Three resistors have the following magnitudes. R1 = 30 Ω ± 5%,... | Technical MCQ | Examoogle

TechnicalElectrical Engineering

Three resistors have the following magnitudes. R1 = 30 Ω ± 5%, R2 = 60Ω ± 5%, R3 = 50Ω ± 5% Determine the magnitude of limiting resultant resistance when all three resistances are connected in series.

140 ± 7Ω

140 ± 15Ω

27.27 ± 15Ω

27.27 ± 7Ω

Correct Answer

⚙️ TE • Technical Concept & PrincipleTechnicalElectrical Engineering

Option A

140 ± 7Ω

Quick Summary:

When resistors are connected in series, the total resistance is the sum of individual resistances ( $R_{eq} = R_1 + R_2 + R_3$ ). The absolute error in the result is the sum of the absolute errors of the individual components ( $\Delta R_{eq} = \Delta R_1 + \Delta R_2 + \Delta R_3$ ).

🔬SCScience SolutionConcept & Principle

💡 Explanation

💡 Everyday Analogy (Real-World Intuition)

If you buy three bags of potatoes with small weight variations, the total uncertainty in weight is the sum of the individual possible errors from each bag.

🔢 Key Formulas

$R_{eq} = R_1 + R_2 + R_3$ — Resultant resistance in series

$\Delta R_{eq} = \Delta R_1 + \Delta R_2 + \Delta R_3$ — Limiting absolute error for summation

⚙️ Working Principle

The magnitude of the resultant resistance is the algebraic sum $R_{eq} = 30 + 60 + 50 = 140 \Omega$ . The individual errors are calculated as: $\Delta R_1 = 5\\% \text{ of } 30 = 1.5$ , $\Delta R_2 = 5\\% \text{ of } 60 = 3$ , and $\Delta R_3 = 5\\% \text{ of } 50 = 2.5$ . Summing these gives the limiting error: $1.5 + 3 + 2.5 = 7 \Omega$ .

📌 Key Points

▸
In series combinations, resistances add up directly.
▸
Errors in additive measurements always sum up, regardless of whether the signs are positive or negative.
▸
The term 'limiting error' refers to the maximum possible variation in the total value.

✅ Advantages

▸
Simple additive calculation for series circuits.

❌ Disadvantages / Limitations

▸
Does not account for statistical (root-mean-square) error reduction; it calculates the worst-case scenario.

🛠️ Applications / Uses

▸
Instrumentation calibration
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Tolerance analysis in circuit design

📄 Additional Information

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The total resistance is $140 \Omega$ and the maximum possible deviation is $7 \Omega$ .
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Option B (140 ± 15) incorrectly sums the percentage values or miscalculates error propagation.
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Options C and D refer to parallel resistance calculations: $R_{eq} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}}$ .

📊 Diagram / Illustration

✅

A is correct — The resultant resistance is the sum of values ( $140 \Omega$ ) and the error is the sum of absolute uncertainties ( $7 \Omega$ ).

Core Concepts Used

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Resistances in Series Error Analysis Propagation of Uncertainty

💡 EXAM TIP

When adding quantities, always sum the absolute errors; when multiplying quantities, sum the relative (percentage) errors.

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