6.2: Graphical and Analytic Representation of Sinusoids
Analyzing two sinusoidal voltages with equal amplitude and period but different phases on an oscilloscope, an instrument used to display and analyze waveforms, involves a three-step process.
The first step is measuring the peak-to-peak value, which is twice the amplitude of the sinusoid. This provides information about the maximum voltage swing of the waveform.
Secondly, the period and angular frequency are determined. The period is the time taken for one complete cycle of the waveform, while the angular frequency (often denoted by the Greek letter omega, ω) is the rate at which the waveform oscillates.
The third step involves measuring the voltage values for both sinusoids at a fixed point in time. This is used to determine the phase angle, which depends on whether the sinusoid has a positive or negative slope at that point.
A graphical representation can be employed to compare the two sinusoids. In this representation, the horizontal axis represents the magnitude of cosine, and the vertical axis represents the magnitude of sine. This arrangement is reminiscent of polar coordinates, where angles measured counterclockwise from the horizontal axis are considered positive, while those measured clockwise are deemed negative.
Interestingly, subtracting 90 degrees from the argument of a cosine function yields the sine function. This property can be leveraged when dealing with sinusoids.
To add two sinusoids of the same frequency - one in sine form and the other in cosine form - the graphical representation comes into play again. The hypotenuse of the right-angled triangle formed by these sinusoids represents the resultant sinusoid. The argument of this resultant sinusoid, measured from the horizontal axis, equals the cosine inverse of the ratio of the base to the hypotenuse.
In conclusion, the use of an oscilloscope coupled with a firm understanding of trigonometric principles provides a powerful toolset for analyzing sinusoidal voltages. This knowledge not only aids in understanding the behavior of AC circuits but also finds applications in numerous fields like telecommunications, signal processing, and power systems.
