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Switching power supply much lower output than LTSPICE model

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So the start of your waveform matches up fairly well, but at 10ms instead of plateauing around 0.6V (0.27A), the input current decreases and starts ringing until stopping completely at 22ms. Try probing other points in the circuit and see if you see any odd behaviors correlating with that.

L3 is probably saturating. Its saturation current is 270mA. LTSpice's default inductor model doesn't model saturation at all. When the current drops low enough that the inductor comes out of saturation, the ringing starts.

I tried shorting out R18 and L3. That's just noise filtering to prevent switch noise from getting back into the USB power source. That eliminates L3 saturation as a factor, and increases final output a bit, but it still takes about 45ms to charge up the 2uF caps. With R18 shorted out, there's no good place to measure a voltage drop and calculate current.

(This is all on a surface mount board, so it's hard to tap in for current. Voltage, no problem,)

Now, if the core is not fully resetting during the "off" time
(and I see nothing explicitly making that so) maybe there
is flux walk / staircasing that leads to eventual saturation.
But flyback converters are kinda outside my scope. Just
a thing to consider.
Is that a possibility? Do I need a diode somewhere to drain out the transformer primary current on each cycle?

The default transformer model in LTSpice is oversimplified for this sort of thing. No saturation, and no hysteresis. If the model diverges badly from the real world, that's the most likely reason. Everything else has a reasonable model. Especially with L3 out of the picture, as noted above.
 

L3 is probably saturating. Its saturation current is 270mA. LTSpice's default inductor model doesn't model saturation at all. When the current drops low enough that the inductor comes out of saturation, the ringing starts.
It's possible that L3 saturating is contributing to the strange behavior... nonlinearity is hard to predict.

I tried shorting out R18 and L3. That's just noise filtering to prevent switch noise from getting back into the USB power source. That eliminates L3 saturation as a factor, and increases final output a bit, but it still takes about 45ms to charge up the 2uF caps. With R18 shorted out, there's no good place to measure a voltage drop and calculate current.
With those components shorted out, does the input current waveform change much? R2 should still work fine for measuring input current.

The sudden change in input current at 22ms is also very strange. I can't tell what the trace offset is on your scope, but it looks like current goes to zero, which should not happen.

Is that a possibility? Do I need a diode somewhere to drain out the transformer primary current on each cycle?

The default transformer model in LTSpice is oversimplified for this sort of thing. No saturation, and no hysteresis. If the model diverges badly from the real world, that's the most likely reason. Everything else has a reasonable model. Especially with L3 out of the picture, as noted above.
Staircase saturation only affects push-pull type converters, flyback not included. If saturation is happening in your transformer, it would be at the beginning of the charging cycle, when Vout is low. Your simulations do show this happening, so it may be a factor. Hard to tell, since coilcraft does not explain how much saturation occurs.

If you were to simulate the circuit with L3 and R18 shorted, you get a much faster charging time, especially at the beginning, and a much higher peak input current (Ip is almost 5A, supply current peaks at 0.76A). But with your real circuit, an overcurrent limit will likely trip before then. If it were me, I would try to implement a soft start so that the duty cycle ramps from zero at the start, in order to keep the inrush current lower.
 
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**broken link removed**
Completely new design

I've gone to a completely new design. It's more complex, but it works.

This circuit uses an LT3750 capacitor charger controller. I'd been avoiding that, because I didn't want to deal with soldering 0.5mm pitch components. But with more practice under the microscope, I can now do it. It's not fun.

The switcher here is the path from Vcc to the top of T1's primary, through MOSFET Q1, through current sense resistor R3 (0.061 ohm), to ground. The charger controller has a flip-flop which controls the MOSFET's gate. There's a comparator which monitors the current through R3 and turns off the flip-flop and gate power when the current reaches a setpoint. (Without that, this thing would short Vcc to ground in a few microseconds.) With the gate turned off, the voltage across the transformer primary drops, and when another comparator detects it's almost zero, the flip-flop and gate are turned on, restarting the cycle. This is a high-efficiency solution.

Over on the secondary side, there's the usual diode D1 and some capacitors, which get charged up to 120V. So that's the capacitor charger.

It takes tight layout and bypass caps to get this to work, but the LT3750 data sheet has good advice on that. This thing generates big spikes, some as narrow as 25ns, while oscillating around a megahertz. Efficiency is good; this can charge caps about 3x as fast as the previous design, while drawing only about 270mA from Vcc (+5). I've successfully run two different Teletype machines with a previous version of this board.

The latest rev eliminates a jumper and resistor used to set the output current, and adds a DMOS, Q2, set up as a current source at the output to make it self-adjust to different loads.

Entire project in KiCAD on Github
 

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