For a DC motor, speed of the armature is inversely proportional to current.
But at the same time, we know that torque of the armature is directly proportional to the armature current.
Since speed can be increased by applying torque, this means that speed is directly proportional to armature current.
How is this possible ? Where have I gone wrong ?
Where you have gone wrong 1
Firstly your question is poorly defined you should clearly state what type of equipment you are referencing. There are numerous types of D.C. Motors and variations on how the excitation can be achieved. Therefore I will ignore differential compound D.C. Motors.
Where you have gone wrong 2 is your statement Since speed can be increased by applying torque. Torque speed curve
In fact this is the exact opposite of what happens. As speed increases the torque reduces. As the the load is reduced and torque approaches zero the armature speed reaches maximum. because AS THE SPEED INCREASES THE BACK EMF INCREASES AND THEREFORE THE CURRENT REDUCES. so the statement speed of the armature is inversely proportional to current is correct.
Tutorials emf = inductance x rate of current change” and a circuit has an inductance of one Henry will have an emf of one volt induced in the circuit when the current flowing through the circuit changes at a rate of one ampere per second.
One important point to note about the above equation. It only relates the emf produced across the inductor to changes in current because if the flow of inductor current is constant and not changing such as in a steady state DC current, then the induced emf voltage will be zero because the instantaneous rate of current change is zero, di/dt = 0.
Therefore it is obvious that as the speed of the armature increases so the rate of change of flux will increase and so the back emf also increase.
However if you were to say that the speed of a Electric locomotive can be increased by applying torque that is correct, at least in the initial stages of movement.
Where you have gone wrong 3
Your statement torque of the armature is directly proportional to the armature current is also incorrect electrical guide
In fact T α Iaφ Torque α Armature current x Field flux and thus the equation T= I? will depend on what type of motor you are referencing.
On light loads, the torque produced by the series motor is proportional to the square of armature current and hence curve drawn between torque and armature current up to magnetic saturation is a parabola. But after magnetic saturation flux φ is independent of excitation current and so torque is proportional to Ia and hence characteristics become a straight line.
The flux of a shunt motor is practically constant. Therefore, T α Ia Torque produced is proportional to the armature current