electricity - alternating current · summary 1 alternating current 2 fresnel diagram 3 complex...

LITTORAL CÔTE D’OPALE

ElectricityAlternating Current

Mathieu Bardoux

IUT du Littoral Côte d’OpaleDépartement Génie Thermique et Énergie

First year

Summary

1 Alternating current

2 Fresnel diagram

3 Complex numbers : a few reminders

4 Complex representation

5 Power

Mathieu Bardoux (IUTLCO GTE) Electricity First year 2 / 34

Alternating current

Summary

1 Alternating currentDefinitionsSummation of signals

2 Fresnel diagram

3 Complex numbers : a few remindersCartesian formTrigonometric form

4 Complex representationDefinitionImpedance

5 PowerComplex powerPower factorBoucherot’s method


Alternating current Definitions

Alternating current

Definition

Alternating current (AC) is an electric current which periodicallyreverses direction, in contrast to direct current (DC) which flows onlyin one direction.

Most often, the alternating current is sinusoidal :

i(t) = I0 cos(ωt +ϕ)

I I0 stands for amplitudeI ω represents angular frequency : ω = 2π · fI ϕ is called phase.



Alternating current

t

i(t)

I0

ϕω

T = 2πω



Alternating voltage

Definition

Alternating voltage is a voltage which periodically reverses direction.

Most of the time, it is a sinusoid :

u(t) = U0 · cos(ωt +ψ)

t

u(t)

U0

ψω

T = 2πω



Some additional definitions

Period : the duration of time of one cycle in a repeating event.Unit : second (s).

Frequency : the number of occurrences of a repeating event per unitof time. Reciprocal of the period. Unit : Hertz (Hz



Root mean square values

Root mean square current

Equal to the direct current that would produce, for the same time, inthe same pure resistance, the same amount of heat.

Irms =I0√2

Root mean square voltage

Equal to the direct voltage that would produce, for the same time, inthe same pure resistance, the same amount of heat.

Urms =U0√

2


Alternating current Summation of signals

Summation of signals : ϕ1 , ϕ2

i1(t) = I1 · cos(ωt +ϕ1)

i2(t) = I2 · cos(ωt +ϕ2)

The sum is :

isum(t) = I1 · cos(ωt +ϕ1)+ I2 · cos(ωt +ϕ2)

t

i(t)

i1

i2

isum

The amplitude of the sum is not equal to the sumof the amplitude.



Summation of signals : ϕ1 = ϕ2Signals are in in phase

i1(t) = I1 · cos(ωt +ϕ) i2(t) = I2 · cos(ωt +ϕ)

itotale(t) = I1 · cos(ωt +ϕ)+ I2 · cos(ωt +ϕ) = (I1 + I2) · cos(ωt +ϕ)

t

i(t)

isum

In this only case, the amplitude of the sumis equal to the sum of the amplitude.



Summation of signals : ϕ1 = ϕ2Signals are in in phase

i1(t) = I1 · cos(ωt +ϕ) i2(t) = I2 · cos(ωt +ϕ)

itotale(t) = I1 · cos(ωt +ϕ)+ I2 · cos(ωt +ϕ) = (I1 + I2) · cos(ωt +ϕ)

t

i(t) isum

In this only case, the amplitude of the sumis equal to the sum of the amplitude.



Summation of signals : ϕ2 = ϕ1 +π and I1 = I2Signals are in phase opposition

i1(t) = I0 · cos(ωt +ϕ) i2(t) = I0 · cos(ωt +ϕ+π)

But cos(a +π) = −cos(a), therefore :

itotale(t) = I0 · cos(ωt +ϕ)+ I0 · cos(ωt +ϕ+π)

= I0 · {cos(ωt +ϕ)− cos(ωt +ϕ)}= 0

t

i(t)

isum

In this case, the amplitude of the sum is zero.Mathieu Bardoux (IUTLCO GTE) Electricity First year 11 / 34

Fresnel diagram

Summary


2 Fresnel diagram





Fresnel diagram

Fresnel diagrammRevolving vectors

Two vectors rotating at angular velocity ω.ϕ is the phase shift, Irms and Urms their length.

Ox

Oy

O

A

Ax

B

Bx

ϕ

Projection of these vectors on the Ox axis :I Ax = Irms cos(ωt)I Bx = Urms cos(ωt +ϕ)

Equivalence between alternating voltage/current and theseprojections.


Fresnel diagram

Fresnel diagrammRevolving vectors

I Adding the vectors adds up the voltages/currentI Very simple for limited operationsI Tedious for more complex configurationsI ⇒ Need for a more powerful tool : complex numbers

representation


Complex numbers : a few reminders

Summary


2 Fresnel diagram





Complex numbers : a few reminders Cartesian form

Cartesian (or algebraic) form

x = a + b , where 2 = −1.a is called real part and b imaginary part.

x can be viewed as a point of the complexe plane, and (a ,b ) as itscartesian coordinates.

<

=

x = a + b

a

b

Complex numbers can be considered as position vectors in atwo-dimensionnal plane.


Complex numbers : a few reminders Trigonometric form

Trigonometric (or polar) form

x can be represented through its absolute value and argument :

<

=

x|x |ϕ


Complex numbers : a few reminders Trigonometric form

From one representation to the other

I From polar to cartesian:a = |x |cosϕb = |x |sinϕ

I From cartesian to polar :|x |=

√a2 + b2

ϕ = atanba


Complex representation

Summary


2 Fresnel diagram





Complex representation Definition

Complex current and voltage

I i(t) =√

2Irms cos(ωt +ϕ)⇒ I = Irmse(ωt+ϕ)

I u(t) =√

2Urms cos(ωt +ϕ)⇒ U = Urmse(ωt+ϕ)

Every rules seen in the previous chapters are still valid in alternatingcurrent, including complex in representation :I Kirchhoff’s lawsI Thevenin’s and Norton’s theoremsI Superposition theoremI etc. . .


Complex representation Definition

Why would we use complex representation ?Because complex is simpler !

I Addition :<(A +B) =<(A)+<(B) and=(A +B) ==(A)+=(B)

I Multiplication : |A ·B |= |A | · |B | and ϕA ·B = ϕA +ϕB

I Derivation :dUdt

= ωU

I Integration :∫Udt =

Uω


Complex representation Impedance

ImpedanceGeneralised resistance

Ohm’s law :U = ZI

Z = Zeϕ

I Resistor : Z = RI Inductor : Z = ωL

I Capacitor : Z =1ωC

Except for pure resistors, Z is a function of ω.


Power

Summary


2 Fresnel diagram





Power Complex power

Power

I Power = Energy per unit of timeI DC power : P = U · II AC power : p(t) = u(t) · i(t)


Power Complex power

Complex power

One can define four different quantities, all of them homogeneous topower :I Complex power : S = U · II Apparent power : S = |S |I Active (or real) power : P =<(S)I Reactive power : Q ==(S)


Power Complex power

Complex power

<

=

S

P

Q


Power Complex power

Active power

Definition

Active power P = Average power consumed by the system during agiven period of time :

P =1T

∫ T

0p(t)dt =

1T

∫ T

0u(t)i(t)dt

If a component has pure real impedance, active power is equal toapparent power.The unit used for active power is W (watt).


Power Complex power

Apparent power

Definition

Apparent power S is the maximum value power can take for givenvoltage and current.It is equal to the magnitude of complex power.

Apparent power is noted S . This is also the nominal power specified onelectrical equipments.The unit for apparent power is V ·A (volt-ampere).


Power Complex power

Reactive power

Definition

Reactive power Q is the imaginary part of complex power.

A component with pure imaginary impedance dissipates zero activepower. In this case, reactive power is equal to apparent power.The unit for reactive power is var (volt-ampere reactive).


Power Complex power

AC power : summary

Apparent, active and reactive power are linked :

S2 = P2 +Q2

I Active power in WI Apparent power in V ·AI Reactive power in var

W, V ·A, var are all homogeneous to power, but their physical meaningis different.


Power Power factor

Power factor

Definition

Ratio of active power to apparent power in a circuit is called the powerfactor.

λ=PS

This is an intrisic caracteristic of the component.Calling ϕ the phase shift between U and I :

λ= cos(ϕ)


Power Boucherot’s method

Boucherot’s method

Boucherot’s theorem

If a circuit contains N components, each of which absorbs activepower Pi and reactive power Qi , then the total active/reactive andpowers are the sums of the active/reactive powers of the circuit:

Ptot =N∑i=1

Pi

Qtot =N∑i=1

Qi



Boucherot’s method

Corollary

Total apparent power is not equal to the sum of all apparent powers :

Stot ,N∑i=1

Qi

It can be determined through total active and reactive powers :

Stot =√P2tot +Q2

tot



Summary

In this chapter, we have :I Described alternative current and voltageI Defined a formalism based on complex numbersI Discoverd the concept of impedanceI Distinguished different forms of power


electricity - alternating current · summary 1 alternating current 2 fresnel diagram 3 complex...

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