[SOUND] Welcome back to Linear Circuits. This is Dr. Weitnauer. This lesson is DC Behavior of Reactive Elements. Our objective is to learn how to simplify and analyze reactive circuits in DC steady state. This builds on v equals L di/dt for inductors, and i equals C dv/dt for capacitors. First, let's define steady state. A circuit in steady state has an ongoing, persistent behavior. There is direct current, or DC steady state. That means that all voltages and currents are constant. Alternating current, or AC steady state, means all voltages and currents are sinusoidal, with the same frequency. This module emphasizes DC steady state, whereas a future module will emphasize alternating current or AC steady state. We assume steady state conditions when all transients have died out, there are other ways to say this. For example, there has been no change to the circuit for quote, a long time or the circuit is, quote, at rest. These all mean some sort of steady state condition. What is the DC Steady State for Inductors? Remember all voltages and currents are constant which means their derivatives are zero. We have v equals L di dt while L di dt is 0. That means v is 0. That means inductors behave like short circuits. So if you have an inductor in your circuit what is good practice is to redraw your circuit. When you analyzing for DC steady state, redraw your circuit to replace the inductors with a short circuit. Now, at a short circuit you going to have zero volts across the terminals but you will have a current generally. For capacitors, since the voltage is constant, the derivative of the voltage is zero, making the current zero. That means capacitors behave like open circuits. So if you have a capacitor in your circuit and you're analyzing the DC steady state behavior, you re-draw your circuit, and replace it with an open circuit. Now an open circuit, the current is zero but in general you'll have a non-zero voltage across the terminals. So let's consider an example of DC steady state. For the circuit below, find the voltage V. That's the voltage across the capacitor and the current i which is the current through the inductor when the circuit is in DC steady state. To analyze this circuit in DC steady state, we'll begin by replacing the capacitor with an open circuit and replacing the inductor with a short circuit. We keep the resistors as they are. I just put a couple of stubs there to remind us that we are still interested in the voltage across those terminals. And just a short circuit represents that inductor in DC steady state. Okay, so let's talk about the voltage first. Notice that this voltage in DC steady state is now completely the voltage across R2 and R1 and R2 are in series and that series combination is in parallel with the voltage source. So we can find v by voltage division. So we have v equals R two over R one plus R two times three volts. Okay, that would be the answer for v. And then i, notice that i is going to flow through both of the resistors in the series and so what we really have is an ohms law problem. To compute i we'll take the three volts and divide by the total resistance and series which is R1 plus R2. Now let's try a quiz. Assume the circuit below is at rest. Select the correct values of the currents i1, which is through the middle branch, and i2, which is the current through the right branch. Okay, here is the solution to this quiz. This capacitor is an open circuit now. All right, and that means that i1 is equal to zero because the capacitor in DC steady state is an open circuit. The inductor is now a short circuit, and that means that all the four amps has to go through this right branch. And that means that i2 is four amps. So the correct answer is B. To sum up to analyze a circuit in DC steady state, you redraw the circuit, you replace capacitors with open circuits, and inductors with short circuits. Thank you. [MUSIC]
