2010 Transformer ckts

2010 Transformer ckts - Transformers EECS 314 lecture notes

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Unformatted text preview: Transformers EECS 314 lecture notes Transformer
circuits

 Why
do
we
need
them?

 Voltages

and
Currents

 Impedance
Matching
 AC
Power
Distribution

 © 2010 A. Ganago 2010 Transformer circuits 1 A very familiar view •  What exactly are those things on the electric pole? •  What does the picture tell us about the power distribution system? •  What would T.A.Edison say about it? © 2010 A. Ganago 2010 Transformer circuits 2 Explanation High‐voltage

 Pole‐to‐pole
wires
 (12
kV
or
more)
 Step­down
 Transformers
 Low‐voltage

 Pole‐to‐house
wires
 (120
V
or
240
V)

 © 2010 A. Ganago 2010 Transformer circuits 3 Explanation, continued Many
Insulators
For

 Each
High‐Voltage

 Wire
 Step­down
 Transformers
 Fewer
Insulators
For

 Each
Low‐Voltage

 Wire
 © 2010 A. Ganago 2010 Transformer circuits 4 Answers to the Questions •  On the electric pole we see insulators for high voltage (many) and for lower voltage (few); the large cylinders are transformers •  The picture reminds us that we distribute AC (not DC) electric power •  T.A.Edison was a powerful opponent of AC power, in favor of DC power, and he lost the ‘battle of powers’ … But why? © 2010 A. Ganago 2010 Transformer circuits 5 DC Power Distribution (1) RLine
 VDC
 IDC
 VRL
 120
V
 RLoad
 −VDC + VRL + 120
V = 0 © 2010 A. Ganago −V + I ⋅ RLine + I ⋅ RLoad = 0 
 DC 2010 Transformer circuits 6 © 2010 A. Ganago 1 Transformers EECS 314 lecture notes DC Power Distribution (2) RLine
 VDC
 IDC
 VRL
 120
V
 RLoad
 VDC
 DC Power Distribution (3) RLine
 IDC
 VRL
 120
V
 RLoad
 Power
losses
in
the
transmission
line
 become
huge
because
the
same
 current
Nlows
through
RLine
and
RLoad
 © 2010 A. Ganago 2010 Transformer circuits 7 Assume
RLine
=
1
Ω
and
Rload
=
240
Ω
is
a
 single
60‐W
lamp.
Then
99.6%
of
the
 power
from
the
source
goes
to
the
load,
 and
VDC
=
120.5
V
 © 2010 A. Ganago 2010 Transformer circuits 8 DC Power Distribution (4) RLine
 VDC
 IDC
 VRL
 120
V
 RLoad
 VDC
 DC Power Distribution (5) RLine
 IDC
 VRL
 120
V
 RLoad
 Assume
RLine
=
1
Ω
and
Rload
=
1
Ω
is
a
set
 of
240
lamps
60‐W
each,
in
parallel.
Then
 only
50%
of
the
power
from
the
source
 goes
to
the
load,
and
VDC
=
240
V
 © 2010 A. Ganago 2010 Transformer circuits 9 Assume
RLine
=
1
Ω
and
Rload
=
0.01
Ω
is
a
 set
of
24,000
lamps
60‐W
each,
in
parallel.
 Then
only
1%
(!)
of
the
power
from
the
 source
goes
to
the
load,
and
VDC
=
12
kV(!)
 © 2010 A. Ganago 2010 Transformer circuits 10 Improved Power Distribution RLine
 M
 M
 VDC
 IDC
 D
 D
 120
V
 RLoad
 VRL
 1 
 2 
 Power
losses
in
the
transmission
line
 can
be
reduced
by
Miraculous
 Devices
MD1
and
MD2
 © 2010 A. Ganago 2010 Transformer circuits 11 What
each
MD
should
do 
 to
improve
power
distribution 
 Convert
low
 voltage
V1
to
 high
voltage
V2
 (or
high
voltage
 V1
to
low
V2),
 conserving
the
 power: 
 © 2010 A. Ganago I1
 I2
 V1
 M
 D
 
 V2
 2010 Transformer circuits V ⋅I = V ⋅I 
1 1 2 2 12 © 2010 A. Ganago 2 Transformers EECS 314 lecture notes Improved Power Distribution RLine
 M
 M
 VS
 IS
 D
 ILine
 D
 120
V
 RLoad
 1 
 2 
 Reduce
the
current
ILine
through
the
 transmission
line
resistance
RLine
 thus
greatly
reduce
losses
(ILine)2
Rline
 © 2010 A. Ganago 2010 Transformer circuits 13 What are those MDs? •  100
years
ago
there
was
only
 one
answer
to
this
question:
 Transformers!
 •  Today,
there
is
a
new
answer:
 

DC­to­DC
converters

 © 2010 A. Ganago 2010 Transformer circuits 14 So what? •  Since
Transformers
work
only
at
 AC,
our
power
distribution
system
 (built
100
years
ago)
is
AC

 •  Today,
a
DC
system
could
be
built
 (may
be
more
stable
but
requires
a
 whole
new
infrastructure)
 © 2010 A. Ganago 2010 Transformer circuits 15 What is a Transformer ? n1 n2 n2 and n2 are the numbers of turns in the coils Two (or more) coils of wire wound on the same magnetic core. Alternating current (AC) in the primary coil induces an AC current in the secondary coil. Any coil can be used as the primary. © 2010 A. Ganago 2010 Transformer circuits 16 Primary Line n I1 1 Secondary Line n 2 What is a Transformer doing ? Primary Line n1 Secondary Line n2 V1 I2 V2 = 
 Due
to
EM
induction:
 V1 n 
1 V2 n2 
 Due
to
power
conservation:
V1 ⋅ I1 = V2 ⋅ I2 © 2010 A. Ganago 2010 Transformer circuits 17 The power in its secondary coil is practically the same as in the primary. Step-up transformers (n1 < n2) increase voltage and decrease current. Step-down transformers (n1 > n2) decrease voltage and increase current. © 2010 A. Ganago 2010 Transformer circuits 18 © 2010 A. Ganago 3 Transformers EECS 314 lecture notes Key facts about Transformers • The ratio (n2 / n1) of the numbers of turns determines by how much the voltages and currents are increased/ decreased. • The same transformer can be used as a step-up or a step-down, depending on how you connect it to your circuit. • Transformers help us reduce the waste of electric power. © 2010 A. Ganago 2010 Transformer circuits 19 More facts about Transformers Operation depends on electromagnetic induction. Recall your Physics: d all formulas include derivatives thus transformers are dt not working with DC currents. A well-built transformer dissipates very little power: the efficiency can exceed 99% © 2010 A. Ganago 2010 Transformer circuits 20 Things to Remember •  In
many
applications,
we
use
 sinusoidal
voltages
and
currents
 •  Use
formulas
for
AC
power
 (average
power
is
most
important)
 even
if
we
neglect
the
phase
shifts
 between
the
voltage
and
the
 current
in
the
circuit
 © 2010 A. Ganago 2010 Transformer circuits 21 Average Power in AC circuits 2 V (t ) = Vm ⋅ cos(ωt + θV ) I(t ) = Im ⋅ cos(ωt + θ I ) 
 P= Vm ⋅ Im ⋅ cos(θV − θ I ) Power Voltage Current Note that Vm and Im are peak values © 2010 A. Ganago 2010 Transformer circuits 22 Average Power in AC circuits P = VRMS ⋅ I RMS ⋅ cos(θV − θ I ) V (t ) = VRMS ⋅ 2 ⋅ cos(ωt + θV ) I(t ) = I RMS ⋅ 2 ⋅ cos(ωt + θ I ) 
 Power Voltage Current Power Factor In
both
sets
of
expressions
for
 Average
Power
(above),
 cos(θV − θ I ) 
 
is
the
power
factor
PF
 For
the
sake
of
simplicity,
we
 assume
PF
=
1
in
the
discussion
 below


 © 2010 A. Ganago 2010 Transformer circuits 24 Note that VRMS
, IRMS are RMS values © 2010 A. Ganago 2010 Transformer circuits 23 © 2010 A. Ganago 4 Transformers EECS 314 lecture notes Use RMS in calculations (1) •  For
sinusoidal
voltages
and
currents,
 we
use
RMS
(Root‐Mean‐Square)
 values:
they
lead
to
simpler
formulas
 •  In
circuits
with
zero
phase
shift
 between
voltage
and
current,
the
 formulas
for
average
AC
power
with
 IRMS
,
VRMS
are
the
same
as
those
for
 DC
power
with
IDC
,
VDC
in

DC
circuits
 © 2010 A. Ganago 2010 Transformer circuits 25 Use RMS in calculations (2) •  You
may
use
either
RMS
or
peak
 values
of
currents
and
voltages,
but
 be
consistent!
 VV •  For
example,
in
a
transformer:
 1 = 2 
Express
both
voltages
as
RMS,

 n1 n2 
 or
Express
both
voltages
as
peak,
 
but
do
NOT

mix
the
two
notations!
 © 2010 A. Ganago 2010 Transformer circuits 26 
 Three main applications of Transformers •  Impedance
matching
in
order
to
 optimize
the
power
transfer
to
the
load
 •  Rising
the
voltage
in
transmission
lines
 in
order
to
reduce
power
losses
 •  Providing
the
desired
voltage
range
for
 power
supply
(electronics,
etc.)
 © 2010 A. Ganago 2010 Transformer circuits 27 Transformers for the these main applications are distinct! •  In
power
distribution,
transformers
can
be
 nearly
as
large
as
a
house

 •  In
transformers
for
impedance
matching
 coils
may
have
relatively
large
resistances

 •  Coils
of
power
transformers
usually
have
 very
small
resistances.
 © 2010 A. Ganago 2010 Transformer circuits 28 Transformers for Impedance Matching •  Problem:
A
welding
machine
 needs
very
high
current
and
very
 low
voltage
(its
resistance
is
low)
 •  If
connected
directly
to
a
source,
 it
can
short
its
output,
burning
 the
source,
etc.
with
high
current
 © 2010 A. Ganago 2010 Transformer circuits 29 Transformers for Impedance Matching, cont’d •  Solution:
Use
a
step‐down
 transformer:
it
will
make
the
 welding
machine
“look”
like
a
 higher
resistance
load
 •  Thus
we
save
the
source
and
 improve
the
power
transfer!
 © 2010 A. Ganago 2010 Transformer circuits 30 © 2010 A. Ganago 5 Transformers EECS 314 lecture notes Impedance matching (1) ZT VT Impedance matching (2) ZT VT I1 n1 n2 I2 ZLOAD I1 n1 n2 I2 ZLOAD V1 V2 V2 I2 31 V1 V1 n 
1 V2 = V2 n2 32 In
the
right
loop:
 Z LOAD = In
the
transformer: 
 Also,
 
V1 ⋅ I1 = V2 ⋅ I2 © 2010 A. Ganago 2010 Transformer circuits 
 © 2010 A. Ganago 2010 Transformer circuits Impedance matching (3) ZT VT Impedance matching (4) ZT VT 2 I1 n1 n2 I2 ZLOAD I1 V1 V1 
 I1 = V2 V2 ⎛ n1 ⎞ ⋅⎜ ⎟ I2 ⎝ n2 ⎠ V1 The
source
“sees”
 the
Apparent
load! 
 ZLOAD, Apparent Thus
in
the
left
loop: 
 © 2010 A. Ganago Z LOAD ,
 Apparent 
 33 © 2010 A. Ganago ⎛ n1 ⎞ = Z LOAD ⋅ ⎜ ⎟ ⎝ n2 ⎠ 2 2010 Transformer circuits 2010 Transformer circuits 34 The Case of Welding Machine Impedance Matching (5) •  Choose
a
step‐down
transformer
 to
match
the
APPARENT
load
 impedance
with
the
source
 impedance
 •  For
example,
match
your
speaker
 to
your
output
audio
ampliNier
 © 2010 A. Ganago 2010 Transformer circuits 36 n1 n 
2 n1 n 
2 Assume
ZLOAD
=
0.1
Ω
 = 10  
ZLOAD,
Apparent

=
10
Ω
 = 100  
ZLOAD,
Apparent

=
1000
Ω
 2010 Transformer circuits 35 © 2010 A. Ganago © 2010 A. Ganago 6 Transformers EECS 314 lecture notes Audio Transformer Specs Impedance Matching (6) •  By
choosing
the
transformer,
we
 can
improve
power
transfer
to
the
 load,
without
altering
impedances
 of
the
load
and/or
the
source!
 •  In
DC
circuits,
this
option
is
not
 available.

 © 2010 A. Ganago 2010 Transformer circuits 37 © 2010 A. Ganago 2010 Transformer circuits 38 Transformers in Power Distribution Systems (1) •  Step‐up
transformers
reduce
the
 current
through
the
transmission
 line:
this
helps
to
reduce
the
 losses
of
power
 •  Voltages
up
to
1
MV
are
used
for
 long
distance
transmission
 © 2010 A. Ganago 2010 Transformer circuits 39 Transformers in Power Distribution Systems (2) •  Step‐down
transformers
reduce
 the
voltage
from
very
high
in
the
 transmission
line
to
reasonably
 low,
safer
for
the
consumer
 •  Several
sets
of
step‐down
 transformers
in
large
systems
 © 2010 A. Ganago 2010 Transformer circuits 40 Transformers in Power Distribution Systems (3) •  Household
voltages
are
120
VRMS
 in
the
US;
240
VRMS
in
Europe
and 
 Australia,
etc.
 •  Step‐down
transformers
are
 used
in
power
supplies
of
 electronic
devices
using
5
–
40
V.

 © 2010 A. Ganago 2010 Transformer circuits 41 AC Power distribution system I1 120

 VRMS
 I2 RLine I3 RLoad 12
kVRMS
 n1 n2 n3 n4 If
I1/I2=100,
the
loss
of
power
in
RLine
 is
reduced
by
a
factor
of
10,000

 © 2010 A. Ganago 2010 Transformer circuits 42 © 2010 A. Ganago 7 Transformers EECS 314 lecture notes Solving a problem (1) I1 VS1 Solving a problem (2) I1 RLoad VS1 I2 RLine I3 120
VRMS
 2 I2 RLine I3 120
VRMS
 RLoad 12
kVRMS
 n1 n2 12
kVRMS
 n1 n2 Begin
with
the
given
 (120
V ) = 120
V RLoad = power
at
the
load
 Power I3 © 2010 A. Ganago 2010 Transformer circuits n3 n4 
 n3 n4 From
I3
and
the
given
voltages,
Nind
I2
 (assume
power
conservation)

 © 2010 A. Ganago 2010 Transformer circuits 44 43 Solving a problem (3) I1 VS1 I1 I2 RLine I3 I2 RLine I3 120
VRMS
 RLoad VS1 n1 n2 12 kV 120 V n3 n4 RLoad 12
kVRMS
 n1 n2 n3 n4 From
I2
and
the
given
RLine
,
Nind
the
 power
wasted
in
the
line
 © 2010 A. Ganago 2010 Transformer circuits 45 Note that, if n3>>n4 the apparent load RLoad, Apparent can greatly exceed the actual resistance of the line RLine thus much more power is transferred to RLoad © 2010 A. Ganago 2010 Transformer circuits 46 Transformer in Household 120/240V Distribution System Primary Line Secondary Line Colors of Black (hot) insulation Reminder:
Any
coil
of
a
 transformer
can
act
as
the
 primary! 
 •  The
next
slide
shows
the
 practical
example
of
a
very
 dangerous
circuit
that
lacks
a
 special
switch
(DPDT
type)
 ©
2010
A.
Ganago
 2010
Transformer
circuits 
 48 
 13,200 V 120 V 120 V White 240 V (neutral) Red (hot) 47 All voltages are RMS © 2010 A. Ganago 2010 Transformer circuits © 2010 A. Ganago 8 Transformers EECS 314 lecture notes I2 RLine I3 12 kV 120 V n3 n4 RLoad VSB Transformers in Power Distribution Systems •  Can
be
‘as
big
as
a
house’
 •  Step‐up
to
1
MV
=
1,000,000
V
 •  Step‐down
to
consumer‐level
 voltages
120
V
RMS
 •  Several
stages
of
step‐up/down
 © 2010 A. Ganago 2010 Transformer circuits 50 A
household
backup
generator
VSB
=
120
V
 (used
in
case
of
power
outage)
 
 can
send
12
kV
to
the
transmission
line,
 life‐threatening
for
the
repair
workers! 
 © 2010 A. Ganago 2010 Transformer circuits 49 Transformers in Power Supplies •  Widely
used
in
many
electronic
 devices
(including
chargers)

 •  Usually,
step‐down
from
120
VRMS
 to
5
–
40
VRMS
 •  Center‐tap
(CT)
transformers
offer
 more
Nlexibility

 © 2010 A. Ganago 2010 Transformer circuits 52 A Very Inexpensive Transformer Notes For the User •  If
you’ve
never
used
transformers,
 call
manufacturers’
tech
support
 and
ask
for
advice

 •  Carefully
read
the
speciNications
 •  Consult
manufacturers’
Application
 Notes
for
standard
solutions
 © 2010 A. Ganago 2010 Transformer circuits 53 © 2010 A. Ganago 2010 Transformer circuits 54 © 2010 A. Ganago 9 ...
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This note was uploaded on 12/06/2010 for the course EECS 314 taught by Professor Ganago during the Spring '07 term at University of Michigan.

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