Thai Nguyen University
Thai Nguyen University of Technology
Faculty of International Training
ECEN-415
Power System Analysis and Design
Chapter 2
Fundamentals of Electric Power Systems
Lecturer:
Nguyen Minh Y, Ph.D.
Contents
1.
2.
3.
4.
5.
2
Phasors
Single-phase AC circuits
Complex power
Balanced three-phase circuits
Per-unit analysis
Department of Electrical Engineering taught in English
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2.1 Phasors
 Sinusoidal voltage
 Amplitude
v (t )  Vmax cos(t   )
 Phase angle
 Effective value
 Root-mean-square (RMS)
 Euler’s identity
V 
Vmax
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e  j  cos   j sin 
 Other way to represent
sinusoidal voltage
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v(t )  Re[Vmaxe j (t  ) ]
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 Phasors representation
 Time-domain
v(t )  Vmax cos(t   )  v(t )
 Re[ 2Ve j (t  ) ]
 Phasors
 Complex number
V  V 
 Rectangular form
 Polar form
 Exponential form
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 Example
 Find the phasor representation of voltage
v(t )  169.7 cos(t  60o ), V
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2.2 Single-phase AC Circuits
 Voltage
v(t )  2V cos(t   )
 Current
i (t )  2 I cos(t   )
 Power
 Instantaneous power
p(t )  v(t )i (t )
 VI cos(t   ) cos(t   )
 VI cos      VI cos  cos(2t     )
Sinusoidal with 2ω
Constant
 V I cos  [1  cos 2(t   )]  V I sin  sin 2(t   )
pR ( t )
 Power angle
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pX (t )
  
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 Active power
 First term of instantaneous power
pR (t )  V I cos   V I cos  cos 2(t   )]
 Define
 The average power in a cycle
P   p  t dt  V I cos 
T
0
 Reactive power
 Second term of instantaneous power
pX (t )  V I sin  sin 2(t   )
 Define
 The amplitude
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Q  V I sin 
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 Power factor
PF  cos  
 cos    
 Apparent power
S  VI
 Example
 The voltage is given by v(t) = 141.4 cos(t) is applied to a load
consisting of a 10  resistor in parallel with an inductive
reactance XL = L = 3.77 . Calculate the instantaneous power
absorbed by the resistor and by the inductor. Also calculate the
real and reactive power absorbed by the load, and the power
factor.
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S  P  jQ
 Complex power
 VI *
 Amplitude
S  P 2  Q 2  VI
 Phase angle
  
V
v

I
i
S
Q

P
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 Example
 The circuit has three loads. The voltage applied to the circuit is
200 V; three loads are Z1 = 60 + j0 Ω, Z2 = 6 + j12 Ω, Z3 = 30 –
j30 Ω. Find the power absorbed by each load and the total
complex power.
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 Problem
Two loads Z1 = 100 + j0 Ω and Z2 = 10 + j20 Ω are connected
across a 200-V rms, 60 Hz source as in the below figure.
(a) Find the total real and reactive power, the power factor at the
source, and the total current.
(b) Find the capacitance of the capacitor connected across the
loads to improve the overall power factor to 0.8 lagging.
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2.4 Balanced Three-phase Circuits
 Three-phase voltage sources
 The same amplitude
 The phase angle is 1200 apart
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 Y-connected loads
Van  V p 0o
Vbn  V p   120o
Vcn  V p   240o
Vab  3 V p 30o
Vbc  3 V p   90o
Vca  3 V p 150o
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Ia 
Van
 Ip  
Zp
Ib 
Vbn
 I p   120o  
Zp
Ic 
Vcn
 I p   240o  
Zp
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 Delta-connected load
I ab  I p 0o
I bc  I p   120o
I ca  I p   240o
I a  I ab  I ca  3 I p   30o
I b  I bc  I ab  3 I p   150o
I c  I ca  I bc  3 I p 90o
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 Equivalent single-line circuits
 Three-phase power
p3  vania  vbnib  vcnic
 3 V p I p cos(   )  3 V p I p cos 
 Complex power
S3  P3  jQ3  3 Vp I p [cos  jsin  ]
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 Example
The line feeds two balanced three-phase loads that are connected in parallel.
The first load is Y-connected and has an impedance of (30 + j40) Ω per phase.
The second load is -connected and has an impedance of (60 – j45) Ω. The
line is energized at the sending end from a three-phase balanced supply of line
voltage 207.85 V. Taking the phase voltage Va as reference, determine:
a) The current, real power, and reactive power drawn from the supply.
b) The line voltage at the combined loads.
c) The current per phase in each load.
d) The total real and reactive powers in each load and the line.
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Per-unit Analysis
 Per-unit quantity
Quantity in p.u =
actual quantity
base value of quantity
 Choose SB and VB
S
IB  B
3VB
S pu 
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(VB )2 (kVB )2
ZB 

SB
MVAB
S
V
I
Z
; Vpu  ; I pu  ; Z pu 
SB
VB
IB
ZB
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 Nameplate
 Resistance, reactance and impedance are usually given in term
of per unit.
 Per Unit
 Base voltage, base current and base impedance
Vbase  Vrated
I base  I rated
Z base
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Vbase Vrated


I base I rated
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Assignment
 Homework 1 is due in the next class.
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