02. 1 Phase Circuits (37)

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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

2

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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.

4

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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

13

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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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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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- ,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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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.

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02. 1 Phase Circuits (37)