Welcome back to our class on linear circuits. Today, we're going to be doing the first part of power factors and power triangles. And with these lessons, we're going to be trying to gain an understanding of how we analyze systems that are power systems in the sinusoidal area. So in the previous lesson, we calculated the root mean squared values. We looked at sinusoidal and triangular functions, and we used these examples to calculate RMS values. But we didn't really use the rms values that we calculated for anything meaningful. We're going to be applying them to our calculations in this section. So today we'll be talking about power factor and power triangles using the tools of RMS calculations we used before. When we understand this we can then go on to talk more about maximum power transfer in sinusoidal systems and see how we can use power factor correction in those types of systems. The objectives for this lesson are to find average power in resistive and reactive devices and to calculate complex power and get some idea behind the meaning of what complex power is. First of all, we look at instantaneous power. Instantaneous power is the power that we are already familiar with. If I look at power being a function of time, that is going to be equal to the voltage at particular time, times the current at particular time. So here, if I look at a resistor, I know that the relationship between current, voltage, and a resistor is v = iR. Here that is illustrated on this graph. I am, on this graph, going to put current, voltage, and power all together even though they're all measured in different units. So don't let that throw you, I just put them all together on the same graph, so you can see the relationship in time between the three. So here I see IM, which is measured in amps, in the amplitude of the blue curve. VM in volts, and that's the red curve is the voltage. And then P, which is measured in watts, being the green curve. We identify that when I multiply the blue curve and the red curve together we get a curve here in green that is at twice the frequency of the blue curve or the red curve. And it's all positive. Well, not negative. And the maximum point of this function is going to be at VM x IM because those are the values. Here, and here, we'll put them together. Turns out that if you take two sinusoids, and you multiply them together, the result is also a sinusoid. It's not centered, but it is sinusoidal. And you can actually do the calculation to show that. Let's look at a different device. This is the capacitor. Again, the instantaneous power, is still defined the same way, multiplying the voltage and the current at any point in time. But the relationship between current and voltage in a capacitor is a little bit different. We know that i is equal to Cdv/dt. Here I'm showing the voltage and the current to have the same amplitudes as in the previous graph. And that again, they're labeled here. Now, my power curve is going to be right here. And it turns out that now, again, it's twice the frequency of the red curve or the blue curve. But we see that this curve is now centered. It's not all above as it was over here. But it's interesting to notice that the amplitude of this green curve, or the distance between the top to this middle region is the same as for this curve over here. If I want to find the average power of the resistor here, we can just take the average over that curve and we find it to be halfway in between the top and the bottom, which is at this green dotted line. Well, I know that this maximum value here is at VmIm. That means that this curve is going to be at VmIm/2, which is halfway between 0 and that maximum point, because again, it's just a sinusoid. So we can calculate P average by taking the magnitude of the voltage times the magnitude of the current and dividing by 2. But it also turns out that this is the same as multiplying the rms voltage times the rms current, remembering that for a sinusoid the rms is going to be equal to the magnitude divided by the square root of 2. And the same thing for the current. If instead I look at the capacitor example, well, remember that sinusoid was centered. So that means that the average power is going to be 0. But we can still see that, again, we can find this amount with Vrms times Irms. Now let's look at something that's just some arbitrary impedance Z. We don't know exactly what it is, but we can see the voltage in red and the current in blue. Now the power in green is not centered like it was for a purely reactive system, like a capacitor. And it's not all completely non-negative as it was in the resistive case. Now it's somewhere in between those two extremes. But again, the distance between the middle of the sinusoid to the top is Vrms Irms. That stays consistent. So what we're going to see is that as we have different types of devices, we're going to get things between these two different extremes, between the completely reactive element which is a centered sinusoid and the completely resistive element, where the power is completely non-negative. And the average power then is one-half of the voltage magnitude times the current magnitude. How do we do the analysis to find these calculations for things that are somewhere in between? One way that we could do that is by using complex power. In this example, we have a voltage expressed as 5 cosine of omega t plus pi thirds. And a current as 2.5 cosine of omega t plus pi twelfths. I graphed them together, again using different units but together on the same plot here. As I multiply these together what we're going to be doing is we're going to use the phasers for our calculation to calculate what's known as the complex power. And the way we calculate that is that the complex power is equal to one-half times the voltage phaser times the current phaser but the current, phaser is going to be the complex conjugate of the phaser, not the phaser itself. So S is equal to one-half VI*. Well, remembering that my phaser for voltage is going to have a form of vm with a angle of theta v And our current Im with an angle of theta I. Multiplying these together in this fashion gives us the complex power which we denote as S = Vm x Im divided by 2 with an angle of theta v- theta i. In this particular example, if I take 5 x 2.5 divided by 2 = 6.25, and if I take pi thirds and subtract pi twelfths, I get pi fourths. So that means that the complex power for this system is equal to 6.25 with an angle of pi fourths. Well, this is all well and good, but what does it really mean? So we're going to look at what complex power actually represents. So again, I put that S = to Im x Vm/2 or VRmS x IRmS, with an angle of theta v- theta i. Plotting this together, which is just like I did before, and then plotting the power curve, we see that this is the power curve. It's important to notice that when I've calculated complex power I've multiplied two phasers together, but my result is not a phaser. That's because phasers are a part of a vector space and you just can't multiply things together that way. It has a different meaning, so just remember that when I multiply these things together to get a complex power I do not get a phaser result. If it were a phaser, the green curve would be centered with the magnitude of Vm Im/2 and it would have a phase shift of theta v- theta i. But we can see that's not exactly the case here. One thing we can do is take this complex value and plot it on a complex plane. So here the complex plane has the horizontal access indicating the real values, and the vertical axis, corresponding to the imaginary values. My value S, my complex power, is this point right there. And f I take this vector and I find the amplitude of it by taking the modular of S which is denoted here, we'll be calling this later on the apparent power. I can just find that by taking the real part which is here, squaring it, adding it to the square of the imaginary part here and then taking the square root. Now this theta corresponds to this theta v- theta i. We're going to be calling this the phase angle. To see what this represents, first of all, we'll look at this apparent power, this modular S. Well, if I measure from the center of the sinusoid to the peak of the sinusoid, it's equivalent to go from this minimal point up to the center. That is equal to the absolute value of S, or the apparent power. So, that's kind of a representation or what is represented by the apparent power. If I now go and find the average power of this curve, which is now this dotted line labeled P, it turns out that this can be found by taking the real part of our complex power. So here P is the real part of this S sector. This is going to be real power, and this is basically corresponding to how much power is being consumed by this device to do work. This would be for making light, heat or moving something, anything that actual power is being consumed to accomplish. And that corresponds to this P here. If I take the imaginary part, we're going to be labeling that as Q, and we'll call that the reactive power. This corresponds to power that's kind of stored up temporarily to be used at a later time. Going back to our curve, we notice that anywhere that power is positive we've identified that to be power being consumed. But anywhere power is negative, that's power being generated. So where is the power being generated? Well, in these regions, where the power is negative, that corresponds to some reactive element, a capacitor or an inductor in the system that is using up the energy that it had stored. And that storage took place somewhere in this positive power area. This value, Q, the imaginary part, corresponds to how much of that power is being temporarily stored and being consumed at a later time. But it doesn't correspond to any power that's actually being used to do any particular work. So to summarize, we calculated the complex power, and we identified the meanings behind the various values of complex power. In the next lesson, we'll look at power triangles which are derived from these complex power vectors that we were looking at. And we'll define the important quantities for power analysis and see what those quantities happen to be for a variety of different loads. Until then.