What is the relationship between the speed, torque, and current of a DC brushless motor?

There is a certain relationship between the speed, torque, and current of a DC brushless motor, which can usually be understood and described through the so-called "motor equivalent circuit model".
Simply put, the equivalent circuit model of a motor can consider the motor as a combination of resistance, inductance, electromotive force, and mechanical load, and the interaction between these components determines the operating characteristics of the motor.
For a given voltage, the output torque and current of the motor can be calculated based on the equivalent circuit model of the motor. Generally speaking, the output torque is directly proportional to the current, and the speed is inversely proportional to the current. This is because under a certain voltage, the characteristics of components such as resistance and inductance of the motor remain unchanged, while an increase in mechanical load will require greater torque to maintain the speed of the motor, resulting in an increase in current.
When the load torque increases to 2 times, according to the above relationship, the motor speed will decrease to 1/2, and the current will increase to 2 times the original. At this point, the input power does indeed increase to twice, but the output power will not increase because the efficiency of the motor will decrease with the increase of load, resulting in the increase in input power being partially lost inside the motor and unable to be fully converted into output power.

In an electric motor, the relationship between torque, current, and speed can be explained by the physical properties of the motor and Ohm's law of the circuit.
We know that the torque (T) of the motor is directly proportional to the current (I), that is, T=kI, where k is the torque constant of the motor.
This is because inside the motor, current generates torque through an electromagnetic field. According to Ohm's law of the circuit, the current (I) is proportional to the voltage (U), that is, I=U/R, where R is the resistance of the motor.
In this way, if we increase the load torque (T) while keeping the voltage constant, then in order to maintain the same torque, the current must increase to satisfy the relationship of T=kI. The relationship between rotational speed (n) and current is not a simple linear relationship.
The speed of the motor is inversely proportional to the torque, because the greater the torque, the slower the motor rotates.
In an ideal situation, if we increase the load torque to twice, the speed will decrease to half of the original. If the load torque increases by 2 times, the speed decreases to 1/2, the current increases by 2 times, the input power increases by 2 times, but the output power remains unchanged? There may be some misunderstandings here.
Although input power (UI) does increase with increasing current, output power (Tn or UI * cos)（ θ)） It may not necessarily increase.
This is because the output power depends not only on the current and voltage, but also on the mechanical efficiency (cos) inside the motor（ θ)）。
If the increase in load torque leads to motor overload or reaches its maximum torque limit, the efficiency of the motor may decrease, resulting in an increase in output power that is not necessarily guaranteed.