How do I determine the terminal voltages and currents that will be produced by turning a dc motor at a known angular velocity and a known torque? I know that for a dc motor, the load torque has a linear relationship with speed (negative slope) and with current (positive slope). But these relationships are for when a voltage and current are being applied. Do the same relationships hold when a torque and angular velocity are being applied?
There are four major (electrical) parameters that are used by engineers when selecting a brushed dc motor: The torque constant (Nm/A), the "back-emf" constant (V/(rad/s)), the armature resistance (ohms), and the armature inductance (H). Normally the inductance is low enough that it is ignored. By some fluke of units it turns out that in the SI system the newton-meter/amp is numerically equal to the "back-emf" which is simply the volts/(radian/second) of rotation. When a motor spins it generates a voltage which is called "back-emf". This happens even when you are using the motor to generate torque by applying a voltage. The result is the faster it turns, the less current flows until an equilibrium is reached. More here:
The result is you can estimate the voltage the motor will produce at a given rotational velocity (Volt/(radian/second)). However, the actual output of the motor is going to vary a ridiculous amount so it really can't be depended on. Your best bet is to find an appropriate dc-dc converter that takes the nominal voltage of the motor as the input and outputs your desired voltage.
Also don't forget about the armature resistance that will limit the total power available.
There is theory and there is practice. I have no doubts about Faraday's / Lenz's laws etc. but expecting a formula for every possible set up is asking too much.
I would suggest you go back to first principles, do a few experiments and plot a few graphs. From those results draw your relationships.
Here's the sort of set up I'm thinking of. The motor and the 'generator' would be the same type of device so you should expect what goes in to be similar in magnitude to what comes out.
note: The 'known torque' is a tricky one as the torque will vary with speed (bearing friction and 'windage') together with any current extracted from the generator.
You would start by calibrating the drive motor without the generator attached, measuring voltage, current and speed with 'no load'
You could then repeat this with known loads (torque calculated from applied force to the rotating shaft) so you have the voltage,current,speed and torque characteristics of the driving system.
You should expect that increasing torque on the motor reduces the rotational speed (at a given input voltage) and increases current.
Now you can attach your 'generator'
Start with the output of the generator open circuit and plot the output voltage against input speed. (From the input measurements you can estimate the 'no-load' torque of the generator). This will give you the maximum voltage you can expect at a given speed.
I would expect to see a value of about 1V per 1000 rpm.
Repeat with different electrical load on the generator (Rload) and see what happens as you take power (draw current) from the system.
There are two basic methods, contacting and non-contacting.
Contacting methods usually involve some type of brake or rubbing contact with the rotating shaft with a variable weight or spring system to apply a force.
Non contacting methods either measure the am(you are measuring three variables so you can calculate the torque at any given speed).ount twisting in a drive shaft or have a means of applying force with no physical contact (e.g. magnetic field).