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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


)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)


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!


θjeA A =!

Complex plane representation


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

=+=

=+=


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



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

•

•


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


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

!


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].


Average Power


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].


Reactive Power


( )

⎩⎨⎧

==→

⇒+=

+=+====

=

+=

==

)( 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



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



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.


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 !! ≈



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


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