RC dynamics shape: the time evolution (rate of charge or discharge) of an ideal first-order capacitor circuit with a step input follows which mathematical function class?

Introduction / Context:
First-order circuits with one energy-storage element (RC) respond to steps in a characteristic way. Identifying the function type is foundational for predicting settling time, bandwidth, and filter behavior.

Given Data / Assumptions:

• Single-pole RC circuit, step input.
• Ideal linear components.
• No additional feedback or higher-order dynamics.

Concept / Approach:
The governing differential equation is dV_c/dt = (1/(R*C)) * (V_in - V_c). Its solution is exponential in time: V_c(t) = V_final + (V_initial - V_final) * exp(-t/(R*C)). The rate (slope) is initially largest and decays as the state approaches the final value, defining the exponential approach to equilibrium.

Step-by-Step Solution:

Verification / Alternative check:
Curve-fitting lab data for RC steps yields straight lines on a semi-log plot (log of the remaining error vs time), confirming exponential behavior.

Why Other Options Are Wrong:
Linear/parabolic/trigonometric/log-only forms do not satisfy the standard first-order homogeneous solution structure for RC steps.

Common Pitfalls:
Confusing the exponential state trajectory with “linear charging” due to constant current (which occurs only with a current source into C).

Final Answer:
An exponential