it is assumed that a core has been selected and that its iron and window areas 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. From Section 2.3.9.2, we see that in the forward converter with a gapped core, flux density moves from about 200 G to some higher valueBmax.
In the push-pull topology as discussed in Section 2.2.9.4, this peak value will be set at 1600 G (for ferrites at low frequencies, where core losses are not a limiting factor). This avoids the problem of a much larger and more dangerous flux swing due to rapid changes in DC
input voltage or load currents. Such rapid changes are not imme- diately compensated because the limited error-amplifier bandwidth can’t correct the power transistor “on” time fast enough.
During this error-amplifier delay, the peak flux density can exceed the calculated normal steady-state value for a number of cycles. This can be tolerated if the normal peak flux density in the absence of a line or load transient is set to the low value of 1600 G. As discussed earlier, the excursion from approximately zero to 1600 G will take place in 80% of a half period to ensure that the core can be reset before the start of the next period (see Figure 2.12b).
Thus, the number of primary turns is set by Faraday’s law at Np= (Vdc−1)(0.8T/2)×10+8
AedB (2.40)
whereVdc=minimum DC input, V T=operating period, s Ae =iron area, cm2
dB=change in flux density, G
2.3.10.3 Secondary Turns Calculation
Secondary turns are calculated from Eqs. 2.25 to 2.27. In those rela- tions, all values except the secondary turns are specified or already calculated. Thus (see Figure 2.10):
Vdc =minimum DC input, V
Ton =maximum “on” time, s(=0.8T/2) Nm, Ns1, Ns2 =numbers of main and slave turns
Np =number of primary turns Vd =rectifier forward drop
If the main output produces 5 V at high current as is often the case, a Schottky diode with forward drop of about 0.5 V is typically used.
The slaves usually have higher output voltages that require the use of fast-recovery diodes with higher reverse-voltage ratings. Such diodes have forward drops of about 1.0 V over a large range of current.
2.3.10.4 Primary rms Current and Wire Size Selection
Primary equivalent flat-topped current is given by Eq. 2.28. That cur- rent flows for a maximum of 80% of a half period per period, so its max- imum duty cycle is 0.4. Recalling that the rms value of a flat-topped
pulse of amplitudeIpisIrms=Ip
Ton/T,the rms primary current is Irms (primary) = 3.12Po
Vdc
√0.4
= 1.97Po
Vdc
(2.41)
If the wire size is chosen on the basis of 500 circular mils per rms ampere, the required number of circular mils is
Circular mils needed = 500 × 1.97Po Vdc
= 985Po Vdc
(2.42)
2.3.10.5 Secondary rms Current and Wire Size Selection
It is seen in Figure 2.11 that the secondary current has the characteris- tic shape of a ramp on a step. The pulse amplitude at the center of the ramp is equal to the average DC output current. Thus, the equivalent flat-topped secondary current pulse atVdc(when its width is a maxi- mum) has amplitudeIdc, width 0.8T/2, and duty cycle (0.8T/2)/Tor 0.4. Then
Irms(secondary) =Idc√ 0.4
=0.632Idc
(2.43) and at 500 circular mils per rms ampere, the required number of cir- cular mils for each secondary is
Circular mils needed = 500 × 0.632Idc
=316Idc (2.44)
2.3.10.6 Reset Winding rms Current and Wire Size Selection The reset winding carries only magnetizing current, as can be seen by the dots in Figure 2.10. WhenQ1 is “on,” diodeD1 is reverse-biased, and no current flows in the reset winding. But magnetizing current builds up linearly in the power windingNp.When Q1 turns “off,”
that magnetizing current must continue to flow. When Q1 current ceases, the current in the magnetizing inductance reverses all winding voltage polarities. WhenD1 clamps the dot end of Nr to ground, the magnetizing current transfers from Np to Nr and continues flowing through the DC input voltage sourceVdc, throughD1, and back into Nr. Since the no-dot end of Nr is positive with respect to the dot end, the magnetizing current ramps downward to zero as seen in Figure 2.10.
The waveshape of this Nr current is the same as that of the mag- netizing current that ramped upward when Q1 was “on,” but it is reversed from left to right. Thus the peak of this triangle of current is Ip(magnetizing)=VdcTon/Lmg,whereLmgis the magnetizing inductance with an air gap as calculated from Eq. 2.39. The inductance without the gap is calculated from the ferrite catalog value of Al,the induc- tance per 1000 turns. Since inductance is proportional to the square of the number of turns, inductance fornturns isLn= Al(n/1000)2. The duration of this current triangle is 0.8T/2 (the time required for the core to reset), and it comes at a duty cycle of 0.4.
It is known that the rms value of a repeating triangle waveform (no spacing between successive triangles) of peak amplitude Ip is Irms=Ip√
3.But this triangle comes at a duty cycle of 0.4, and hence its rms value is
Irms= VdcTon Lmg
√0.4
√3
=0.365VdcTon Lmg
and at 500 circular mils per rms ampere, the required number of cir- cular mils for the reset winding is
Circular mils required = 500 × 0.365VdcTon
Lmg (2.45) Most frequently, the magnetizing current is so small that the reset winding wire can be No. 30 AWG or smaller.