Second order systems are, by definition, systems whose input-output relationship is a second order differential equation. A second order differential equation contains a second order derivative but no derivative higher than second order. Second order systems contain two independent energy storage elements, per our comments in Chapter 7 pertaining to the relationship between the number of energy storage elements in a system and the system order

Second order systems, like first order systems, are an extremely important class of systems. In previous chapters, we saw that the natural response of first order systems decays exponentially with time – the natural response decays monotonically to zero. The natural response of second order systems can, however, oscillate with time – we will see that a second order systems response can contain sinusoidal components. The motion of a pendulum, for example, can be modeled by a second order system. These oscillations are due to the transfer of energy between the two energy storage mechanisms; a pendulum, for example, oscillates because of the cyclic exchange of potential and kinetic energy of the mass.

Since the natural response of second-order systems can oscillate with time, their response can be fundamentally different than the response of first order systems. In the introduction to chapter 7, we noted that it is common to approximate higher-order systems as first order systems (at the time, we said that such a system has dominant first order modes). This approximation is not possible if the natural response of the higher order system oscillates. However, it may be possible to approximate the response of such a system as a second order system. Systems which behave approximately as second order systems have what are called dominant second order modes. In fact, the natural response of any higher order system can be considered in terms of the responses of multiple first and/or second order systems¹⁾. This is why an understanding of first and second order system responses is so crucial to the engineer – these responses provide the building blocks for understanding the responses of all linear systems.

In this textbook, of course, we are interested in the response of electrical circuits. Thus, we begin this chapter with a presentation of two simple second order electrical circuits: the series RLC and parallel RLC circuits. In section 8.1, we derive the governing equations for these circuits and use the results to write the general form of the differential equation governing second order systems. This equation is in terms of two very important parameters: the system natural frequency and the system damping ratio. The homogeneous solution of this general equation is determined in sections 8.2 and 8.4. In section 8.2, we develop the form of the solution (in terms of the natural frequency and damping ratio). Since the response of second order systems contains complex exponential functions, we provide some material (in section 8.3) relative to complex exponentials and sinusoidal signals. This material will provide us the necessary background to allow us to determine the natural response. Section 8.3 is optional for readers who are comfortable with complex exponential and sinusoidal signals. The overall natural response is developed in section 8.4, using the solution form presented in section 8.2 and the background material on complex exponentials in section 8.3.

• Write differential equations governing second order circuits
• Define damping ratio and natural frequency from the coefficients of a second order differential equation
• Express the form of the natural response of an arbitrary second order system in terms of complex exponentials, the damping ratio, and the natural frequency
• Summarize the behavior of the complex exponentials in the system natural response for the damping ratio ranges below:
  • Damping ratio greater than one
  • Damping ratio less than one
  • Damping ratio equal to one
• Write complex numbers in terms of complex exponentials
• Express sinusoidal signals in terms of complex exponentials
• Classify overdamped, underdamped, and critically damped systems according to their damping ratio
• Identify the expected shape of the natural response of over-, under-, and critically damped systems
• State from memory the definition of an underdamped second order system’s overshoot, rise time, and steady-state response
• Use the coefficients of a second order system’s governing equation to estimate the system’s overshoot, rise time, and steady-state response

We will develop our discussion of second order systems in the context of two electrical circuits examples.

Consider the circuit shown in Figure 8.1 below, consisting of a resistor, a capacitor, and an inductor (this type of circuit is commonly called an RLC circuit). The circuit contains two energy storage elements: an inductor and a capacitor. The energy storage elements are independent, since there is no way to combine them to form a single equivalent energy storage element. Thus, we expect the governing equation for the circuit to be a second order differential equation. We will develop equations governing both the capacitor voltage, $v_C(t)$ and the inductor current, $i_L(t)$ as indicated in Figure 8.1.

In order to determine the governing equations for $v_C(t)$ and $i_L(t)$ we will attempt to write two first-order differential equations for the system and then combine these equations to obtain the desired second order differential equation. To facilitate this process, the circuit of Figure 8.1 is repeated in Figure 8.2 with the node and loop we will use labeled. Note that we also label the current through the capacitor in terms of the capacitor voltage and the voltage across the inductor in terms of the inductor current.