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LITTORAL CÔTE D’OPALE
ElectricityAlternating Current
Mathieu Bardoux
IUT du Littoral Côte d’OpaleDépartement Génie Thermique et Énergie
First year
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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
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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
Mathieu Bardoux (IUTLCO GTE) Electricity First year 3 / 34
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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.
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Alternating current Definitions
Alternating current
t
i(t)
I0
ϕω
T = 2πω
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Alternating current Definitions
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
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Alternating current Definitions
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
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Alternating current Definitions
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
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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.
Mathieu Bardoux (IUTLCO GTE) Electricity First year 9 / 34
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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.
Mathieu Bardoux (IUTLCO GTE) Electricity First year 9 / 34
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Alternating current Summation of signals
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.
Mathieu Bardoux (IUTLCO GTE) Electricity First year 10 / 34
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Alternating current Summation of signals
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.
Mathieu Bardoux (IUTLCO GTE) Electricity First year 10 / 34
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Alternating current Summation of signals
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
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Fresnel diagram
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
Mathieu Bardoux (IUTLCO GTE) Electricity First year 12 / 34
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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.
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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
Mathieu Bardoux (IUTLCO GTE) Electricity First year 14 / 34
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Complex numbers : a few reminders
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
Mathieu Bardoux (IUTLCO GTE) Electricity First year 15 / 34
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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.
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Complex numbers : a few reminders Trigonometric form
Trigonometric (or polar) form
x can be represented through its absolute value and argument :
<
=
x|x |ϕ
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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
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Complex representation
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
Mathieu Bardoux (IUTLCO GTE) Electricity First year 19 / 34
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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. . .
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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
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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.
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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.
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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.
Mathieu Bardoux (IUTLCO GTE) Electricity First year 22 / 34
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Power
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
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Power Complex power
Power
I Power = Energy per unit of timeI DC power : P = U · II AC power : p(t) = u(t) · i(t)
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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)
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Power Complex power
Complex power
<
=
S
P
Q
Mathieu Bardoux (IUTLCO GTE) Electricity First year 26 / 34
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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).
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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).
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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).
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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.
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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)
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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
Mathieu Bardoux (IUTLCO GTE) Electricity First year 32 / 34
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Power Boucherot’s method
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
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Power Boucherot’s method
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
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