E_b = V-I_aR_a

But E_b=PФZN/60A

PФZN/60A  = V- I_aR_a

Or  N = (V- I_aR_a)/ Ф ×  60A/ PZ

Or N = K (V- I_aR_a)/ Ф

But         V- I_aR_a = E_a

Therefore N= K E_b/ Ф

Or N α E_b/ Ф

Therefore, in a d.c. motor, speed is directly proportional to back e.m.f. E_b and inversely proportional to flux per pole Ф.

Speed Relations

If a d.c. motor has initial values of speed, flux per pole and back e.m.f. as N₁ ,Ф₁ and E_b1 respectively and the corresponding final values are N₂ ,Ф₂ and E_b2 then,

N₁ α E_b1/ Ф₁ and N₂ α E_b2/ Ф₂

Therefore N₂/ N_{1 =} (E_b2/ E_b1) ×( Ф₁ / Ф₂)

(i) For a shunt motor, flux practically remains constant so that Ф1 = Ф₂.

therefore  N₂/ N_{1 =} E_b2/ E_b1

(ii) For a series motor, Ф α I_a prior to saturation.

therefore N₂/ N_{1 =} (E_b2/ E_b1) × (I_a1/I_a2)

where I_a1 = initial armature current
I_a2 = final armature current

Speed Regulation

The speed regulation of a motor is the change in speed from full-load to no-load and is expressed as a percentage of the speed at full-load i.e.

% Speed regulation = [( N.L. speed - F.L.speed)/F.L.speed ] × 100

=[(N_o -N)/N] × 100

where N_o = No – load .speed
N = Full – load speed

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