Transcript Document

Magnetics Design
Primary Constraints:
Peak Flux Density (B field) in the core : Bmax (T or Wb/m2)
Core losses
Saturation
Peal Allowable Current density in any winding : Jmax (A/m2)
Resistive losses
Heat
Wire with cross section Acond , carrying
Irms amperes of current, has current density :
Wire Cross Section:
Acond (m2)
If we have a limit on J < Jmax, then for each
winding we must choose a wire gauge with :
J
I rms
Acond
Acond
I rms

J max
Core Window Area: AW
(window through which all windings must pass)
Torroid
AW
E Cores
N Turns
of wire
with
cross
section
Acond
Fill Factor kw, the fractional part of
the window actually occupied by
conductor cross sections.
Applying our previous constraint to
Acond, for a single winding (inductor):
AW
Aw  NAcond
kw Aw  NAcond
NI rms
Aw 
kw J max
Inductor Core Cross Section: Acore
The core cross section must accommodate the
peak induced flux without exceeding the
maximum allowable flux density.
For an inductor, the peak flux is proportional
to peak current and inductance:
Acore
LIˆ

NBmax
Acore 
ˆ
Bmax
ˆ
LI
ˆ 
N
Inductor Air Gap: lgap
The total reluctance of the magnetic path must be:
RT = Rcore  Rgap
lgap
lcore
N2



core Acore 0 Agap L
The inductor air gap length is computed to
provide the appropriate reluctance in the
magnetic path for the desired inductance
with the chosen number of turns:
Since Ag ~ Acore, and 0 << core, the gap
length can be approximated as:
lgap
0 Agap
N2

0 Agap  lcore
L
core Acore
N 2 0 Acore
lg 
L
Transformers (multiple windings)
The different windings often must accommodate different currents, in
which case they must have different gauges (different Acond):
kw Aw  N1 Acond ,1  N2 Acond ,2 
  N y Acond , y
y
However, maximum current density must not be exceeded for
any winding.
Recall: Acond
Thus:
I rms

J max
N I
y rms , y
Aw 
y
kw J max
Applies to all
windings.
Rated Current for
winding “y”.
Transformer Core Cross Section: Acore
In a transformer, the flux density may not exceed +/- Bmax over one AC
operating cycle at the worst case operating condition.
Let T be the maximum excitation time for the primary. This will be at most, half the
period, or kc /fs , where kc is parameter reflecting the operating duty cycle of a power
converter, typically 0.5 for a bipolar square wave (worst case flux density).
For a constant (bipolar square
wave) primary voltage, Faraday’s
Law tells us that the change in
magnetizing flux will be :
Therefore, the maximum change in
flux density (-Bmax to +Bmax) for a
voltage applied in one polarity is:
This places a lower limit on the
core cross section:
m 
Vpri
N pri
Vpri kc
T 
N pri f s
Vpri kc
m

 2Bmax
Acore N pri f s Acore
Acore 
kcVpri
2 f s N pri Bmax
Additional Notes on Transformer Cores
In conventional transformer applications, the flux in the core reverses completely during
each half cycle of the primary voltage waveform, as reflected in the above discussion.
The author chose to derive the relations and present an example using the forward
converter topology. The forward converter is an unconventional transformer
application, in that the core is magnetized in one direction and then completely
demagnetized during each cycle, as illustrated in Figure 9.2 of the text.
As a result, each cycle begins with zero flux in the core, increases to a maximum, and
decreases back to zero. Note that power is delivered to the secondary circuit by
transformer action only while the flux is increasing, and the maximum allowable
change in flux is only half of that allowable in a conventional transformer application.
For sinusoidal AC operation, and Vpri is RMS, a value of kconv = 0.45 provides the
equivalent magnetizing volt-seconds. Since author omitted the “2” in the denominator
of 9.9, he suggests a value of kconv = 0.225 be used in equations 9.6, 9.9, 9.12, and 9.14
In problem 9.8, if the author’s convention is followed, the values of kconv will be half of
the values obtained when the above convention is used.
Area Product
We now have expressions for the minimum core window and cross section,
as functions of independent maximum ratings and operating parameters,
which tells us how big the device must be. These expressions are used to
express a useful magnetic core design parameter we call the Area Product:
 N y I rms , y  kc  Vy I rms , y
 kcV pri   

y
y
Ap  Acore,min A w,min  



 2 f s N pri Bmax   k w J max  2 f s k w Bmax J max


ˆ
 LIˆ   NI rms 
LII
rms
Ap  Acore ,min A w,min  



NB
k
J
 max   w max  kw Bmax J max
For conventional
transformer
applications
For Inductors
Note that Area product is independent of the number of turns!!
** For sinusoidal operation, since Vpri is RMS, the Vy are also RMS.
Core Selection
From vendor data, we select a core with Ap, Aw, and Acore which meet our
minimum requirements.
For an inductor, the number of turns is
computed using:
For a conventional transformer, the
number of turns for each winding is
computed using :
LIˆ
N
Acore Bmax


kc
N y  Vy 

2
A
B
f
 core max s 
Note typo, eqn 9.14
The inductor air gap length is computed
using the approximation derived previously:
N 2 0 Acore
lg 
L