Sinusoids are an extremely important category of time-varying functions (or signals). Here are some examples of their uses:
The alternating current (or AC) signals used in power transmission are, in fact, so pervasive that engineers commonly refer to any sinusoidal signal as “AC”. (By contrast, power which is delivered by constant voltages and currents are referred to as DC power; DC stands for direct current. Any signal which does not vary with time is commonly called a DC value.) We will present the basic definitions associated with the parameters of sinusoidal signals and provide some physical insight into these specifications.
For now, we will consider sinusoidal signals to be any time-varying function with the following form:
(1.)
\[f(t) = A\cos (\omega t + \theta )\]Where:
A provides the peak value of the sinusoid, or the maximum value of the function. (A also provides the minimum value of the function, which is simply the negative value of A.) ω governs the rate of oscillation of the signal—smaller values of ω mean that the signal is oscillating more slowly, or that there is a longer time between peaks in the sinusoid. θ affects the translation of the sinusoid in time. A typical sinusoidal signal is shown in Fig. 1.
Note that equation (1) above represents an arbitrary sinusoid since sine functions can be represented in terms of cosines with addition of a change in phase. The general relationship is:
(2.)
\[A\cos (\omega t + \theta ) = A\sin (\omega t + \theta + \frac{\pi }{2})\]If θ is in radians.
(3.)
\[A\cos (\omega t + \theta ) = A\sin (\omega t + \theta + 90^\circ )\]If θ is in degrees.
Sinusoids are commonly represented in terms of their frequency in Hertz (abbreviated Hz), or cycles/second, rather than their frequency in radians/second. As the units imply, frequencies in Hertz provide the number of cycles that the sinusoid goes through in one second. The inverse of this, called the period of the sinusoid, is typically abbreviated as T. The period is simply the number of seconds required for one cycle of the sinusoid—it is the time between successive peaks in the sinusoidal signal graph. Figure 2 below shows a typical sinusoid with its period labeled.
Mathematically, the frequency (in Hertz) of a sinusoid is simply the inverse of its period:
(4.)
\[f = \frac{1}{T}\]Since there are 2π; radians in one cycle, the relationship between frequency in Hertz and radian frequency is:
(5.)
\[f = \frac{\omega }{{2\pi }}\]Or
(6.)
\[\omega = 2\pi f\]Where f is the frequency of the sinusoid in Hz. Thus, the sinusoid of equation (1) can be written as:
(7.)
\[f(t) = A\cos (2\pi ft + \theta )\]Sinusoidal experimental data are often expressed with frequency in Hertz; this is especially true of most function generators (which create time-varying signals, including sinusoids). Likewise, measured frequencies are most often expressed in Hertz. However, it should be emphasized that any mathematical analysis of these signals must be performed in radians/second—make sure you keep track of the factor of 2π present in equation (2)^{1}.
It is very common for an experimentally-generated sinusoidal signal to contain an offset. This simply means that a constant (or DC) value has been added to the sinusoid. Thus, based on equation (1), a sinusoidal signal with some offset can be represented as:
(8.)
\[f(t) = C + A\cos (\omega t + \theta )\]Where C is the offset—or the DC component of the signal. Figure 3 shows a sinusoidal signal with a DC offset, C. Note that the maximum value of this signal is (C+A) and the minimum value is (C-A). The average value of the signal is the DC component, C.
\(f = \frac{1}{T}\) And \(\omega = \frac{{2\pi }}{T}\)