c 2014–2016 Y. Ding
ELEC3300: Electrical Energy Conversion & Utilisation
Three Phase Systems
Advantages
• Single phase systems requires both forward and
return conductors to have current capacity
load current.
Wye
• Line current equal to Phase current:
• Balanced three phase systems can deliver same
power to three single phase loads, with only half
the number of conductors.
IL = IP
• Line-to-line voltage has magnitude
magnitude of phase voltage:
L|
=
p
3|VP |,
\VL
3 times
• Line losses potentially reduced by 50%
• Reduced capital, costs of transmission and distribution
\VP = +30
L
E
|VL
p
• Better voltage control/regulation
Delta
• Total instantaneous power delivered nearly constant when balanced
• Line-to-line voltage equal to Phase voltage:
L
= VP
• Line current has magnitude
tude of phase current:
|IL | =
p
3|IP |,
\IL
p
• Single phase systems has constant component
and double frequency component, causing:
PL
VL
3 times magni-
– Variation in mechanical power to generator
and shaft torque.
\IP =
Wye $ Delta
– Shaft vibration and noise, and even shaft
failure
Per Unit (p.u.)
General cases
M
1
ZY = Z
3
30
Power (both configurations)
✓P is the phase angle of Z in each phase.
• Real power:
SA
PT = 3PP = 3|VP ||IP | cos(✓P )
p
p
= 3PL = 3|VL L ||IL | cos(✓P )
per-unit quantity =
actual quantity
base value quantity
• Ohm’s law still applies:
Ibase =
Sbase
Vbase
Zbase =
V2
Vbase
= base
Ibase
Sbase
• p.u. electrical quantities:
Sper-unit =
P + jQ
= Pper-unit + jQper-unit
Sbase
V\ v
|V |
=
\
Vbase
Vbase
I\ i
Iper-unit =
Ibase
Z\ z
Zper-unit =
Zbase
• Reactive power:
Vper-unit =
QT = 3QP = 3|VP ||IP | sin(✓P )
p
p
= 3QL = 3|VL L ||IL | sin(✓P )
• Complex power:
Three Phase Systems
ST = 3SP = 3|VP ||IP |
p
p
= 3SL = 3|VL L ||IL |
• Base power:
ST, base = 3SP, base
• Instantaneous power:
• Base voltage:
3
PT = Vm Im cos(✓P )
2
VL, base =
1
p
3VP, base
v
c 2014–2016 Y. Ding
ELEC3300: Electrical Energy Conversion & Utilisation
Efficiency
⌘=
Open Circuit Test
Poutput
Poutput
⇥ 100% =
⇥ 100%
Pinput
Poutput + ⌃(losses)
• Equivalent impedance:
Rc (jXm )
Rc + jXm
ZOC = Zm
where ⌃(losses) =:
• core losses + copper losses
• Equivalent magnetising impedance: |Zm | =
VOC
IOC
V2
• Equivalent core resistance: Rc = OC
POC
• Equivalent magnetising reactance:
• core losses + (I12 R1 + I22 R2 )
• core losses + (I12 Re1 )
• core losses + (I22 Re2 )
1
E
Practical considerations
Xm = q
• For constant power factor load, maximum efficiency achieved if total copper losses = core
losses
1
1
Rc2
|Zm |2
• Alternative: QOC = VOC IOC sin ✓,
✓
◆
V2
POC
1
✓ = cos
! Xm = OC
VOC IOC
QOC
PL
• High voltage transformers operating continuously near rated capacity, designed for maximum efficiency at or near rated load
• Rc1 = a2 Rc2 , Xm1 = a2 Xm2
• Distribution class transformers always connected but with significant load variations designed for maximum efficiency at or near aver- Short Circuit Test
age load
• Series impedance of transformer, referred to HV
side:
|VSC |
Equivalent circuit
|ZE1 | =
|ISC |
jX1
jX2
R2 I 2
N1 :N2
I10
I 1 R1
Ic
V1
+
+
+
E1
E2
M
Ie
Im
jXm1
Rc1
V2
Load
SA
Cantilever (approximate equivalent) circuit
• Equivalent series resistance, referred to HV side:
Referred to primary
I2
a
jXe1
Re1
I1
+
V1 Rc1
jXm1
RE1 =
+
• Equivalent series reactance, referred to HV side:
q
2 = X + a2 X
XE1 = |ZE1 |2 RE1
1
2
aV2
• Hence:
Referred to secondary
aI1
Re2
jXe2
+
V1
a
R 1 = a 2 R2 =
I2
+
Rc2
PSC
= R1 + a 2 R 2
2
ISC
RE1
,
2
X1 = a 2 X 2 =
Three Phase Transformers
jXm2 V2
3 ⇥ single-phase transformers
• Easy to replace failed units
3
XE1
2
c 2014–2016 Y. Ding
ELEC3300: Electrical Energy Conversion & Utilisation
Blocked Rotor Test
• Rotor copper losses – power dissipated in the
rotor windings:
Rotor blocked, restricting it from rotating, i.e. !m =
0 and s = 1. Machine is like a short-circuited
transformer, and e↵ectively bypasses the magnetising branch.
A reduced three phase voltage is applied to the
stator so the rated current flows. When the machine
operates near rated load (s = 0), currents in rotor
have low frequency. Ths frequency of voltage applied
in the blocked rotor test is reduced, recommended at
25% rated conditions by IEEE.
RCL = Pcu2 = 3I22 R2
• Power available for conversion from electrical to
mechanical form – developed power:
Pconv = Pdev = Pag RCL
Rs
1 s
= 3I22
3I22 R2 = 3I22 R2
s
s
• Output power available to shaft of motor is
found by subtracting mechanical losses from developed power. This includes frictional losses
and rotational losses: Pout = Pdev Pmech
Pout
• Efficiency: ⌘ =
⇥ 100%
Pin
•
•
PL
•
Distribution of Power
with R1 measured by the DC test
Total impedance of windings measured by the
test:
Vbl
Zbl = p
3Ibl
Block rotor reactance, adjusted for ratio between operational frequency and test frequency:
q
frated
2
Xbl =
⇥ Zbl2 Rbl
ftest
Assuming stator and rotor leakage reactances
equal:
X1 = X2 = 0.5Xbl
Shunt magnetising reactance:
SCL
Pin
Xm = Xnl
X1
Performance Analysis
Power
SA
• Input power to the induction motor, in terms of
single-phase parameters:
Pdev
Shaft
Just below the synchronous speed where speed falls
almost linearly as the torque increases. The rated
torque is, by definition, 1 pu at rated speed, and the
efficiency is maximal at around rated speed.
• Mechanical torque at motor shaft: Tmech =
Pout
!m
• Electrical torque:
Telec =
Pin = 3P1 = 3V1 I1 cos(✓1 )
(1 s)Pag
Pag
Pdev
=
=
!m
(1 s)!s
!s
Maximum Torque
Firstly, losses occur in the stator side, dissipated
as SCL and core losses.
• Stator copper losses – power dissipated in the
stator windings:
SCL = Pcu1 =
Pag
Pcore
RCL
Torque
M
•
E
• Blocked rotor resistance, obtained from input
power required for the test. This also represents
combined stator and rotor resistances:
Pbl
Rbl = 2 = R1 + R2 ! R2 = Rbl R1
3Ibl
R2
smax = p 2
R1 + (X1 + X2 )2
Te,max =
3I12 R1
h
3V12
i
p
R1 + R12 + (X1 + X2 )2
2!s
• Power passed to rotor across air-gap, maximum
power that can be used by the rotor for torque
Speed/Torque Control
production, or power absorbed by equivalent ro• Variation of rotor resistance: external varitor resistance Rss :
able resistance in series with each phase of the
Rs
Pag = Pin SCL Pcore = 3I22
rotor winding. s / R2 . Reduces efficiency.
s
• Double cage rotor: inner and outer cage
The power transferred across the air gap is furpresent. At low speed, R2 dominated by high
ther dissipated as RCL, and what remains is the
resistance of outer cage, at near synchronous
developed power.
speed, rotor resistance reduces.
6
Pmech