Electronic – arduino – H-Bridge Power Supply Erratic Behavior

You are right in that a switcher makes a lot more sense for your application (12V in, 5V 1.5A out) than a linear regulator. A linear would waste 7V * 1.5A = 10.5W in heat, which would be challenge to get rid of. For linear regulators, current in = current out + operating current. For switchers power in = power out / efficiency.

I haven't looked up the TI part you mention (I might have if you had supplied a link). There are two broad classes of switching regulators, those with internal switches and those that drive external switches. If this regulator is the second kind, then dissipation in the part won't be a problem since it's not handling the power directly.

If it is a fully integrated solution, then you do have to look at dissipation. You can compute this dissipation from the output power and the efficiency. The output will be 5V * 1.5A = 7.5W. If the switcher is 80% efficient, for example, then the total input power will be 7.5W / 0.8 = 9.4W. The difference between the output power and the input power is the heating power, which in this case is 1.9W. That's way better than what a linear regulator would do, but is still enough heat to require some thought and planning.

80% was just a number I picked as a example. You need to look at the datasheet carefully and get a good idea what efficiency is likely to be at your operating point. Good switcher chips have lots of graphs and other information about this.

Once you know how many Watts will be heating the chip, you look at its thermal spec to see what the temperature drop from the die to the case is. The datasheet should give you a degC per Watt value. Multiply that by the Watts dissipation, and thats how much hotter the die will be than the outside of the case. Sometimes they tell you the thermal resistance from the die to ambient air. This is usually the case when the part is not intended to be used with a heat sink. Either way, you find how many deg C hotter the die will be than anything you can cool or deal with.

Now you look at the max die temp, then subtract off the above temp drop value. If that's not at least a little above your worst case ambient air temperature, then you have a problem. If so, it gets messy. You either need a heat sink, forced air, or use a different part. Higher power switchers are usually designed for external switch elements because power transistors come in cases intended to be heat sunk. Switcher chips usually don't.

I don't want to go on speculating, so come back with numbers about your particular situation, and we can continue from there.

What you describe is workable - it's essentially a buck converter - but your exact circuit is not specified and there are many ways to go from a somewhat correct general concept to an incorrect circuit.

The design note 188 that you reference is about 15 years old and its time has passed. While it is a marvel of improvisation, you can get better results for less cost using modern ICs.
I'm aware that you only cited it as an example, and that you wanted a switch mode version, but using a dedicated switch mode converter IC would be a good idea at minimum, and using a LiIon charger IC would be an even better one.

An LTC1541 - datasheet here costs about $3 in 1's. It's quite a useful IC if you need its features. With a 5uA typical supply current it could not fight its way out of a wet paper bag, and a slew rate of 8 mV/uS means that you have to be clever to use it and the characteristics of the device can greatly affect circuit performance.

AN188 loosely uses the term "float voltage". A Lithium Ion battery must not be "floated" in the sense the term is usually used. They intend it to mean "hold at CV while charge current ramps down due to battery chemistry action" - but the applied voltage should be removed once current has fallen to some preselected percentage of Imax.

Once you have terminated charging there will usually not be any need to restart charging until there has been some load current drawn. This is because an unloaded LiIon cell will maintain a close to full state of charge over considerable periods. If it is going to stand unloaded for weeks or months then testing battery voltage for possible restart of charging may be in order - but the load from the voltage measuring circuit can easily exceed the self discharge rate.

If you are trying to produce a low cost design with a high element of DIY in it you could try the venerable MC340-63 switching regulator IC. These are about as cheap as you can buy a SMPS IC for and can implement almost any topology. As you have noted, to make a constant current driver you compare the drop across the current sense resistor with a reference voltage. The MC34063 has an unfortunately high reference voltage (about 1.2V) so you can use an opamp or comparator. This adds cost and complexity, but using an eg LM358 dual opamp is almost as cheap as the MC34063. Scale current sense resistor with opamp up to reference voltage. Figure 6. in this applicationnote give the general idea - in this case opamp inverts and it's a CUK converter, but the principle is the same.