Revision as of 19:19, 7 October 2018

Here several useful generator equations are derived.

Electrical Efficiency

Often, because maximum power transfer occurrences when impedance is match, there are designs where the generator load equals the internal source resistance. Sure maximum possible power is transferred, but 1/2 of the mechanical input becomes heat inside the generator. Instead, oversize the generator, and use a fraction of the possible power to keep efficiency.

Electrical efficiency as a fraction input mechanical power that is available as electricity a 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)   (efficiency)

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

E = Rl / (Rl+Rm)

let Rl = k * Rm then  (define the load resistance as a fraction of the motor series resistance)

E = k/(1+k)

Thus to keep the efficiency high, then keep the load resistance
higher than the motor source resistance, or:

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

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

Efficiency resulting from under loaded generator

For a load resistance 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 (but 1/2 of the input power becomes heat.) Here we calculate k (from above) based on how undersized the load is.

Let C be the generator load rating reduction factor. C = Pload/Pmax

below it is shown the resulting 'k' is : k = [2-C +- 4sqrt(1-C)] / C

Found a mistake in the original solution so I rework below. Lots of math, probably I'm missing the easy method....

load power : Pload = Pin - Ploss    (1)

Pin = Vemf * Im  (2)
Ploss = Im^2 * Rm  (3)

substituting into (1)
Pload = Vemf * Im - Rm*Im^2  (4)

Im = Vemf / (Rm + Rload) (5)

let Rload = k*Rn (6)

substituting 6 into 5:

Im = Vemf / Rm (1+k)  (7)

substituting 7 into 4

Pload = Vemf * Vemf / Rm(1+k) - Rm * Vemf^2 / [Rm(1+k)]^2

Pload = Vemf^2/Rm(1+k) - Vemf^2/Rm(1+k)^2   (8)

collecting common factors :
Pload = Vemf^2/Rm * [ (1+k)^-1 = (1+k)^-2] (9)

knowing  Vemf = Ks * omega (10)  (Ks units are volts/RPM)

replace Vemf in 9 with 10:
Pload = Ks^2 omega^2 / Rm  * [ (1+k)^-1 = (1+k)^-2] (11)

From elsewhere, the equation for maximum possible generator power:
Pmax = omega^2 Ks^2 / 4*Rm (12)    [note in derivation of Pmax below I have Ks units of RPM/volt]

Let C be the generator load rating reduction factor.
C = Pload/Pmax
or
C * Pmax = Pload (13)

substituting (12) into (13) LHS, and (11) into (13) RHS.

C*omega^2 Ks^2 / 4*Rm = Ks^2 omega^2 / Rm  * [(1+k)^-1 - (1+k)^-2] (14)

we can can cancel : omega^2,  Ks^2 and Rm

C/4 = [ (1+k)^-1 - (1+k)^-2]  (15)

making RHS denominator in (15) the same

C/4 =   [(1+k) - 1]/(1+k)^2 

simplify and multiply by 4
C/4 = k /(1+k)^2  (16)

C*(1+k)^2 = 4k (17)

C*(k^2 + 2k + 1) -4k = 0;
C*k^2 + 2Ck + C - 4k = 0;

C*k^2 + k(2C-4) + C = 0;  (18)

solve for k with quadratic equation with  :

k = -b +- sqrt(b^2 - 4ac) / 2a;   (19)

a = C
b = 2C-4
c = C  


k = [4-2C +- sqrt(4C^2 - 8C -8C + 16 - 4*C*C)] / 2C

simplify :
k = [2-C +- sqrt(16-16C)] 

k = [2-C +- 4sqrt(1-C)] / C (20)

Comments : k is the ratio Rload to Rm (6) If C > 1 (load greater than maximum possible power, imaginary solution) The negative root results in a negative resistance for Rload in (6)

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