Electrical – floating inverter output causing over circuit protection

If you really must avoid the use of the inverter IC, an alternative is to create your own inversion in software, using more of the Arduino pins:

Each of the outputs generated by your Arduino code is being used to generate a positive and a matching negative signal via the inverter, to drive opposing halves of the H-Bridge in the L293.

To achieve the same result in software, if your code uses something like DigitalWrite(pin1, x) on a given pin, you will have to assign another pin and add another DigitalWrite(pin2, !x) immediately following the previous line of code. This will generate the inverted signal, which you would feed in where the inverter output line goes in your current circuit.

If, however, PWM and AnalogWrite() are used as your method of motor control, some slightly more involved coding will be needed to generate PWM that is exactly opposite in phase to the existing PWM outputs. This may well be within your skills (or not), but I'll refrain from coding it for you in this answer.

If you need help with that, it deserves a separate question.

Yes, you can do this, and it may or may not be fairly easy for you. If you supply the Uno with 12 volts, you're halfway there.

In principle, this will work

^{simulate this circuit – Schematic created using CircuitLab}

but there are a few caveats.

1) This assumes the PWM output of your Uno actually goes to 5 volts, and for a load like this, it probably will. The R1/C1 combination filters out the PWM frequency to produce an approximately DC voltage level of 0 - 5 volts. Then the op amp, with a gain of 2, boosts this to 0 - 10 volts, and Q1 provides the necessary current boost to drive 40 mA.

2) The output may not quite reach 10 volts, since the LM324 needs to provide 10.7 volts when you include the base-emitter voltage needed, and this is right at the edge of what the LM324 can do with 12 volts. If this doesn't work, and you've GOT to have the last volt of drive (9 - 10 volts), you'll need to increase the 12 volts to 15. Or, you can replace the op amp with a rail-to-rail amplifier, and not worry about it.

3) The values of R1/C1 produce a compromise - low ripple of the DC value vs speed of response to a PWM change. You'll have to determine for yourself which compromise is acceptable. Increasing R1 or C1 (or both) will provide a smoother drive voltage, but at the expense of slower response times. I assume an inverter control doesn't have to respond all that quickly, but you need to determine that for yourself.

4) I've shown a 2N3904 as Q1. If you use this, it may well need to be heat-sinked. Assuming the inverter analog input behaves like a simple resistor drawing 40 mA at 10 volts says that you can treat it like a 250-ohm resistor. In that case, Q1's maximum power dissipation will be about 0.15 watts. This ought to be OK, but the transistor should get up to nearly 60 degrees C, which is the "ouch" threshold. Just something to keep in mind.

5) R4 is just there as a bit of insurance that the output will remain stable when the inverter is not connected - and you do want to test this before you hook it up, right?