Write the differential equation for a series RLC circuit with input voltage v(t) and current i(t).

Write the differential equation for a series RLC circuit with input voltage v(t) and current i(t).

• A. v(t) = R i(t) + L di/dt + i(t)/C
• B. v(t) = R i(t) + L di/dt + (1/C) ∫ i(τ) dτ
• C. v(t) = R i(t) + L d^2i/dt^2 + (1/C) ∫ i(τ) dτ
• D. v(t) = (1/R) i(t) + (1/L) di/dt + C ∫ i dt

Answer: (B)

In a series RLC circuit, the same current flows through every element, so the input voltage is the sum of the voltages across the resistor, inductor, and capacitor. The resistor’s voltage drop is R i(t). The inductor’s voltage drop is L di/dt. For the capacitor, use i = C dv_C/dt, which gives v_C = (1/C) ∫ i(τ) dτ (ignoring initial voltage for now). Putting these together gives the relationship v(t) = R i(t) + L di/dt + (1/C) ∫ i(τ) dτ. This matches the differential equation form that relates the input voltage to the current in a series RLC circuit (with the capacitor term expressed as an integral of the current). The other forms misplace the capacitor term or misstate the inductor term, so they don’t correctly represent the circuit’s behavior. If you differentiate both sides, you also get dv/dt = R di/dt + L d^2i/dt^2 + i(t)/C, which is another valid way to express the same relationship.

In a series RLC circuit, the same current flows through every element, so the input voltage is the sum of the voltages across the resistor, inductor, and capacitor. The resistor’s voltage drop is R i(t). The inductor’s voltage drop is L di/dt. For the capacitor, use i = C dv_C/dt, which gives v_C = (1/C) ∫ i(τ) dτ (ignoring initial voltage for now).

Putting these together gives the relationship v(t) = R i(t) + L di/dt + (1/C) ∫ i(τ) dτ. This matches the differential equation form that relates the input voltage to the current in a series RLC circuit (with the capacitor term expressed as an integral of the current). The other forms misplace the capacitor term or misstate the inductor term, so they don’t correctly represent the circuit’s behavior. If you differentiate both sides, you also get dv/dt = R di/dt + L d^2i/dt^2 + i(t)/C, which is another valid way to express the same relationship.