Three-phase power systems
Split-phase power systems achieve their
high conductor efficiency and low safety risk by splitting up the
total voltage into lesser parts and powering multiple loads at those
lesser voltages, while drawing currents at levels typical of a
full-voltage system. This technique, by the way, works just as well for
DC power systems as it does for single-phase AC systems. Such systems
are usually referred to as three-wire systems rather than
split-phase because "phase" is a concept restricted to AC.
But we know from our experience with
vectors and complex numbers that AC voltages don't always add up as we
think they would if they are out of phase with each other. This
principle, applied to power systems, can be put to use to make power
systems with even greater conductor efficiencies and lower shock hazard
than with split-phase.
Suppose that we had two sources of AC
voltage connected in series just like the split-phase system we saw
before, except that each voltage source was 120o out of phase
with the other:
Since each voltage source is 120 volts,
and each load resistor is connected directly in parallel with its
respective source, the voltage across each load must be 120 volts
as well. Given load currents of 83.33 amps, each load must still be
dissipating 10 kilowatts of power. However, voltage between the two
"hot" wires is not 240 volts (120 ∠ 0o - 120 ∠ 180o)
because the phase difference between the two sources is not 180o.
Instead, the voltage is:
Nominally, we say that the voltage
between "hot" conductors is 208 volts (rounding up), and thus the power
system voltage is designated as 120/208.
If we calculate the current through the
"neutral" conductor, we find that it is not zero, even with
balanced load resistances. Kirchhoff's Current Law tells us that the
currents entering and exiting the node between the two loads must be
zero:
So, we find that the "neutral" wire is
carrying a full 83.33 amps, just like each "hot" wire.
Note that we are still conveying 20 kW of
total power to the two loads, with each load's "hot" wire carrying 83.33
amps as before. With the same amount of current through each "hot" wire,
we must use the same gage copper conductors, so we haven't reduced
system cost over the split-phase 120/240 system. However, we have
realized a gain in safety, because the overall voltage between the two
"hot" conductors is 32 volts lower than it was in the split-phase system
(208 volts instead of 240 volts).
The fact that the neutral wire is
carrying 83.33 amps of current raises an interesting possibility: since
it's carrying current anyway, why not use that third wire as another
"hot" conductor, powering another load resistor with a third 120 volt
source having a phase angle of 240o? That way, we could
transmit more power (another 10 kW) without having to add any
more conductors. Let's see how this might look:
A full mathematical analysis of all the
voltages and currents in this circuit would necessitate the use of a
network theorem, the easiest being the Superposition Theorem. I'll spare
you the long, drawn-out calculations because you should be able to
intuitively understand that the three voltage sources at three different
phase angles will deliver 120 volts each to a balanced triad of load
resistors. For proof of this, we can use SPICE to do the math for us:
120/208 polyphase power system
v1 1 0 ac 120 0 sin
v2 2 0 ac 120 120 sin
v3 3 0 ac 120 240 sin
r1 1 4 1.44
r2 2 4 1.44
r3 3 4 1.44
.ac lin 1 60 60
.print ac v(1,4) v(2,4) v(3,4)
.print ac v(1,2) v(2,3) v(3,1)
.print ac i(v1) i(v2) i(v3)
.end
VOLTAGE ACROSS EACH LOAD
freq v(1,4) v(2,4) v(3,4)
6.000E+01 1.200E+02 1.200E+02 1.200E+02
VOLTAGE BETWEEN "HOT" CONDUCTORS
freq v(1,2) v(2,3) v(3,1)
6.000E+01 2.078E+02 2.078E+02 2.078E+02
CURRENT THROUGH EACH VOLTAGE SOURCE
freq i(v1) i(v2) i(v3)
6.000E+01 8.333E+01 8.333E+01 8.333E+01
Sure enough, we get 120 volts across each
load resistor, with (approximately) 208 volts between any two "hot"
conductors and conductor currents equal to 83.33 amps. At that current
and voltage, each load will be dissipating 10 kW of power. Notice that
this circuit has no "neutral" conductor to ensure stable voltage to all
loads if one should open. What we have here is a situation similar to
our split-phase power circuit with no "neutral" conductor: if one load
should happen to fail open, the voltage drops across the remaining
load(s) will change. To ensure load voltage stability in the even of
another load opening, we need a neutral wire to connect the source node
and load node together:
So long as the loads remain balanced
(equal resistance, equal currents), the neutral wire will not have to
carry any current at all. It is there just in case one or more load
resistors should fail open (or be shut off through a disconnecting
switch).
This circuit we've been analyzing with
three voltage sources is called a polyphase circuit. The prefix
"poly" simply means "more than one," as in "polytheism" (belief
in more than one deity), polygon" (a geometrical shape made of
multiple line segments: for example, pentagon and hexagon),
and "polyatomic" (a substance composed of multiple types of
atoms). Since the voltage sources are all at different phase angles (in
this case, three different phase angles), this is a "polyphase"
circuit. More specifically, it is a three-phase circuit, the kind
used predominantly in large power distribution systems.
Let's survey the advantages of a
three-phase power system over a single-phase system of equivalent load
voltage and power capacity. A single-phase system with three loads
connected directly in parallel would have a very high total current
(83.33 times 3, or 250 amps:
This would necessitate 3/0 gage copper
wire (very large!), at about 510 pounds per thousand feet, and
with a considerable price tag attached. If the distance from source to
load was 1000 feet, we would need over a half-ton of copper wire to do
the job. On the other hand, we could build a split-phase system with two
15 kW, 120 volt loads:
Our current is half of what it was with
the simple parallel circuit, which is a great improvement. We could get
away with using number 2 gage copper wire at a total mass of about 600
pounds, figuring about 200 pounds per thousand feet with three runs of
1000 feet each between source and loads. However, we also have to
consider the increased safety hazard of having 240 volts present in the
system, even though each load only receives 120 volts. Overall, there is
greater potential for dangerous electric shock to occur.
When we contrast these two examples
against our three-phase system, the advantages are quite clear. First,
the conductor currents are quite a bit less (83.33 amps versus 125 or
250 amps), permitting the use of much thinner and lighter wire. We can
use number 4 gage wire at about 125 pounds per thousand feet, which will
total 500 pounds (four runs of 1000 feet each) for our example circuit.
This represents a significant cost savings over the split-phase system,
with the additional benefit that the maximum voltage in the system is
lower (208 versus 240).
One question remains to be answered: how
in the world do we get three AC voltage sources whose phase angles are
exactly 120o apart? Obviously we can't center-tap a
transformer or alternator winding like we did in the split-phase system,
since that can only give us voltage waveforms that are either in phase
or 180o out of phase. Perhaps we could figure out some way to
use capacitors and inductors to create phase shifts of 120o,
but then those phase shifts would depend on the phase angles of our load
impedances as well (substituting a capacitive or inductive load for a
resistive load would change everything!).
The best way to get the phase shifts
we're looking for is to generate it at the source: construct the AC
generator (alternator) providing the power in such a way that the
rotating magnetic field passes by three sets of wire windings, each set
spaced 120o apart around the circumference of the machine:
Together, the six "pole" windings of a
three-phase alternator are connected to comprise three winding pairs,
each pair producing AC voltage with a phase angle 120o
shifted from either of the other two winding pairs. The interconnections
between pairs of windings (as shown for the single-phase alternator: the
jumper wire between windings 1a and 1b) have been omitted from the
three-phase alternator drawing for simplicity.
In our example circuit, we showed the
three voltage sources connected together in a "Y" configuration
(sometimes called the "star" configuration), with one lead of each
source tied to a common point (the node where we attached the "neutral"
conductor). The common way to depict this connection scheme is to draw
the windings in the shape of a "Y" like this:
The "Y" configuration is not the only
option open to us, but it is probably the easiest to understand at
first. More to come on this subject later in the chapter.
- REVIEW:
- A single-phase power system is
one where there is only one AC voltage source (one source voltage
waveform).
- A split-phase power system is
one where there are two voltage sources, 180o phase-shifted
from each other, powering a two series-connected loads. The advantage
of this is the ability to have lower conductor currents while
maintaining low load voltages for safety reasons.
- A polyphase power system uses
multiple voltage sources at different phase angles from each other
(many "phases" of voltage waveforms at work). A polyphase power system
can deliver more power at less voltage with smaller-gage conductors
than single- or split-phase systems.
- The phase-shifted voltage sources
necessary for a polyphase power system are created in alternators with
multiple sets of wire windings. These winding sets are spaced around
the circumference of the rotor's rotation at the desired angle(s).
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