**A.C. Waveforms**

**I. Basic AC Waveforms**

The Basic circuit on the left shows an unspecified signal

source which is not a battery. Inside the circle for the signal

source Vs there should appear the "~" sign which stands for

an A.C. (Alternating Current) signal instead of a D.C signal.

D.C. stands for direct current, i.e. the signal is always a fixed value. In contrast an A.C. signal has a maximum

value and a minimum value and it alternates between these two values during some period of time.

The graph at the side shows a D.C voltage source with a

value of 5 volts which does not change with time, but

remains constant.

A.C. signal sources can be made up of different types of Waveforms. The most common waveforms are the

Sinusoidal waveform, the Square waveform and the Triangle waveform. There is other kinds of A.C

waveforms which will be encountered as one becomes more involved with electronics.

An example of each of these A.C. waveforms is shown below :

1. __ The Sinusoidal Waveform. __ (Horizontal
time is in milliseconds, vertical axis -5 to +5 volts)

2. __
The Square Waveform. __ ( Horizontal time is
in microseconds, vertical axis -2.5 to +2.5V)

3. __ The
Triangle Waveform.__ ( Horizontal time is in
microseconds, vertical axis -2.5 to 2.5V)

Of course the Waveforms shown are all "ideal" and in the case of the Square waveform the rise and fall times

are not zero, meaning it takes a finite amount of time for the voltage to rise to the maximum value and

subsequently to go to the minimum value. These rise and fall times are normally very small and for the "ideal"

case we will ignore them. In the case of the Sinusoidal and Triangle waveforms other anomalies can appear

such as noise (which are unwanted signals "riding" on either waveform) and even waveforms that are not

completely symmetrical. One will find these problems as he/she progresses in the study of electronics and

signals.

The most common of the waveforms is the Sinusoidal type and hence we will be dealing most of the time

with these type of waveform in our studies of A.C. electronics.

__II. Period and
Frequency of Sinusoidal Waveforms.__

The common thing with all of the AC Waveforms is that they are __periodic __
which means that they repeat

themselves after a __period of time T. __ Said in a different way the
__period__ of a signal is the __time T __that it takes

for
the signal to traverse a whole __cycle__, after which the signal repeats
itself.

We can see the meaning of period and cycle better if we take a look at the sinusoidal waveform below:

At the start the waveform is at 0V. After 0.25ms we reach a peak at 1V. After 0.5ms we are back at

0 volts. We have just completed what is called "the positive alternation".

From 0.5ms to 0.75ms the waveform goes to a minimum value of -1V. Then the waveform goes back up

again until reaches 0V at 1ms. We have just completed "the negative
alternation". The time after we have traversed one positive and one
negative "alternations" or __one whole cycle __of the waveform is called the

__Period T of the signal. __ In this example, the period T = 1ms.
After this period we start a new cycle of the

signal with exactly the same waveform as the first cycle. Hence, we call the sinusoidal signal a periodic

signal.

It is important to know well the characteristics of a Sine or Cosine function to understand better what these functions represent for us in Electronics, hence if you need a review please go to the School's Library to review some Math. In particular I recommend the AC Waveforms Tutorial at : www.sweethaven.com/acee/forms/toc01.htm

In brief, it is important to note that if the horizontal axis was changed from
time to degrees the following will correspond to the data-points discussed
already: The __first peak__ will occur at an __angle of 90 degrees;__

the __first positive alternation__ will occur at an __angle of 180 degrees__.
The __first minimum point__ at -1V will happen

when the __angle is 270 degrees__. The __first negative alternation__
and the__ first cycle __will occur at __360 degrees__.

Since at 360 degrees we complete a whole circle, it is only obvious that the function will repeat itself going around one more circle.

__ What about Frequency? __Once we have understood
what the Period and the Cycle of a function

represent the concept of Frequency shall follow up more easily. __ Frequency__
is defined to be as __the number__

__of cycles per unit time of a function__ expressed as __ f =
cycles/time. __When time is given in seconds, frequency has the standard unit
of __ #cycles/second or Hertz = Hz. __
Taking the example we were discussing, we have 1cycle in a complete period of
1ms. If we wanted to increase the frequency this will mean that we want to
include more cycles in that same period of 1ms. Initially the frequency
is: f = 1cycle/1ms = 1MHz (1MegaHertz). (If you still need help with Units
review DC Basics under DC Electronics.)
Suppose we want to include three cycles in 1ms, then the frequency would be f =
3cycles/1ms or 3MHz.

Finally, notice what happens to the period of the signal if we include 3cycles in 1ms! Initially the Period

T was equal to 1ms, now we cover 3 cycles in 1ms and the Period has been reduced by a factor of 3

such that T(new) = 1/3 T(original) = 0.33ms. In other words, if we increase the frequency by 3 we will

reduce the Period of the signal by the same factor of 3 and:

__Hence the key relationship between Period T and Frequency F is:__

T = 1/f or f = 1/T

__The Period and Frequency of a waveform are the reciprocal of each other!!__

** Note: **This relationships holds for ANY periodic function and
not just sinusoidal functions!

__III. Magnitude and Phase of a Sinusoidal Waveform__

For AC sinusoidal waveforms, besides having to

deal with frequency and Period, we need to be able

to have a measurement for the __
varying Magnitude__,

and also account for the fact that as we traverse

through an AC circuit the __Phase
__of the waveforms will be different than the Phase of the Input waveform.

To illustrate both factors we shall take a look at the circuit above which will also give us some idea of what is

coming up in later lessons.

First, we realize that the input is a 60Hz frequency waveform, given by a description V1: -170/170V.

The numbers -170V and 170V stand for the minimum and maximum voltages of the sinusoidal waveform.

This waveform is shown in the graph below.

__Magnitude:__
Clearly then, one of the ways of specifying the Magnitude of the Signal
is by stating

its maximum or peak voltage, in this
case the__ Peak Voltage or Vp = 170V__.

A second way of specifying the Signal's Magnitude is by giving its Amplitude which is the voltage from

its maximum of 170V to its minimum of
-170V, this is also called the__ Peak-to-Peak Voltage__ or

Vp-p which in this case is two times
170V or __Vp-p = 340V.__

The third and most common way of giving the Magnitude of the Waveform is by using a traditional

measure called__ RMS or
Root-Mean-Square value__. The RMS value is obtained by considering the

average power of the Signal and deriving it will take a lengthy explanation. The RMS value is

equivalent to the Peak value of the
signal divided by a factor of √2, in brief:
__Vrms = Vp/__√2__.__

Thus, for our example Vrms = Vp/1.4142 = 0.7071*Vp = 0.7071*170V = 120Vrms, which is the

RMS voltage value of an AC signal coming directly from an outlet of our home or building!!!

Note
that __the RMS value is smaller than the Peak Value, since it is an average of
the signal's Power__!

__Phase:
__Finally, a last consideration for our AC signal Measurements is the
concept of the Phase

relationship between different signals. As a reminder, a standard Sine wave as we discussed earlier

on this same page start at 0V at an angle of 0 degrees. It goes to a Maximum or Vp at 90 degrees,

goes back to 0V at 180 degrees, then reaches a Minimum or -Vp at 270 degrees and finally completes

a whole period or cycle at 360 degrees. Thus, when thinking that the x-Axis is given by a time (t) basis,

get used to also relating the different portions of the waveform to an x-Axis that would be given in degrees.

In the circuit that we are using for illustration the waveform at the output looks as shown below:

Compare this Waveform to the previous Input Waveform. Note that the output signal is smaller in

Magnitude and also note that the output signal has its first Maximum or Peak Value later than the

input signal. Therefore, there is a time lapse between the input and output signals, or equivalently

there is a Phase Difference or Phase lapse between both signals. The input signal reaches its first Peak

voltage at about 5ms. The output signal reaches its first Peak at about 7ms. The time lapse between

both signals is then 2ms

Since the frequency of the input and the output need to stay the same for building a comparison between

both, the period is __T = 1/f =
1/60Hz = 0.01667 seconds = 16.67ms__. We know that one cycle is given in

360 degrees, so for a whole cycle or 360 degrees the equivalent period is 16.67ms.

We need to answer, what is the Phase Difference for 2ms?

Using a little Math, the Phase
Difference between both signals is __Phase Diff = (2/16.67) * 360 degrees__

__Phase Diff = 43.19 degrees.
__ It's just so easy!---Just kidding, if you didn't get the
first time, you may

want to review this first lesson on AC Electronics.

Well, and how about the difference in Magnitude between the input and output signals? And what if

we were to change the frequency? These and many more questions will be answered in the following

AC Electronics lessons.