Sine Wave Characteristics
We're in Section 6.2 Sine Wave Characteristics. Sine waves may present voltage, current or some other parameter. Most commonly we'll be looking at voltage. The period is the sine wave is the time from any given pint on the cycle to the same point on the following cycle. The period is measured in time and in most cases, it is measured in seconds or fractions thereof. The time of a 60-cycle signal is 16.667 milliseconds. If e we said one second and we divided it into 60 parts, the n we would get the time. Here, this would be the time of one cycle and hence one period.
We could look at a circuit simulation and here we would have our single, the period of one cycle. This actually came from our previous lesson. Remember we started out with 1volt at 60hz and this is the resultant sine wave and the time from right here to right here. That represents one period. We can see the time read out here 16, it's about 16.5 milliseconds. Should be 16.6, I guess we could have, our measurement points were not quite perfect, but you get the idea. That one period, if we're going at 60Hz is 16.6 milliseconds.
The frequency of a sine wave is the number of complete cycles that occur in one second. Frequency is measured in a quantity called Hertz. One hertz corresponds to one cycle per second. Frequency and period have an inverse relationship. We can say that time = 1/frequency and that frequency=1/time. We just had that from the previous screen we did this. We said, 1/60=0.01666, actually that's how we would read 16.6 milliseconds. But on a calculator, it would come up looking like that. That's the fractional equivalent of 16.666 millisecs. Also, frequency equals 1/time. If we took that value and we said 1/ that value 0.01666, we would get 60 Time=1/frequency and frequency=1/time.
Frequency-to-period and period-to-frequency conversions are common in electronic calculations.
The peak value of a sine wave is the maximum value or current it reaches. This is the value you would read on an oscilloscope and we had our, when we look at that signal, the peak value would be at this point. That would be the positive peak and this would represent the negative Peak. Peak voltages occur at two different points in the cycle. One is positive, one is negative. The positive peak occurs at 90° and the negative at 270°. The positive and negative have equal amplitudes and we can quickly look again at our simulation.
We can see that the, here we have it. We see this is the peak value at this point and then this is the negative peak. This is at 90° and this is at 270°. This is 360° and this is at 0°.
The average value of any measured quantity is the sum of all the intermediate values. The average value of a full sine wave is 0. The average value of one-half cycle of a sine wave is, this is the average of one. If you're looking at the sine wave, and we look at average, every single point on this entire wave going both positive and negative, well, we're going to resolve in this. It's going to be 0. But if we average every point on the wave on this, the positive side, or just the negative side, we would find that the average of all of these values is this fractional value right here, 0.637. If we wanted to calculate the average value of the peak wave, we would simply take it again if we started out with that voltage of 1volt. If we took 0.637*1volt peak, well then, our average value's 0.637.
One of the most important characteristics of the sine wave is this RMS or often times we refer to as effective value. The RMS value describes the sine wave in the RMS of an equivalent DC voltage.
The RMS value of a sine wave produces the same heating effect in a resistance as an equal value of DC. The abbreviation RMS stands for root mean square, and is determined by, here we have a formula, 0.707*the peak value. That is in voltage if it's current it'll be the same thing. The peak current*0.707 will give us the RMS current. The derivation of this formula is addressed in upper-level courses. For our purposes, we will simply use it as a method of converting between RMS and peak. Let's take a look at our simulation. In our simulation, we showed the sine wave of 1volt peak and we had 1volt from 0 to the peak value here and 0 going to the negative peak.
We also have a multi-meter connected into the circuit, this is for discussion purposes of RMS and notice the oscilloscope is connected across this 1ohm resistor and we're measuring 1volt peak and 1volt negative. That goes to be 2volts, peak-to-peak if we were looking at it that way. But typically we say, 1volt peak. We put our volt meter across the same place in the circuit across the same component. Notice we're measuring this with selected AC and you'll notice the reading is 0.707, 0.011 millivolts. That correlates to our calculation that are the values that we just looked at.
Remember we said the RMS value id .0707*the peak value. We had looked at that previous signal and we saw that we had 1volt peak-to-peak or 1volt peak. Then we took 0.707*that value and that gave us the RMS value.
RMS Value for Voltage or Current
This is just to review. if you take peak*0.707 will give you the RMS value. Or you can take RMS value divide it by 0.707 and that will give you the peak value. Or you can say RMS*1.414 and that will give you peak. We said with this, we already did this when we started out with 1 volt*0.707. Obviously, we got .0707 volts. Then we too, or we could take the RMS value and that will be 0.707 in this case and divide it by 0.707 and that would equal the peak.
0.707/0.707 that's 1 volt which was our original value in peak. Or we could take RMS*1.414. And if we took 0.707*1.414, we would get 1 volt. Most voltmeters are calibrated to measure in RMS. When you measure, when you take a voltmeter, a vom or a digital multimeter and you measure an AC value, you will measure the RMS value. Oscilloscopes display peak or peak-to-peak values.
This is a, I've eluded to it already, but another measurement use to describe sine waves are their peak-to-peak values. The peak-to-peak value is the difference between the two peak values. Here we have, the first peak, and this goes to be our positive peak, this is our negative peak and if we're measuring from here to here, then that would be our peak-to-peak value.
When you're doing oscilloscope readings, typically you have to read either the peak value or the peak-to-peak and obviously, peak-to-peak is going to be twice the value of the peak. That concludes section 6.2.