An apparatus for controlling speed in induction motor, including a detector detecting a rotation speed of an induction motor, the apparatus comprises: a first calculation unit calculating a diameter of a material using the rotation speed, a minimum diameter of the material, a maximum speed of the induction motor and a line speed command; a second calculation unit calculating a command speed using the diameter of the material calculated by the first calculation unit, the line speed command, the minimum diameter of the material, and the maximum speed of the induction motor; a first adding unit outputting a final command speed by adding a speed bias to the command speed; a speed controller generating a torque current command by compensating an error between the final command speed and the rotation speed; the apparatus characterized by: a third calculation unit calculating a torque limit using the diameter of the material calculated by the first calculation unit, a tension command, the minimum diameter of the material and friction loss compensation; and a limiter limiting the torque current command to a predetermined scope in response to the torque limit, wherein the third calculation unit calculates the torque limit using the following equation: torque limit = tension command × current diameter D cur minimum diameter D min ± friction loss compensation where an output is made by adding the friction loss compensation in a winder mode, and an output is made by deducting the friction loss compensation in an un-winder mode a first calculation unit calculating a diameter of a material using the rotation speed, a minimum diameter of the material, a maximum speed of the induction motor and a line speed command; a second calculation unit calculating a command speed using the diameter of the material calculated by the first calculation unit, the line speed command, the minimum diameter of the material, and the maximum speed of the induction motor; a first adding unit outputting a final command speed by adding a speed bias to the command speed; a speed controller generating a torque current command by compensating an error between the final command speed and the rotation speed; the apparatus characterized by: a third calculation unit calculating a torque limit using the diameter of the material calculated by the first calculation unit, a tension command, the minimum diameter of the material and friction loss compensation; and a limiter limiting the torque current command to a predetermined scope in response to the torque limit, wherein the third calculation unit calculates the torque limit using the following equation: torque limit = tension command × current diameter D cur minimum diameter D min ± friction loss compensation where an output is made by adding the friction loss compensation in a winder mode, and an output is made by deducting the friction loss compensation in an un-winder mode a third calculation unit calculating a torque limit using the diameter of the material calculated by the first calculation unit, a tension command, the minimum diameter of the material and friction loss compensation; and a limiter limiting the torque current command to a predetermined scope in response to the torque limit, wherein the third calculation unit calculates the torque limit using the following equation: torque limit = tension command × current diameter D cur minimum diameter D min ± friction loss compensation where an output is made by adding the friction loss compensation in a winder mode, and an output is made by deducting the friction loss compensation in an un-winder mode