Q3Electric Drives
Question
Q.3. Derive the expression for torque in an induction motor from its equivalent circuit. Explain the torque-speed characteristics and their implications for motor selection in various industrial applications.
Answer
The torque developed by an induction motor is derived from its per-phase equivalent circuit as T=(3/ωs)×(I2'²×R2'/s), where I2' is the referred rotor current found from the Thevenin-equivalent circuit seen by the rotor branch; the resulting torque-slip characteristic shows torque rising nearly linearly from zero at synchronous speed, reaching a maximum (breakdown torque) at slip smax=R2'/X2', then decreasing toward standstill, with the specific shape of this characteristic determining motor suitability for different industrial load types (constant-torque, quadratic-torque/fan-pump, or high-starting-torque applications).
Derivation of Torque Expression from the Equivalent Circuit
The standard per-phase equivalent circuit of an induction motor (referred to the stator side) comprises the stator resistance R1 and leakage reactance X1 in series with the magnetizing branch (Rc parallel Xm), followed by the referred rotor resistance R2'/s (representing both the actual rotor resistance and the mechanical power conversion, split conventionally as R2' + R2'(1-s)/s) and referred rotor leakage reactance X2', all supplied from the per-phase stator voltage V1.
Using Thevenin's theorem to simplify the circuit as seen from the rotor branch (looking back from the rotor branch terminals toward the stator supply, through the stator impedance and magnetizing branch), the equivalent Thevenin voltage VTh and Thevenin impedance (RTh+jXTh) are found, and the referred rotor current is then:
The air-gap power (power transferred across the air gap from stator to rotor, for all 3 phases) is Pag = 3×I2'²×(R2'/s), of which the fraction s is dissipated as rotor copper loss (3×I2'²×R2') and the remaining fraction (1-s) is converted to mechanical power output (Pmech = 3×I2'²×R2'×(1-s)/s). Since developed torque T = Pmech/ωm (mechanical angular speed) = Pag/ωs (synchronous angular speed, since Pmech/ωm = Pag/ωs, using the relation ωm=(1-s)ωs), the torque expression is most conveniently written directly in terms of air-gap power divided by synchronous speed:
This is the standard expression for induction motor developed torque as a function of slip s, showing torque depends on the supply voltage (squared, VTh² — hence induction motor torque is highly sensitive to supply voltage variation, roughly as the square of any voltage change), the rotor resistance R2', the slip s, and the various stator/rotor leakage reactances.
Torque-Slip (Torque-Speed) Characteristic
Maximum (breakdown) torque and its slip: differentiating the torque expression with respect to slip and setting dT/ds=0 gives the slip at which maximum torque occurs:
and substituting this back gives the maximum (breakdown) torque value, which is notably independent of rotor resistance R2' (though the slip at which it occurs is directly proportional to R2', as discussed elsewhere in this paper regarding the effect of external rotor resistance in slip-ring motors).
Linear region (low slip): as discussed in an earlier answer, near synchronous speed (small s), torque increases approximately linearly with slip, giving a nearly linear torque-slip relationship in the normal operating region between no-load and full-load, similar in character to the separately-excited DC motor's speed-load characteristic.
Beyond breakdown torque (high slip): as slip continues increasing beyond smax, torque decreases (despite continuing increase in rotor current), since the rotor circuit's increasing reactive impedance at higher slip increasingly shifts rotor current out of phase with the torque-producing flux component, as discussed in detail in an earlier answer.
Implications for Motor Selection in Industrial Applications
Constant-torque loads (conveyors, positive-displacement pumps, compressors, hoists) require a motor whose available torque remains adequate across the full required operating speed range, generally favoring motors with a moderate rotor resistance (or, in slip-ring motors, adjustable external rotor resistance) giving a good balance of starting torque and running efficiency; quadratic-torque loads (centrifugal fans and pumps, whose torque demand rises with the square of speed) can generally use standard squirrel-cage induction motors with lower starting torque requirements, since the load itself demands little torque at low speed during starting; and high-starting-torque applications (cranes, hoists, and other loads requiring high torque from standstill) favor either slip-ring induction motors with external starting resistance (as discussed elsewhere), or specially-designed high-slip/high-starting-torque squirrel-cage motor designs (such as NEMA Design D motors), illustrating how a correct understanding of the derived torque-slip relationship directly informs the appropriate motor design/type selection for a given industrial load's specific torque-speed demand profile.