### Lecture 18 Power in AC Circuits

```Lecture 18
Power in AC Circuits
Hung-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
Resistor with
resistance R
P  iv  i R 
2
v
2
A +
v - B
R
i
reference current should flow from “+” to “-”
Negative Communed Power
= Supplied Power
Power for AC Circuits
• Consumed Instantaneous Power
A +
p t   v t   i t 
• If p(t) is periodic with period T
• Average Power
P 
1
T
t T
 p t dt
t
v t 
i t 
- B
Average Power for AC Circuits
i t 
v t 
Network
Z
i t   I m cos  t   i 
v t   V m cos  t   v 
p t   v t i t   V m I m cos  t   v  cos  t   i 

1
2
I m V m cos  v   i   cos  2  t   v   i 
cos  cos  
1
2
cos     
1
2
cos   

Average Power for AC Circuits
p t  
1
2
2
2
2
I m V m cos  v   i   cos  2  t   v   i 
Absorb power
I m V m cos  v   i   1 
1
1
1
I m V m cos  v   i 
I m V m cos  v   i   1 
Supply power
 1  cos  v   i   1
Average Power for AC Circuits
i t 
v t 
Network
Z
p t  
P 
1
2
P 
1
2
i t   I m cos  t   i 
I  Im  i
v t   V m cos  t   v 
V  Vm   v
I m V m cos  v   i   cos  2  t   v   i 
I m V m cos  v   i 
1
2
I  I m   i
*
V I  V m I m   v   i 
*

Re V I
*

  V
Re V I
*
m
I m cos  v   i 
Average Power for AC Circuits
i t 
v t 
Network
Z
i t   I m cos  t   i 
I  Im  i
v t   V m cos  t   v 
V  Vm   v
Vm
Z  R  jX
Z
V

I
R 
Vm
Im
P 
cos  v   i 
1
2
X 
Im
Vm
Im
I m V m cos  v   i  
  v   i 
sin  v   i 
1
2
RI
2
m

Z 
Im
1 R
2 Z
Vm
2
2
Vm
Summary
i t   I m cos  t   i 
I  Im  i
v t   V m cos  t   v 
V  Vm   v
P 
1
2
P 
1
2
I m V m cos  v   i 

Re V I
*

P 
Z  R  jX
1
2
P 
2
RIm
R
2Z
2
2
Vm
Revisit
Maximum Power Transfer
Review: Maximum Power Transfer
for DC Circuits
power
Device
Power consumed
by Rs
Real source
RL  Rs
2
vs
4RL
Power consumed
by RL
Rs is fixed, increase RL
Maximum Power Transfer
for AC Circuits
Zs
ZL
Real source
Device
Real source
Source Impedance: Z s  R s  jX s
Find ZL such that the device can obtain
maximum power
Z L  R L  jX
L
Device
Maximum Power Transfer
for AC Circuits
Voltage Source: V
VL 
I
PL 
1
2
ZL
ZL  Zs
V

R e VL I
*

Z L  R L  jX
ZL
V
*
ZL  Zs
s
L
Zs
V
I 
Z s  R s  jX

*
ZL  Zs
*

VL
*
Real source
*


1
ZL
V
 Re
V *
* 
2
ZL  Zs 
 ZL  Zs
2



R

jX
V
1
L
L
 Re
2
2
2




R

R

X

X
 L
s
L
s
Device
2
 1
RLV

2
2
 2  R L  R s    X L  X s 
Maximum Power Transfer
for AC Circuits
PL 
1
RLV
2
2 R L  R s   X L  X s 
2
ZL  Zs
*
Z L  R L  jX
Find ZL that maximize P
2
Find XL and RL that maximize P
XL = -Xs
RL = Rs
Z s  R s  jX
P max 
1 V
2
2 4 RL

1 V
2
2 4 Rs
s
L
Maximum Power Transfer
for AC Circuits
Z L  R L  jX
PL 
RLV
1
L
Z L  Z L cos   j Z L sin 
2
2 R L  R s   X L  X s 
2
Only |ZL| can be tuned
2
θ is fixed (R/X is fixed)
PL 
| Z L | cos  V
1
2
2 | Z L | cos   R s   | Z L | sin   X s 
2
 PL
 | ZL |
0
PL is maximized when
Special case:
If
Z  R
2
| Z L |
R 
R s  X s | Z s |
2
2
Rs  Xs
2
2
Maximum Power Transfer
for AC Circuits – Example 1
• Determine ZL that maximize the power drawn from the
circuit. What is the maximum power?
ZT
V oc
ZL  Z
*
T
Z L  R  jX
2
P max 
1 V oc
2 4R
Maximum Power Transfer
for AC Circuits – Example 1
• Network
Find ZT
Z T  j 5  4 || 8  j 6 
 2 . 933  j 4 . 467
ZT
Z L  Z T  2 . 933  j 4 . 467
*
2
P max 
1 V oc
2 4R
R  2 . 933
Maximum Power Transfer
for AC Circuits – Example 1
• Network
V oc 
V oc
8  j6
4  8  j 6 
10
 7 . 454   10 . 3

Maximum Power Transfer
for AC Circuits – Example 2
• Find RL that can absorb maximum power
ZT
V oc
R L  ZT
RL 
9 . 412
2
 22 . 35
Z T   40  j 30  || j 20  9 . 412  j 22 . 35
2
Maximum Power Transfer
for AC Circuits
Power Transfer Efficiency:
E ff 
PL 
Ps 
1
Z L  R L  jX
Zs
PL
ZL
Ps  PL
RLV
Real source
2
2 R L  R s   X L  X s 
2
1
Z s  R s  jX
RsV
2
E ff 
2
2 R L  R s   X L  X s 
2
Device
2
RL
Rs RL
s
L
Example 7.4
Zs
Z s  R s  jX
Z L  R L  jX
s
ZL
L
Real source
E ff 
RL
Rs RL
Device
Find Power
Transfer Efficiency
E ff 
20
2  20
 91 %
Real source
Example 7.4
Real source
Zs  2
Z L    10 j  || 20  4  8 j
E ff 
RL
Rs RL

4
24

RL  4
2
3
Homework
• 7.4, 7.14, 7.20
Thank you!