Filter behavior of classic building blocks Consider the standard small-signal approximations for first-order networks. An RL integrator (output taken appropriately so the circuit performs time integration) behaves as what type of simple filter, and…

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Filter behavior of classic building blocks  Consider the standard small-signal approximations for first-order networks. An RL integrator (output taken appropriately so the circuit performs time integration) behaves as what type of simple filter, and…

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Introduction / Context:First-order RC and RL circuits can act as crude differentiators and integrators when operated within limits (small or large time-constant assumptions relative to signal period). Recognizing their corresponding filter “shapes” helps when sketching Bode plots or predicting pass/stop behavior.Given Data / Assumptions:Ideal passive components.Small-signal, linear operation near the intended use regime.Standard topologies: RL integrator output across R; RC differentiator output across R in series with C (or equivalent forms yielding the classic responses).Concept / Approach:An RL integrator is commonly realized with a series RL input and output taken across the resistor. Its transfer magnitude is |Vout/Vin| = R / √(R^2 + (ωL)^2), which rolls off with frequency and passes low frequencies better than high—this is low-pass behavior. An RC differentiator is often a series RC with output across the resistor; its high-frequency gain approaches that of the divider, while low frequencies are attenuated, giving a high-pass response. When used strictly as integrator/differentiator, additional constraints on frequency range are imposed so that the intended operation dominates.Step-by-Step Solution:Write RL integrator transfer: H_RL(ω) = R / (R + jωL) → low-pass magnitude trend.Write RC differentiator transfer: H_RC(ω) = jωRC / (1 + jωRC) when output is across R → high-pass magnitude trend.Relate time-domain roles: integration emphasizes low-frequency content; differentiation emphasizes high-frequency content.Conclude mapping: RL integrator → low-pass; RC differentiator → high-pass.Verification / Alternative check:Bode slopes: both are first-order with 20 dB/decade slope. The RL integrator magnitude decreases with frequency (low-pass), while the RC differentiator magnitude increases with frequency up to the asymptote (high-pass).Why Other Options Are Wrong:high-pass, low-pass: reverses the correct pairing.low-pass, low-pass / high-pass, high-pass / band-pass, band-stop: do not match the canonical behaviors for these elementary blocks.Common Pitfalls:Confusing whether the output is taken across R or L/C; the terminal choice flips the filter type. Also, forgetting that “integrator” and “differentiator” descriptions are valid only over a restricted frequency band.Final Answer:low-pass, high-pass

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