Introduction
If I needed to describe the distance
between two cities, I could provide an answer consisting of a single
number in miles, kilometers, or some other unit of linear measurement.
However, if I were to describe how to travel from one city to another, I
would have to provide more information than just the distance between
those two cities; I would also have to provide information about the
direction to travel, as well.
The kind of information that expresses a
single dimension, such as linear distance, is called a scalar
quantity in mathematics. Scalar numbers are the kind of numbers you've
used in most all of your mathematical applications so far. The voltage
produced by a battery, for example, is a scalar quantity. So is the
resistance of a piece of wire (ohms), or the current through it (amps).
However, when we begin to analyze
alternating current circuits, we find that quantities of voltage,
current, and even resistance (called impedance in AC) are not the
familiar one-dimensional quantities we're used to measuring in DC
circuits. Rather, these quantities, because they're dynamic (alternating
in direction and amplitude), possess other dimensions that must be taken
into account. Frequency and phase shift are two of these dimensions that
come into play. Even with relatively simple AC circuits, where we're
only dealing with a single frequency, we still have the dimension of
phase shift to contend with in addition to the amplitude.
In order to successfully analyze AC
circuits, we need to work with mathematical objects and techniques
capable of representing these multi-dimensional quantities. Here is
where we need to abandon scalar numbers for something better suited:
complex numbers. Just like the example of giving directions from one
city to another, AC quantities in a single-frequency circuit have both
amplitude (analogy: distance) and phase shift (analogy: direction). A
complex number is a single mathematical quantity able to express these
two dimensions of amplitude and phase shift at once.
Complex numbers are easier to grasp when
they're represented graphically. If I draw a line with a certain length
(magnitude) and angle (direction), I have a graphic representation of a
complex number which is commonly known in physics as a vector:
Like distances and directions on a map,
there must be some common frame of reference for angle figures to have
any meaning. In this case, directly right is considered to be 0o,
and angles are counted in a positive direction going counter-clockwise:
The idea of representing a number in
graphical form is nothing new. We all learned this in grade school with
the "number line:"
We even learned how addition and
subtraction works by seeing how lengths (magnitudes) stacked up to give
a final answer:
Later, we learned that there were ways to
designate the values between the whole numbers marked on the
line. These were fractional or decimal quantities:
Later yet we learned that the number line
could extend to the left of zero as well:
These fields of numbers (whole, integer,
rational, irrational, real, etc.) learned in grade school share a common
trait: they're all one-dimensional. The straightness of the
number line illustrates this graphically. You can move up or down the
number line, but all "motion" along that line is restricted to a single
axis (horizontal). One-dimensional, scalar numbers are perfectly
adequate for counting beads, representing weight, or measuring DC
battery voltage, but they fall short of being able to represent
something more complex like the distance and direction between
two cities, or the amplitude and phase of an AC waveform. To
represent these kinds of quantities, we need multidimensional
representations. In other words, we need a number line that can point in
different directions, and that's exactly what a vector is.
- REVIEW:
- A scalar number is the type of
mathematical object that people are used to using in everyday life: a
one-dimensional quantity like temperature, length, weight, etc.
- A complex number is a
mathematical quantity representing two dimensions of magnitude and
direction.
- A vector is a graphical
representation of a complex number. It looks like an arrow, with a
starting point, a tip, a definite length, and a definite direction.
Sometimes the word phasor is used in electrical applications
where the angle of the vector represents phase shift between
waveforms.
Vectors and AC waveforms
Okay, so how exactly can we represent AC
quantities of voltage or current in the form of a vector? The length of
the vector represents the magnitude (or amplitude) of the waveform, like
this:
The greater the amplitude of the
waveform, the greater the length of its corresponding vector. The angle
of the vector, however, represents the phase shift in degrees between
the waveform in question and another waveform acting as a "reference" in
time. Usually, when the phase of a waveform in a circuit is expressed,
it is referenced to the power supply voltage waveform (arbitrarily
stated to be "at" 0o). Remember that phase is always a
relative measurement between two waveforms rather than an absolute
property.
The greater the phase shift in degrees
between two waveforms, the greater the angle difference between the
corresponding vectors. Being a relative measurement, like voltage, phase
shift (vector angle) only has meaning in reference to some standard
waveform. Generally this "reference" waveform is the main AC power
supply voltage in the circuit. 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.
This concept of a reference point is not
unlike that of the "ground" point in a circuit for the benefit of
voltage reference. With a clearly defined point in the circuit declared
to be "ground," it becomes possible to talk about voltage "on" or "at"
single points in a circuit, being understood that those voltages (always
relative between two points) are referenced to "ground."
Correspondingly, with a clearly defined point of reference for phase it
becomes possible to speak of voltages and currents in an AC circuit
having definite phase angles. For example, if the current in an AC
circuit is described as "24.3 milliamps at -64 degrees," it means that
the current waveform has an amplitude of 24.3 mA, and it lags 64o
behind the reference waveform, usually assumed to be the main source
voltage waveform.
- REVIEW:
- When used to describe an AC quantity,
the length of a vector represents the amplitude of the wave while the
angle of a vector represents the phase angle of the wave relative to
some other (reference) waveform.
Simple vector addition
Remember that vectors are mathematical
objects just like numbers on a number line: they can be added,
subtracted, multiplied, and divided. Addition is perhaps the easiest
vector operation to visualize, so we'll begin with that. If vectors with
common angles are added, their magnitudes (lengths) add up just like
regular scalar quantities:
Similarly, if AC voltage sources with the
same phase angle are connected together in series, their voltages add
just as you might expect with DC batteries:
Please note the (+) and (-) polarity
marks next to the leads of the two AC sources. Even though we know AC
doesn't have "polarity" in the same sense that DC does, these marks are
essential to knowing how to reference the given phase angles of the
voltages. This will become more apparent in the next example.
If vectors directly opposing each other
(180o out of phase) are added together, their magnitudes
(lengths) subtract just like positive and negative scalar quantities
subtract when added:
Similarly, if opposing AC voltage sources
are connected in series, their voltages subtract as you might expect
with DC batteries connected in an opposing fashion:
Determining whether or not these voltage
sources are opposing each other requires an examination of their
polarity markings and their phase angles. Notice how the polarity
markings in the above diagram seem to indicate additive voltages (from
left to right, we see - and + on the 6 volt source, - and + on the 8
volt source). Even though these polarity markings would normally
indicate an additive effect in a DC circuit (the two voltages
working together to produce a greater total voltage), in this AC circuit
they're actually pushing in opposite directions because one of those
voltages has a phase angle of 0o and the other a phase angle
of 180o. The result, of course, is a total voltage of 2
volts.
We could have just as well shown the
opposing voltages subtracting in series like this:
Note how the polarities appear to be
opposed to each other now, due to the reversal of wire connections on
the 8 volt source. Since both sources are described as having equal
phase angles (0o), they truly are opposed to one another, and
the overall effect is the same as the former scenario with "additive"
polarities and differing phase angles: a total voltage of only 2 volts.
The resultant voltage can be expressed in
two different ways: 2 volts at 180o with the (-) symbol on
the left and the (+) symbol on the right, or 2 volts at 0o
with the (+) symbol on the left and the (-) symbol on the right. A
reversal of wires from an AC voltage source is the same as
phase-shifting that source by 180o.
Complex vector addition
If vectors with uncommon angles are
added, their magnitudes (lengths) add up quite differently than that of
scalar magnitudes:
If two AC voltages -- 90o out
of phase -- are added together by being connected in series, their
voltage magnitudes do not directly add or subtract as with scalar
voltages in DC. Instead, these voltage quantities are complex
quantities, and just like the above vectors, which add up in a
trigonometric fashion, a 6 volt source at 0o added to an 8
volt source at 90o results in 10 volts at a phase angle of
53.13o:
Compared to DC circuit analysis, this is
very strange indeed. Note that it's possible to obtain voltmeter
indications of 6 and 8 volts, respectively, across the two AC voltage
sources, yet only read 10 volts for a total voltage!
There is no suitable DC analogy for what
we're seeing here with two AC voltages slightly out of phase. DC
voltages can only directly aid or directly oppose, with nothing in
between. With AC, two voltages can be aiding or opposing one another
to any degree between fully-aiding and fully-opposing, inclusive.
Without the use of vector (complex number) notation to describe AC
quantities, it would be very difficult to perform mathematical
calculations for AC circuit analysis.
In the next section, we'll learn how to
represent vector quantities in symbolic rather than graphical form.
Vector and triangle diagrams suffice to illustrate the general concept,
but more precise methods of symbolism must be used if any serious
calculations are to be performed on these quantities.
- REVIEW:
- DC voltages can only either directly
aid or directly oppose each other when connected in series. AC
voltages may aid or oppose to any degree depending on the phase
shift between them.
Polar and rectangular notation
In order to work with these complex
numbers without drawing vectors, we first need some kind of standard
mathematical notation. There are two basic forms of complex number
notation: polar and rectangular.
Polar form is where a complex number is
denoted by the length (otherwise known as the magnitude,
absolute value, or modulus) and the angle of its
vector (usually denoted by an angle symbol that looks like this: ∠). To
use the map analogy, polar notation for the vector from New York City to
San Diego would be something like "2400 miles, southwest." Here are two
examples of vectors and their polar notations:
Standard orientation for vector angles in
AC circuit calculations defines 0o as being to the right
(horizontal), making 90o straight up, 180o to the
left, and 270o straight down. Please note that vectors angled
"down" can have angles represented in polar form as positive numbers in
excess of 180, or negative numbers less than 180. For example, a vector
angled ∠ 270o (straight down) can also be said to have an
angle of -90o. The above vector on the right (5.4 ∠ 326o)
can also be denoted as 5.4 ∠ -34o.
Rectangular form, on the other hand, is
where a complex number is denoted by its respective horizontal and
vertical components. In essence, the angled vector is taken to be the
hypotenuse of a right triangle, described by the lengths of the adjacent
and opposite sides. Rather than describing a vector's length and
direction by denoting magnitude and angle, it is described in terms of
"how far left/right" and "how far up/down."
These two dimensional figures (horizontal
and vertical) are symbolized by two numerical figures. In order to
distinguish the horizontal and vertical dimensions from each other, the
vertical is prefixed with a lower-case "i" (in pure mathematics) or "j"
(in electronics). These lower-case letters do not represent a physical
variable (such as instantaneous current, also symbolized by a lower-case
letter "i"), but rather are mathematical operators used to
distinguish the vector's vertical component from its horizontal
component. As a complete complex number, the horizontal and vertical
quantities are written as a sum:
The horizontal component is referred to
as the real component, since that dimension is compatible with
normal, scalar ("real") numbers. The vertical component is referred to
as the imaginary component, since that dimension lies in a
different direction, totally alien to the scale of the real numbers.
The "real" axis of the graph corresponds
to the familiar number line we saw earlier: the one with both positive
and negative values on it. The "imaginary" axis of the graph corresponds
to another number line situated at 90o to the "real" one.
Vectors being two-dimensional things, we must have a two-dimensional
"map" upon which to express them, thus the two number lines
perpendicular to each other:
Either method of notation is valid for
complex numbers. The primary reason for having two methods of notation
is for ease of longhand calculation, rectangular form lending itself to
addition and subtraction, and polar form lending itself to
multiplication and division.
Conversion between the two notational
forms involves simple trigonometry. To convert from polar to
rectangular, find the real component by multiplying the polar magnitude
by the cosine of the angle, and the imaginary component by multiplying
the polar magnitude by the sine of the angle. This may be understood
more readily by drawing the quantities as sides of a right triangle, the
hypotenuse of the triangle representing the vector itself (its length
and angle with respect to the horizontal constituting the polar form),
the horizontal and vertical sides representing the "real" and
"imaginary" rectangular components, respectively:
To convert from rectangular to polar,
find the polar magnitude through the use of the Pythagorean Theorem (the
polar magnitude is the hypotenuse of a right triangle, and the real and
imaginary components are the adjacent and opposite sides, respectively),
and the angle by taking the arctangent of the imaginary component
divided by the real component:
- REVIEW:
- Polar
notation denotes a complex number in terms of its vector's length and
angular direction from the starting point. Example: fly 45 miles ∠ 203o
(West by Southwest).
- Rectangular
notation denotes a complex number in terms of its horizontal and
vertical dimensions. Example: drive 41 miles West, then turn and drive
18 miles South.
- In rectangular notation, the first
quantity is the "real" component (horizontal dimension of vector) and
the second quantity is the "imaginary" component (vertical dimension
of vector). The imaginary component is preceded by a lower-case "j,"
sometimes called the j operator.
- Both polar and rectangular forms of
notation for a complex number can be related graphically in the form
of a right triangle, with the hypotenuse representing the vector
itself (polar form: hypotenuse length = magnitude; angle with respect
to horizontal side = angle), the horizontal side representing the
rectangular "real" component, and the vertical side representing the
rectangular "imaginary" component.
Complex number arithmetic
Since complex numbers are legitimate
mathematical entities, just like scalar numbers, they can be added,
subtracted, multiplied, divided, squared, inverted, and such, just like
any other kind of number. Some scientific calculators are programmed to
directly perform these operations on two or more complex numbers, but
these operations can also be done "by hand." This section will show you
how the basic operations are performed. It is highly recommended
that you equip yourself with a scientific calculator capable of
performing arithmetic functions easily on complex numbers. It will make
your study of AC circuit much more pleasant than if you're forced to do
all calculations the longer way.
Addition and subtraction with complex
numbers in rectangular form is easy. For addition, simply add up the
real components of the complex numbers to determine the real component
of the sum, and add up the imaginary components of the complex numbers
to determine the imaginary component of the sum:
When subtracting complex numbers in
rectangular form, simply subtract the real component of the second
complex number from the real component of the first to arrive at the
real component of the difference, and subtract the imaginary component
of the second complex number from the imaginary component of the first
to arrive the imaginary component of the difference:
For longhand multiplication and division,
polar is the favored notation to work with. When multiplying complex
numbers in polar form, simply multiply the polar magnitudes of
the complex numbers to determine the polar magnitude of the product, and
add the angles of the complex numbers to determine the angle of
the product:
Division of polar-form complex numbers is
also easy: simply divide the polar magnitude of the first complex number
by the polar magnitude of the second complex number to arrive at the
polar magnitude of the quotient, and subtract the angle of the second
complex number from the angle of the first complex number to arrive at
the angle of the quotient:
To obtain the reciprocal, or "invert"
(1/x), a complex number, simply divide the number (in polar form) into a
scalar value of 1, which is nothing more than a complex number with no
imaginary component (angle = 0):
These are the basic operations you will
need to know in order to manipulate complex numbers in the analysis of
AC circuits. Operations with complex numbers are by no means limited
just to addition, subtraction, multiplication, division, and inversion,
however. Virtually any arithmetic operation that can be done with scalar
numbers can be done with complex numbers, including powers, roots,
solving simultaneous equations with complex coefficients, and even
trigonometric functions (although this involves a whole new perspective
in trigonometry called hyperbolic functions which is well beyond
the scope of this discussion). Be sure that you're familiar with the
basic arithmetic operations of addition, subtraction, multiplication,
division, and inversion, and you'll have little trouble with AC circuit
analysis.
- REVIEW:
- To add complex numbers in rectangular
form, add the real components and add the imaginary components.
Subtraction is similar.
- To multiply complex numbers in polar
form, multiply the magnitudes and add the angles. To divide, divide
the magnitudes and subtract one angle from the other.
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.
Some examples with AC circuits
Let's connect three AC voltage sources in
series and use complex numbers to determine additive voltages. All the
rules and laws learned in the study of DC circuits apply to AC circuits
as well (Ohm's Law, Kirchhoff's Laws, network analysis methods), with
the exception of power calculations (Joule's Law). The only
qualification is that all variables must be expressed in complex
form, taking into account phase as well as magnitude, and all voltages
and currents must be of the same frequency (in order that their phase
relationships remain constant).
The polarity marks for all three voltage
sources are oriented in such a way that their stated voltages should add
to make the total voltage across the load resistor. Notice that although
magnitude and phase angle is given for each AC voltage source, no
frequency value is specified. If this is the case, it is assumed that
all frequencies are equal, thus meeting our qualifications for applying
DC rules to an AC circuit (all figures given in complex form, all of the
same frequency). The setup of our equation to find total voltage appears
as such:
Graphically, the vectors add up in this
manner:
The sum of these vectors will be a
resultant vector originating at the starting point for the 22 volt
vector (dot at upper-left of diagram) and terminating at the ending
point for the 15 volt vector (arrow tip at the middle-right of the
diagram):
In order to determine what the resultant
vector's magnitude and angle are without resorting to graphic images, we
can convert each one of these polar-form complex numbers into
rectangular form and add. Remember, we're adding these figures
together because the polarity marks for the three voltage sources are
oriented in an additive manner:
In polar form, this equates to 36.8052
volts ∠ -20.5018o. What this means in real terms is that the
voltage measured across these three voltage sources will be 36.8052
volts, lagging the 15 volt (0o phase reference) by 20.5018o.
A voltmeter connected across these points in a real circuit would only
indicate the polar magnitude of the voltage (36.8052 volts), not the
angle. An oscilloscope could be used to display two voltage waveforms
and thus provide a phase shift measurement, but not a voltmeter. The
same principle holds true for AC ammeters: they indicate the polar
magnitude of the current, not the phase angle.
This is extremely important in relating
calculated figures of voltage and current to real circuits. Although
rectangular notation is convenient for addition and subtraction, and was
indeed the final step in our sample problem here, it is not very
applicable to practical measurements. Rectangular figures must be
converted to polar figures (specifically polar magnitude) before
they can be related to actual circuit measurements.
We can use SPICE to verify the accuracy
of our results. In this test circuit, the 10 kΩ resistor value is quite
arbitrary. It's there so that SPICE does not declare an open-circuit
error and abort analysis. Also, the choice of frequencies for the
simulation (60 Hz) is quite arbitrary, because resistors respond
uniformly for all frequencies of AC voltage and current. There are other
components (notably capacitors and inductors) which do not respond
uniformly to different frequencies, but that is another subject!
ac voltage addition
v1 1 0 ac 15 0 sin
v2 2 1 ac 12 35 sin
v3 3 2 ac 22 -64 sin
r1 3 0 10k
.ac lin 1 60 60 I'm using a frequency of 60 Hz
.print ac v(3,0) vp(3,0) as a default value
.end
freq v(3) vp(3)
6.000E+01 3.681E+01 -2.050E+01
Sure enough, we get a total voltage of
36.81 volts ∠ -20.5o (with reference to the 15 volt source,
whose phase angle was arbitrarily stated at zero degrees so as to be the
"reference" waveform).
At first glance, this is
counter-intuitive. How is it possible to obtain a total voltage of just
over 36 volts with 15 volt, 12 volt, and 22 volt supplies connected in
series? With DC, this would be impossible, as voltage figures will
either directly add or subtract, depending on polarity. But with AC, our
"polarity" (phase shift) can vary anywhere in between full-aiding and
full-opposing, and this allows for such paradoxical summing.
What if we took the same circuit and
reversed one of the supply's connections? Its contribution to the total
voltage would then be the opposite of what it was before:
Note how the 12 volt supply's phase angle
is still referred to as 35o, even though the leads have been
reversed. Remember that the phase angle of any voltage drop is stated in
reference to its noted polarity. Even though the angle is still written
as 35o, the vector will be drawn 180o opposite of
what it was before:
The resultant (sum) vector should begin
at the upper-left point (origin of the 22 volt vector) and terminate at
the right arrow tip of the 15 volt vector:
The connection reversal on the 12 volt
supply can be represented in two different ways in polar form: by an
addition of 180o to its vector angle (making it 12 volts ∠
215o), or a reversal of sign on the magnitude (making it -12
volts ∠ 35o). Either way, conversion to rectangular form
yields the same result:
The resulting addition of voltages in
rectangular form, then:
In polar form, this equates to 30.4964 V
∠ -60.9368o. Once again, we will use SPICE to verify the
results of our calculations:
ac voltage addition
v1 1 0 ac 15 0 sin
v2 1 2 ac 12 35 sin Note the reversal of node numbers 2 and 1
v3 3 2 ac 22 -64 sin to simulate the swapping of connections
r1 3 0 10k
.ac lin 1 60 60
.print ac v(3,0) vp(3,0)
.end
freq v(3) vp(3)
6.000E+01 3.050E+01 -6.094E+01
- REVIEW:
- All the laws and rules of DC circuits
apply to AC circuits, with the exception of power calculations
(Joule's Law), so long as all values are expressed and manipulated in
complex form, and all voltages and currents are at the same frequency.
- When reversing the direction of a
vector (equivalent to reversing the polarity of an AC voltage source
in relation to other voltage sources), it can be expressed in either
of two different ways: adding 180o to the angle, or
reversing the sign of the magnitude.
- Meter measurements in an AC circuit
correspond to the polar magnitudes of calculated values.
Rectangular expressions of complex quantities in an AC circuit have no
direct, empirical equivalent, although they are convenient for
performing addition and subtraction, as Kirchhoff's Voltage and
Current Laws require.
Contributors
Contributors to this chapter are listed
in chronological order of their contributions, from most recent to
first.
Jason Starck
(June 2000): HTML document formatting, which led to a much
better-looking second edition.
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