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Ch 7. Alternating Current Circuits

• Lecture outcomes (what you are supposed to learn):

– Root Mean Square (tehollisarvot)

– Phase and Phase shift (vaihekulma ja vaihesiirto)

– Phasors as vectors and complex numbers (osoitin)

– Real, Reactive, Complex power (Pätö- , Lois-, Näennäisteho)

– Power factor (tehokerroin)

– Electric energy

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• Electricity discovered around 600 BC• Extensive use started at the end of 19th century

• Batteries produce DC-voltage• DC generator or Dynamo at beginning of 19th century

• DC power system for lighting 1882– Low voltage (100 V)– Large losses and voltage drop in wires

• AC motor and generator + transformer• AC power system with high voltage

Introduction

Courtesy of CRC Press/Taylor & Francis Group

Courtesy of CRC Press/Taylor & Francis Group

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• Voltage level can be adjusted with transformers

• High-voltage allow for long transmission lines– Low losses– Low wire cross-section

• AC system produce rotating field necessary to spin motors

• Is the battle over? Rectifiers, inverters, HVDC, local DC

Why AC won the battle ?

Courtesy of CRC Press/Taylor & Francis Group

Where dc-system won?

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• Waveform

• Frequency (taajuus):

• Root Mean Square (tehollisarvo)– Makes it possible to quantify distorted waves

Basic quantities

maxsinv V tϖ<

2f

ϖο

<

2

0

1T

rmsV v dt

T< ⟩

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• Waveform

• Resistive load:

• Inductive load:

Phase shift

maxsinv V tϖ<

max sinV

i tR

ϖ<

max

0

1cos

t Vi vdt t

L Lϖ

ϖ< < ,⟩

Reactance

LL Xϖ <

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• Waveform

• Capacitive load:

• Complex load:

Phase shift

maxsinv V tϖ<

maxcos

dvi C CV t

dtϖ ϖ< < Reactance

1C

XCϖ

<

∋ (max

sini I tϖ π< ,

What is theunit of

reactance

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• Graphical representation with mathematical basis

• Give quick information on magnitude and phase shifts

• Vector length proportional to rms value

• Angle with respect to x-axis equals to phase shift

Concept of Phasors

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• Resistive load

• Inductive load

• Capacitive load

• Complex load

Concept of Pahsors

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Current lagging voltage Current leading voltage

Lagging and leading phase shifts

∋ (max

sini I tϖ π< ,max

max

cos

sin( )2

i CV t

CV t

ϖ ϖοϖ ϖ

<

< ∗

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• Phasors are good visuals but not convenient to compute

• Vectors can be represented as complex numbers

• Calculus with complex number is handy.

Complex representation of Phasors

0V V< ∉ ↓

I I π< ∉ , ↓

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• Real component

• Imaginary component

• Conjugate

Complex numbers calculus

cos sinA A A j X jYπ π π < ∉ ↓ < ∗ < ∗

cosA Xπ <

sinA Yπ <

*A X jY A π< , < ∉ , ↓

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• Basic impedances

– Resistance

– Inductance

– Capacitance

• Complex– Series and parallel connections of basic impedances

Impedances

00

0R R

V VR R

I I∉

< < < ∉∉

090

90L LL L

V VX X

I I∉

< < < ∉ ↓∉ , ↓

090

90C CC C

V VX X

I I∉

< < < ∉ , ↓∉ ↓

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• Series connected impedances are additive

• Parallel connected impedances are not additive• Parallel connected admittances are additive

Admittance1

YZ

<

1 10G

R R< < ∉ ↓

1 190

LL L

BX X

< < ∉ , ↓

1 190

CC C

BX X

< < ∉ ↓

What is theunit of

admittance

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• Instantaneous power

• For a complex load

Electric power

p vi<

∋ ( ∋ (

∋ ( ∋ (Ζ ∴max max

sin sin

cos cos 2

p V t I t

VI t

ϖ ϖ π

π ϖ π

< , < , ,

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Instantaneous power

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• Energy is the time integral of the power

– Constant power

– Discrete power

– Variable power

Electric energy

0

T

E pdt< ⟩

E pT<

i ii

E pT<

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• The power that produces energy is the average of

• This is called Active or Real power

• The Complex or Apparent power is defined as

Active, Reactive and Complex Power

p vi<

∋ (2

0

12

cos( )

P pd t

VI

ο

ϖο

π

<

<

⟩

*S V I≠Why not?

It is a 100-years oldconvention

S V I≠

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• Part of the instantaneous power does not produce energy, it is calledReactive or Imaginary power

• In case of sinusoidal quantities the reactive power is well defined but notobvious in case of distorted quantities !

Active, Reactive and Complex Power

∋ (sinQ VI π≠

∋ ( ∋ (* cos sinS V I VI jVIπ π< < ∗

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• The power factor is defined as the ratio of real to apparent power

• In an electric circuit it can be calculated as

Power factor

∋ (cosP

pfS

π< <

2 2

R Rpf

Z X R< <

∗

2 2

P Ppf

S P Q< <

∗

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• Increases losses in the transmission line

• Reduces the sparse capacity of transmission line

• Reduces the voltage across the load

• Need for power factor correction

Problems with power factor

( )s

wire wire load

VI

R j X X<

∗ ∗

2loss wire

P I R<

2 2

1

sload

wire wire

L L

VV

R X

X X

<∑ ⌡ ∑ ⌡ ∗ ∗

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• Sinusoidal voltage and current represented by phasors

• Phasor has amplitude and phase complex number

• The nature of load (resistive, reactive) produces phase shift betweenvoltage and current power factor

• Real power produces work, reactive power is a parasitic effect thatincreases the line losses

• Real mean square useful measure of AC-quantities even under distortion.

Summary of the lecture