The topic of this problem is AC Steady State Power. The problem is to determine the power supplied by the power company to the given 480 V load. So we have a load, which is external to our company, which has a voltage of 480 V at the load wall outlet, if you will. And the power associated with that load is 88 KW. That's the average power. It's the real power associated with the AC steady state load. We also have a power factor, and the power factor is 0.707 lagging. Lagging means that the current lags a voltage. The phase angle of the current lags the phase angle of the voltage by a certain amount. Remember that the power factor definition is, power factor is equal to cosine of the angle of the voltage minus the angle of the current. So we can use this information to determine what that difference in the two phase angles are. Our voltage is supplied by the power company and a certain current is supplied by the power company. There's a line voltage of 0.08 ohms associated with the path from the power company to the customer, and we'll take that into the count as well. So, the first thing we want to find if we're looking for the power supplied by the power company to the given 480 V load, is we need to find the current Is. And we're going to work in RMS terms. Remember, the RMS, in the case of current, if we have a sinusoidal current with an amplitude of I sub m, then our RMS value is going to be I sub m divided by squared root of 2. So it gives us our RMS value. We can write a similar expression for V sub m, and VRMS is going to be equal to V sub m divided by square root of 2. So now if we want to find IRMS, we're going to take information that we know, namely that first of all, if we want to go back to our equation for average power. We know that average power is 1/2 V sub m I sub m cosine ( theta V- theta sub i). And so, if we plug in RMS values for V and I, and this VRMS IRMS times the power of factor. Again, the power of factor consign of theta V minus theta I. So we can use this information to find IRMS. Therefore, IRMS is going to be equal to power average for a load, which is (88 KW) divided by our VRMS, which is 480 V for a load time times our power factor, which is 0.707. So if we take that expressions and solve for IRMS then we have IRMS which is 259.3 A. And again, this is RMS current, root mean square current. So the power. Supplied is going to be a combination Of the power that's absorbed in the source, Resistance, plus the power and the load. So the power of the source from the lead lines that attach the power company to its customers is 0.08 times I squared RMS. And that's simply using our equation that we have up above. We know the power factor for the source, which is a resistance, the power factor is 0. The angle between the voltage and the current is 0. We know that IRMS is IRMS. And we know VRMS is IRMS times R, which is the 0.08. So we get this first term from our expression for the average power. In this case, it's for our source resistance. Then we're going to add to it the power to the load. And the power of the load is given to us, it's 88 KW. So, the total power that's supplied is going to be equal to 93.38 KW. Now, if we take the same problem and we change the power factor, right now, our power factor is 0.707. Let's say, that we improve the power factor and we bring it closer to an ideal factor of one. Ideally, our power company would be delivering its power to a purely resistive load, meaning the community. But in fact, that's not the way it works. In practice we know that there is some capacitive and inductive, which in totality end up shifting our load away from a purely resistive load to an impedance base load. But the closer we can get to a purely resistive network, the more efficient our power company can be with delivering power to the customer. Because now we're just delivering real power, or average power, and we're not going to be delivering any of the reactive power that is ultimately lost in heat or electromagnetic radiation or other lost mechanisms. So let's say that we increase our power factor, we improve our power factor. And we're going to raise it to 0.9 lagging. It means, the current lags the voltage, and the cosine of the angle between those two is 0.9. So in this case, IRMS using the same approach that we had above to finding IRMS. Instead of 0.707 for it, putting in 0.9 for it and solving, we end up with IRMS which is 203.7 A. And again, this is RMS. So first of all, if we look at this we first of all increase or improve our power factor from 0.7 to 0.9. And what's that do to the amount of current that the power company has supplied to the supplier? It goes down, and was originally 259 A required. Now it's only 203 A required to satisfy the same load at the output, the same average power load. We've improved the power factor once again. So, what is the power that has to be supplied in this case? So if we look at the power supplied. The power supplied is going to be the 88 KW from the load again, because we know that that's part of what's being absorbed at the far end of the circuit. Plus we have, again, our 0.08 line resistance, we haven't improved our lines. And we're only using 203.7 A of current now. And so we end up supplying 91.32 KW. So the power company has a reduction in the amount of power that they have to supply at their end in order to satisfy the needs of the load in terms of real power if the power factory increases from 0.707 to 0.9.
