Doesn’t matter whether it’s series or shunt wound, the model’s the same.
It’s not that difficult conceptually.
A d.c. motor will have a locked rotor torque:
T[sub]0[/sub] = K V[sub]s[/sub] / R, where
K = a constant for a particular motor,
V[sub]s[/sub] = supply voltage,
R = locked rotor resistance.
It will have a no-load speed of:
w[sub]0[/sub] = V[sub]s[/sub] / K
and in between locked rotor speed (0) and no-load speed, the torque will be:
T = T[sub]0[/sub] (1 - w / w[sub]0[/sub])
It will draw a current of:
I = T / K
Let’s say we have a 12 V motor, which draws 2 A with a locked rotor, and has a no-load speed of 3,000 rads[sup]-1[/sup]. Then:
R = 12 / 2 = 6 ohms,
K = 12 / 3000 = 0.004 NmA[sup]-1[/sup],
T[sub]0[/sub] = 0.004 x 12 / 6 = 0.008 Nm.
What will happen if we reduce the supply voltage to 8 V? No-load speed will drop to 8 / 0.004 = 2,000 rads[sup]-1[/sup]. This doesn’t tell us much about current, though, because in this model, no-load current is zero (i.e., frictional losses are treated as a load).
So let’s have the thing drive a fan. We’ll assume that with a 12 V supply, it’ll drive the fan at 2,700 rads[sup]-1[/sup]. We’ll further assume that the torque required to drive the fan is of the form:
T[sub]f[/sub] = a w
where a is a constant for the fan.
So for our fan,
a x 2,700 = 0.008 (1 - 2,700 / 3,000)
a = 2.963 x 10[sup]-7[/sup],
and the motor will draw a current of:
0.008 (1 - 2,700 / 3,000) / 0.004 = 0.2 A.
What happens now when we reduce the supply voltage to 8 V?
2.963 x 10[sup]-7[/sup] x w = 0.004 x 8 / 6 (1 - w / 2,000)
w = 1,800 rads[sup]-1[/sup]
I = 0.004 x 8 / 6 (1 - 1,800 / 2000) / 0.004 = 0.133 A
So speed dropped and current dropped as the supply voltage dropped.
The above result will hold for any monotonically increasing load torque function, i.e., provided it always takes more torque to make the thing go faster. This will generally hold for any type of load, once the initial static friction has been overcome.
That’s the end of the story for d.c. motors.
We’re not talking about negative resistance devices, here, so reducing current through the thing necessarily implies reducing the voltage across it.