[MUSIC] Welcome back to Linear Circuits. This is Dr. Weitnauer. This lesson is Introduction to Second-Order Circuits. Our objectives are to define a second-order circuit, to define the natural and forced responses, and to specify the procedure for obtaining the complete response. There are two equivalent definitions of the second order circuit. First, a linear circuit with effectively two distinct storage elements. Now while there's two capacitors here, they can't be combined. They're not in parallel because of that resistor, so this is a second-order circuit. The other definition is a circuit with voltages and currents that are described by a second-order differential equation, for example what's shown here. There are a couple of standard examples of second-order circuits. The first is the parallel RLC. For example, you might be asked to find the current through the inductor. The other example is the Series RLC, and for instance, you might be asked to find the voltage across the capacitor. There's a standard form for the differential equation shown here. On the left, notice that there are only terms involving the variable of interest, which is y, as I've shown here. Y will represent the desired current or voltage. The derivative orders are arranged from the highest, which is the second order derivative, to the lowest, which is the zero order derivative. And there's a unity coefficient, that is a coefficient of one, on the second order derivative term. Now when it's in this form, everything else is on the right-hand side of the equation, and that's called the forcing function. The solution to the differential equation has two parts. They will be the natural and forced responses. Shown here, we usually use a subscript n to indicate the natural response and f to indicate the forced response. The natural response describes the circuits reaction to stored energy at t=0. The forced response is a result of the independent sources in the circuit that are active at t>0. Now here's some examples of natural responses. There's actually three categories of them. One category is overdamped, then you have critically damped, and then underdamped. There will be another lesson that focuses on natural responses but for now, let me just indicate that each one of these three has two undetermined coefficients. For example, the A1 here and A2. And so these will come up later in this lesson. Because of the e to the -3t type of factor, the natural response eventually dies out, and for this reason it's also called the transient response. Now let's talk about forced response. In this lesson, we're going to be emphasizing DC sources. DC sources yield a constant forcing function. For example, in this differential equation that's in standard form, you see the forcing function is just 21. And in response to a constant forcing function, you have the forced response which is also constant, shown here as K. Now you'll identify K through substitution into the differential equation. So since it's a constant, the second order and first order differential terms are going to go to 0. And you're just going to be left with the 7k = 21 which you can solve to get k = 3. So that makes the forced response y sub f equal to 3. Now here's the procedure for obtaining the complete response and there'll be other lessons that deal with different parts of this. First, you find the differential equation in standard form. You get the natural response. Do not evaluate the coefficients. Remember there's two of them. Do not evaluate these coefficients yet. Then you find the forced response and you do evaluate the coefficients through substitution. You find the initial conditions. Then you add the natural and forced responses and you evaluate the remaining coefficients using the initial conditions. There's two parts of this procedure which are the most challenging and they're also the ones that are related to circuit analysis, finding the differential equation and finding the initial conditions. We'll have lessons on each of those. So summing up, the key concepts are second order circuits are defined by either having two distinct storage elements, or equivalently, that they're voltages of currents are described by a second order of differential equation. The voltages and currents each have two parts, the natural response to stored energy and the forced response to the active independent sources. The complete response is the sum of natural and forced responses and you'll use the initial conditions to evaluate those last two coefficients. Okay, thank you. [MUSIC]