Sinusoidal Signals

Signals

Sinusoidal Signals

Introduction

Sinusoids are an extremely important category of time-varying functions (or signals). Here are some examples of their uses:

In the electrical power industry sinusoids are the dominant signal used to transfer power.
In communication systems (cellular telephones, radio signals, etc.) the so-called carrier signals are sinusoidal.
Vibrations in mechanical systems are a common source of failure—the stresses caused by these vibrations are commonly analyzed in terms of sinusoids.

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.

Sinusoidal Signals

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 is the amplitude of the sinusoid.
ω is the frequency (in radians/second) of the signal.
And θ is the phase angle of the signal.

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.)

\[f = \frac{\omega }{{2\pi }}\]

(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.

Figure 2. Period, T, of a sinusoidal signal.

Sinusoids with Offsets

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.

Figure 3. Sinusoidal signal with a DC component (or offset).

Important Points

The term signal is generally used to denote any time-varying function.

Sinusoidal signals are commonly referred to as AC signals. AC stands for alternating current.

Constant signals (signals which do not change with time) are commonly referred to as DC signals. DC stands for direct current.

The period of the sinusoid is the time between successive peaks of the sinusoid. It is generally abbreviated as T.

A sinusoidal signal is represented mathematically as:
\[f(t) = A\cos (\omega t + \theta )\]

The amplitude of a sinusoid provides the maximum and minimum values that the sinusoid achieves. The amplitude is the value A in the above expression. The maximum value is the amplitude; the minimum value is the negative of the amplitude.

The frequency of the sinusoid is generally represented in terms of Hertz (abbreviated Hz, and generally represented as f) or radians/second (generally referred to as radian frequency, and represented as ω). These parameters are related to the signal's period according to:
\(f = \frac{1}{T}\) And \(\omega = \frac{{2\pi }}{T}\)

The radian frequency and the frequency in Hertz are related by:
\[\omega = 2\pi f\]

Sinusoids commonly have an added DC component. A sinusoidal signal with an offset C is represented mathematically as:
\[f(t) = C + A\cos (\omega t + \theta )\]

¹ Not accounting for the difference between Hertz and radians/second is probably the most common error seen in sophomore and junior level electrical engineering labs (in the experience of this instructor, anyway).