Transcript Introduction to Technical Mathematics

10 – Three Phase Theory
10 – Three Phase Theory
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The intent of this presentation is to present enough information to provide the reader with a
fundamental knowledge of Three Phase Alternating Current and to better understand basic
Michelin system and equipment operations.
10 – Three Phase Theory
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10 – Three Phase Theory
Characteristics of Three Phase Systems
A three phase (3) system is a combination of three single phase (1) systems. In a 3 balanced system, the
power comes from an AC generator that produces three separate but equal voltages, each of which is out of
phase with the other voltages by 120(illustrated below). When any of the three voltages is at its maximum value
(negative or positive), each of the other two voltages will be at 50 percent of their maximum value in the opposite
direction. Although single phase circuits are widely used in electrical systems; most generation and distribution
of alternating current is 3.
10 – Three Phase Theory
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10 – Three Phase Theory
Definitions
Single Phase System
- System energized by a single alternating voltage.
Poly Phase System
- System energized by two, three or more alternating voltages.
Three Phase System
- System energized by three equal alternating voltages 120 apart.
Line Voltage (VL)
feeder wires.
- The voltage measured between terminals of an alternator, motor or branch circuit
Phase
load.
- Each winding of an alternator, of a motor, or of a branch circuit forming part of a
Phase Voltage (Vph)
- The voltage measured across a phase of an alternator, motor or load.
10 – Three Phase Theory
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10 – Three Phase Theory
Advantages of a 3 phase system versus a single phase system
•
3 circuits require less weight of conductors than 1 circuits of the same power rating.
•
3 circuits permit flexibility in the choice of voltages.
•
3 circuits can be used for 1 loads.
•
3 equipment is smaller in size, lighter in weight, and more efficient than 1 machinery of the same
rated capacity.
10 – Three Phase Theory
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10 – Three Phase Theory
The three phases of a 3 system may be connected in two ways.
In most industrial applications, the 3 phase generation is done by the power company and distributed through a
power distribution system. The voltage is stepped down along the way and delivered to the facilities. The facility
uses three phase transformers to supply the loads in the facility. The following representations are of the
secondary windings of a 3 phase transformer not a standalone generator.
1.
If the three common ends of each secondary phase winding are connected together at a common terminal
marked N for neutral, then the other three ends become the A, B, and C 3 line conductors.
A
This system is Wye or Y-connected.
This configuration can be related as being connected in
series. Sometimes the neutral connection is grounded
and brought out to form a 3, 4-wire system.
Phase Winding
Neutral
N
B
C
10 – Three Phase Theory
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10 – Three Phase Theory
The three phases of a 3 system may be connected in two ways.
2. If the three phases are connected to form a closed loop, the system is delta or Δ-connected. This
configuration can be related as being connected in parallel.
A
Phase Winding
B
C
10 – Three Phase Theory
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10 – Three Phase Theory
Three Phase Wye Relationships
Voltage Relationship in a Wye Configuration - Since the line voltage is connected across two windings and
the phase voltage is only connected across one winding, the line voltage (VL) is much larger than the phase
voltage (Vph). At first, you would think that the line voltage is twice as large as the phase voltage, this is not the
case. The actual relationship is as follows:
IL
Current Relationship in a Wye Configuration - Since
the line current does not split as it leaves the phase
winding, the line current (IL) is equal to the phase
current (Iph). The relationship is as follows:
IPh
A
VPh
VL
IPh
IL
IPh
B
VL
I L = I ph
VL
IL
10 – Three Phase Theory
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10 – Three Phase Theory
Three Phase Wye Relationships
Power Relationship in a Wye Configuration - As discussed in AC concepts, there are three different power
relationships for AC circuits. Real, reactive and apparent power still applies for three phase circuits. There
are some minor differences as shown with the relationships below:
IL
P Real = 3  V L  I L  cos
IPh
A
VPh
VL
QReactive= 3  V L  I L  sin 
S Apparent = 3  V L  I L
IPh
IL
IPh
B
VL
VL
IL
10 – Three Phase Theory
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10 – Three Phase Theory
Three Phase Delta Relationships
Voltage Relationship in a Delta Configuration - Since the line voltage is connected across one winding just as
is the phase voltage, the line voltage (VL) is equal to the phase voltage (Vph). The actual relationship is as
follows:
V L = V ph
IL
Current Relationship in a Delta Configuration - Since
the line current is a combination of current from two
windings and the phase current only comes from one
winding, the line current (IL) is much larger than the
phase current (Iph). At first, you would think that the
line current is twice as large as the phase current, this
is not the case. The actual relationship is as follows:
I L = 3  I ph or
10 – Three Phase Theory
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IPh
IPh
VPh
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VL
VPh VL
IL
B
IPh
VPh
I
I ph = L
3
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VL
IL
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10 – Three Phase Theory
Three Phase Wye Relationships
Power Relationship in a Delta Configuration - As discussed in AC concepts, there are three different power
relationships for AC circuits. Real, reactive and apparent power still applies for three phase circuits. There
are some minor differences as shown with the relationships below:
P Real = 3  V L  I L  cos
QReactive= 3  V L  I L  sin 
S Apparent = 3  V L  I L
10 – Three Phase Theory
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10 – Three Phase Theory
Three Phase Generation
The generation of 3 AC and the associated relationships has been discussed. We now need to talk about the
3 loads and their associated relationships. Remember, without a load connected to the supply, there will be no
current flow. The voltage, current and power relationships that apply to the generation of 3 AC also applies to
3 loads. There are two different types of loads: balanced and unbalanced. We will only be discussing the
balanced loads.
In actual industrial applications there are several different configurations for loads:
Resistive, Inductive and Capacitive, as well as, many combinations of each. To better understand the basic
concepts for line and phase voltages and currents we will only use loads that are resistive.
Balanced 3 loads have identical resistances in each winding. The phase voltage Vph and the phase
current Iph are the same for each winding.
10 – Three Phase Theory
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10 – Three Phase Theory
Three Phase Generation
Once the phase voltage of the load Vph and phase current of the load Iph are established, the circuit can be
evaluated just as a single phase AC circuit. You can then evaluate the circuit just as we did in AC Theory.
IPh
VPh
RPh
10 – Three Phase Theory
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10 – Three Phase Theory
Three Phase Wye Connected Loads
IL
A
RPh
VPh
IPh
VL
Ph
R
IL
B
R Ph
VL
IPh
IPh
VL
IL
C
10 – Three Phase Theory
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10 – Three Phase Theory
Three Phase Delta Connected Loads
IL
A
IPh
VL
VPh
IPh
R
R Ph
Ph
VL VPh
IL
B
RPh
IPh
VL
IL
VPh
C
10 – Three Phase Theory
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10 – Three Phase Theory
Three Phase Calculation Example
R
R Ph
Ph
RPh
Given:
Find:
Supply VPH
= 120v
freq.
= 60Hz
Load RPH
= 10Ω
Load IPH
VL
IL
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10 – Three Phase Theory
1. First label the given information on the circuit, then what you need to find.
IL
Load RPH = 10Ω
Supply VPH = 120v
R
VL
R Ph
Ph
IPh Load
RPh
2. Next do the Supply calculation for VL.
3. Then do the Load calculation for VPh.
10 – Three Phase Theory
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Y

Supply V  V  3  120v  3  207.846v
L
Ph
Load V  V  207.846v
L
ph
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10 – Three Phase Theory
4. Now we can draw a single phase circuit that represents only one phase of the 3 phase delta load.
Iph Load
VPh Load = 207.846v
RPh
Load RPH
= 10Ω
5. Next do the single phase calculation.
Load Single PhaseCalculation I ph Load 
Vph
Load
Rph
Load

207.846v
 20.785A
10
6. Then do the IL calculation.
I L  I ph
Load
10 – Three Phase Theory
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 3  20.785A  3  36.001A
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10 – Three Phase Theory
Practice Exercises
R
R Ph
Ph
RPh
1. Given:
Vph 120v (Supply)
freq. = 60Hz
RPh = 12
10 – Three Phase Theory
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Find:
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Preal =
IL =
VL =
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10 – Three Phase Theory
Practice Exercises
Vph 208v (Supply)
freq. = 60Hz
RPh = 100
10 – Three Phase Theory
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Find:
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R
R Ph
2. Given:
Ph
RPh
Preal =
IL =
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10 – Three Phase Theory
Practice Exercises
VL = 480v
f = 60Hz
RPh Load = 20Ω
10 – Three Phase Theory
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R
R Ph
3. Given:
Ph
RPh
IPH Load =
VPH SupplyIL =
Preal =
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