Environmental Engineering Reference
In-Depth Information
90°
A
q
0°
180°
270°
Figure A.10
The Argand diagram of complex numbers
of numerous combinations of such elements and a more approprite tool is required for their
analysis.
The kind of information that expresses a single dimension, such as linear distance or tem-
perature, is called a
scalar
quantity. The voltage produced by a battery, for example, is a
scalar quantity. However, when alternating current circuits are analysed, it is found that
voltage and current are not the familiar one-dimensional quantities. Rather, these quantities,
because they are dynamic (alternating in direction and amplitude), possess two dimensions,
i.e. amplitude and phase. Therefore there is a need to work with mathematical techniques
capable of representing two-dimensional quantities. A
complex number
is a single mathemati-
cal quantity able to express these two dimensions of amplitude and phase at the same time.
A graphic representation of a complex number is known as a
phasor
.
It must be obvious that there must be some common frame of reference for angles to have
any meaning. In this case, directly right in Figure A.10 is considered to be 0 ° and angles are
counted in a positive direction going counterclockwise.
The length of the phasor represents the
magnitude
(or amplitude) of the AC waveform.
The greater the magnitude of the waveform, the greater the length of its corresponding phasor.
The angle of the phasor represents its
phase
, i.e. the phase shift in degrees between the
waveform in question and another waveform acting as a 'reference' in time. In this topic a
phasor is denoted by a bold capital letter, for example
A
. Phasor
A
in Figure A.10
has a magnitude
A
and a phase angle
. The representation of a phasor by magnitude and
phase is known as its
polar
form. The polar form of phasor
A
in Figure A.10 is written as
A
=
A
θ
. The phase of a waveform in a circuit is usually expressed with regard to the
power supply voltage waveform (arbitrarily stated to be 'at' 0 °). If there is more than one
AC voltage source, then one of those sources is arbitrarily chosen to be the phase reference
for all other measurements in the circuit. Phase is always a
relative
measurement between
two waveforms rather than an absolute property.
In Figure A.11, A and B could represent the voltage and current waveforms, (a) for a
resistor, (b) for an inductor and (c) for a capacitor; (d) could represent the antiphase relation-
ship of power in an inductor and a capacitor.
∠
θ
A.7 Phasor Addition
If phasors that are not in-phase or antiphase are added, their magnitudes add up quite differ-
ently to that of scalar quantities. Figure A.12 shows an example of such an addition.
