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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
