Characteristics of a Sine Wave

As stated before, by graphing the current vs. time, we end up with a pattern known as a SINUSOIDAL WAVE, or SINE WAVE for short. We say that the sine wave has positive and negative peaks at 90^o and 270^o respectively.
A sine wave has several important characteristics.

1. One complete revolution of a generator, from 0^o to 360^o, is known as a CYCLE.
2. The DISTANCE between the beginning of one cycle, and the beginning of the next cycle is called one WAVELENGTH. The symbol for WAVELENGTH in electronics is the Greek letter lambda (l).
3. The TIME in which it takes to complete one cycle is known as a PERIOD.
4. The number of complete sine wave cycles generated in one second, is called FREQUENCY, and is measured in CYCLES PER SECOND (cps), PERIODS PER SECOND (pps) or more often HERTZ(Hz).

5. The height of the sine wave is called the AMPLITUDE, and is measured in Voltage. The highest point of any wave is called the PEAK AMPLITUDE or PEAK VOLTAGE.
6. The difference in amplitude between the highest positive voltage, and the highest negative voltage is called the PEAK TO PEAK VOLTAGE, which is equal to twice the peak Voltage.

Now without going into a whole bunch of math and physics...(yea, RIGHT!)... you learned sometime in your life that there is a relationship between distance, time, and speed. If you are driving down a road at 50 miles per hour, for exactly one hour, you know from basic math that you will have traveled exactly 50 miles. Why?

Distance = Speed x Time

This is a basic law of physics. It also applies to electricity and waves.

If a Wavelength is a length (read: Distance) then we can use time to compare the speed at which it travels through space. Now electricity travels at the speed of light, but what exactly is that? Well, it differs slightly from one medium to another, but for our purposes, we will assume the speed of light in a vacuum.

Wavelength = Speed x Time
We know that

1. Speed of light = 299,792,458 meters/second (In a perfect vacuum)


2. Time = (299,792,458) Speed / Distance (l )

If we assume a signal with a 2 meter long wavelength, we can use the following formula to find the time that it takes to complete one cycle:

Using this formula we can plug in the numbers
Now by simple division, we get the following answer:
Now if a 2 meter long signal is equal to 6.67 nanoseconds, we can find the frequency of the signal as follows:
Let's plug in the numbers...

Since we know that 149MHz = 2 meters in length, we can deduce that frequency and wavelength have a relationship such that:

Wavelength (in meters) = 298 / Frequency (in MHZ)

or if you prefer: Wavelength (in feet) = 936 / Frequency (in MHZ)


Let us test this theory. The 11 meter band, otherwise known as the CB Radio, or Citizen's Band, resides at (approx.) 27 MHZ. Does the math work out?
The formulas you can look up in the future, should you need them. The basic knowledge we have tried to stress here is, that WAVELENGTH is a DISTANCE, a PERIOD is an amount of TIME, and that FREQUENCY is the number of wavelengths in a second. Furthermore, we stressed that all of these terms are related, but not quite interchangeable.
So far, we have discussed that Sine waves have FREQUENCY, WAVELENGTH, PERIOD, and AMPLITUDE.

They also have one more important characteristic: PHASE

Phase is the timing relationship between two different sine waves. If two generators are connected across a given load in series, and if their armatures begin rotating together at exactly the same time and speed, two different alternating voltages will be produced. In the example to the left, one is a 4 Volt sine wave, and the second is a 3 Volt sine wave.

If we examine the picture closely, we find that both sine waves meet up at the 0^o and 180^o points. Furthermore, they both peak out at 90^o and 270^o respectively. We say, then, that both of the two waves produced by the two different generators are IN PHASE with each other. Whenever two waves are in phase, like these are, the voltage resulting from the two waves will not be the same as either of the two voltages. The resulting voltage will be the SUM of the two voltages. In this case, we have 3 and 4 volts being produced by the generators, and the resulting output voltage would be 3+4 or 7 Volts. This is because the energy in the two voltages work together, and combine to add up to 7 Volts. But what happens if the generators are NOT in phase?

Whenever two waves are combined out of phase, the resultant waveform is not so simple to figure out. Look at the picture on the right. The 3 Volt generator was started later than the 4 Volt generator. We say that the 3 Volt wave LAGS behind the 4 volt wave. In this case, the 3 Volt wave LAGS by 90^o.

Voltages that are out of phase can not be added simply by adding them together, as we do with in phase waves. We must resort to a sort of "high math" called VECTOR ADDITION. To make it simpler to understand, Vector math simply means that we break out a piece of graph paper, and plot the 3 volt wave horizontally (left and right), while we plot the 4 volt wave vertically (up and down).

Examine the chart to the left to see how this works. Using our 3 and 4 volt waves, we draw the two lines on the graph paper. Then we draw a "mirror image" of the same two lines. When we are finished drawing the mirror image, it should form a parallelogram. Now we draw a diagonal line from the "0" point in the middle of the graph, to its opposite corner of the parallelogram. The distance from the 0 point to its opposite corner will be the vectorial sum of the two voltages. This method works, no matter what the Phase difference is between the two voltages, but does require a little modification if it is different from 90^o.

Let us assume now, that instead of the two waves being 90 degrees out of phase, that the 4 Volt wave is lagging the 3 volt wave by 45 ^o this time. The method of finding their vectorial sum is basically the same. First we plot the 3 Volt wave horizontally on the chart. Next we measure 45^o to draw the second line. We draw the 4 Volt line 45^o from the 3 Volt wave. Now again we draw a parallelogram, reflecting the original two plotted voltages. We draw a line bisecting the parallelogram from the 0 point to its opposite corner. The length of the resultant line indicates the vectorial sum of the two original voltages.
Ok, so now that you think you've had enough with math, you find that AC has more complicated math than DC does. But the fun isn't quite over yet. You've got to be able to convert AC voltage to their DC equivalent voltages, and visa versa. The main problem is with Voltage. DC Voltage is straightforward. If it's 10 Volts, it's 10 Volts - period.

But with AC, Voltage becomes more difficult to define. Looking at an AC wave, we actually have 3 different voltages to compare. The voltage from the 0 line to the positive peak of the AC curve is called the PEAK VOLTAGE. If we measure the Voltage from the top of the positive peak, to the bottom of the Negative peak, we call it the PEAK TO PEAK VOLTAGE, which is equal to 2 times the peak voltage. Finally, when we try to do work with an AC Voltage, we find out that a 10 Volt peak voltage wont turn a motor as fast as a 10 Volt DC Voltage. Reason? Because 10 Volts DC is 10 Volts all the time. A 10 Volt peak AC Voltage is only 10 Volts for an instant. The rest of the time it is swinging higher and lower in Voltage level. So at what Voltage level does the AC wave do as much work as a pure DC Voltage?

It was found that it takes a 141 Volt AC wave to do the same amount of work as a 100 Volt DC source. The EFFECTIVE value of a 141 Volt AC source then is only 100 Volts. Another term for EFFECTIVE voltage is RMS, which stands for Root Mean Square.

Often, electricians and electronics technicians find that they need to be able to convert AC voltages to DC voltages. They need to know what the effective voltage is. Based on the 141:100 ratio of AC to DC, the following formulae were conceived:

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Where E_peak equals the peak voltage of an AC signal and E_eff equals its effective (RMS) DC equivalent.

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Just when you thought it was safe to get back into the water, I'm gonna throw one more formula at you. What happens if we take all the instantaneous voltage values of a sine wave, add them all up, and then take the average of them? Well, it doesn't quite come up to the effective voltage. When working with rectifier circuits (we'll discuss them in a later section), we must sometimes use what is known as the AVERAGE VOLTAGE of a given AC sine wave. The average voltage is found by the following formula:

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Now at this point, one might pose the question - WHY do we go into such detail about different voltage levels (peak, RMS, and average). The reason is because you ABSOLUTELY need to know which you are working with at any particular time. For instance - 100 Volts PEAK voltage may be the threshold at which your $30,000 piece of lab equipment (say a 6 trace digital oscilloscope)gets destroyed. If you measure it with a multimeter first, and it shows 90 Volts, then think you can put your scope on it - you just blew the front end of your scope. Why? Because 90 Volts RMS is greater than 100 Volts Peak by 27 Volts!!! Multimeters measure in RMS typically, and most aren't accurate enough to measure in "True RMS". You have to know the parameters of the device being tested, as well as the limitations of your equipment, or you'll wind up doing a lot of expensive damage! 
k, a human being can only be subjected to so much math before their head explodes. I don't know about you, but I've about reached my limit for now. So let's discuss something a little less math intensive.

We spoke earlier of the ability of a wire to produce a magnetic field when a current is flowing through it. We say that the current is INDUCING the magnetic field. We also stated that this inductance becomes stronger when the wire is wound up into a coil. But this is not the only factor which effects the amount of inductance within a coil. There are actually 4 factors which determine the inductance of a coil of wire: The number of turns in the coil. The material that the CORE of the coil is made of. The length of the coil. The diameter of the coil

NUMBER OF TURNS:

If we have two coils of identical length and diameter, but one has 4 turns of wire, while the second has 8 turns of wire, the coil with 8 turns will have more inductance than the one with 4 turns.

CORE MATERIAL:

A coil wrapped around an iron core will have a much greater inductance than one wound with nothing in the middle but air. This is because iron has a much higher PERMEABILITY than air.

LENGTH OF COIL:

If the coil has the same diameter, and the same number of turns, but is made longer (say, by stretching it out), the inductance of the coil will decrease.

DIAMETER OF COIL:

If the coil is the same length and number of turns, but smaller in diameter, the inductance will also be smaller.

Now we could go into the math and physics of why the coil increases or decreases with size, shape, number of turns, etc, but that is not the scope of this course. A basic general understanding of inductance is all you need at this point. So let us first explain what inductance is and how it works. When we first created our coil of wire, we were working with DC. Now we are learning a little about AC, and as you've found from the math.... the game changes some. When we attach a battery to our coil, it induces a magnetic field. This magnetic field does not, however, appear instantly, as one might believe.

You see, electricity, as we have found, moves at a given speed (the speed of light). It travels the distance of the wire, depending upon its length. Now we know that there is a special relationship between speed and distance, in that it takes TIME for something to travel a distance at a given speed.

Prior to our attaching the battery (we'll assume a 12 Volt battery for our discussion), the potential difference between the two ends of the wire is 0 Volts, and the current flowing through the wire is zero. As we attach the battery, the wire, which has resistance, enforces Ohm's Law. The Voltage (now 12V), divided by the resistance of the wire sets up a current within the wire. That current begins at one end of the wire, moving through the wire until it reaches the other end. During the time period of time that the current begins movement (0 speed), to the time it reaches its maximum speed, it is accelerating. Of course, when we disconnect the battery, we have an opposite effect, and for some time, the current is decelerating. The important point here is that the current speeds up and slows down within the coil whenever we attach or detach the voltage source.

Now for AC:
As we use an ALTERNATING current source, we know that the Voltage is in a constant state of change, fluctuating from 0 Volts, up to a maximum positive peak, back down to 0, continuing to a negative peak, then returning again to its starting point of 0 Volts. It takes TIME for this to happen. Now if we attach this fluctuating voltage source to a coil, the voltage is in a constant state of change. As the AC generator begins its upward curve, the voltage in the coil is 0 Volts. As the voltage reaches its positive peak, recall that it takes time for the electric current to flow, the voltage in the coil is beginning to rise, and a NORTH magnetic pole is beginning to be formed. As the coil reaches its peak of current, the alternator has rotated to its point of 0 Volts and is beginning to swing negative. (Can you predict what is coming?)

As the alternator swings negative, it forces current through the wire in the opposite direction. At the same time, the magnetic field in the coil attempts to collapse, by inducing its magnetic field into the wire in the same direction it was originally moving. The current that the magnetic field is producing, is in direct opposition to the current that the generator is now producing. Of course, the magnetic field is small, and has less staying power than the generator, which has force and momentum behind it. So the generator wins and pushes the current through the coil, now forming a SOUTH pole on the electromagnetic coil.

So as the generator is swinging from positive to negative, the coil is swinging from North to South, expanding and collapsing between Voltage peaks. In addition, the coil is slightly out of phase (remember that word?) with the generator, and hence, is opposing the flow of electric current. Isn't that the definition of RESISTANCE? The coil RESISTS the flow of AC, but allows DC to pass freely. INDUCTANCE is defined as the opposition to any change in current flow. It is NOT the opposition to current flow, but the opposition to CHANGE of the flow. It is a form of light resistance caused by the collapsing magnetic field opposing the CHANGE in current within the coil.

Did we say that a coil acts like a resistor in some circumstances? Yup, sure did! Then it must follow Ohm's Law, huh? Yup, sure does! Not only does the wire itself have resistance, but the coil resists AC current flow. But Ohm's law doesn't EXACTLY apply in the same way, and for now, you've had enough math, so I won't throw that at you right away.

What you should consider, though, is that inductors can be act like resistors in parallel circuits. This means that you solve for inductance, the same way you solve for resistance in a series circuit.... you add the value of the inductors. Along the same lines, in a parallel circuit, you would "reciprocate the sum of the reciprocals" of individual inductances