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EEL 203EEL 203
Professor Bhim SinghProfessor Bhim Singh
Department of Electrical EngineeringDepartment of Electrical Engineering
Indian Institute of Technolo DelhiIndian Institute of Technolo DelhiHauz Khas, New DelhiHauz Khas, New Delhi--10016, India10016, India
. . . ,. . . ,[email protected]@gmail.com
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LectureLecture IIII
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3
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Learning GoalsLearning Goals
Be familiar with single phase system using R, R-L and R-
C loads.
, ,
power and apparent power.
What is power factor.
Phasor representation of voltage and current
What is complex impedance
Sin le- hase s stem with non-sinusoidal volta e source.
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What is Direct Current/Alternating CurrentWhat is Direct Current/Alternating Current
The electricity flowing in constant direction and/ or possessing voltage
with constant direction is known as direct current (DC). DC is a kind of
.
Certain sources of electricity like rotary electro-mechanical generatorsnaturally produces voltages alternating in polarity, reversing positive and
negative over the time. Either voltage switching polarity or current
switching polarity back and fourth , this kind of electricity is known as
alternating current (AC).
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Load
IL
Vdc
DC Supply System
DC Supply System Component:
o age or urren source.Load impedance (resistance, inductance or capacitance).
6
.
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AC Supply System
AC Supply System Component
Voltage or current source
Load (resistance, inductance, and Capacitance)
The components are connected in series or in parallel.
7
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Single-Phase Supply
The voltage source produces a sinusoidal voltage wave
=
Where Vrms value of the source voltage in volt and wt isthe an ular fre uenc of the sinusoidal function in
rms
(rad/sec)
w=2f; and f=1/T,
T is the cycle time period in seconds.
w is the su l fre uenc .
The peak value (max value) of the voltage is 2m rmsV v=T
8
0
( )rms
v v t dt T
=
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The current is also sinusoidal and is given as,.
where: Irms is the rms value of the current.
rms=
s e p ase-s e ween curren an vo age.
The rms current is calculated by the Ohms Law:
rmsrms
VI
Z
=
where: Z is the impedance
The impedances (in Ohms) are :
Resistance (R)
Inductive reactance L X wL=
9
Capacitive reactance cXwc
=
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Phase Representation of a Sinusoidal Current
A sinusoidal quantity is taken as an examplesinmi I t=
Length OP along the x-axis represents the maximum value of the
current
It is being rotated in the counter-clockwise direction at an
m
10
angular speed w, and takes up a position OA after some time t. Here
t =
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The vertical position of OA is plotted in the right hand side with
. .
Since OA is at angle with respect to x-axis.
The vertical projection of OA along y-axis is OC=AB= sinmi I t=
Which is instantaneous value of the current i at any time t.
IThe line OP can be taken as rms value ,then the
vertical projection of OA does not represent exactly the2
mrmsI =
,
2
11
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Phasor Representation of Voltage & Current
The volta e and current waveforms are iven b
and
As shown in Fig. voltage lags the current by an angle .
sin( )mi I t = +sinmv V t=
In phasor notation the voltage and current are represented by OP
and OQ.
a ema ca y e wo p asor can e represen e n po ar orm
as 0 0, cos sin
o
v V V i i I I jI = = + = = +
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Sing Purely resistive circuit (R only) le
i V
R
t
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The instantaneous value of the current through the circuit is given by,
sin sinm mVv
i t I t R R
= = =
rms value of current is givewnI
byV V
The
=2 2
In hasornotation
R= =
0 0V 0 (1 0); 0 (
1 0)V V j I I I j= = + = = +
an0
0
e mpe ance o e c rcu s o ane as,V 0V
= = =0I 0
I
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Sing Purely inductive circuit (L only)
iV
L
t
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For the Circuit , the current i, is obtained as,
mv=L =V sin 2 si
2=
n tdt
V
t V =
0 0 0int ,2 2
cos sin( 90 ) sin( 90 ) 2 sin( 90 )m
L
egratinV V
i tg t I wt I wL
tL
= = = =
0
0
0
V= 90
wL
I I=
= = = =
impedance of the circuit isThe
00
0
V V 0Z = 0 90
I I -90L
V j L jX L
jI = = = = + =
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Sing Purely capacitive Load (C only)
i
t
V
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m
, , ,
v = V s in 2 s in t , i i sd v
i= C ;d t
t V =
0
r m s v a lu e I i s
i= C 2 s in ( ) 2 c o s s in ( 0 )9md
V t C
T h e
V t I t d t = = +
09 0
1 / ( )
V I C V I
C
= = =
i m p e d a n c e
0 0 ; I 9 0
o
0
f
V V V j I
T
I j
h e
= = + = +=
t h e c i rc u i t i s
0
Z 9 0I C = =
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Sing Resistive-inductive Load (R-L Load)
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for the R-L series circuit is asThe voltage balance equation
2 si
v=Ri+Ld
n i= 2 sin( )
(1)
and current
t
v V t I t where =
2 sin . 2 sin( ) . 2 cosin
( )equation (1)
V t R I substituing
t L tI = +
sol iv above equations leads to
e ma nitude and hase an le of the curre
V=(Rcos + L.sin ).I and 0=(-Rsin + Lco
nt
s )
ng the
rom these uations the
I are derived
tan =( L/R) cos =(R/Z)
as
and sin =( L Z)/
and Z= 2 2
2 2
R ( )
V VI
L
= =
+
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t h a t t h e c u r re n t la g s t h e v o l t a g e b y a n a n g le .N o t e
0 0
1 1W =
2 s i
n
* 2 s i n ( )v id V t I t d
=
0
1
[ c o s c o s
t h e a b
]
o v e
( 2 )
s o l v in
V I d
= e u a t io n
W = V Ic o s
N o t e p o w e r i s o n ly c o n s u m e d i n r e s is ta n c e R ,
2W =
b u t n o t i n i n d u c t a n c e ,
I .
L s o ,
R
a v e r a g e p o w e r c o sa p p a r e n t p o w e r
P o w e r F a c to rV I
R R
= =
2 2 c o s
( )Z R L= = =
+
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Resistive-ca acitive circuit R-C Load
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v o lta g e b a la n c e e q u a t io n f o r th e R -C s e r ie s c i rc u i t isT h e
v = R i+ 2 s inC
c u r re n t i s
id t V t
T h e
=
i= 2 s in (
im p e d a n c e o f th e s e r ie s (R -C ) c i r c u i t
)
1
is ,
I t
T h e
+
cZ - = R - X
,
R jC
w h er e
=
2 2 1; ta n ( ) tacc
Z R X R
= + = =
0
1
0 0
n ( )
V V j
C R
+
2 2
(1 / ) Z R jV
CV
I
= =
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Complex Power, Volt-Amperes (VA) and Reactive Power
The Complex power is the product of the voltage and complex
0
, .
For the inductive circuit,
0
the current I (cosI = sin ) is lagging the voltage byj
*S=VI
an ang e .
The complex power is
Q=Im(S)=VIsthe active power
the reactive powerP=Re(S)=VIco
iss
inand
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Power Definitions under Sinusoidal Conditions
ideal signal phase system with a sinusoidal voltage
source and a linear (ressitive-inductive) load has
n
vo age an curren a are ana y ca y re
v(t)= 2 sin( ) a
presen e a
nd i(t)= 2
s,
i ( )s nV t I t
.
The instantaneouspower is given by the product of the
instantaneousvolta e and current that is
p(t)=v(t)i(t)=2VIsin( ) si
p(t)=VIcos -VIc
n(
s(2 )
)
o t
t t
it shows that the instantaneous power of the single-phase
system is not constant. it has an oscillating component at
twice the line frequency added to a DC value given by VIcos .
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Power Definitions under Sinusoidal Conditions
Decomposing the oscillating component and rearranging
the above e uation ield the followin e uation with two terms
[ ]p(t)=VIcos 1-cos(2 t) -VIsin sin(2 )t .
the unit of measurment in the international system is Watt(W)
,
has a peak va VIsin .lue to
, .
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Concept of Power Under Non-Sinusoidal
Supply Conditions
The concept of Power under non-sinusoidal conditions are not
unique.
Two sets of power definitions are normally used ; one in the
frequency domain established by Budeanu and the other in the
.
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Power definitions by Budeanu
T e e n t ons are esta s e n t e requency oma n. So t eycan be applied only in steady-state analysis.
steady-state, its voltage and current waveforms can be decomposedin Fourier series. Then the corresponding phasor for each
armon c componen can e e erm ne , an o ow ng e n ons
of power can be derived.
A arent ower -
S=VI
It is identical to the apparent power given with sinusoidal
conditions. But the difference is that V and I are the rms values ofgeneric, periodic voltage and current waveforms, which are
calculated as
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Power definitions by Budeanu
2 2
10
( )1
n
n
V v t dt V T
=
= =
2 2
10
1 ( )n
T
n I i t dt I T
== =
Here Vn and In correspond to the nth harmonic components of the
Fourier series, and T is the period of the fundamental component.The displacement angle of each pair of the nth harmonic voltage
and current components is represented by n.
1 1
cosn n n nn n
Active PowerP P V I
= =
= =
1 1
Re Power Q= sinn n n n
n n
act Q Vive I = =
=
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Power definitions by Budeanu
- ,apparent power can not characterize satisfactorily the issue of power
quality.
It is due to the fact that above defined reactive power does not
include cross product between voltage and current harmonics at
.
It is noted that neither the active power nor the reactive power
includes the products of harmonic components at differentfrequency.
Further the algebraic sum of harmonic reactive power components
,
several displacement factors n.
The loss of power quality under non-sinusoidal conditions can be
better characterized by another power definition, the distortion
factor
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Power definitions by Budeanu
2 2 2 2 D S P Q=
The power defined from above equations are well known and
widely used in the circuit analysis of circuits operating under non-
.
The active power defined above represents the average value of
the instantaneous active power or the average of energy transferbetween two electric subsystems.
In contrast the reactive power and apparent power are just
.
Another limitations of this definitions is that a common instrument
used for power measurement based on the power definitions in the
frequency domain can not indicate loss of power quality in practical
cases.
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Power Tetrahedron and Distortion Factor
-conditions, graphical power representations is given on the three-
dimensional reference frame, instead of a power triangle as
described earlier .Fig shows the new graphical power representation
that is well known as a power tetrahedron
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Power Tetrahedron and Distortion Factor
PQ
1 1
releation between the apparent power S and the complex power S
PQ n n
n n
The
S P jQ P j Q= =
= + = +
2 2 2 2 2
power factor is defined as the ratio of the active power with respect
PQS V
The
I P Q D S D
= = + + = +
o e apparen power, e s equa o cos n e power e ra e ron.
cosP
S
= =
p
cos =PQ
SDisplacement factor
factor cos =
relation is valid
PQ
SDistortion
The following
cos cos .cosP
S
= = =
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Power Definitions by Fryze
voltage and current. the basic equations according to the Fryzes
approach are given as
w0 0
power1 1
P ( ) ( ) ( )w w
p tActive dt v t i t dt V I VI T T
= = = =
are the active voltage and current as defined below. the rms value of
voltage and c
w w
urrent are calculated as,
2 2
0 1
1( )
T
n
n
V v t dt V T
=
= =
2 2
10
1( )
T
n
n
t I I i t d T
=
= =
wTogether with the active power P , these rms values from the basis of
theFryze's approac .h
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Power Definitions by Fryze
s
w w
power P
P P=
pparant VI =
=S
2 2
q
P VI
Re power ps w q q
active P P V I VI = = =
q q
2
V and I are the reactive voltage and current as defined below
Reactive ower Factor 1
where
=
wvoltage VActive wand active current I :
. I .V V I = =
q qRe voltage V and reactive current I :
. I .
active
V V I = =
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Fryze defined reactive power as comprising all the portions of
voltage and current, which does not contribute to the active power .
Fr ze verified that the active ower factor reaches its maximum
(=1) if and only if the instantaneous current is proportional to the
, .
However under non-sinusoidal conditions, the fact of having
current proportional to the voltage does not ensure an optimal
power flow from the electromechanical point of view.
The above set of definitions does not need any decompositions of
,
still requires the calculations of rms values of voltage and current.
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References
1. I Mckenzie Smith, HUGHES Electrical Technology, VII edition, Pearson Education,
Asia,2001.
2. Brian Moore John Dona h Electrical machines Basic rinci les series Pitman
1988.
3. McLaren, Peter "Elementary Electric Power and Machines" Ellis Horwood.(1984).4. I .J. Nagrath, Basic Electrical Engineering, Tenth Reprint, Tata Mcgraw-Hill
u s ng o. ., .
5. A. Sudhakar and S. P. Shyammohan, Circuits And Networks, Analysis and Synthesis
TMH Publishing Co. Ltd, New Delhi .
6. www.n tel.ac.in.