AC Power Analysis 1-Phase

Prepared for

Electrical Engineering Laboratory II, ECE L302 by

Mohammed Muthalib Center for Electric Power Engineering

Drexel University (http://power.ece.drexel.edu)

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Outline

Complex numbers

AC voltage and current

Power Instantaneous, Average

RMS

Power Real, Reactive, Complex

Power factor

Power factor Correction

Summary

Power lab safety

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Complex Numbers

The complex plane is used to represent numbers on a set of orthogonal axis (real and imaginary)

Facilitates addition and subtraction of complex numbers

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( )

( )

j

2 2

1

z j z e

z x y complex modulus/ magnitude

ytan complex argument/ phasex

where j is the imaginary unit j

y

1

x θ

−

= + =

= +

θ =

= −

j

Using Euler's formulaz e z cos j z sin

jx y

θ = θ + θ

= +

Complex Numbers

For two complex numbers z1 and z2:

Multiplication and Division are (easily) done in polar coordinates

Addition and Subtraction are (easily) done in rectangular coordinates

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1 21 1 2

j j1 1 2 22z j z e ,x y z z ex jyθ θ= + = = + =

( ) ( )1 2 1 2j j111 2 1 2

2 2

zzz z z z e , ez z

θ +θ θ −θ= =

( ) ( )( ) ( )

1 2 1 2

1 2 2

1 2

2 11

z z j

z

x x y y

x xz j y y

+ = + + +

− = − + −

AC Voltage and Current

Power is characterized through two quantities; voltage and current

AC voltage and current Sinusoidal time varying waveforms

Phasor representation (time invariant)

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( ) ( )( ) ( )

peak V

peak I

V

I

v t V sin t

i t I sin t

voltage phase anglecurrent phase angle

= ω + θ

= ω + θ

θ −

θ −

V

I

jV

jI

peak

V V e

I I e

Where V and I are rms values

V (amplitud ) V

V

I

e 2

θ

θ

= ∠θ =

= ∠θ =

=

Voltage and current phasors (polar coordinates)

AC Voltage and Current

Voltage and current waveforms

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Voltage and current phasors (polar coordinates)

Power

Instantaneous power: Product of v(t) and i(t)

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( ) ( )( ) ( )( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( )

V

I

V I

V I

V I V I

v t 2 V sin t

i t 2 I sin t

p t v t * i t

2 V I sin t sin tuse trigonometric identity2sin usin v cos u v cos u vu t , v tp t V I cos V I cos 2 t

= ω + θ

= ω + θ

=

= ω + θ ω + θ

= − − +

= ω + θ = ω + θ

= θ − θ − ω + θ + θ

time invariant (constant)

time varying with frequency 2ω

Power

Instantaneous power

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( ) ( ) ( )p t v t * i t=

Power

Average power: Average of the instantaneous power DC equivalent

|V| and |I| are rms quantities, what does that mean?

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( )

( ) ( ){ }

( ) ( ){ }

( )

T

AVG0T

V I V I0

2

V I V I0

AVG V I

1P p t dtT

1 V I cos V I cos 2 t dtT

1 V I cos V I cos 2 t d t2

P V I cos

π

=

= θ − θ − ω + θ + θ

= θ − θ − ω + θ + θ ωπ

= θ − θ

∫

∫

∫ Time invariant portion of instantaneous power

RMS quantities

The root-mean-square (rms) value of a periodic current or voltage is a dc equivalent that will deliver the same average power (PAVG) to a resistance R.

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( )T 2

2 rmsAVG rms

0

V1P p t dt RIT R

= = =∫T2

2rms

0

T2

rms0

rms voltage:

V 1 1 v dtR T R

1V v dtT

=

=

∫

∫

T2 2rms

0

T2

rms0

rms current:

1RI R i dtT

1I i dtT

=

=

∫

∫

RMS quantities

Vrms and Vpeak √2|V|=√2Vrms = Vpeak

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( )

( )( )

peak

22

0

22 2peak

0

22peak

0

2peak

peak

v t V sin t

1V v t d t2

1 V sin t d t2

1 1 cos 2 tV d t2 2

1 V2

V

2

π

π

π

= ω

= ωπ

= ω ωπ

− ω= ω

π

=

=

∫

∫

∫

T22rms

0

T2

rms0

rms voltage:

V 1 1 v dtR T R

1V v dtT

=

=

∫

∫

T2 2rms

0

T2

rms0

rms current:

1RI R i dtT

1I i dtT

=

=

∫

∫

Power

Real power: Real power = Average power

Denoted P with units of Watts Power absorbed by the resistive components of the system

Reactive power: Denoted Q with units of VAR Power absorbed by the reactive components of the system Example, Inductance and Capacitance

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( )AVG V IP P V I cos= = θ − θ

( )V IQ V I sin= θ − θ

Power

Complex power: Is a representation of power in a complex vector space Denoted S with units of VA

|S| - apparent power

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2 2S P Q V I= + =

( )( ) ( )

( ) ( )

V I

*

jV I

V I V I

S VI *denotes complex conjugate

S V I V I eusing Euler's formulaS V I cos j V I sin

P jQ

θ −θ

=

= ∠ θ − θ =

= θ − θ + θ − θ

= +

Power Factor

Power Factor: Is a measure of how effectively a system component draws real

power. It is the ratio between real power and apparent power

PF is presented as a real number between 0 and 1 with a leading/lagging denotation for the PF angle o Lagging - current angle lags the voltage angle, θV>θI

o Leading - current angle leads the voltage angle, θV<θI

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( )V IPPF cosS

= = θ − θ

Power Factor

Power Factor:

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( )V IPPF cosS

= = θ − θ

Power Factor

Power Factor:

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V

I

4530

PF cos150.9659lagging

θ = °

θ = °= °=

Power Factor Correction

PF=1 indicates that all the power consumed in the system is real power. A load with PF=1 emulates a resister Reactive power draw is zero

Some loads (electronic devices etc.) have low power factors, which demand more reactive power from the grid.

To compensate for these loads power factor correction (PFC) is necessary (to bring the PF towards 1.0)

PFC is done by injecting reactive power into the system

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Power Factor Correction

PFC by injecting reactive power

Reactive power injection is done by adding capacitive and inductive loads to the system. Capacitive – supply reactive power Inductive – consume reactive power

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Power Factor Correction

Complex power circle Higher power factor yields more real power for the same apparent

power o More desirable

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Power Factor Correction

Apparent power is the same in both cases

|S|=1200 VA

Lower PF reduces the amount of available real power

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Power Factor Correction

Injecting reactive power

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Reactive power Elements

Capacitors and Inductors

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j2

C

j0

CjC 2

j2

C C

2

1 1Z ej C C

V eViZ 1 e

C

V Ce

Q V I sin 02

V C

π−

π−

π

= =ω ω

= =

ω

= ω

π = −

= − ω

j2

Lj0

LjL 2

j2

C L

2

Z j L Le

V eViZ

LeV

eL

Q V I sin 02

VL

π

π

π−

= ω = ω

= =ω

=ω

π = +

=ω

Capacitive load supplies reactive power

Inductive load consumes reactive power

Summary

Application to a simple circuit

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Load side quantities

( )

( )

( )

1 LV I

L

L L V I

L L V I

Power Factor:

QPF cos cos tanP

Real Power:P V I cos WattsReactive Power:Q V I sin VA

− = θ − θ =

= θ − θ

= θ − θ

Lab Experiment

Simulating a 2-bus power system Measure voltages and currents; calculate power Observe effects of loading

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Lab Experiment

Test system with 3 load types Resistive (2 – 20 bulbs) RL (20 bulbs + 1-5 inductors) RC (20 bulbs + 1-4 capacitors)

Power Factor Correction Set RL load Determine necessary caps to correct load PF

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R-Load RL-Load RC-Load

Lab Experiment

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Lab safety

You will be dealing with voltages and currents of 120V and 25A

Adherence to safety and conduct guidelines is imperative Please read the Power Lab Safety document Please watch lab safety video

http://power.ece.drexel.edu/videos/

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Thank you

Questions are most welcome

"He who asks a question is (might look like) a fool for five minutes; he who does not ask a question remains a fool forever.“ – Chinese proverb