Welcome back to linear circuits where today we're doing part two of our power factors and power triangles lecture. In the previous lecture we talked about calculating complex power and identifying what the various aspects of that complex power represented. Today we're going to continue on talking about the power factors and power triangles, particularly about power triangles. And as we understand those things we'll then be able to talk more about maximum power transfer in AC systems. The objectives for this lesson are to have you use power triangles and to calculate power angle and power factor, real and reactive power, and apparent power for a system. When we calculated the complex power, we plotted it on a complex plane. Where we had the real part here and the imaginary part here and we labeled all the various values. What we are going to do is take the triangle out of this complex plane and draw it like this, we see that we have all the same basic measurements. We have the P, the Q and the absolute value of S corresponding the real power, reactive power, and the parent power, and the hour angle beta. All right, our power angle, so, to remind us of the equation, we use proof finding the, complex power. I've listed it here and I've put the power triangle, and what we're going to do is find out how to calculate all these things from the equations. So here we have the voltage equation and the current equation with the m cosine of omega T, plus state E and Im cosine of omega T plus theta I. Now we'll go through one by one all of the different important things that we're going to be defining for these power triangles. First of all, the power angle, the power angle theta is the single right here. And that corresponds to theta v minus theta i which we can pull immediately from our equations for voltage and current. We are going to call the cosign of theta the power factor. The reason we are able to say that, is if you calculate the length of P, given we know theta and the apparent power and absolute value of S, as P is equal to modular s, cosign of theta, and so that's why we call it the power factor. Because it's how much we're multiplying by our apparent power to find the average power that's being consumed. So that means that we can calculate average power by taking the RMS voltage multiplying it by the RMS current, and multiplying that by cosine theta, and this just follows immediately, because of our calculation of how we found the apparent power. For reactive power, Q we want to find the length of this leg, and you think again basic trigonometry. It was going to be equal to the apparent power times the sine of beta. Now, here's where it gets a little bit hairy as far as units are concerned. How much power is going to be measured in watts because we are doing work, and watts is the rate by which energy is being used. But, when we are talking about reactive power it corresponds to power that is temporarily being stored up to be used at a later time, and it is not actually doing any real work. So, to help distinguish it, we are going to be using a different unit. The units are Volt-Amperes Reactive, sometimes distinguished as VARs, or VARs. It's basically measuring the same thing, because we're taking a voltage, multiplying it by a current, and then multiplying it by something that's unitless, which is basically the same thing we have for power. And so we're just going to call it something different to help distinguish that it's not really being used, it's just a temporary storage type of thing. And apparent power we're again going to use a different unit, we're going use volt amperes or va and we've all ready discussed how to calculate that apparent power. So now we can see that we can calculate all these different values from the voltage and the current equations. And that they have a relationship that corresponds to the trigonometric relationship that we see in a triangle. Now we can look at the relationships between the apparent power, the real power, and the complex power, or the reactive power, brother for different devices. So first of all, the resistance case, we've already looked at. The voltage and the current multiply together to give something that is not negative or power, they're in the green, and [INAUDIBLE] is in the same, B is red, i is blue, and green is p. Again, all different units here, I here illustrate the impedance of this device. Because it's all real, it's just a real line out here and this is a triangle in a degenerative sense where this phase angle, this angel here in this corner is zero. And we all see that the impedance triangles are similar in the mathematical sense similar to the complex power triangles. All my power, all my apparent power is real power, and there is no reactive power for a resister, as I move this around, get more of a proper triangle. Now I have a slightly inductive load because I have a resistor, I have an inductor. Now you'll notice that, on these plots, my voltages stay the same, I'm moving currents with respect to the voltages, and then obviously the powers are changing as well. No longer are the voltage in the current in phase. Now the voltage leads the current a little bit, or the current lags behind the voltage. This is my impedance triangle for this system where the real part comes from the resistor, the imaginary part comes from the inductor. So the real part can be imaginary part giving us this and this theta corresponds to this theta, because these are two similar triangles. For systems that have an inductive load, my phasing goal's going to be between 0 and pi 1/2 for my power angle. And the power is going to be similar between the extremes of zero and it would be value of the apparent power. But so is my reactive power, somewhere between zero and the apparent power. I'm going to say that the system is lagging, because it's inductive and we say that it lags or is lagging because the current lags behind the voltage. The voltage hits its maximum point before the current. If I go to the extreme case where it is a pure inductor, now my triangles are again degenerative because my power angles are pi halves. There is no real power being consumed. All the apparent power goes to reactive power. I can do the same thing with capacitive loads like this. Now, we see that the current leads the voltage, so it's going to be leading. This is my impedance triangle, wich is similar to my power triangle. My theta, my power angle, is between minus pi halves and zero. My power is between zero, and the apparent power, and my reactive power is now going to be negative because it's down here in this quadrant. And it's going to be between the minus value of the apparent power and zero, we can see all of the relationships here. And finally, looking at the purely capacitive case, all of the power is reactive power because again the capacitor stores up the energy temporarily to be used at a later time. What are the implications of this? Well first of all only real power is being transformed as heat or light or moving the motor ro doing real work, and typically that's what we're most interested in. We don't really care about how much power is being temporarily stored up in the system, we want to know how much is being used to do real work, but that doesn't mean we can neglect the reactive power. Reactive power causes increased currents. This slide means that the more reactive power we have in our system, the more currents need to be used in the lines connecting to your device. And so you have more loss in the transmission lines leading to that device. So if you're a power company, Bill Walton give it some heat to that reactive power. Now private customers for their private homes and residences typically only are charged for the real power that they consume. The power company doesn't really care too much about how much reactive power there is because the residential home doesn't have a lot. But if your big industrial system with big motors, there's a lot of inducting rules in those types of systems. And so, consequently, industrial consumers are often charged at a lower rate for the reactive power. Because the power company needs to have more equipment to have more loses in their transmission lines, and they need to design their systems with extra transformers and extra devices to be able to handle those increased loads. So because of this, big industrial users might want to give some consideration to their systems, and change their systems a little bit to reduce the amount of reactive power that they have to reduce the costs that they pay to the power company. In summary we defined power angle and power factor, real and reactive power and apparent power. And showed how they all correlate in a triangle behaviour using power triangle we can see the relationship between all those different values. In the next lesson will see how we can use reactive elements Like capacitors and conductors to control the amount of reactive power that we have in the system. And we can analyze maximum power transfer for AC systems and look at how that is different from the maximum power transfer that was calculated in DC systems. Until then.
