A second-order circuit may have two storage elements of different type or the same type (provided elements of the same type cannot be represented by an equivalent single element).
A second-order circuit is characterized by a second-order differential equation. It consists of resistors and the equivalent of two energy storage storage elements.
An op amp circuit with two storage elements that cannot be combined into a single equivalent element is second-order. Such circuits find a wide range of applications in devices such as filters and oscillators.
Typical examples of second-order circuits are RLC circuits, in which the three kinds of passive elements (resistors, inductors, and capacitors) are present. Other examples are RL and RC circuits.
General procedures to find the complete response \(x(t)\) of second-order circuits:
- Determine the initial conditions \(x(0)\) and \(dx(0)/dt\) and the final (steady state) value \(x(\infty)\).
- Turn off the independent sources and obtain the second-order differential equation by applying KCL and KVL.
- Once a second-order differential equation is obtained, determine its characteristic roots.
- Obtain the transient response \(x_{t}(t)\) depending on whether the response (damping) is overdamped (real and distinct roots), critically damped (real and repeated roots), or underdamped (complex roots).
- Obtain the steady-state response as \(x_{ss}(t) = x(\infty)\).
- The complete response \(x(t)\) is now found as the sum of the transient response \(x_{t}(t)\) and steady-state response \(x_{ss}(t)\).
- Determine the constants associated with the transient response \(x_{t}(t)\) by imposing the initial conditions \(x(0)\) and \(dx(0)/dt\) to the complete response \(x(t)\).
Second-order circuits can also be solved by using Laplace transforms.