What type of equation governs second-order circuit behavior?

• A. First-Order Differential Equation
• B. Second-Order Differential Equation
• C. Algebraic Equation
• D. Integral Equation

Answer: (B)

In circuits with energy storage elements, the dynamics are described by differential equations in time. When both an inductor and a capacitor are present, the highest time derivative that appears is second order, so the governing equation is second-order. For a series RLC circuit with a source, Kirchhoff’s voltage law gives v_s(t) = L d^2q/dt^2 + R dq/dt + q/C, where q is the capacitor charge. Since current i = dq/dt, this can be written as L d^2q/dt^2 + R dq/dt + q/C = v_s(t). This is a second-order differential equation in q (and, equivalently, a second-order equation in i if you differentiate appropriately). The two energy storage elements cause this second-order behavior, analogous to a mass-spring-damper system in mechanics. If the circuit had only one energy storage element, you’d see a first-order equation instead.