Transients in electric circuits normally disappears within a time equal to

Question

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Option A: $4 	imes 	ext{time constant}$

Explanation & Concepts:
In an electric circuit containing resistance and energy storage elements (inductors or capacitors), the transient response follows an exponential decay characterized by the time constant (	au). Theoretically, the transient component decays to zero as time approaches infinity, but for all practical engineering purposes, it is considered to have vanished after 4 to 5 time constants, where the value reaches approximately 98.2% to 99.3% of its steady-state value.

Working Principle: The transient response is governed by a first-order differential equation. For an RL circuit, the current is given by $i(t) = I_f + (I_0 - I_f)e^{-t/	au}$, where $	au = L/R$. At $t = 4	au$, the exponential term $e^{-4} \approx 0.018$. Thus, the remaining transient deviation is less than 2% of the initial value, which is generally within the tolerance of standard engineering measuring instrumentation.

Key Formulas: $v(t) = V_f + (V_i - V_f)e^{-t/	au}$ — General transient equation; $	au = R_{eq}C$ or $\frac{L_{eq}}{R_{eq}}$ — Definition of time constant

Key Points: At $t = 1	au$, the circuit completes ~63.2% of its transition.; At $t = 4	au$, the transient is considered negligible (~98% complete).; At $t = 5	au$, the transient is approximately 99.3% complete.; Transients arise due to sudden changes in circuit configuration (switching).

Advantages: Provides a standardized threshold for relay operation timing.; Simplifies analysis of power system stability after fault clearing.

Disadvantages: Does not account for higher-order systems (oscillatory transients).; Neglects non-linear load behaviors during transient periods.

Applications: Protective relay setting coordination.; Determination of circuit breaker re-closing times.; Power system stability studies.

Additional Info: The 4	au rule is an industry standard for simplifying analysis in electrical power systems.. Option B (2	au) represents only ~86.5% completion, which is insufficient for most practical system stability applications.

Answer: A is correct — transients are considered effectively decayed after 4 time constants because the remaining magnitude is less than 2% of the initial value.

Exam Tip: Always remember that in power systems, DC offset transients in fault currents typically decay within 3-5 cycles, which is related to the circuit's X/R ratio and its resultant time constant.

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Option B: $2 	imes 	ext{time constant}$

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Option C: $8 	imes 	ext{time constant}$

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Option D: $1 	imes 	ext{time constant}$

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