Which expression correctly describes the RC low-pass differential equation relating Vin, Vc, R, and C?

Which expression correctly describes the RC low-pass differential equation relating Vin, Vc, R, and C?

• A. Vin = RC dVc/dt + Vc
• B. Vin = Vc / RC + dVc/dt
• C. Vin = Vc + RC dVc/dt
• D. Vin = RC dVc/dt - Vc

Answer: (A)

In a series RC circuit with the output across the capacitor, the same current flows through both components. The current through the capacitor is i = C dVc/dt, so the voltage across the resistor is Vr = iR = RC dVc/dt. Applying Kirchhoff’s voltage law around the loop, the input voltage equals the sum of the drops: Vin = Vr + Vc = RC dVc/dt + Vc. This time-domain equation shows how Vin drives the capacitor voltage and defines the RC time constant that governs the response. The form Vin = Vc + RC dVc/dt is the same equation, just written in a different order. The option with Vc/RC would mix units (current vs. voltage) and the minus-sign version would describe a different circuit or polarity, so it’s not correct for this setup.

In a series RC circuit with the output across the capacitor, the same current flows through both components. The current through the capacitor is i = C dVc/dt, so the voltage across the resistor is Vr = iR = RC dVc/dt. Applying Kirchhoff’s voltage law around the loop, the input voltage equals the sum of the drops: Vin = Vr + Vc = RC dVc/dt + Vc. This time-domain equation shows how Vin drives the capacitor voltage and defines the RC time constant that governs the response. The form Vin = Vc + RC dVc/dt is the same equation, just written in a different order. The option with Vc/RC would mix units (current vs. voltage) and the minus-sign version would describe a different circuit or polarity, so it’s not correct for this setup.