1
AC Circuit Analysis
Sinusoidal quantity
A 0,707 2
A dt )t(cos AT1 A
[Hz]; 2
T1
[s]; 2 T
number; real :[1] number; positive real :[1/s]
number; positive real a is and a(t) of valuemaximal:A
) t ( cosA a(t)
Tt
t
22 ==+=
==
=
+=
∫+
θω
πω
ωπ
θω
θω
e
f frequency natural
period
phasefrquency radian
amplitude
-θ/ω
A
t
T = 2π/ω=1/f
) t ( cos A 2 ) t ( cosA a(t)θω
θω+=
+=
e
Effective value - rms value(root-mean-square value)
Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna
a(t) = A cos (ω t + θa )b(t) = B cos (ω t + θb ) ϕ = θa - θb phase shift angle (a and b are ϕ out of phase)
ϕ= 0 → a(t) and b(t) are in phase;ϕ > 0 → a(t) leads b(t) by ϕ (advanced);ϕ < 0 → a(t) lags b(t) by ϕ (delayed).
t
a(t)
b(t)
ϕ = ±π → a(t) and b(t)in opposition of phase
t
a(t)b(t)
ϕ = ±π /2 → a(t) and b(t)90° out of phase
t
a(t)
b(t)
ϕ = 0 →a(t) and b(t)in phase
AC Signals Iso-frequentcy Quantities
a(t) leads b(t) by ϕ
a(t) = A cos ω t b(t) = B cos (ω t −ϕ)
t
ϕ/ω = (θa− θb) /ω
b(t)a(t) ϕ = −θb
0
2
A e j(ω t+θ ) = A cos (ω t + θ ) + j A sin (ω t + θ )⎡⎣ ⎤⎦a(t) = A cos (ω t + θ ) = Re A e j(ω t+θ )⎡⎣ ⎤⎦ = Re A e jω t⎡⎣ ⎤⎦ where : A = A e jθ = A e jθ = A cos θ + j sin θ( )
ααα sin j cos e : j +=Identity Euler
AC Signals: Sinusoids and Phasors
Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna
)t( θω +
AM
t ℑm
ℜe
a(t)
A
[ ] [ ] [ ]( ) θθθ
θω
θ
ωωθθω
A sin j cosA eA A :Acomplex by thegiven voltageaor current a is the where
e A e eA eA ) t ( cosA a(t)
j
tjtjj)tj(
=+==
===+= +
!!
!phasor
ReReRe
There is a biunique correspondence
between sinusoids at the same frequency and
phasors.
( ) ( ) t sin sin A - t cos cosA
) t cos(A
ωθωθ
θω
+=
=+
θ
θθθ
A AeA A
sin A j cosA Aj
==
+=
!!!
)t( θω +
A
t ℑm
ℜe
a(t)
AC Signals: Sinusoids and Phasors
Rectangular form
Exponential formPolar form
Sinusoid (time domain) Phasor (frequency domain)
3
Re
Im
Aθ
a
b
( )
0 a ,ab tan
b a A
sin A b cosA a
b j a
sin j A eA A
1-
22
j
>⎟⎠⎞⎜
⎝⎛=
+=
==
+=
+==
where
where
cos
θ
θθ
θθθ!
AC Signals: Sinusoids and Phasors
θjeA A =!
Complex plane representation
Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna
Uniqueness: Two sinusoids at the same frequency are equal if and only if they are represented by the same phasor
a(t) = b(t) ⇔ A = B
As e jωt is a complex number which rotates on the complex plane when t increases, a(t) is equal to b(t) at any t, if both the real and the immaginary parts of A and B are equal :a(t) = b(t) ⇒ Re A e jω t⎡⎣ ⎤⎦ = Re B e jω t⎡⎣ ⎤⎦ Re cosω t + j sinω t( ) A⎡⎣ ⎤⎦ = Re cosω t + j sinω t( ) B⎡⎣ ⎤⎦ Re cosω t A⎡⎣ ⎤⎦ + Re j sinω t A⎡⎣ ⎤⎦ = Re cosω t B⎡⎣ ⎤⎦ + Re j sinω t B⎡⎣ ⎤⎦ cosω t Re A⎡⎣ ⎤⎦ -sinω t Im A⎡⎣ ⎤⎦ = cosω t Re B⎡⎣ ⎤⎦ -sinω t Im B⎡⎣ ⎤⎦ ⇒a(t) = b(t) ⇔ Re A⎡⎣ ⎤⎦ = Re B⎡⎣ ⎤⎦ and Im A⎡⎣ ⎤⎦ = Im B⎡⎣ ⎤⎦ .
AC Signals: Sinusoids and Phasors[ ][ ]tj
b
tja
e B ) t ( cos B b(t)
e A ) t ( cosA a(t)ω
ω
θωθω
!
!
Re
Re
=+=
=+=
A = a + jb j A = ja - bRe j A⎡⎣ ⎤⎦ = − Im A⎡⎣ ⎤⎦
4
Linearity:
Bc Ac b(t) c a(t) c 2121!! +⇔+
frequency same the at sinusoids of ncombinatio linearsame the represents phasors of ts)coefficien
real and constant (with ncombinatio linearThe
[ ] [ ] ( )[ ]tj21
tj2
tj1 eBc Ac e B c e A c ωωω !!!! +=+ ReReRe
:relationfollowing the from results It
AC Signals: Sinusoids and Phasors[ ][ ]tj
b
tja
e B ) t ( cos B b(t)
e A ) t ( cosA a(t)ω
ω
θωθω
!
!
Re
Re
=+=
=+=
Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna
If A is the phasor of a(t) = Acos (ωt), the fasor
of the time derivative of a(t), da(t)
dt= d
dtAcos (ωt)⎡⎣ ⎤⎦ ,
is given by jω A
da(t)dt
⇔ jω A;
d2a(t)
dt2 ⇔ jω jω A( ) = - ω 2 A
Derivative:
[ ]tja e A ) t ( cosA a(t) ωθω !Re=+=
[ ]{ } [ ]{ } [ ][ ] [ ]tj) t j(
) t j(tj
e Aj eA j ) t sin(A -
) t cos(A dtd eA
dtd e A
dtd
ωθω
θωω
ωωθωω
θω!
!
ReRe
ReRe
==+=
=+==+
+
:relationfollowing the from results It
AC Signals: Sinusoids and Phasors
Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna
Re j A⎡⎣ ⎤⎦ = − Im A⎡⎣ ⎤⎦
5
In the circuit of the figure from the element equation it results :
v(t) = L didt
+ 1C
i(t')dt' + R i(t) −∞
t
∫
⇒ dvdt
= L d2idt2 + R d i
dt + 1
C i
⇒ Re jω Vejω t⎡⎣ ⎤⎦ = Re (jω )2 LIe jω t⎡⎣ ⎤⎦ +
+ Re jωRIe jω t⎡⎣ ⎤⎦ + Re 1C
I e jω t⎡
⎣⎢
⎤
⎦⎥
⇒ Re jω Vejω t⎡⎣ ⎤⎦ = Re -ω 2LIe jω t + jωRIe jω t + 1C
I e jω t⎡
⎣⎢
⎤
⎦⎥
⇒ Re jω Vejω t⎡⎣ ⎤⎦ = Re -ω 2LI + jωRI + 1C
I⎛⎝⎜
⎞⎠⎟
e jω t⎡
⎣⎢
⎤
⎦⎥
⇒ jω Vejω t = -ω 2LI + jωRI + 1C
I⎛⎝⎜
⎞⎠⎟
e jω t ⇒ jω V = -ω 2LI + jωRI + 1C
I⎛⎝⎜
⎞⎠⎟
In a linear circuit a sinusoidal current i(t) corresponds to a sinusoidal voltage v(t) at the same frequency and different phase.
v(t) = V cos(ω t +θv ) = Re Vejω t⎡⎣ ⎤⎦i(t) = I cos(ω t +θi ) = Re I e jω t⎡⎣ ⎤⎦
where : V = V e jθV ; I = I e jθI .
AC Circuit Analysis•
i(t) R
•
L
v(t)C
•
⇒ V = R + j ωL − 1ωC
⎛⎝⎜
⎞⎠⎟
⎡
⎣⎢
⎤
⎦⎥ I
V = Z I or Ve = Z Ie - Element equation in the freqency domain
From the definition of the effective phasors it is Ve =V
2 and Ie =
I2
AC Circuit Analysis
•i(t) R
•
L
v(t)C
•
•
-
The impedence (complex number) is the ratio of the pasor voltage V and the phasor current I.
Z = R + j ωL − 1ωC
⎛⎝⎜
⎞⎠⎟
or Z = R + j X
where R resistance
X reattance, X = XL + XC; XL = ωL, XC= - 1ωC
The impedence is measured in ohm [Ω].
The reciprocal of impedence is the admittance: Y = 1Z
SI unit : siemens [S]( )
reactance,
6
Re
Im
I!V!
AC Circuit Analysisv(t) = V cos(ω t +θV ) = Re Vejω t⎡⎣ ⎤⎦i(t) = I cos(ω t +θI ) = Re I e jω t⎡⎣ ⎤⎦where V = V e jθV ; I = I e jθI
V = Z I where Z = R+jX
or Z = Z e jθZ ; Z = R2 + X2 , θZ = tan-1 XR
⎛⎝⎜
⎞⎠⎟
I = VZ
= VZ
e j(θV - θZ ) = I e jθI
⇒
I = VZ
= V
R2 + X2 =
V
R2 + ωL- 1ωC
⎛⎝⎜
⎞⎠⎟
2
θI = θV - θZ = θV - tan-1 XR
⎛⎝⎜
⎞⎠⎟
= θV - tan-1ωL- 1
ωCR
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
⎧
⎨
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
•i(t) R
•
L
v(t)C
AC Circuit Analysis
Phase shift
v(t) = V cos(ω t +θV ) = Re Vejω t⎡⎣ ⎤⎦i(t) = I cos(ω t +θI ) = Re I e jω t⎡⎣ ⎤⎦where V = V e jθV ; I = I e jθI
VI
= VI
e j(θV -θI ) = VI
e jϕ = Z
⇒ Z = R + jX = Z e jθZ = VI
e jϕ
Therefore the phase shift angle ϕ = θV -θI between current and voltage phasors is also the angle of the load impedence.
Re
Im
Z!ϕ
X
R
•
•
Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna
7
AC Circuit AnalysisWhen assuming as the reference : θv = 0 (this without any loss of generality, since all phase angles will be referenced to the source voltage's angle)
ϕ = θv - θi = - θi ⇒v(t) = V cos ω t = Re Vejω t⎡⎣ ⎤⎦ , V = V i(t) = I cos(ω t − ϕ ) = Re I e jω t⎡⎣ ⎤⎦ , I = I e-jϕ
V = Z I where Z = R + j ωL − 1ωC
⎛⎝⎜
⎞⎠⎟= R + jX
It is also : Z = Z e jθZ ; Z = R2 + X2 ; θZ =ϕ = tan-1 XR
⎛⎝⎜
⎞⎠⎟
I = VZ
= VZ
e j(θV - θZ ) = I e jθI = I e-jϕ
⇒
I = VZ
= V
R2 + X2 = V
R2 + ωL- 1ωC
⎛⎝⎜
⎞⎠⎟
2
ϕ = θZ = tan-1 XR
⎛⎝⎜
⎞⎠⎟
= tan-1ωL- 1
ωCR
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
⎧
⎨
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
Re
Im
I!
V!
ϕ
Circuit Analysis
Integro-differential Equations
Transformation from the
time domain to the
frequency domain
(Steinmetz transform)
Algebraic equations
Solutionand
InverseTransformation
from the frequency domain
to thetime domain
)i(v
0v
0i
rr
m r
n r
rf=
=
=
∑∑
)(
0
0
rr
m r
n r
IV
V
I
!!
!
!
rf=
=
=
∑∑
AC Circuit Analysis
Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna
8
The Resistor: Z = R
v(t) = Vcos ω t = Re Vejω t⎡⎣ ⎤⎦ , V = Vi(t) = Icos(ω t − ϕ ) = R I e jω t⎡⎣ ⎤⎦ , I = I e-jϕ
V = Z I where Z = R
⇒ I = VZ
= I e-jϕ where I =
VR
ϕ = tan-1 XR
⎛⎝⎜
⎞⎠⎟
= 0
⎧
⎨⎪⎪
⎩⎪⎪
⇒ I = VR
i(t) = I cos(ω t − ϕ ) = VR
cos ω t Re
Im I! V!
•
•
R
I!
V!
t
v(t)
i(t)
v(t) and i(t) in phase
AC Circuit Analysis
v(t) = Ri(t) ⇔ V = RI
The Inductor: Z = j ωL
v(t) = Vcos ω t = Re Vejω t⎡⎣ ⎤⎦ , V = V i(t) = Icos(ω t − ϕ ) = Re I e jω t⎡⎣ ⎤⎦ , I = I e-jϕ
V = Z I where Z = jωL
⇒ I = VZ
= I e-jϕ where I = V
ωL
ϕ = tan-1 ωL0
⎛⎝⎜
⎞⎠⎟= π
2
⎧
⎨⎪⎪
⎩⎪⎪
⇒ I = VωL
e-jπ
2
i(t) = I cos(ω t − ϕ ) = VωL
cos ω t - π2
⎛⎝⎜
⎞⎠⎟
Re
Im
I!
V!
•
•
L
I!
V!
i(t) and v(t) are π/2 out of phasei(t) lags v(t) by π/2 (delayed)
t
v(t)
i(t)
AC Circuit Analysis
v(t) = L%&%' ⇔ V = ȷ L𝜔 I
9
The Capacitor: Z = - j 1ωC
v(t) = Vcos ω t = Re Vejω t⎡⎣ ⎤⎦ , V = V i(t) = Icos(ω t − ϕ ) = Re I e jω t⎡⎣ ⎤⎦ , I = I e-jϕ
V = Z I where Z = -j 1ωC
⇒ I = VZ
= I e-jϕ where
I = V ωC
ϕ = tan-1−1
ωC0
⎛
⎝⎜⎜
⎞
⎠⎟⎟= -π
2
⎧
⎨⎪⎪
⎩⎪⎪
⇒ I = V ωC ejπ
2
i(t) = I cos(ω t − ϕ ) = V ωC cos ω t + π2
⎛⎝⎜
⎞⎠⎟
•
•
C
I!V!
Re
ImI!
V!
t
v(t)
i(t)
i(t) and v(t) are π/2 out of phasei(t) leads v(t) by π/2 (advanced)
AC Circuit Analysis
v(t) = ,- ∫ i(t’)dt′'34 ⇔ V = - ȷ ,56 I
Series Resonance
v(t) = V cos ω t = Re Vejω t⎡⎣ ⎤⎦ , V = V
i(t) = I cos(ω t − ϕ ) = Re I e jω t⎡⎣ ⎤⎦ , I = I e-jϕ
I = VZ
= I e-jϕ con
I ω( ) = V
R2 + ωL- 1ωC
⎛⎝⎜
⎞⎠⎟
2
ϕ ω( ) = tan-1ωL- 1
ωCR
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
• At the resonance frequency ω0 , for which the inductive and the capacitive reactance in serie are equal in magnitude, the total reactance is zero. Therefore it follows that:- the impedence is only resistive,- the current amplitude/effective value I(ω0 ), Ie (ω0 ) are maximual (this is the motivation of the term "resonance"). - the voltage and the current are in phase.
ω0L-1
ω0C = 0 ⇒ ω0 = 1
LC
•i(t) R
•
L
v(t)C
ω0 ω
ϕ
ω0 ω
I (ω)
•
•AC Circuit Analysis
10
( )
IX j IX j I R I Z V
X- X isit LC1 At
X X j R C1 - L j R Z
:
CL
CL0
CL
!!!!!!
!
++==
===
++=⎟⎠⎞⎜
⎝⎛+=
. ωω
ωω
Resonance Series •i(t) R
•
L
v(t)C
I!V!
IX j C!
I R !
IX j L!
I!
V!IX j L!
I R !
IX j C!
ω < ω0 ω = ω0 ω > ω0
I!I R V !! =
IX j C!IX j L
!
Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna
AC Circuit Analysis
For a given �� the voltage phasor �� is given by the vector sum R��+ jXL��+jXC��
Parallel Resonance:
Z = ZLZC
ZL + ZC
, dove ZL = j ωL, ZC = -j 1ωC
⇒ Z = -j L C
ωL - 1ωC
⇒ I ω( ) = V L C
ωL - 1ωC
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
• At the resonance frequency ω0 , as the inductive and the capacitive reactance are equal in magnitude, the parallel of a capacitor with an inductor has a total reactance equal to infinite. Therefore it follows that:- the total impedence is infinite;- the total current flowing through the circuit is zero.- the currents in the inductor and the capacitor are: IL = -IC
ω0L-1
ω0C = 0 ⇒ ω0 =
1
LC
and at ω =ω0 ⇒ IL = - IC = - j CL
V
•
•
L CV!
I!
LI! CI!
ω0 ω
XLC
AC Circuit Analysis
•
•
11
•
1Z!
2Z!
nZ!∑=
k keq Z1
Z1
!!
∑=k
keq Z Z !!1Z! 2Z! nZ!
� • •
•1
3
2ZΔ1
ZΔ2ZΔ3
ZY1
ZY3ZY2 ZY1 =
ZΔ1ZΔ2
ZΔ1 + ZΔ2 + ZΔ3
ZΔ1 =ZY1 ZY2 + ZY2 ZY3 + ZY3 ZY1
ZY3
Series of impedances
Parallel of impedances
Wye DeltaConnection
Impedance ConnectionsAC Circuit Analysis
� �� � �
� �
�
� �
�
•
•
i(t)
v(t)
)tcos(I i(t)tcosV v(t)
M
M
ϕ−ω=ω=
p(t) = v(t) i(t) With the assumption : θv = 0 v(t) = Vcosω t; i(t) = Icos(ω t −ϕ )
i(t) = I cos ω t cos ϕ + sin ω t sin ϕ( ) = I cos ϕ cos ω t + I sin ϕ sin ω t = ia (t) + ir (t)
ia (t) = I cos ϕ cos ω t; in phase current , ir (t) = I sin ϕ sin ω t; reactive current (π / 2 out of phase).The istantaneous power is the sum of the in phase istantaneous power pa (t) and the istantaneous reactive power pr (t). p(t) = v(t) i(t) =
= v(t) ia (t) + v(t) ir (t) = = pa (t) + pr (t)
t
v(t)
i(t)
Power in AC Circuits
12
♦ pa (t) = v(t) ia (t ) = Vcosω t Icos ϕ cos ω t
= VI cosϕ cos2ω t
pa (t) is the in phase istantaneous power. It is always positive. Thereby it is a power flowing into the circuit element and is dissipated by the resistance of the element. This power is consumed and therefore it is utilized by the circuit element that is the cicuit load.
♦ pr (t) = v(t) ir (t) = Vcosω t I sen ϕ sen ω t
= 12
V I sin ϕ sin 2ω t = VeIe sin ϕ sin 2ω t
pr (t) is the istantaneous reactive power. Its average value is equal to zero. It corresponds to an energy flowing into the circuit element for half period [the period of pr is T/2 = π /(2ω )] and outside of it for the next half period and hence in and out of the memary elements (inductance and capacitors).
t
v(t)ia(t)
pa(t)
t
v(t)ir(t)
pr(t)
2ap (t) V I cos cos tϕ ω=
Istantaneous in phase power :
r
:1p (t) V I sin sin 2 t 2
ϕ ω=
Istantaneous reactive power
Power in AC CircuitsInstantaneous in Phase and Reactive Power
The average power P (said also real power) is the average of the istantaneosus power over one period T = 2π /ω .
P = 1T
p(t')t0
t0+T
∫ dt' = 1T
pa (t') + pr (t')⎡⎣ ⎤⎦t0
t0+T
∫ dt' = 1T
pa (t')t0
t0+T
∫ dt'
as pa (t) = V I cos ϕ cos2ωt, it is :
P = 12
V I cos ϕ = Ve Ie cos ϕ
where Ve and Ie are the effective values of the tension and the current respectively [they are the rms values of v(t) and i(t). "cosϕ " is said to be the power factor]. The average power is the power utilized by the circuit loads, it is consumed by the circuit.
In the SI System of Units the average power is measured in watt [W].
Power in AC Circuits
Average Power
Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna
13
The reactive power Q is the maximal value of the istantaneous reactive power pr (t):
Q = pr (t)⎡⎣ ⎤⎦Max⋅sign ϕ( ) =
12
VI sin ϕ = VeIe sin ϕ
where pr (t) =12
V I sinϕ sin2ω t = VeIesinϕ sin2ω t
Q is the maximal value of the power exchanged by a circuit element, or generally by a load, with the circuit into which the circuit element is inserted (or with the network to which the load is connected). Q can be positive or negative depending on the sign of ϕ. For an inductive load Q is positive, for a capacitive load Q is negative. In the SI Sistem of Units Q is measured in volt - ampere reactive [VAR].
Power in AC Circuits
Reactive Power
Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna
( )
⎩⎨⎧
==→
⇒+=
+=+====
=
+=
==
)( I X Q)( I R P
jQ P N
I X j I R I X j I R I Z I I Z I V N
I Z V
jQ P N
IV I V 21 N
2
2
22222**
**
reactancethe on depends powerreactive the resistancethe on depends poweraverage the
is itSince
e
e
eeeeeeeee
ee
ee
!
!!!!!!!
!!!
!
!!!!! ( )
⎪⎪⎩
⎪⎪⎨
⎧
+=
=+==
====
jQ P
sin IjV cos IV e IV
e IV e I e V *I V Nj
-j-jj IVIV
ϕϕϕ
θθθθ
eeeeee
eeeeee!!!
Complex Power
Power in AC Circuits
Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna
14
When two loads Z1 and Z2 are in serie with a voltage source (top figure), KTL yelds
Ve = Ve,1 + Ve,2 ⇒ N = VeIe
* = Ie* Ve,1 + Ve,2( ) =
= Ve,1Ie* + Ve,2 ,Ie
* = N1 + N2
When the loads are in parallel with the source (midle figue), KCL yelds
Ie = Ie,1 + Ie,2 ⇒ N = VeIe
* = VeIe,1* + Ie,2
*( ) = = Ve
Ie,1* + Ve
Ie,2* = N1 + N2
For a circuit of l elements, whether the loads are connected in serie or in parallel or both connections are present (bottom figure), the total power supplied by the source equals the amount of power delivered to the circuit elements.
N = Ve Ie* = Ve,k Ie,k
*
k=1
l
∑ = Nk k=1
l
∑ N = P + jQ = Pk + j Qk( )
k=1
l
∑
•
• Rk
Xk
I!V!
+-
Power in AC CircuitsAC Power Conservation
⇒ P = Rk Iek
2
k=1
l
∑ (P is due to the resistances)
Q = Xk Iek2
k=1
l
∑ (Q is due to the reactances)
⎧
⎨⎪⎪
⎩⎪⎪
+-
I!V!
1Z! 2Z!•
•
+-
I!V!
1Z! 2Z!•
•
The apparent power is the product of the effective voltage (rms value) and the effective current (rms value).
N = Ve Ie = 12
V I
In the SI sistem of units N is measured in volt-ampere [VA].
From the definition of the average power P it is:
P = N cosϕ or N = P
cosϕ
These relations indicate the relevance of the power factor cosϕ. When the load angle encreases above zero cosϕ decreases below 1 and the average power P, which is the power utilized by the load, decreases below the apparent power N, which is the power capacity needed by the sources (for example the electrical power capacity of the generators of the electrical power stations).
Apparent Power
Power in AC Circuits
Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna
15
The power factor is the cosine of the phase difference between voltage and current. Hence it is the cosine of the angle of the load impedence: cosϕ = cos(θV−θI) .
ϕϕ cos Ncos IVP == ee NP
IVP cos ==ee
ϕ
Power in AC CircuitsPower Factor
The value of the power factor cosϕ ranges between 0 and unity. Ø For a pure resistive load the voltage and current are in phase, so that
θV−θI = ϕ = 0 and the power factor cosϕ = 1. Therefore the apparent power is equal to the average power.
Ø For a purely reactive load θV−θI = ϕ = ±π/2 and the cosϕ = 0. In this case the average power is zero .
Ø In between these two extreme cases the power factor is said to be leading or lagging. Leading power factor means that the current leads the voltage (capacitive load). Lagging power factor means that the current lags the voltage (inductive load).
ϕϕϕ tan
cos IVsen IV
PQ ==
ee
ee ⎟⎠⎞⎜
⎝⎛=
PQtan cos cos 1-ϕ
ϕϕ cos Ncos IVP == ee NP
IVP cos ==ee
ϕ
Power in AC CircuitsPower Factor
↑↑⇒↓= N I cos V P cos V
P I , eee
e ϕϕ
andfixed at
Ø In the electrical systems the power factor has to be as big as possible to utilize the maximal part of the apparent power generated (P → N) and to reduce the current at a given power utilized (P) and at a given voltage. Many utilities request to pay the power factor utilized when its value is below 0.9.
Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna
16
Motivations for the reduction of the phase shift angleØ If the current is reduced, the power lost along the line decreases (PL=2Rln I2).Ø As constant voltage is required by the net and as , in
order to reduce the voltage losses along the line the current has to be as low as possible.
Ø The apparent power, that has to be generated by the electrical power station, decreases with the decrease of the current.
G∼ U
•
•
•
•
V!
I!
' V!
Rln
I2R V 'V ln!!! += V 'V !! ≈
Power in AC Circuits
Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna
P = VeIe cos ϕ Q = VeIe sin ϕ
⎫⎬⎭
⇒ Q = P tan ϕ
Most loads are inductive and operate at a low lagging power factor. In order to reduce ϕ to a determined ϕ' (usually cosϕ' = 0,9 )a capacitor is set in parallel with the load. The capacity C is given by the the capacitive reactive power which compensates the inductive power : Q = P tan ϕ Q +QC = P tan ϕ ' ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ QC = P tan ϕ' - tan ϕ( )where Q is the reactive power to be corrected and QC = -ωC Ve
2 (IC = ωCV) is the reactive power of the capacitor utilized for the load factor correction. Hence it is
C = Pω Ve
2 tan ϕ - tan ϕ '( )
•
•
C
LI!
V!CI!
I!
U
•
•
CI!
V!
I!LI!
CI!
ϕϕ’
Power in AC CircuitsShift Angle Correction
C = -:;<=;>
= ?
<=;>tan 𝜑 − tan 𝜑′
17
Department of Electrical Engineering – University of Bologna33
admittance 电导
amplitude 振幅
derivative 微分的
frequency domain 频率范围
to lag 滞后
to lead 超前
impedence 阻抗
linearity 线性
inverse transform 逆变换
natural frequency 自然频率,固有频率
opposition of phase 反相
parallel resonance 并列共振
phase 相位
phase shift angle 相移角度
phasor 矢量
out of phase 异相
90° out of phase 90异相
period 周期
radian frequency 角频率
reactance 电抗
resonance frequency 共振频率
root –mean-‐‑square value, rms value
均方根值
series resonance 串联谐振
sinusoid 正弦
sinusoidal quantity 正弦量
Steinmetz transform 斯坦梅兹变换
time domain 时域
uniqueness 单值性
Terminology English – Chinese
34
Terminologyadmittance ammettenza
amplitude ampiezza
derivative derivazione, derivata
frequency domain dominio delle frequenze
to lag essere in ritardo di fase
to lead essere in anticipo di fase
impedence impedenza
linearity linearità
inverse transform trasformata inversa
natural frequency frequenza ciclica, frequenza
opposition of phase opposizione di fase
parallel resonance antirisonanza
phase fase
phase shift angle angolo di sfasamento
phasor fasore
out of phase sfasato di
90° out of phase sfasato di 90°, in quadratura di fase
period periodo
radian frequency pulsazione, frequenza angolare
reactance reattanza
resonance frequency frequenza di risonanza
root –mean-‐‑square value, rms value
valore efficace
series resonance risonanza
sinusoid sinusoide
sinusoidal quantity grandezza sinusoidale
Steinmetz transform trasformata di Steinmetz
time domain dominio del tempo
uniqueness unicità
Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna