Sinusoidal AC Waveform Definitions

The sinusoidal waveform of Fig. 1 with its additional notation will now be used as a model in defining a few basic terms. These terms, however, can be applied to any alternating waveform. It is important to remember as you proceed through the various definitions that the vertical scaling is in volts or amperes and the horizontal scaling is always in units of time.
Important parameters for a sinusoidal voltage.
Fig. 1: Important parameters for a sinusoidal voltage.
Waveform: The path traced by a quantity, such as the voltage in Fig. 13.3, plotted as a function of some variable such as time (as above), position, degrees, radians, temperature, and so on.
Instantaneous value: The magnitude of a waveform at any instant of time; denoted by lowercase letters ($e_1$, $e_2$).
Peak amplitude: The maximum value of a waveform as measured from its average, or mean, value, denoted by uppercase letters (such as $E_m$ for sources of voltage and $V_m$ for the voltage drop across a load). For the waveform of Fig. 1, the average value is zero volts, and $E_m$ is as defined by the figure.
Peak value: The maximum instantaneous value of a function as measured from the zero-volt level. For the waveform of Fig. 1, the peak amplitude and peak value are the same, since the average value of the function is zero volts.
Peak-to-peak value: Denoted by $E_{p-p}$ or $V_{p-p}$, the full voltage between positive and negative peaks of the waveform, that is, the sum of the magnitude of the positive and negative peaks.
Periodic waveform: A waveform that continually repeats itself after the same time interval. The waveform of Fig. 1 is a periodic waveform.
Period (T ): The time interval between successive repetitions of a periodic waveform (the period $T_1 = T_2 = T_3$ in Fig. 1), as long as successive similar points of the periodic waveform are used in determining T.
Cycle: The portion of a waveform contained in one period of time. The cycles within $T_1$, $T_2$, and $T_3$of Fig. 2 may appear different in Fig. 2, but they are all bounded by one period of time and therefore satisfy the definition of a cycle.
Defining the cycle and period of a sinusoidal waveform.
Fig. 2: Defining the cycle and period of a sinusoidal waveform.
Frequency ( f): The number of cycles that occur in 1 s. The frequency of the waveform of Fig. 3(a) is 1 cycle per second, and for Fig. 3(b), $2{1\over 2}$ cycles per second. If a waveform of similar shape had a period of 0.5 s [Fig. 3(c)], the frequency would be 2 cycles per second.
Demonstrating the effect of a changing frequency on the period of a sinusoidal waveform.
Fig. 3: Demonstrating the effect of a changing frequency on the period of a sinusoidal waveform.
The unit of measure for frequency is the hertz (Hz), where $$\bbox[10px,border:1px solid grey]{1\, hertz (Hz) = 1\, cycle \, per \, second \,(c/s)} \tag{1}$$ The unit hertz is derived from the surname of Heinrich Rudolph Hertz, who did original research in the area of alternating currents and voltages and their effect on the basic R, L, and C elements. The frequency standard for North America is 60 Hz, whereas for Europe it is predominantly 50 Hz.
As with all standards, any variation from the norm will cause difficulties. In 1993, Berlin, Germany, received all its power from eastern plants, whose output frequency was varying between 50.03 and 51 Hz. The result was that clocks were gaining as much as 4 minutes a day. Alarms went off too soon, VCRs clicked off before the end of the program, etc., requiring that clocks be continually reset. In 1994, however, when power was linked with the rest of Europe, the precise standard of 50 Hz was reestablished and everyone was on time again.
Using a log scale, a frequency spectrum from 1 Hz to 1000 GHz can be scaled off on the same axis, as shown in Fig. 4. A number of terms in the various spectrums are probably familiar to the reader from everyday experiences. Note that the audio range (human ear) extends from only 15 Hz to 20 kHz, but the transmission of radio signals can occur between 3 kHz and 300 GHz.
Fig. 4: Areas of application for specific frequency bands
The uniform process of defining the intervals of the radio-frequency spectrum from VLF to EHF is quite evident from the length of the bars in the figure (although keep in mind that it is a log scale, so the frequencies encompassed within each segment are quite different). Other frequencies of particular interest (TV, CB, microwave, etc.) are also included for reference purposes. Although it is numerically easy to talk about frequencies in the megahertz and gigahertz range, keep in mind that a frequency of 100 MHz, for instance, represents a sinusoidal waveform that passes through 100,000,000 cycles in only 1 s-an incredible number when we compare it to the 60 Hz of our conventional power sources.
The new Pentium II chip manufactured by Intel can run at speeds up to 450 MHz. Imagine a product able to handle 450,000,000 instructions per second-an incredible achievement. The new Pentium IV chip manufactured by Intel can run at a speed of 1.5 GHz. Try to imagine a product able to handle 1,500,000,000,000 instructions in just 1 s-an incredible achievement.
Since the frequency is inversely related to the period-that is, as one increases, the other decreases by an equal amount-the two can be related by the following equation: $$\bbox[10px,border:1px solid grey]{f = {1 \over T}} \,(Hz)\tag{2}$$ $$\bbox[10px,border:1px solid grey]{T = {1 \over f}} \,seconds (s)\tag{3}$$