I have a doubt on relation between torque and speed in a DC motor. Please correct me wherever I am wrong. As we know that if we increase load the armature speed will decrease so that back emf also decrease or say armature current will increase and we know that torque depends on armature current Ia ( in a dc shunt motor so the flux could be constant) Torque proportional to flux × Ia Sience Flux =constant =Torque proportional to Ia So when armature current will increase torque will increase and thus speed should also increase. So my question is that if rotor slows down because armature is not producing enough torque for increased load then why does it not regain the same speed (on which it was rotating) after the armature produces sufficient torque (due to increase in current)
At constant supply voltage, if you increase the load, that that slows the motor down, mechanically. The reduction in speed reduces the back emf, which leaves a higher voltage left over to push more current through the windings. The higher current generates more torque, which allows the motor to come to equilibrium with a higher torque into the higher load.
If you increase the current flowing through the motor by increasing the supply voltage, which increases the voltage across the winding resistance, that will increase the torque and the motor will speed up. Its load will probably require higher torque to drive it at the higher speed, so the motor will come to a new equilibrium at the higher speed, with a higer current and torque.
You have one too many unknowns or not enough information to answer this question.
You are experiencing some kind of rotational speed hysteresis with assumed same dynamic power resuming after dropping out. E.g. Solar powered motor.
DC motors have Torque proportional to Current ( like PV's have current proportional solar power input ) and then No-Load RPM proportional to Voltage. Maximal Power may be around 80% of maximum no-load speed but maximal torque is always from 0 RPM and declines to 0 available torque at some voltage with no-load speed.
You have not defined the motor load (torque vs RPM profile.)
Hypothetically, if you have a load profile that is inversely proportional to speed or has positive feedback or friction is high initially and builds up momentum then you get a hysteresis effect.
The torque is proportional to the current and with no torque the speed is proportional to the voltage. In the simple model, there is a resistor, the armature resistance, and a voltage source, the back-emf, which is proportional to speed.
In the ideal case when there is no load torque, the speed is proportional to the input voltage and the current is zero. If a load torque is added, the current increases, dropping voltage over the armature resistance, so less voltage is across the back-emf, so the speed has to be lower.