Tema  2.1.  Circuitos  de  corriente  alterna  

Energía  y  Telecomunicaciones  

Alberto  Arroyo  Gu;érrez  Mario  Mañana  Canteli  Raquel  MarBnez  Torre  Jesús  Mirapeix  Serrano  

Cándido  Capellán  Villacián  

Departamento  de  Ingeniería  Eléctrica  y  Energé5ca  

Este  tema  se  publica  bajo  Licencia:  Crea5ve  Commons  BY-­‐NC-­‐SA  4.0  

a a = 1!120

BCfGiIIj j = 1!90

LMPQR !SSuUUW JX !YZ !Z !!" !m# S/m$ U I%

+

+

+u

i

1

x(t) = X!2 cos(2&ft+ $0)

x(t) tXf$0 t = 0

X!2

x(t)

1f

f

xmedio xmedio = 1T

!T x(t)dt

xrms xrms ="

1T

!T x2(t)dt

xp xp = max{x(t)}xpp xpp = max{x(t)} "min{x(t)}

FC = xp

xrms

FF = xrmsxmedio

#

$%

$0

X"

X$0 = 0 % = 2&f

Urms Upico!2

R

u(t) = Ri(t)

U = RI

u(t)

+

i(t)

R

I U

L

u(t) = Ldi(t)

dt

U = j%LI

C

u(t) =1

C

#i(t)dt

u(t)

+

i(t)

LI

U

L

U =1

j%CI

u(t)

+

i(t)

C I

U

C

Z$

u(t) =!2U cos(%t)

i(t) =!2I cos(%t" $)

u(t)+

i(t)

Z

P

QS

$

Z

pi(t) = u(t)i(t) = 2UI cos(%t) cos(%t" $) = UI(cos(2%t" $) + cos($))

= UI(cos($)cos(2%t) + sin($) sin(2%t) + cos($))

= UI cos($)(1 + cos(2%t)) + UI sin($) sin(2%t)

= P (1 + cos(2%t)) +Q(sin(2%t))

P (1 + cos(2%t))Q sin(2%t)

P Q

P = UI cos($)

Q = UI sin($)

S

S =$P 2 +Q2

P Q S

$I!!!

S = U I"= U!0I!! = UI!! = UI cos($) + jUI sin($) = P + jQ

(%)

cos!cos" = 12 (cos(!+ ") + cos(!" "))

cos(!" ") = cos!cos" + sin!sin"

R S = U I"= U!0 I!0 = UI!0 &

P = UI;Q = 0

L S = U I"= U!0 I!90 = UI!90 &

P = 0;Q = UI

C S = U I"= U!0I!!90 = UI!!90 &

P = 0;Q = "UI

U

I % %

S =%N

i=1 Pi + j%N

i=1 Qi S1 = P1 + jQ1

S2 = P2 + jQ2

SN = PN + jQN

!" #$R1 R2

Ug

+

I R1 C

R2 L

P = 100+200 = 300 Q = 50"50 = 0 S =!3002 + 02 = 300

FPcos$

FP = cos$ =P

S=

P$P 2 +Q2

FPFP

FP

!" #$FP

FP FP = P!P 2+Q2

= 100#1002+502

= 0, 89

FPFP

S1

U+

I

C

S1 Red

FP

P Q

FPS1,abierto =P$

P 2 +Q2

S1 CQC Q > 0

QC < 0 FP

FPS1,cerrado =P$

P 2 + (Q +QC)2

Q$C FP FP $

FP $ =P$

P 2 + (Q +Q$C)

2

Q$C

Q$C = P

&1

FP !2" 1"Q

!" #$C

P Q % = 2&f = 2&50

FP $ C =100!1000

!112

!1

2"50·2302 = 6 µF

TE

TE

EACTIV A =

#

TP (t)dt

EREACTIV A =

#

TQ(t)dt

TE

TE

F

F = FPU + FEU

FPU

FEU

PotTPU

FPU = TPU · Pot

Pot

Pot

Pot

Pot

FEU ='

p

(Ep · TEUp)

Ep p

TEUp p

i(t) 45 e(t)R

e(t) = 230!2 cos(2&50t)

+

i(t) R 0, 1 H

10 µF

e(t) = 230!2 cos(2&50t)

+

i(t) 20 ! 20 mH

Zx 10 !

Zx C Zx FPFP

e(t) = 230!2 cos(2&50t)

+

i(t) 2 ! C

Zx

R1 R2 CC R1 = 10 ! C = 1 µF R2 = 10 !

E = 230 V+

i(t) R1

C R2

FPS1 C

S1 FP

E = 230 V+

i(t)

S1

C

20 !

0, 1 H

R = 286, 9 ! uR(t) = 230 cos(2&50t + 45 ) uL(t) = 25, 1 cos(2&50t + 135 ) uC(t) =255, 1 cos(2&50t" 45 )

Pg = 2407, 1 Qg = 756, 3 var S = 2523, 4

Zx = 6 + j8 ' 10!53,1 C = 157, 9 µF

UR1 = 115!0,09 UR2 = 114, 99!!0,09 IC = 0, 036!89,91

FP = 0, 54 (i) C = 72, 1 µF

TP = 16, 96 e E = 320 kWh TE = 36, 71 e Impuesto electricidad = 2, 74 eEquiposmedida = 1, 51 e Total = 67, 19 e

e

e

ee

e "