• 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 previous
For this reason, it makes sense that (derivatives) => (energy storage elements). The reason why the order determines the number of energy storage elements is more mathematical. Imagine
A second-order circuit is characterized by a second-order differential equation. It consists of resistors and the equivalent of two energy storage elements. Finding Initial and Final Values. First, focus on the variables that cannot change
A 1st order system has one energy storage element and requires just one initial condition to specify the unique solution to the governing differential equation. RC and RL circuits are 1st
Second-Order Circuit To find the response of the second- order circuit, Represent the circuit by a second-order differential equation. Find the general solution of the homogeneous differential
Typical examples are the spring-mass-damper system and the electronic RLC circuit. Second-order systems with potential oscillatory responses require two different and independent types of energy storage, such as the inductor and the capacitor in RLC filters, or a spring and an inert mass.
A physical system that contains two energy storage elements is described by a second-order system model. Examples of second-order systems include an RLC circuit and an inertial mass with position output. The following examples illustrate second-order system models. Example 1.6: Series RLC circuit
We then consider second-order electrical, thermal, and fluid systems. In our consideration of second-order systems, the natural frequencies are in general complex-valued. We only need a limited set of complex mathematics, but you will need to have good facility with complex number manipulations and identities.
The second-order system which we will study in this section is shown in Figure 1.19. As shown in the figure, the system consists of a spring and damper attached to a mass which moves laterally on a frictionless surface. The lateral position of the mass is denoted as x. As before, the zero of Figure 1.19: Second-order mechanical system.
Circuits including two different types of energy storage elements, an inductor and a capacitors. Circuits where there are two energy storage elements of the same type (inductor or capacitor) which cannot be reduced to a single equivalent. Very different topologies can be found for second order circuits.
For this second-order system, initial conditions on both the position and velocity are required to specify the state. The response of this system to an initial displacement x(0) = x0 and initial velocity v(0) = x ̇(0) = v0 is found in a manner identical to that previously used in the first order case of Section 1.1.
