﻿ Signals, Sinusoidal Signals

# Signals

Thursday, December 19th, 2019 - Analog Electronics

Signals can be categorized in a number of different ways. For example, one can distinguish between analog and digital signals. An analog signal is one which can take any value (generally between two limits imposed by practical considerations). A digital signal is restricted to a set number of distinct values. The binary signals in such widespread use in our current technology have one of only two possible values, the well known 0 and 1. The conversion of information from the analog to the digital form, and vice versa.

## Types Of Signals

This text deals primarily with analog signals and circuits used to process them. Predictable, repetitive analog signals are usually specified in terms of their waveform, the plot of magnitude against time (sinusoid, sawtooth, square wave, etc.). Others are more easily described in terms of their statistical properties such as the average, average power, probabihty density and others (gaussian noise, speech, etc.). One of the most important, if not the most important, analog signal for electronic engineers is the sinusoid as explained below.

### Sinusoidal Signals

The sine wave, or sine function, has a special place in electrical and electronic engineering because of its particular properties. Some of these are:

• The sum of two or more sine waves of the same frequency f is also a sine wave at that frequency f.
• The derivative and the integral of a sine wave of frequency f are also sine waves at the same
frequency f.
• Any arbitrary waveform can be expressed as a sum (possibly of an infinite number) of sine
waves of harmonically related frequencies.
• A sinusoidal voltage and current are induced in a coil rotating in a uniform magnetic field
(generator) or a stationary coil in a sinusoidally alternating magnetic field. This is, basically, why most of the world’s electrical energy is generated and distributed in the form of (sinusoidally) alternating current, a.c. The origin of the sine function

The original mathematical purpose of the sine and cosine functions was to describe the relationship, in fact the ratio, of one side of a right angled triangle and the hypotenuse as illustrated in picture above. It is written as

sin   and cos

The plot of these functions for (Φ) = 0 to 360° (or 0 to radians) is the familiar sine ‘wave’ shown in picture below. For values of (Φ) greater than 360° the pattern repeats itself as can be deduced by a simple consideration of the original triangle. The full range of values for sine and cosine functions Two sinusoidally alternating voltages

Voltages and currents which vary in time according to the same sinusoidal function as the sides of the triangle described above are frequently used in electrical and electronic engineering. Some of the reasons for their use are outhned above. A plot of two sinusoidally alternating voltages (of the same frequency) against time is shown in picture above. The voltages may alternate through several full cycles of the sine wave shown in picture above in a second. The number of cycles of repetition in a second is called the frequency/which is measured in units of hertz (Hz) or in some old fashioned texts cycles/second or c/s. The time taken for one cycle, or one period, is the periodic time T where

sec

In a continuously changing sine wave Φ changes continuously by radians per cycle. Since this takes place f times per second the rate of change of Φ is called the angular velocity, or angular frequency, ω. This is measured in the units of radians per second, and is given by

The value of Φ at any particular time t is given by

Φ = ωt + θ

where the m term represents the change of Φ since t = 0. The θ term, called the phase angle, represents the fact that the sine wave may not have started at Φ = 0 when r = 0 in the particular scale of time used here. In other words the two sine waves reach a particular point in their cycle, such as zero, positive peak, etc., at different times. The picture above shows two sinusoids of different phases as an illustration. It is important to remember that phase is a relative measure, always measured with respect to a particular time reference or relative to another sine wave defined as the reference. Phase, or phase difference, is measured as an angle, as a proportion of the full cycle of radians or 360°. The voltage v2 in the picture above reaches its zero crossing time t1 after that of v1. Therefore the phase θ1 of V2 relative to v1 is found as the proportion

or

or

So, the general expression for finding the instantaneous value of a sinusoidally alternating quantity, using a voltage v as an example here is

V = Vp sin(ωt + θ) V

Since the maximum value of the sine function is 1, Vp is the maximum, or peak, value of the alternating voltage waveform. Note that in some applications the peak-to-peak value, Vpp of the waveform is of interest. Of course Vpp = 2Vp.

Therefore a sinusoidal signal is fully defined by the following three quantities:

1. The peak value of the magnitude. This is also known as the amplitude.
2.  The frequency, either as ω or as f . The same information is given by both since ω = 2πf.
3. The phase angle, relative to a chosen and defined reference of the same frequency.

Note that it is not possible to specify the phase difference between two sine waves of different frequencies. An analogy may be drawn with two cars moving along a road. If they travel at exactly the same speed (frequency) then the distance between them (phase) is constant and can be given as a single number. If their speeds are different the distance between them changes with time and can not be described by a single number.