We're continuing in section 6.2 Phase in the subject of Phase.

Phase is a relative term to compare two or more sine waves that have the same frequency. Sine waves that are in phase have various, identical points occurring at exactly the same time. The terms leading and lagging are used to describe the relative positions of sine waves with respect to each other. In image, A two different alternators are generating a signal. Note that the loops are shifted one-quarter turn or 90?. This accounts for the 90? phase shift. If we look over here we can see that here we have the two alternators and you will notice that this one is at, this particular cross-section one is cutting maximum magnetic flux lines and this one is in parallel with the North and the South pole.

Now, this one is actually shifted one-quarter turn which equates to 90?. If we look at it on this scale here the black dots here are going to equate to the black phase signal here and this is at the 90? a point which is right here. The blue one, this equates to 0? which is going to be and in parenthesis, we see the 0? which is right here.

We see that these are shifted 90?. Now, remember the mention here was made of leading and lagging and this would be the leading sine wave ad this would be called the lagging sine wave because it is lagging in time. Remember if this was let's pretend this is our 60Hz signal and remember that the time of the 60Hz signal we said it was what? 16.66 milliseconds. If that is the case then we can look at the time from here to here as well and so that is one-fourth of the signal and that's going to be about 4.167 milliseconds.

This signal starts and then 4.167 milliseconds later we have this signal starting and so this will be the leading and this will be the lagging. In this particular case, the reason that it is lagging is because the generator that is generating this signal it is 90? out of phase with the leading signal. Let's see down here I have a picture this is not from your text. This is showing three-phase industrial power and if you've ever come across that will be the three-phase are actually 90? apart.

### Phase

Okay, phase angles are measured in degrees or radians. Make sure your calculator was set up for degrees for this particular course we will be dealing with degrees, we will refer to radians but most of our calculations will be done in degrees. Let's look at like here calculator and calculators will come in many sizes and shapes. Under options here notice we have something here for angle and we need to make sure we are set here for degrees this would be in radians if you were calculating angles and you have to separate these you're going to get the wrong answer, so make sure that you are set out to measure degrees.

Okay, and we will say okay there. Our calculator is now set up to measure in degrees and we can move on here.

### Instantaneous Values

We can express the voltage or current at any instant during a sine wave's cycle. The instantaneous voltage of a sine way is zero at 0? and instantaneous voltages and currents are found by using this formula. The instantaneous voltage can be found by taking VpSin0 and wherever we are in the angle based on the degrees to illustrate this let's draw a sine wave let's see if I can draw I kind of have this as a sine wave and it's not too bad.

Here we would have 0?, this would be 90 and this would be 180 and this would be 270 and this would be 360. What this is saying is we can calculate the voltage at any point in the signal. Let's give ourselves a value let's say that this signal is the amplitude is 10 volts peak. But we want to know what the signal is going to be at let's say at 45? right here, we will calculate there. Let's have another one let's say 70?. Let's have another one down here let's say we will have it at 210? and well, let's take a look at those.

With our calculator let's bring up our calculator and let's input some values here. Let's start out with 45?, so we are going to say 45 and I'll click on sine and take that times notice times is the peak values which is 10 volts and we'll say equals here. What we just calculated was the instantaneous value what is the ... in this particular wave at 45? what is the voltage? It is 7.07 volts at 45?.

We could likewise calculate it for 70, so let's go and say 70 sine times our peak value of 10 and that will be that's very close to our peak of 10 at 70? we are at 9.39. Let's confirm that this actually works, let's say at 90? if we went 90 sine times 10 that would yield our 10-volt peak signal.

My question let's confirm at 180, so let's say at 180 sine value was zero so times 10 it's going to be zero. Zero there. Then let's go to 210 so if we not 210 sine times our 10 volts value and notice it is -5 volts and that correspond that's what it should be because we are looking at a negative voltage now, now we are on the negative portion of the sound wave. At 270 if we went 270 sine times 10 that's where we would see our peak value our negative peak at 10 and finally if we did 360 sine you know this is zero so that is going to be 10 again.

This is how we would calculate instantaneous values on a given sine wave.

### Time and Frequency Domains

All signals can be viewed from either of two perspectives; time domain or the frequency domain. Fourier developed mathematical principles that link together time and frequency domain. Now we will not be viewing four-year analysis in this course but it is a subject for advanced topics in courses that would follow.

A time-domain signal is one whose instantaneous voltage changes over time. An Oscilloscope displays signals in the time domain. We've been talking about this when we look at a signal on an Oscilloscope we look at how long is this sound wave in relation to time? Any given waveform can be shown to be composed of one or more sinusoidal signals at specific frequencies and phases. That is the frequency domain. Spectrum analyzer displays the frequency domain. Now, notice any given waveform can be shown to compose of one or more sinusoidal signals.

Now, if you have purely a pure waveform you won't many harmonics. You will begin to see harmonics when you deal with non-sinusoidal signals like a square wave.

### Harmonics

Any repetitive and this gets into the subject of harmonics. Any repetitive non-sinusoidal waveform can be shown to be composed of a fundamental frequency and some combination of harmonic frequencies.

Notice that it is non-sinusoidal if you have a pure sine as I mentioned that you won't see harmonics but when you see a square wave they will be quite obvious. The fundamental frequency is the basic frequency of the waveform as determined by its period. Harmonic frequencies are exact multiples of the original frequency and they can be odd or even. From your texts, you have a few images that depict this. Here in this particular screed the blue wave here. This is the fundamental frequency, let's just pretend that is 10Khz.

Then we have the third Harmonic. Notice this signal here. There's been black here, here we have one cycle, we have two cycles and here we have three cycles. If this original was 10Khtz then the third harmonic would be at 30 KHz. The first harmonic, let's look at it, here we have one, we have two, we have three, we have four and we have five cycles in the time of the fundamental so that would be at 50Khz if the fundamental is at 10Khz. Here we have slide b. Now, this shows the algebraic sum of the fundamental third and first harmonic and in this case, it's beginning to look like as square wave.

Here we are looking at just the individual frequencies that make up a square wave. Here we are seeing how it would actually display if you only had the fundamental, the third and the fifth. This is adding in the seventh and then here we've added in the ninth and the 11th and you see that it is beginning to take on the shape of a square wave. It will not be a square wave until you have all the harmonics which would be kind of infinite and then it would take on and it would look kind of like this.

I have a simulation of a spectrum analyzer we'll be looking at shortly and it will aid in how that you actually can view the harmonics. This is the demo using electronics work bench of a device called a spectrum analyzer and in your texts, there was mention named of the time domain and the frequency domain. The time domain is viewed with a device called an Oscilloscope and the frequency domain is viewed with a device called the spectrum analyzer. Spectrum analyzers are very precise pieces of equipment's and they can be extremely expensive. The entry level prices on these are probably around $3,000 to $5,000 and you can easily spend upwards over $100,000 and it really is high-end spectrum analyzer. For purposes of our course, we won't be needing one of this but I did want to demo just because a brief mention of it was made in your textbook.

The reference in your book had to do with harmonics and here I have a circuit set up and this is a function generator and we are generating a square wave at 10kHz and the amplitude 10 volts peak-to-peak and that signal were put across this resistor right here. The spectrum analyzer is set up across this resistor and we are looking at the output that the spectrum analyzer displays. I have this setup, it doesn't go instantly and display I have to go in and set all this stuff up I'm not going to go into that because we are not going to be doing that but I did want to show you the outcome.

Now what you see here is the fundamental frequency and that is going to be the largest peak here and then all of these others are the harmonics. If I scroll over here we can actually look at what frequency is that. We can lock in on a fairly close look. There is 9.9 so that would be our fundamental of 10kHz and if we scroll over to the next one, it should be about the third harmonic … that's very close to 30kHz. What we represent the third and then over a little further tom, it should be 50 the way we are. 50kHz would be the fifth and then her to almost 70, 90, that would be the ninth. About 110 that would be the 11th. Let's see what's next one here 130.

So on and so forth you get the idea here and this would go on indefinitely. We are graphing out to 242kHz and like I can say there was this limitation of how much I can display. If I put this up really high then things get so crunched together that you can't. It's hard to decipher what's what. Anyway, this has to do … we just wanted to display the ability to look at the fundamental and all of the harmonics. By the way, I don't believe the spectrum analyzer is available in your student or textbook version of Workbench. I believe you have to get at least the student upgrade to be able to use it. I don't think you have access to this but not to worry we won't be using this. This is simply a show and tells item.

In this lesson we looked at phase relationships, we looked at what does leading and lagging mean and we look at these particular waveforms here. We signed here, we saw that the black hair form is leading the blue waveform by 90? and we looked at our calculators to make sure they were set up for degrees, we calculated instantaneous values using the sine function on a calculator. We discussed time and frequency domains, we would primarily dealing with the time domain, the frequency domain is very interesting. We had mentioned that it is subject for advanced courses. We looked at the subject for harmonics, we looked at the spectrum analyzer output to better understand harmonics.