lecture 18 power in ac circuits hung-yi lee. outline textbook: chapter 7.1 computing average power...

Lecture 18Power in AC

CircuitsHung-yi Lee

Outline

• Textbook: Chapter 7.1• Computing Average Power• Maximum Power Transfer for AC circuits• Maximum Power Transfer for DC circuits has

been discussed in Chapter 3.1

Average Power for AC Circuits

Review: Power for DC Circuits • Consumed Power for DC Circuits

ivP A Bv

i

+ -

reference current should flow from “+” to “-”

Negative Communed Power = Supplied Power

Resistor with resistance R

Ri2R

2v

Power for AC Circuits

• Consumed Instantaneous Power

• If p(t) is periodic with period T• Average Power

titvtp A B tv

ti

+ -

Tt

t

dttpT

1P

Average Power for AC Circuits

ti

tv NetworkZ

im tti cosI

vm ttv cosV

ivmm ttItitvtp coscosV

cos2

1cos

2

1coscos

ivivmm tI 2coscosV2

1

Average Power for AC Circuits

ivivmm tItp 2coscosV2

1

1cosV2

1 ivmmI

1cosV2

1 ivmmI

ivmmI cosV2

1

1cos1 iv

Absorb power

Supply power

Average Power for AC Circuits

ti

tv NetworkZ

im tti cosI

vm ttv cosV

im II

vm VV

ivmmI cosV2

1P

*R2

1P IVe

im II*

ivmm IVIV *

ivmme cosIVIVR *

ivivmm tItp 2coscosV2

1

Average Power for AC Circuits

ti

tv NetworkZ

im tti cosI

vm ttv cosV

im II

vm VV

ivm

m I

V

I

VZ jXRZ

ivm

m

I cos

VR iv

m

m

I sin

VX

ivmmI cosV2

1P 2R

2

1mI 2

2 VR

2

1m

Z

m

m

I

VZ

Summary

ivmmI cosV2

1P

*R2

1P IVe

im tti cosI

vm ttv cosV

im II

vm VVjXRZ

2R2

1P mI

22 V

2

RP m

Z

Revisit Maximum Power

Transfer

Review: Maximum Power Transfer for DC Circuits

Real source

Device (Load)

power

Rs is fixed, increase RL

Power consumed by Rs

Power consumed by RL

sRR L

L

s

R

v

4

2

Maximum Power Transfer for AC Circuits

Real sourceDevice (Load)

Source Impedance: sss jXRZ

LLL jXRZ

Find ZL such that the device can obtain maximum power

Real source Device (Load)

sZ

LZ

Real source Device (Load)

sZ

LZ

Maximum Power Transfer for AC Circuits

sZZ

L

VI

VVoltage Source:

*LL R

2

1P IVe

**L

** V

IsZZ

**

L

*

L

L

ZZZZ

ZR

2

1

ss

VVe

VVL

LL

sZZ

Z

sss jXRZ

LLL jXRZ

LV

2

sL2

sL

2

LL

XXRR

jXRR

2

1 Ve

2sL

2sL

2

L

XXRR

R

2

1

V

Maximum Power Transfer for AC Circuits

2sL

2sL

2

LL

XXRR

R

2

1P

V

Find XL and RL that maximize P

XL = -Xs

RL = RsL

2

max 42

1P

R

V

Find ZL that maximize P

sss jXRZ

LLL jXRZ

s

2

42

1

R

V

*sL ZZ

Maximum Power Transfer for AC Circuits

LLL jXRZ

sinZcosZ LLL jZ

Only |ZL| can be tuned

θ is fixed (R/X is fixed)

2sL

2sL

2

LL

Xsin|Z|Rcos|Z|

cos|Z|

2

1P

V

0|Z|

P

L

L

|Z|XR|Z| s22

L ss

22 XRR ss

PL is maximized when

Special case: RZ If

2sL

2sL

2

LL

XXRR

R

2

1P

V

Maximum Power Transfer for AC Circuits – Example 1• Determine ZL that maximize the power drawn from the

circuit. What is the maximum power?

TZocV

*TL ZZ

R

Voc

42

1P

2

max jXRZL

Maximum Power Transfer for AC Circuits – Example 1• Network

Find ZT

TZ

68||45ZT jj

467.4933.2ZZ *TL j

933.2R R

Voc

42

1P

2

max

467.4933.2 j

Maximum Power Transfer for AC Circuits – Example 1• Network

ocV 10684

68V

j

joc

3.10454.7

Maximum Power Transfer for AC Circuits – Example 2• Find RL that can absorb maximum power

TZocV

TL ZR 22L 35.22412.9R

35.22412.920||3040 jjjZT

Maximum Power Transfer for AC Circuits

Real source Device (Load)

sZ

LZ

sss jXRZ

LLL jXRZ

2sL

2sL

2

LL

XXRR

R

2

1P

V

2sL

2sL

2

ss

XXRR

R

2

1P

V

Power Transfer Efficiency:

Ls

L

PP

Pff

E

Ls

LffRR

RE

Example 7.4

Real source Device (Load)

sZ

LZsss jXRZ

LLL jXRZ Ls

LffRR

RE

Real source Device (Load)

202

20E

ff

%91

Find Power Transfer Efficiency

Example 7.4

Real source Device (Load)

2sZ

jjZ 8420||10L 4R L

Ls

LffRR

RE

3

2

42

4

Homework

• 7.4, 7.14, 7.20

Thank you!

Answer

• 7.4: Irms = 5.65mA, 160mW• 7.14: R=141Ω, C=0.707μF• 7.20: (a) (12-5j)k Ω, 120 μW, 50%; (b) (10.4-7.8j)k Ω,

90 μW, 46%; (c) 13k Ω, 115 μW, 52%

Acknowledgement

• 感謝 吳東運 (b02)• 指出投影片中的錯誤