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Thank you to everyone that answered.

Regarding the concern of the thermal mass of the Peltier, the Petier is 2cm by 2cm in dimension:

[URL unfurl="true"]https://lairdthermal.com/products/thermoelectric-cooler-modules/peltier-multistage-series/MS2-192-14-20-11-18-11-W8/pdf[/URL]

To investigate the rate of temperature change that could be supported, I did a rudimentary test by trying to change the temperature of the Peltier by manually changing the settings on the DC power supply. I definitely see the concern now, as if I try to, say, step the temperature up by 1 degree every 10 seconds, the peltier's surface could not 'catch up' with the temperature that I'm trying to achieve, even if I up the power to the max. safe threshold. So I definitely see where betwixt is coming from when he said driving the microcontroller at a slow speed, as finer temporal resolution won't be very practical.

And also please correct me if I am wrong, but from what I understand, an H-bridge to swap the polarity of the DC current won't be necessary if I use two DC power supplies that are providing the 'opposite' voltage from each other, right?

I have attached a very crude circuit diagram (hand drawn). Are there any glaring issues that you see with this design? I am a bit iffy on the placement of the DC-DC power converter, because since a 'buck' converter only steps voltage 'down', if the relays route the 'negative' DC current to a buck converter for cooling the peltier, then it will cause problems, right?

I've also searched up some DC-DC power converters, but the number of varieties are overwhelming. Aside from the current/voltage range of the input and output, are there any other parameters I should keep an eye out for (at least for my intended use)?

Finally, do you recommend any software (that are hopefully free or low cost) that are available for circuit diagram illustration / circuit simulation?

I appreciate all the help.