DC-DC converter with some limitations

[MrMoore]

I need a way to boost the voltage in a circuit in which I have a limiting resistor that cannot be removed, and a limited max current. The input is 5vdc coming from a 9v battery put through a 7805 ic regulator, and the limiting resistor is 120ohms. The purpose of all of this is to make an 18ohm motor (with a start up current described to be around 40mA) run as quickly as possible within the provided parameters. An additional power source can be used to power another circuit, such as a PWM or something of that sort, so long as none of that power makes it to the motor.

Here is an image of the circuit as it stands with no attempt to boost anything:
**broken link removed**

I have run many tests, and discovered the following:
- Before the start up requirements are met for the motor (before it's running), the current is at its maximum, around 37mA
- After start up requirements are met (when it is running), the current is kicked down to somewhere between 14mA and 16mA. This leads me to believe that the equivalent resistance of the motor as it is running is somewhere in the 270ohm area.

With this information in mind, I scoured the internet looking for different ways to boost my voltage using what was left of my current; keeping in mind that whatever current I lost would be irrelevant, so long as I left enough for the motor to run on, and that I could always use programing to start the boost after the motor was already running. I've tried nearly everything I could find, including but not limited to home made boost converters, switch mode power supply ICs that I had purchased online, charge pumps, different types of transistors, and a few other more abstract ideas that don't have names yet. All I've managed to do so far is heavily frustrate myself. I'm willing to try just about anything that someone else has made work before.

 

[Mr RB]

This looks the same problem as your other thread with the 5v source and 120 ohm source impedance?

At least this time you have said more about what the actual load is.

I would explore options of running the motor direct from the 9v battery. Just because you have 5v regulator does not mean you have to run the motor from 5v, especially when your 5v has a high source impedance.

There are a number of ways you can run the motor from the 9v battery with good starting current and good running efficiency.

 

[MrMoore]

Obviously using the full 9 volts would yield better results than using 5v. This is a challenge to do it using the 5 volts.

 

[Mr RB]

I don't need a challenge.

 

[MrMoore]

I think perhaps you misunderstand me. I'm not asking for a way to make this happen for the fun of it; I'm in a competition that requires me to make this work within the given parameters.