What is the capacitor voltage during charging in an RC circuit after a step input?

What is the capacitor voltage during charging in an RC circuit after a step input?

• A. Vc(t) = V_final (1 - e^{-t/τ}).
• B. Vc(t) = V_final e^{-t/τ}.
• C. Vc(t) = V_final (1 + e^{-t/τ}).
• D. Vc(t) = V_final (1 - t/τ).

Answer: (A)

When a DC step is applied to an RC charging circuit, the capacitor voltage rises exponentially toward the supply voltage with a characteristic time constant τ = RC. The correct form is Vc(t) = V_final [1 - e^{-t/τ}]. This matches the intuition: at the moment the step occurs, the capacitor is uncharged so Vc(0) = 0, and as time goes on the exponential term decays to zero, letting Vc approach the final value V_final. The other forms don’t fit the charging behavior: e^{-t/τ} would start at V_final and decay to zero (discharging), a linear ramp 1 - t/τ isn’t the true RC response, and 1 + e^{-t/τ} would keep the voltage above the final value.

When a DC step is applied to an RC charging circuit, the capacitor voltage rises exponentially toward the supply voltage with a characteristic time constant τ = RC. The correct form is Vc(t) = V_final [1 - e^{-t/τ}]. This matches the intuition: at the moment the step occurs, the capacitor is uncharged so Vc(0) = 0, and as time goes on the exponential term decays to zero, letting Vc approach the final value V_final. The other forms don’t fit the charging behavior: e^{-t/τ} would start at V_final and decay to zero (discharging), a linear ramp 1 - t/τ isn’t the true RC response, and 1 + e^{-t/τ} would keep the voltage above the final value.