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Sinusoidal waveforms, AC, and DC signals.

A sine wave

The whole purpose for almost all electronics is to modify changing signals. Even in the case of our thermistor circuit, there would be no reason to be interested in the temperature if it did not change. This is our introduction to signals and AC.

The type of waveform that is always discussed when learning about signals is a sine wave.  There are several good reasons for that.  The most important reason is a fundamental rule that any repeating waveform can be considered the sum of sinusoidal waves and a DC offset.  Later we will be talking about how to generate a sinusoid, but for now just accept there is such a thing.

A circle from sine and cosine.

Way back there in Episode 35, I talked about the relationship of a circle to the sine and cosine functions.  The circle in the picture to the right is generated by using the cosine function to determine the X position on the graph and the sine function to generate the Y position on the graph.  This circle which as the radius of 1 is called “the unit circle.”

Imagine a line drawn from the center of the circle and intersecting the circle.   If this line is horizontal and intersects the right hand side of the circle (3 o’clock position) this is defined as 0 degrees. As the line rotates in the counterclockwise direction the angle between the horizontal line and the new position defines the degrees of rotation.  Using the clock numbers as examples gives:  2 o’clock = 30 degrees rotation; 12 o’clock = 90 degrees rotation; 9 o’clock = 180 degrees rotation; and 6 o’clock = 270 degrees rotation. At 360 degrees rotation we are back at 0.   Using these numbers and measuring the vertical  or Y position along the circle we would get: sine(angle).  That is how I drew the waveform in the first picture.

Sine and Cosine waveforms.

Taking this one step further, we can draw the cosine wave exactly the same way except we are looking at the horizontal or X position.  Notice that these two curves look exactly the same, except there is a 90 degree difference.  This will become important in the future.  We would say the sine wave lags the cosine wave by 90 degrees.  That will take a little thinking about because we normally think from left to right and the picture looks the opposite of what I just said. The sine wave achieves the same value the cosine had 90 degrees before, so the sine wave lags the cosine wave.

A sine wave plotted against time.

Normally when we are talking about waveforms we are looking at the change compared to time and often this is displayed on a device called an oscilloscope.  Now is the time to define names of values on the wave form.  That is shown in the picture to the right.  The period is the time for the waveform to complete one cycle.  One cycle can be thought of as one revolution around the circle we talked about earlier.   The period can be measure from one peak to the next, one trough to the next, or from a zero crossing to the next zero crossing in the same direction.   The period of this waveform is 1 millisecond .  The frequency of the wave is the number of complete cycles per second or 1/period.   The frequency of this wave would be 1000 Hz or 1KHz.

So far I have not dealt with AC or DC with respect to waveforms.   AC means alternating current and that the waveform goes both positive and negative.  DC means direct current and although the waveform can change values through the cycle, the polarity remains the same.   All of the waveforms I have drawn in this post are AC waveforms.  If I added +1 to all of the waveforms the waves would go from 0 to 2 and it would be a changing DC signal.   Commonly people would say that signal has both an AC and DC component.

We have talked about the time portion of the waveform, but we have yet to talk about the amplitude portion of it.   The amplitude would be measured on an oscilloscope by measuring the peak value or the peak-to-peak value.  In a sine wave the peak-to-peak value is 2 times the peak value.  Another important value for a sine wave is the RMS value and this is the way AC power values are defined.  For example, here in the US the voltage at power receptacles is defined to be 115 Volts A.C. (Vac) at 60 Hz.  (I know some of you may call it 110 and some may call it 120… there is a long story about that, for now just go with 115.)  The 115 Vac is actually the RMS value.  The peak voltage is 162 Volts or 115 * square root of 2.

The question on your mind right now is: “Why RMS and where does this come from?”   The RMS value is the voltage of a DC source that would produce the equivalent equivalent power in the same load resistor.  The way it is calculated is the following:

  • Take samples through one complete cycle at regular intervals.
  • Square each of these value.  (The S part)
  • Take the average of each of the squares. (The M part for Mean)
  • Take the square root of the average. (The R part for Root)

The reasons for all of these steps.  Squaring does two things. First, it makes all values positive so they can be added together to find the average.  Second, and probably most importantly, if you remember the power equation for DC.  P = V2/R, this means that voltage near the peak will provide much more power than voltage near the zero crossing. The mean simply averages all these squared values and then taking the square root makes it back equivalent to the DC value.   I have a spreadsheet in excel format that shows this being done.   For sine waves:  The RMS value is the peak value/ squareroot of 2 and the peak value is the RMS value X squareroot of 2.

This is also probably a good time to point out that while humans normally refer to degrees most calculations are actually done in radians.   There are 2Π (2 pi) radians in a complete cycle.   Very soon you will be seeing the term 2Πf showing up, this partly explains where some of that comes from.

As normal, this become long and complicated.  If you have any question, please contact me via e-mail or in a comment.

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Gary.


 

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