For a first-order process characterized by a resistance R and a capacitance C (electrical or analogous thermal/fluid storage), what is the time constant τ expressed in terms of R and C?
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AR + C
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BR - C
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CRC
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D1/RC
Answer
Correct Answer: RC
Explanation
Introduction / Context:First-order dynamics appear in RC circuits, well-mixed tanks with linear resistances, and thermal systems with lumped capacitance. The time constant τ determines how quickly the system output approaches its new steady state after a step input.Given Data / Assumptions:
- Linear first-order model with a single energy/mass storage element.
- R denotes resistance to flow/heat/electric current.
- C denotes capacitance (storage capacity).
Concept / Approach:In standard form, the first-order differential equation is τ dy/dt + y = K u, where τ is the time constant. For an RC circuit, τ = R * C in seconds. This generalizes via analogies: hydraulic (R = flow resistance, C = tank capacity), thermal (R = thermal resistance, C = heat capacity), etc. The product RC sets the exponential rise/decay speed: y(t) reaches about 63.2% of its final value in one τ.Step-by-Step Solution:
Write the governing equation for an RC charge/discharge: v(t) = V(1 - e^{-t/(RC)}).Identify τ as the denominator in the exponential → τ = RC.Conclude that the time constant equals the product of resistance and capacitance.Verification / Alternative check:Dimensional analysis: [R]*[C] gives seconds in both electrical and analogous domains, confirming consistency.
Why Other Options Are Wrong:
R + C or R - C: mixing unlike units; not dimensionally valid.1/RC: appears in the exponent numerator; τ is the reciprocal of that coefficient.Common Pitfalls:Confusing τ with 1/(RC); the exponential form must be carefully read to avoid inversion errors.
Final Answer:RC