Design of a low voltage and high current DC-DC converter

A few additional thoughts here:

1. A push-pull topology as you have suggested using is great for passing high power, but it must be run with a current mode feedback controller.  Do not attempt to run a push-pull transformer using voltage mode, or any, and I mean any, imbalance in MOSFET, winding, or trace parameters will cause a DC current to build through the transformer over time...sometimes it takes a while, but it will happen eventually.  When this DC current is sufficiently great, your transformer core will enter saturation which is basically a short circuit (technically the DC resistance of the transistors, wiring, and PCB traces) which will lead to catastrophic failure of your primary side.
2. If you're targeting somewhere near 50% duty cycle, look into using a current doubler type secondary side.  This topology has the unique characteristic that as you approach your 50% duty cycle, your output capacitor requirement goes to zero (obviously nothing is perfect and you will need a minimum amount): https://www.ti.com/lit/an/slua121/slua121.pdf
3. If you really need fine control (you mention 1mA accuracy), you may need to consider a linear post regulator.  To limit heating you may need to couple the secondary current regulator feedback to the primary regulator in order to reduce output voltage.  I would do this with a voltage difference amplifier across the input and output of the current linear regulator circuit so a greater voltage drop on the current regulator Vin to Vout (which generates more heat) would be scaled appropriately and added to the feedback pin of the primary controller to drive the effective output voltage from the first stage lower.
4. I agree with previous comments using a full H-bridge is probably your best bet for polarity management.
5. Another option altogether and is quite possibly simpler would be to connect the H-bridge directly across your rectified output.  The solenoid itself is an inductor which you can use to your advantage.  PWM all four FETs with constant alternating duty cycle say 50% for "positive path" and 50% for "negative path".  In this arrangement, the net average current is 0A!.  Slightly tweak the duty cycle balance between the positive and negative paths to get the DC average you want.  Have a current sensor in series with your solenoid to provide the necessary feedback, note you will need a high CMRR part as this is the noisy switching node.  Note you should run the H-bridge at a suitably high frequency to meed your ripple requirement; also do not run it at a multiple of your primary side driver or you may get some weird behavior if the frequencies line up just right.

A few additional thoughts here:

1. A push-pull topology as you have suggested using is great for passing high power, but it must be run with a current mode feedback controller.  Do not attempt to run a push-pull transformer using voltage mode, or any, and I mean any, imbalance in MOSFET, winding, or trace parameters will cause a DC current to build through the transformer over time...sometimes it takes a while, but it will happen eventually.  When this DC current is sufficiently great, your transformer core will enter saturation which is basically a short circuit (technically the DC resistance of the transistors, wiring, and PCB traces) which will lead to catastrophic failure of your primary side.
2. If you're targeting somewhere near 50% duty cycle, look into using a current doubler type secondary side.  This topology has the unique characteristic that as you approach your 50% duty cycle, your output capacitor requirement goes to zero (obviously nothing is perfect and you will need a minimum amount): https://www.ti.com/lit/an/slua121/slua121.pdf
3. If you really need fine control (you mention 1mA accuracy), you may need to consider a linear post regulator.  To limit heating you may need to couple the secondary current regulator feedback to the primary regulator in order to reduce output voltage.  I would do this with a voltage difference amplifier across the input and output of the current linear regulator circuit so a greater voltage drop on the current regulator Vin to Vout (which generates more heat) would be scaled appropriately and added to the feedback pin of the primary controller to drive the effective output voltage from the first stage lower.
4. I agree with previous comments using a full H-bridge is probably your best bet for polarity management.
5. Another option altogether and is quite possibly simpler would be to connect the H-bridge directly across your rectified output.  The solenoid itself is an inductor which you can use to your advantage.  PWM all four FETs with constant alternating duty cycle say 50% for "positive path" and 50% for "negative path".  In this arrangement, the net average current is 0A!.  Slightly tweak the duty cycle balance between the positive and negative paths to get the DC average you want.  Have a current sensor in series with your solenoid to provide the necessary feedback, note you will need a high CMRR part as this is the noisy switching node.  Note you should run the H-bridge at a suitably high frequency to meed your ripple requirement; also do not run it at a multiple of your primary side driver or you may get some weird behavior if the frequencies line up just right.

pctechnocrat

Just to let you know, the original post was not mine. I was simply offering my best advise as you did.

But in reply to you:

1.  Your statement is not quite accurate. I have 45+ years hands-on design experience in dc-dc converters and while one must take steps to avoid saturating the transformer due to imbalance (especially in a push-pull), there are many ways to accomplish this, even using voltage-mode. Interestingly, mosfets have much better matching in such a circuit compared to bipolars. One way is to place a small gap in the transformer to flatten-out the BH curve. Another would be to not run the converter at full duty cycle, but perhaps 85% of full duty cycle. This gives the core time to reset despite imbalances.

2. In order to regulate the output current, the converter would have to control (adjust) the duty cycle and considerably less than 50%. Otherwise, if a fixed, 50% duty cycle is chosen, a second stage for current control & regulation would be needed. But this would add complexity and cost without any additional benefit.

3. A linear post regulator doesn't imply finer control than a switching regulator could provide. Other design variables would control the maximum practical output current resolution.

4. Good, we have agreement!

5. I agree. But since I don't have the solenoid specs and don't know why he wants to have +/-10A into the 0.8 mOhm solenoid, I can't intelligently comment on that. Again, it was not my post. If it was controlled that way, there would be a high ripple, which may or may not be an issue. Also, you would still need to convert his 5V, 3A input into a nominally 0.5V, 30A output before the H-bridge you mentioned. Since the current would be bipolar, it would be a bit harder to use the current feedback to control the duty cycle. It might make more sense to keep the output LC filter to have unidirectional control of the 0A to 10A, and then control the polarity with a H-Bridge. It's 2 stages, but the first stage could easily be off-the-shelf and inexpensive. The second stage would not be that hard to design, a simple H-Bridge with only 2 or 3 states, positive Iout, negative Iout, and possibly an off or tri-state output (optional).

I really enjoyed the discussion. None of my comments were intended as criticism of your comments. Thank's for the debate.

HI Kamran, I appreciate your insights as well.  My post was in response to Simone, the original poster (I did not "reply" to you).  My comments were generalizations since, as you say, we have limited specs so just opening the door to lots of solutions for a vague problem.

Cheers.