Lecture 22 Mutual Inductance & Transformers
Transcript of Lecture 22 Mutual Inductance & Transformers
Lecture 22Lecture 22Mutual Inductance &
Transformers
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Flux Linkages and Faraday’s LLaw
Magnetic flux passing through a surface area A:
φ d⋅= ∫ ABA∫
For a constant magnetic flux density perpendicular
BA=to the surface:
The flux linking a coil with N turns:
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
φλ N=
Faraday’s Lawy
Faraday’s law of magnetic induction:
dtde λ
=dt
The voltage induced in a coil whenever its gflux linkages are changing. Changes occur from:
• Magnetic field changing in time
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
• Coil moving relative to magnetic field
Lenz’s Law
Lenz’s law states that the polarity of the induced voltage is such that the voltagewould produce a current (through an external resistance) that opposes the original change in flux linkages.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Mutual Inductance & TransformersMutual Inductance & Transformers
1 Determine the inductance and mutual1. Determine the inductance and mutual inductance of coils given their physical parametersparameters.
2. Understand ideal transformers and solve circuits that include transformerscircuits that include transformers.
3. Use the equivalent circuits of real transformers to determine their regulationstransformers to determine their regulations and power efficiencies.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Inductance and Mutual InductanceDefinition of inductance L:
linkagesFluxL λ==
icurrentL ==
φλ N=Substitute for the flux linkages using φλ N=Substitute for the flux linkages using
Nφi
NL φ=
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Inductance and Mutual Inductance
NiSubstituting
RNi
=φ
2NL =R
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Faraday’s Lawy
Voltage is induced in a coil when its fluxVoltage is induced in a coil when its flux linkages change:
diLLidde )(λdt
Ldtdt
e ===
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Mutual InductanceSelf inductance
Self inductance
11Lλ
= ← 22Lλ
= ←
inductance for coil 1
inductance for coil 2
11
11 i
L
λ=
=
22
22 i
L
λ
=
1i=
2i=
Mutual inductance between coils 1 and 2:
12212112M λλλλ==== ←←
Mutual inductance between coils 1 and 2:
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
2211 iiiiM ====
Mutual Inductance
Total fluxes linking the coils:
12111 λλλ ±=
21222
12111
λλλ ±= 21222 λλλ ±=
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Mutual Inductance
Currents entering the dotted terminals produce aiding fluxes
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Circuit Equations for Mutual Inductance
iLMiMiiL 2111
+±=±=
λλ
diMdiLde
iLMi
211
2212
±
+±=λ
λ
dididdt
Mdt
Ldt
e 211
11 ±==
λdtdiL
dtdiM
dtde 2
212
2 +±==λ
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Example
WeberturnsampereR /)(107 −=
mHNL 11007
221
1 ===R
p )(
Self inductance:
mHNL 420010
7
222
2
71
===
R
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
1072 R
Example
15
7111
1 1010
100 iiiN −===R
φ iN 10200 15
1221 ×== −φλ
mHi
M 21
21 ==λ
Mutual inductance:
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
i1
Example
Does the flux produced by i aid or opposeDoes the flux produced by i2 aid or oppose the flux produced by i1?
didi 21
dididtdiM
dtdiLe
12
2111 −=
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
dtdiM
dtdiLe 12
22 −=
Exercise 15.13
• •• •
Mark the dot for coil 2
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Fluxes from 1 and 2 aid each other in paths 1&2, oppose in path 3
Transformers
Can be used to step up or step down ac voltages
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Can be used to step up or step down ac voltages
Electrical Power Distribution
)cos(d l d IVP = θ )cos(rmsrmsdelivered IVP = θ
2rmslineloss IRP = rmslineloss IRP
Minimize loss by increasing voltage and decreasing current. Modern transmission grids
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
use ac voltages up to 765,000 Volts
War of the Currents
In the "War of Currents" era in the late 1880s, Nikola Tesla and Thomas Edison became adversaries due to Edison's promotion of direct current (DC) for electric power distribution over the more efficient alternating current (AC) advocated by Tesla.efficient alternating current (AC) advocated by Tesla.
During the initial years of electricity distribution, Edison's direct current was the standard for the United States and Edison was not disposed to lose all his patent royalties. Direct current worked well for the incandescent lamps that were the principal load of the day From his work with rotary magnetic fields Teslaload of the day. From his work with rotary magnetic fields, Tesla devised a system for generation, transmission, and use of AC power. He partnered with George Westinghouse to
i li hi
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
commercialize this system.
War of the Currents
Edison's propagandaEdison s propaganda
Edison went on to carry out a campaign to discourage the use of alternating current. Edison personally presided over several g p y pexecutions of animals, primarily stray cats and dogs, to demonstrate to the press that his system of direct current was safer than that of alternating current Edison's series of animalsafer than that of alternating current. Edison s series of animal executions peaked with the electrocution of Topsy the Elephant. He also tried to popularize the term for being electrocuted as being "Westinghoused".
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
War of the CurrentsTopsy the Elephant (circa 1875 - January 4, 1903) was a member of a domesticated herd at Coney Island's Luna Park. She had been a part of the Forepaugh Circus Topsy was deemed an ill tempered and dangerousForepaugh Circus. Topsy was deemed an ill-tempered and dangerous animal since she had killed three men in as many years, including an abusive trainer who tried to feed her a lit cigarette.
i l h d id d h d hBecause Topsy was so violent, her owners decided to put her to death. A proposal of hanging was abandoned after the American Society for the Prevention of Cruelty to Animals protested. The elephant was offered a carrot poisoned with cyanide but did not eat it Later Thomas Edisoncarrot poisoned with cyanide, but did not eat it. Later, Thomas Edison suggested electrocution, using the Westinghouse alternating current system of electricity transmission, which Edison, a backer of direct current, argued was more dangerous than DC The ASPCA found this suggestionwas more dangerous than DC. The ASPCA found this suggestion acceptable, viewing electrocution as a more humane form of killing. Electrocution killed Topsy quickly. Edison recorded the execution with a motion picture camera, and showed his film to audiences around the
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
p ,country as part of his unsuccessful attempt to discredit AC.
Ideal Transformers
Law sFaraday'by )cos()( 111 dtdNtVtv mφω ==
)sin()cos()(1
1
1
1
101
1t
NV
dttN
Vdttv
Nφ(t) m
t tm ω
ωωφ∫ ∫ ==+=
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
10 011 NNN ω
Ideal Transformers
[ ])sin()(1
1222 t
dtd
NV
NdtdNtv m== ω
ωφ
)()cos( 12
12
1
tvNNtV
NN
dtNdt
m == ω
ω
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
11 NN
Ideal Transformers
( ) ( )tNt 2( ) ( )tvN
tv 11
22 =
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Ideal Transformers
0 since02211 iNiN ≈≈=−= RRF φThe magneto-motive force (mmf) applied to the core:
)()( 1
2211
tiNti
iNiN
=
=
If the voltage is stepped up
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
)()( 12
2 tiN
ti = g pp pthe current is stepped down
Ideal Transformers
)()()()()( 12 tiNtvNtitvtp ==
)()()(
)()()()()( 12
11
222
ttit
tiN
tvN
titvtp ==
)()()( 111 tptitv ==
( ) ( )( ) ( )tptp 12 =
Net power is neither generated nor consumed by an ideal transformer
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
an ideal transformer
Ideal Transformers
( ) ( )tvNtv 2=( ) ( )tvN
tv 11
2 =
)()( 11
2 tiNti = )()( 12
2 N
( ) ( )tptp 12 =
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Mechanical Analogg
d11
d2
( ) ( ) 12
212
2 dlldtv
NNtv ==
11
211
2
11
)()( FllFti
NNti
lN
==
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
22 lN
Transformer SummaryTransformer Summary
1. We assumed that all of the flux links all of the windings of both coils and that the resistance of the coils is zero. Thus, the voltage across each coil is proportional to the number of turns on the coil.
( ) ( )N 2( ) ( )tvNNtv 1
1
22 =
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
2 We assumed that the reluctance of the core is2. We assumed that the reluctance of the core is negligible, so the total mmf of both coils is zerozero.
( ) ( )tiNNti 2
12 =
N 2
3. A consequence of the voltage and current q grelationships is that all of the power delivered to an ideal transformer by the ysource is transferred to the load.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Analysis of a Circuit Containing anAnalysis of a Circuit Containing an Ideal Transformer
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Example
Find the rms values of the currents and voltages with the switch open and closed. With the switch open:
2211011
22 === VVVNV rmsrms
sourcethefromtakenispowerNo0
5
22
1
11
2
→==
rmsrms
INI
N rmsrms
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
sourcethefrom taken ispower No021
1 →==rmsrms
IN
I
Example
With the switch closed:V
AVR
VI rms
rmsL
2.210222
2 =Ω
==
AAINI
L
44.02.210
22
1 =×==
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
N rmsrms 521
1
Impedance Transformations
LZ2 =IV
NNNN 112
2
)/()/(
=IV
I
LLLL ZNZZNZNN
NN2
1''2
2'12
121
)/(
)/(
⎟⎟⎞
⎜⎜⎛
=→⎟⎟⎞
⎜⎜⎛
==
I
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
LLLL ZN
ZZN
ZNN 2121 )/( ⎟⎟
⎠⎜⎜⎝
=→⎟⎟⎠
⎜⎜⎝
==
Impedance Transformations
ZNZ2
11⎟⎟⎞
⎜⎜⎛
==′V
LL ZN
Z21⎟⎟⎠
⎜⎜⎝
==I
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Example 15.11
Find the phasor voltages and currents. First, transform ZL to the primary side of the transformer:
2010
22
⎟⎞
⎜⎛
+=
N
jZ L
200010001000
20001000)2010()10('2
2
1
+++
+=+=⎟⎟⎠
⎞⎜⎜⎝
⎛=
jZRZ
jjZNNZ LL
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.452828452200020002000
200010001000'1
∠=∠=+=
++=+=
j
jZRZ Ls
Example 15.11
oo
o
453536.045282801000
1 −∠=∠∠
== AZ
sVI
o
o
)20001000)(453536.0(
452828
'11
1
+−∠==
∠
jZ
Z
L
s
IVoo )43.632236)(453536.0(
))((11
∠−∠=
jL
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
o43.186.790 ∠=
Example 15.11
We can now calculate the current andWe can now calculate the current and voltage phasors on the secondary side using the turns ratio:
oo 45536.3)453536.0(1011
2 −∠=−∠== II N
t e tu s at o:
oo 43186079)43186790(1
45536.3)453536.0(10
2
12
2
∠∠
∠∠
VV
II
NN
oo 43.1860.79)43.186.790(101
1
22 ∠=∠== VV
N
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Example 15.12
Th lt d R l b t f d t th dThe voltage source and R1 can also be transformed to the secondary side:
1N∠== 01000
101
1
2' VNNV ss
o
Ω=⎟⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛= 10)1000(
101 2
1
2
1
2'1 R
NNR
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
⎠⎝⎠⎝ 101N
Example 15.12
o )0100(20202010
2010102010 '
2 ∠++
=++
+=
jjV
jjV s
o
o)0100(
45220)10/20(400100
000001
∠∠
∠+=
−Tan
jj
ooo
o
43.1806.79)0100(43.6351045220
∠=∠∠
=
∠
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
o43.1806.79)0100(
45220∠∠
∠
Exercise 15.17
Find the values of V2 and the power delivered to the load. To find I2, transform Vs and Rs from the primary to the secondary idside:
∠=∠== 0400)0100(14
1
2'ss N
N ooVV
Ω==⎟⎟⎞
⎜⎜⎛
= 64016
12
2'
1
RRNR
N
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Ω==⎟⎟⎠
⎜⎜⎝
= 640161
SSS RRN
R
Exercise 15.17
RVs 0385.0
4006400400
400'
'
2 ∠=+∠
=Ω+
= oo
I
R
R
L
S
08.153)400)(0385.0(
400640400
22 ∠=Ω∠==
+Ω+ooIV
WRIP Lrms6.29)400(
2385.0)(
22
2 =Ω⎟⎟⎠
⎞⎜⎜⎝
⎛==
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
rms 2 ⎟⎠
⎜⎝
Exercise 15.18
Find the turns ratio that maximizes the power transfer to the load. What turns ratio transforms the 400Ω load resistance into a
i t f 40Ω th i id ?resistance of 40Ω on the primary side?
40)400(2
12
1' Ω=Ω⎟⎟⎞
⎜⎜⎛
=⎟⎟⎞
⎜⎜⎛
=NRNR LL
1140
40)400(
12
1
22
Ω⎟⎞
⎜⎛
ΩΩ⎟⎟⎠
⎜⎜⎝
⎟⎟⎠
⎜⎜⎝
NN
NR
NR LL
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
101
101
40040
2
1
2
1 =→=ΩΩ
=⎟⎟⎠
⎞⎜⎜⎝
⎛NN
NN
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.