Before we take up the description of a sinusoidal signal, it is better to define first what a sinusoidal function is. The equation of a cosine function is displayed now on the left side of window, and the waveform of the cosine function is displayed on the right side of window. For every 2p radians or 360o, the function repeats itself. For example, cos(30o) has the same value of cos (390o).
The waveform of ac sinusoidal signal is illustrated on the right side of window. It can be described by equation (1). Since the cosine function repeats every 2p radians, we get equation (2). Here the sinusoidal signal is a function of time, and the signal repeats itself after a cycle period, where T is the period of a cycle. Since the sinusoidal signal repeats itself every 2p radians or T seconds, we get equation (3). From equation (3), it can be seen that the product of w and the cycle period is 2p radians. If the time period of a cycle is T seconds, the number of cycles per second, called the frequency, is the reciprocal of cycle period. Hence the value of w , which is equal to 2p radians over T, can be expressed by equation (4), as the product of frequency and 2p. Here w is called the angular frequency, and it is expressed in radians per second.
Equation (5) defines a sinusoidal signal. The sinusoidal signal is completely defined by three parameters, which are its amplitude, its frequency, and its phase. The amplitude of the waveform displayed is indicated by letter, E. The second parameter is its frequency. The frequency is the reciprocal of the cycle period, marked as T. Since frequency is the number of cycles per second, the reciprocal of cycle period yields frequency. The product of frequency and cycle period is one second. The third parameter is the phase.
Equation (5) defines a sinusoidal signal. The phase can vary from - p radians to p radians. The magnitude of phase can be obtained from the value of signal at time T equal to zero, as shown by equations (6) and (7). But it is difficult to determine the sign or polarity from this definition, since the cosine function is an even function. It is better to define phase in another way. Let the sinusoidal signal with zero phase be called the reference signal. On the right side of window, the reference signal is displayed in black color, whereas the phase-shifted signal is shown in red color. The phase is the angle by which the phase-shifted signal lags, or leads the reference signal. If the phase-shifted signal leads the reference signal, then the phase is a positive value. It is the angle at which the phase-shifted signal has its peak value. When the phase-shifted signal leads the reference signal, as shown here, the positive peak of the phase-shifted signal occurs at time, prior to zero.
It can be seen from equation (5) that the phase-shifted signal has the maximum value when (wt + f ) is equal to zero. Hence when phase f is positive, the peak occurs prior to t = 0. When phase f is negative, the peak of phase-shifted signal occurs for a positive value of time. If the phase-shifted signal lags the reference signal, then phase is a negative value. As stated earlier, phase is the angle at which the phase-shifted signal has its peak value. If it occurs for values of time t, greater than zero, then the phase-shifted signal lags the reference signal, and the phase is negative.
Next we learn about complex numbers.