Which of the following formulas is used to find the value of different Stud resistance?

Question

Examoogle · Accepted Answer

Option D: All of these

Explanation & Concepts:
In DC motor starters (like faceplate starters) or field regulators, the total resistance is divided into several steps called studs. The resistance of each individual stud or total resistance up to a given stud ($R_n$) can be expressed using different equivalent mathematical formulations based on geometric progression or additive stud values. All given expressions correctly relate the stud resistance values under different indexing and configuration notations.

Working Principle: During the starting of a DC motor, starting current must be limited by gradually cutting out resistance steps as the motor speeds up and develops back EMF. The resistance sections between consecutive contacts (studs) are designed such that the upper and lower limits of starting current remain constant across steps, forming a geometric progression where $R_n = \alpha R_{n-1}$ or $R_n = \alpha^{n-1} R_1$. Alternatively, expressed in terms of individual step resistances $r_n$, the total resistance at stud $n$ is given by $R_n = r_n + R_{n+1}$.

Key Formulas: $R_n = r_n + R_{n+1}$ — Total resistance at stud $n$ as the sum of element resistance $r_n$ and remaining resistance $R_{n+1}$; $R_n = \alpha R_{n-1}$ — Resistance at stud $n$ using the geometric progression ratio $\alpha$; $R_n = \alpha^{n-1} R_1$ — Total resistance at stud $n$ expressed in terms of the first stud resistance $R_1$

Key Points: Stud resistance steps in DC motor starters are graded in Geometric Progression (G.P.) to keep maximum and minimum current limits uniform across all steps.; The constant ratio $\alpha = \frac{I_{min}}{I_{max}}; The total starter resistance $R_1$ decreases step-by-step as the starter handle moves from Stud 1 to the final running stud.

Advantages: Ensures equal peak current during each switching step.; Prevents current surges and torque spikes during motor acceleration.

Disadvantages: Requires precise calculation and custom resistance element sizes for each stud.

Applications: Design of 3-point and 4-point starters for DC shunt and compound motors.; Rotor resistance starters for slip-ring induction motors.

Additional Info: Option A ($R_n = r_n + R_{n+1}$) represents the resistance breakdown per step.. Option B ($R_n = \alpha^{n-1} R_1$) gives the total resistance at stud $n$ from the start of G.P. sequence.. Option C ($R_n = \alpha R_{n-1}$) represents the recurrence relation between consecutive total resistance values.

Answer: D is correct — All listed formulas are valid expressions representing stud resistance relations in stepped starters and regulators.

Exam Tip: In DC starter design questions, remember that stud resistances form a G.P. ($R_1, R_2, R_3, \dots, R_n$) whereas current limits ($I_{max}$ and $I_{min}$) are kept constant for optimal starting.

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: $R_n = \alpha R_{n-1}$

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