Revision as of 01:07, 13 January 2009

Here several useful generator equations are derived.

Electrical Efficiency

Electrical efficiency as a fraction input mechanical power that is available as electricity the output terminals. Of course the voltage of the available power might not be optimal - but thats what switching power supplies are for.

Rl = load resistance
Rm = motor series (internal) resistance

E = Pout / (Pout + Ploss) 

Pout = I^2 * Rl and  Ploss = I^2 * Rm thus

E = I^2 * (Rl / (Rl+Rm)) 

let Rl = k * Rm then  

E = k/(1+k)

for a given desired efficiency keep

Rl >= Rm * E/(1-E)

some results:
k = 4 -> 80% 
k = 9 -> 90%

Power Efficiency

For a load that is a multiple of the generator resistance, a fraction of the maximum possible generator output power is obtained. If the load equals the internal generator resistance, then the maximum output power (Pmax) is obtained.

Let the power efficiency be:

Pe = Pout / Pmax

some math...

Pe = 4k/(1+k)^2

64% -> k = 4   (80% electrical efficiency)
36% -> k = 9   (90% electrical efficiency)

Maximum Power

Its also possible to size a generator based on two fundamental properties of the generator. Here we derive the maximum generator power as a function of rotational velocity omega.

For a generator / motor:

omega = Ks * Vemf    (Vemf is back emf, and Ks is the speed constant)
or Vemf = omega/Ks

Vm = Vemf - I * Rm  (Vm is voltage at terminals)

I = Vm / Rl          
let Rl = Rm for max output power

then Vm = Vemf - (Vm/Rm)*Rm
or
Vm = omega/Ks - Vm

so at max output power
Vm = omega/2Ks

output power is V^2/R so

Pmax = Vm^2 / Rm 
or
Pmax = omega^2 / 4 Rm Ks^2

lets define the power constant as 
 
Pk = 1/4RmKs^2   

so Pmax = Pk * omega^2

For example:
A generator might produce 70 volts at 2500 rpm.  The internal resistance is 7.7 ohms.

ks = 2500/70 = 35.7 (rpm per volt) 

Pk  = 25e-6   units are v^2/(ohm rpm^2)
at 1000 rpm maximum power is 25.5 watts
at 2000 rpm 101.8 watts.

A 2nd example is this 490g outrunner motor:

Rs = 0.03, ks = 295 rpm per volt
Pk = 1 / 4*0.03*295^2 = 96e-6 
at 1000 rpm, max power is 95 watts

3rd example : Himax HC6332-230 Outrunner:

Rs = 0.043, ks = 230 rpm per volt
Pk = 109e-6
at 1000 rpm, max power is 109 watts

Powermax EC30-48V :

Rs = .386, ks = 348 
Pk = 5.3e-6
at its peak speed of 25,000 rpm - this little motor could produce 3300 watts 
(and overheat in a few millisconds!)  thus this is a great high speed motor 
to use at fraction of its possible power.

• AXI Gold 5330/24 Brushless Outrunner Motor Kv=197, Rs=0.057 = Pk 111e-6
• E-flite Power 160 BL Outrunner Motor, 245Kv 0.03ohm, Pk=139e-6
• AXI 5345/18 GOLD LINE , Kv = 171, Rs = 42mO, Pk=203e-6
• AXI Gold 5360/20 Kv=120, Rs=.068, Pk = 255e-6
• GNM70130 Rs=26mO->5.7ohm, Kv=9.47->139 Pk=~500e-6 PDF