Which of the following formula only used during grading of starting resistance for the 3-Phase Slip Ring Induc

Question

Which of the following formula only used during grading of starting resistance for the 3-Phase Slip Ring Induction motor starter?

Examoogle · Accepted Answer

Option D: \alpha = (S_m)^{\frac{1}{n}}

Explanation & Concepts:
In a 3-phase slip ring induction motor starter, the ratio of maximum resistance to minimum resistance per phase step is denoted by $\alpha$. For a starter with $n$ resistance sections, the parameter $\alpha$ is determined by the relation $\alpha = (S_m)^{1/n}$, where $S_m$ is the slip at maximum torque.

Working Principle: During the starting process, external resistance is inserted into the rotor circuit to limit the starting current and maximize starting torque. The total resistance per phase decreases geometrically across $n$ steps from $R_1$ to $R_{n+1}$ (where $R_{n+1} = R_r$, the rotor winding resistance). The step ratio $\alpha$ ensures that the upper limit of rotor current remains equal for all resistance steps.

Key Formulas: $\alpha = (S_m)^{\frac{1}{n}}$ — Step grading ratio in terms of slip at maximum torque; $\alpha = \left(\frac{R_1}{R_r}\right)^{\frac{1}{n}}$ — Step grading ratio in terms of total initial resistance $R_1$ and rotor resistance $R_r$; $R_n = \alpha^{n-1} R_1$ — Total resistance per phase at the $n$-th step; $r_n = R_n - R_{n+1}$ — Resistance value of the $n$-th section

Key Points: The external resistances of the rotor starter are graded in a Geometric Progression (G.P.).; The constant ratio of the progression is $\alpha = \frac{I_1}{I_2} = \frac{E_2}{E_2'} = (S_m)^{1/n}$.; By choosing $\alpha = (S_m)^{1/n}$, the starting current fluctuates between fixed upper limit $I_1$ and lower limit $I_2$ across all $n$ steps.

Advantages: Provides uniform current peaks during starting.; Ensures high starting torque with smooth acceleration.

Disadvantages: Requires precise calculation of external resistance sections.; Power loss occurs in the external resistor banks during startup.

Applications: Rotor resistance starters for 3-phase slip ring (wound rotor) induction motors.; Heavy-duty applications requiring high starting torque like cranes, hoists, and lifts.

Additional Info: Option A: $r_n = R_n - R_{n+1}$ gives the individual section resistance, which is common to both DC and AC motor resistance starters.. Option B: $r_n = \alpha^{n-1} r_1$ describes the general G.P. term relation for individual resistance sections.. Option C: $R_n = \alpha^{n-1} R_1$ expresses total rotor circuit resistance at step $n$.. Option D: $\alpha = (S_m)^{1/n}$ is unique specifically to 3-phase slip ring induction motors because $S_m$ (slip at maximum torque) is an induction motor parameter.

Answer: D is correct — $\alpha = (S_m)^{\frac{1}{n}}$ uniquely incorporates the slip at maximum torque $S_m$, making it exclusive to the resistance grading of 3-phase slip ring induction motor starters.

Exam Tip: Always remember that slip parameters ($S_m$, $s$) appear only in AC induction motor starter equations, distinguishing them from DC motor starter resistance formulas.

Examoogle · Answer

Option A: $r_n = R_n - R_{n+1}$

Examoogle · Answer

Option B: $r_n = \alpha^{n-1} r_1$

Examoogle · Answer

Option C: $R_n = \alpha^{n-1} R_1$

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