More on AC "polarity"
Complex numbers are useful for AC circuit
analysis because they provide a convenient method of symbolically
denoting phase shift between AC quantities like voltage and current.
However, for most people the equivalence between abstract vectors and
real circuit quantities is not an easy one to grasp. Earlier in this
chapter we saw how AC voltage sources are given voltage figures in
complex form (magnitude and phase angle), as well as polarity
markings. Being that alternating current has no set "polarity" as direct
current does, these polarity markings and their relationship to phase
angle tends to be confusing. This section is written in the attempt to
clarify some of these issues.
Voltage is an inherently relative
quantity. When we measure a voltage, we have a choice in how we connect
a voltmeter or other voltage-measuring instrument to the source of
voltage, as there are two points between which the voltage exists, and
two test leads on the instrument with which to make connection. In DC
circuits, we denote the polarity of voltage sources and voltage drops
explicitly, using "+" and "-" symbols, and use color-coded meter test
leads (red and black). If a digital voltmeter indicates a negative DC
voltage, we know that its test leads are connected "backward" to the
voltage (red lead connected to the "-" and black lead to the "+").
Batteries have their polarity designated
by way of intrinsic symbology: the short-line side of a battery is
always the negative (-) side and the long-line side always the positive
(+):
Although it would be mathematically
correct to represent a battery's voltage as a negative figure with
reversed polarity markings, it would be decidedly unconventional:
Interpreting such notation might be
easier if the "+" and "-" polarity markings were viewed as reference
points for voltmeter test leads, the "+" meaning "red" and the "-"
meaning "black." A voltmeter connected to the above battery with red
lead to the bottom terminal and black lead to the top terminal would
indeed indicate a negative voltage (-6 volts). Actually, this form of
notation and interpretation is not as unusual as you might think: it's
commonly encountered in problems of DC network analysis where "+" and
"-" polarity marks are initially drawn according to educated guess, and
later interpreted as correct or "backward" according to the mathematical
sign of the figure calculated.
In AC circuits, though, we don't deal
with "negative" quantities of voltage. Instead, we describe to what
degree one voltage aids or opposes another by phase: the
time-shift between two waveforms. We never describe an AC voltage as
being negative in sign, because the facility of polar notation allows
for vectors pointing in an opposite direction. If one AC voltage
directly opposes another AC voltage, we simply say that one is 180o
out of phase with the other.
Still, voltage is relative between two
points, and we have a choice in how we might connect a voltage-measuring
instrument between those two points. The mathematical sign of a DC
voltmeter's reading has meaning only in the context of its test lead
connections: which terminal the red lead is touching, and which terminal
the black lead is touching. Likewise, the phase angle of an AC voltage
has meaning only in the context of knowing which of the two points is
considered the "reference" point. Because of this fact, "+" and "-"
polarity marks are often placed by the terminals of an AC voltage in
schematic diagrams to give the stated phase angle a frame of reference.
Let's review these principles with some
graphical aids. First, the principle of relating test lead connections
to the mathematical sign of a DC voltmeter indication:
The mathematical sign of a digital DC
voltmeter's display has meaning only in the context of its test lead
connections. Consider the use of a DC voltmeter in determining whether
or not two DC voltage sources are aiding or opposing each other,
assuming that both sources are unlabeled as to their polarities. Using
the voltmeter to measure across the first source:
This first measurement of +24 across the
left-hand voltage source tells us that the black lead of the meter
really is touching the negative side of voltage source #1, and the red
lead of the meter really is touching the positive. Thus, we know source
#1 is a battery facing in this orientation:
Measuring the other unknown voltage
source:
This second voltmeter reading, however,
is a negative (-) 17 volts, which tells us that the black test
lead is actually touching the positive side of voltage source #2, while
the red test lead is actually touching the negative. Thus, we know that
source #2 is a battery facing in the opposite direction:
It should be obvious to any experienced
student of DC electricity that these two batteries are opposing one
another. By definition, opposing voltages subtract from one
another, so we subtract 17 volts from 24 volts to obtain the total
voltage across the two: 7 volts.
We could, however, draw the two sources
as nondescript boxes, labeled with the exact voltage figures obtained by
the voltmeter, the polarity marks indicating voltmeter test lead
placement:
According to this diagram, the polarity
marks (which indicate meter test lead placement) indicate the sources
aiding each other. By definition, aiding voltage sources add
with one another to form the total voltage, so we add 24 volts to -17
volts to obtain 7 volts: still the correct answer. If we let the
polarity markings guide our decision to either add or subtract voltage
figures -- whether those polarity markings represent the true
polarity or just the meter test lead orientation -- and include the
mathematical signs of those voltage figures in our calculations, the
result will always be correct. Again, the polarity markings serve as
frames of reference to place the voltage figures' mathematical signs
in proper context.
The same is true for AC voltages, except
that phase angle substitutes for mathematical sign. In
order to relate multiple AC voltages at different phase angles to each
other, we need polarity markings to provide frames of reference for
those voltages' phase angles.
Take for example the following circuit:
The polarity markings show these two
voltage sources aiding each other, so to determine the total voltage
across the resistor we must add the voltage figures of 10 V ∠ 0o
and 6 V ∠ 45o together to obtain 14.861 V ∠ 16.59o.
However, it would be perfectly acceptable to represent the 6 volt source
as 6 V ∠ 225o, with a reversed set of polarity markings, and
still arrive at the same total voltage:
6 V ∠ 45o with negative on the
left and positive on the right is exactly the same as 6 V ∠ 225o
with positive on the left and negative on the right: the reversal of
polarity markings perfectly complements the addition of 180o
to the phase angle designation:
Unlike DC voltage sources, whose symbols
intrinsically define polarity by means of short and long lines, AC
voltage symbols have no intrinsic polarity marking. Therefore, any
polarity marks must by included as additional symbols on the diagram,
and there is no one "correct" way in which to place them. They must,
however, correlate with the given phase angle to represent the true
phase relationship of that voltage with other voltages in the circuit.
- REVIEW:
- Polarity markings are sometimes given
to AC voltages in circuit schematics in order to provide a frame of
reference for their phase angles.
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