Three-Phase Circuits

1
Three-Phase Circuits

• In three-phase circuits the 3 voltages sources
  are 120 apart
• Polyphase generation and transmission of
  electricity is more advantageous and economical
• (1) three-phase instantaneous power is constant
  over time
• (2) single-phase line losses are 50 greater
  than three-phase losses (for the same load power,
  voltage, pf), i.e., PSingle3/2PThree

2
Balanced System

• A balanced system is one in which the 3
  sinusoidal voltages have the same magnitude and
  frequency, and each is 120 out-of-phase with the
  other two

3
Three-Phase Voltages

• Balanced If
• VanVrms ?0 VbnVrms ?-120 VcnVrms ?-240

4
Balanced Circuit

• A balanced three-phase circuit is one in which
  the loads are such that the currents produced by
  the voltages are also balanced

5
Three-Phase Instantaneous Power

• Recall that the instantaneous power is for one
  phase is p(t) i(t) v(t)
• The total instantaneous power for three phases is
• the instantaneous power is a constant over time!

6
Three-Phase Connections

• Standard notation specifies the phase sequence as
  abc (called positive phase sequence) where
• VanVp ?0 VbnVp ?-120 VcnVp ?-240
• VbnVan ?-120 VcnVan ?-240
• Vbn lags Van by 120
• where Vp is the phase voltage which is the
  magnitude of the phasor voltage from the neutral
  to any line
• The balanced voltage set means that
• Van Vbn Vcn 0

7
Wye (Y) Connected Load
Load
a
a
ZY
b
b
ZY
ZY
ZY
c
ZY
ZY
c
n
n
Load
8
Delta (?) Connected Load
a
a
Z?
Z?
Z?
b
Z?
b
Z?
Z?
c
c
Load
Load
9
Source/Load Connections

• The equivalent load may be in an equivalent delta
  (?) or wye (Y) configuration
• the delta load configuration has no neutral line
• the current is zero in the neutral line for a
  balanced wye load configuration
• Source/Load connections can be connected in any
  of the possible combinations
• Ysource-Yload ?source-Yload
• Ysource-?load ?source- ?load
• Analysis prefers the first connection (Y-Y)

10
Balanced Wye-Wye Connection
a
ZY
Van

b
ZY

Vbn
c
ZY

Vcn
n
Load
11
Balanced Wye-Wye Connection

• The line-to-line voltages (or line voltages) are
• Vab VP?3 ?30
• Vbc Vab ?-120
• Vca Vab ?-240
• where VP is the magnitude of the phasor voltage
  between any two lines
• The magnitude of the line voltage is
• VL VP?3

12
Balanced Wye-Wye Connection

• The line current for phase a is
• Ia Van / ZY Vp ?0 / ZY
• In a Y-Y connection the magnitude of the line
  current (IL) equals the magnitude of the load
  current (IY)
• The neutral current In is
• In Ia Ib Ic 0
• Balanced Y-Y three-phase ac circuits may be
  analyzed on a per phase basis

13
Balanced Wye-Delta Connection
Van
a

Z?
Vbn
b

Z?
n
Z?
Vcn
c

Load
14
Three-Phase Power Relationships

• The real and reactive power per phase are
• The total real, reactive, and complex powers are