March 14,
1997

Spreadsheet
simplifies switch-mode power-supply flyback-transformer design

Brooks
Leman, Power Integrations

Designing
flyback transformers for switch-mode power supplies involves many
calculations. A spreadsheet can make short work of this
otherwise-tedious task.

When you develop flyback power
supplies, especially continuous-mode supplies, transformer design is
usually the biggest stumbling block. Reference 1 uses an Excel spreadsheet as the centerpiece of simple
flyback-transformer design technique. With the spreadsheet, continuous or
dis- continuous-mode flyback-transformer design takes less than 10
minutes. Compare this operation with the tedious, time-consuming task
of manually number-crunching more than 50 independent parameters and
dependent variables.

An improved spreadsheet incorporates many
new features, including a means of accurately estimating magnet-wire
gauge and current capacity. An energy equation, which uses loss
allocation, now determines the continuous-mode primary inductance.
The improved spreadsheet also calculates output-capacitor ripple
current and estimates MOSFET switch-drain peak-stress voltage. To
simplify design and iteration, the peak switch current and duty
cycle are now dependent variables instead of independent parameters,
as in Reference 1. For multiple-output supplies, the new spreadsheet calculates
auxiliary-diode peak-inverse voltage and transformer auxiliary-winding
turns. The remaining calculations remain unchanged.

Contact Power Integrations Inc for copies
of Reference 3, which contains the following application notes: AN-14:
"TOPSwitch tips, techniques, and troubleshooting guide," AN-16:
"TOPSwitch flyback design methodology," AN-17: "Flyback transformer
design for TOPSwitch power supplies," and AN-18: "TOPSwitch flyback
transformer construction guide." You can also obtain copies of the
Excel spreadsheet from Power Integrations or download it.

Figure 1 shows a typical flyback power supply using the TOP204 TOPSwitch.
This device combines an integrated, high-voltage MOSFET switch with
a complete switching-power-supply controller and protection
circuitry in a single three-pin TO220 package. The power supply
operates from 85 to 265V ac and delivers 30W at 15V output.

Figure 2 shows typical voltage and current waveforms taken with the
flyback power supply delivering 30W from a 115V-ac input. The
primary current, IPRI, increases linearly with a rate of
change (di/dt) that varies directly with the dc input voltage and
inversely with the primary inductance. The ripple current,
IR, is defined as the incremental linear current rise
(di) over the switch's entire on-time, tON. The peak
primary current, IP, is the final value that occurs just
before the switch turns off. The transformer core's magnetic field
stores energy proportional to IP2, just as if
the primary winding were a simple inductor. The secondary winding
carries a reflected voltage that is related to the primary voltage
by the transformer turns ratio and has the same "dot" polarity.

While the switch is on, the output diode,
D2, and bias diode, D3, are reverse-biased,
preventing secondary-current flow. When the switch turns off, the
collapsing magnetic field induces an abrupt voltage reversal on all
transformer windings, so the dot side is now higher in potential than the
no-dot side. Diodes D2 and D3 become
forward-biased, and the secondary current rises quickly to a peak
value of IP/n, where n is the transformer turns ratio.
IPRI immediately drops to zero. The switch-drain voltage
quickly rises to the sum of the dc input voltage and the reflected
output voltage. The secondary-winding current, ISEC, now
linearly decreases at a rate that varies directly with the output
voltage and inversely with secondary inductance. The duty cycle is
defined as the ratio of tON to the switching period, T.

Figure 2 shows the drain and diode voltages as well as the trapezoidal
current waveforms characteristic of continuous-mode operation.
Secondary current is still flowing when the switch turns on at the
beginning of the next cycle. The stored energy is not completely
delivered to the load. Because of the nonzero magnetic field, energy
remains in the core when the switch turns on again. This energy
causes the initial step in the switch current.

The ripple-to-peak-current ratio,
KRP, determines how continuously the transformer operates.
Transformers designed for discontinuous operation have a KRP
of 1. Practical continuous designs have lower peak currents and a
KRP that is less than 1 but usually greater than 0.4.
KRP is inversely proportional to the primary inductance, so a
continuous design with lower KRP has a higher inductance.

Flyback-transformer
design now begins with the specification of the spreadsheet's three
groups of independent variables (Figure 3).

Application variables

The output power, PO; output
voltage, VO; ac-mains frequency, fL; switching
frequency, fS (fixed at 100 kHz); and the minimum and maximum
ac-mains voltage, (VACMIN and VACMAX), come
directly from the application specification. For
efficiency, leta,
start with an estimate based on measurements in similar power
supplies, or use a value of 0.8 if data is unavailable.

You can use efficiency to calculate the
power supply's total power loss, PL. Some power losses
that occur in series primary components, such as the bridge
rectifier, common-mode choke, and switch, are not directly
associated with energy stored in the flyback-transformer core. The
remaining power losses, which occur in the output rectifier and
clamp zener diode when the flyback transformer releases energy, are
now defined as secondary power loss PLS. The
loss-allocation factor, Z, defined as the ratio of PLS to
PL, is a scale factor that distributes the power losses between
the primary and secondary. Z is typically 0.5, which means
PLS is usually 50% of PL.

PL=PO·((1-leta)/leta);

Z=PLS/PL.

Bias voltage, VB, depends on the
feedback circuitry and is usually 10 to 30V. For a bridge rectifier,
a conduction time, tC, of 3 msec is typical.

For a universal or 115V-ac input, start
with a standard input-filter-capacitor value, CIN (in
microfarads), of two to three times the output power in watts. For a
230V-ac input, use a standard value in microfarads equal to the
output power in watts. For example, 60 to 90 µF is a suitable
capacitance range for a universal-input, 30W supply; 68 and 82 µF
are both standard values within this range.

The reflected output voltage,
VOR, appears across the transformer primary when the
switch is off and current is flowing through the secondary and the
output-rectifier diode. Transformers optimized for TOPSwitch
applications should have a maximum VOR of 60V or less for
the TOP1xx series (115V-ac input) and 135V or less for the TOP2xx
series (universal or 230V-ac input). VDS is the on-state
TOPSwitch voltage (typically, 10V).

Output-rectifier forward-voltage drop
VD depends on the type of diode you select. Use 0.4V for
Schottky diodes and 0.7V for ultra-fast-recovery pn-junction diodes.
Bias-winding-diode forward-voltage drop (VDB) is also typically
0.7V.

KRP determines how far into the
continuous mode a flyback transformer operates. Continuous-mode
transformers optimized for TOPSwitch applications that operate from 100 to
115V-ac or universal-input voltage should have a minimum
KRP of 0.4. Applications that operate from 230V ac should
have a minimum KRP of 0.6. Discontinuous-mode
transformers optimized for TOPSwitch applications always have a
KRP of 1.0.

Core and bobbin manufacturers' data sheets
specify the following effective parameters: cross-sectional area
AE (in centimeters squared), path length LE (in
centimeters), ungapped inductance AL (specified in either
millihenries/(1000 turns)2 or nanohenries/T2), and
physical bobbin-winding width BW (in millimeters). Margin width M,
determined by insulation methods and regulatory requirements, is
usually set at 2.5 to 3.0 mm for margin-wound transformers or to
zero for transformers wound with insulated wire.

Primary windings normally use one or two
layers, L. Higher numbers of layers reduce coupling and increase cost,
capacitance, and leakage inductance. The number of secondary turns,
NS, is a key iteration variable. One turn per volt of
output voltage is a good starting value for NS. (For
example, start with five turns for a 5V output.)

Current-waveform shape parameters

DMAX is the actual duty cycle
that occurs when the power supply delivers the maximum output power
from the minimum input voltage, VMIN. DMAX has
an upper limit equal to the minimum value of the TOPSwitch
data-sheet parameter, DCMAX (64%).

DMAX=(VOR)/(VOR+VMINVDS),

IAVG=PO/(leta·VMIN),

IP=IAVG·(2/((2KRP)·DMAX)),
and

IRMS=IP·DMAX·(((KRP2)/3)KRP+1)0.5,

where IRMS is the rms value of
the primary current.

The flyback-transformer energy equation
defined below determines the primary inductance, LP, in
microhenries. The flyback transformer stores energy proportional to
the square of the primary current. When the switch is on, the
primary current linearly ramps up over a current range,
IR, and increases the energy stored in the
flyback-transformer core. When the switch turns off, the stored-energy
increment associated with IR transfers to the load and the
secondary losses in the rectifier and clamp diode. You can now
calculate LP from PO; leta, Z;
IP; switching frequency fS; and
KRP, which determines IR.

PO·((Z·(1leta))+leta)/leta
(fSIP2·KRP·1(KRP/2))

The number of primary turns, NP,
depends on NS, VO, VD, the
effective primary voltage (VMINVDS), and
DMAX:

NP=NS·(VMINVDS)·DMAX/((VO+VD)·(1DMAX)).

Determine the primary insulated-wire
diameter, OD, in millimeters from the effective bobbin width,
BWE, and NP:

OD=BWE/NP.

The actual magnet-wire outside diameter is
slightly larger than the diameter, DIA, of the bare-copper conductor.
Insulation thickness varies inversely with the AWG size of the
bare-copper conductor, which means that smaller diameter conductors
have thinner insulation. Data from several manufacturers yields an
empirical expression for total insulation thickness, INS, in
millimeters, as a function of heavy insulated-magnet-wire outside
diameter in millimeters.

INS=(0.0594·log(OD))+0.0834.
DIA=ODINS.

Another empirical equation determines the
AWG for magnet wire with a given DIA. Integer AWG values are
standard sizes, so before proceeding with the current-capacity or
circular-mils-per-ampere (CMA) calculation, you should always round
up the calculated AWG value to the next integer or standard value,
the next-smaller standard-conductor diameter.

AWG=9.97·(1.8277(2·log(DIA))).

The cross-sectional area of magnet wire for
transformer windings is usually specified in circular mils (CM). A
circular mil is the cross-sectional area of a wire that has a diameter
of 1 mil (0.0254 mm). The following simple expression relates a
bare-conductor wire's effective cross-sectional area in circular mils to
its standard AWG size.

CM=2(50AWG)/3.

Circular mills per ampere is a convenient
way to specify a winding's current capacity. CA, which is the
inverse of current density, is simply the ratio of cross-sectional
area in circular mils to IRMS. CA should be 200 to 500.

CMA=CM/IRMS.

This calculation completes all necessary
primary-winding calculations. You must now calculate the secondary peak
current, rms current, average output current, output-capacitor
ripple current, and the minimum and maximum conductor diameter of
the secondary windings.

The peak secondary current, ISP,
is a simple function of IP, NP, and
NS.

ISP=IP·NP/NS.

You can find the secondary rms current,
ISRMS, from DMAX, ISP, and
KRP. (KRP is identical for the primary and
secondary.)

ISRMS=ISP·((1DMAX)·(((KRP2)/3)KRP+1))0.5;
IO=PO/VO.

The ripple current, IRIPPLE, in
output capacitor C2 is not a true transformer parameter,
but you need to know it for capacitor selection. You can easily
calculate IRIPPLE from other transformer parameters:
ISRMS and IO.

IRIPPLE=(ISRMS2IO2)0.5.

The minimum secondary bare-conductor
diameter, DIAS, (in millimeters) is based on the
previously calculated circular mils per ampere and secondary rms
current. You can calculate the minimum secondary bare-conductor
circular mils per ampere from the primary circular mils per ampere
and the secondary ISRMS:

CMS=CMA·ISRMS.

You can then calculate the minimum
secondary wire gauge, AWGS, from another empirical
equation. Always round AWGS down to the next integer
value. Doing so selects the next larger standard wire size.

AWGS=9.97·(5.017log(CMS)).

Now, determine DIAS in
millimeters.

DIAS=((4·(2(50AWG)/3)/(1.27·3.14))0.5)·25.4/1000.

You must also calculate the maximum outside
wire diameter, ODS, in millimeters for a single layer
based on the number of secondary turns and the bobbin width:

ODS=(BW(2·M))/NS.

You can now calculate the secondary-wire
insulation thickness from the outside diameters of the bare conductor
(determined by circular mils per ampere) and the insulated wire
(determined by the number of turns and the effective bobbin width). Note
that the secondary insulation thickness, INSS, in
millimeters is the insulation-wall thickness rather than the total
insulation thickness you use in the primary-winding calculation.

INSS=(ODSDIAS)/2.

Obviously, if INSS is not a
positive number, you must perform another transformer-design
iteration with either more secondary layers, a smaller number of
secondary turns, or a transformer core with a wider bobbin.

For insulated-wire secondaries,
INSS must equal or exceed the selected wire's insulation
thickness. Parallel combinations of wire with half the diameter may
be easier to wind and terminate, but the effective secondary
circular mils per ampere will be half the value of the single
winding.

The maximum drain voltage,
VDRAIN, is the sum of the maximum dc-input voltage,
VMAX, an estimated drain clamp-voltage term based on
VOR, and an estimated voltage term related to typical
blocking-diode forward recovery. (Refer to application note AN-14
for more detail.)

VDRAIN=VMAX+(2.1·VOR)+20.

You can easily calculate the parameters of
auxiliary windings for additional outputs. VX is the
output voltage, VDX is the diode voltage drop,
NX is the number of turns, and PIVX is the
peak-inverse voltage associated with the auxiliary winding.

NX=NS·(VX+VDX)/(VO+VD).
PIVX=VX+(VMAX·(NX/NP)).

You can now use spreadsheet iteration to
reach a final and acceptable solution for the flyback-transformer
design. Iterate NS or KRP until the maximum
flux density, BM, is between indicated limits (2000 to
3000 gauss), and check that the gap length, LG, is higher
than the indicated minimum value (0.051 mm). BM decreases
and LG increases as you increase NS or
KRP.

Examine circular mils per ampere. If
circular mils per ampere is less than the specified lower limit of
200, consider increasing the number of primary layers from one to
two or using the next-larger core size and performing a new
iteration. If CMA is greater than 500, consider using the
next-smaller core size. (A greater-than-500 CMA simply means that
the wire diameter is oversized for the expected rms current.)

The transformer design is now complete. The
improved spreadsheet contains all transformer-design parametric
information that a transformer vendor requires.

References

Leman, Brooks, "Finding the keys to flyback power supplies
produces efficient design." EDN, April 13, 1995, pg 101.

Bisci, J, "Magnet wire: Selection
determines performance," PCIM, October 1994, pg 37.

1996/97 data book and design guide,
Power Integrations Inc, Sunnyvale, CA, 1996.

Author's biography

Brooks Leman is a principal application
engineer at Power Integrations Inc (Sunnyvale, CA), where he has worked
for eight years. He develops low-cost flyback and
power-factor-correction circuits using highly integrated
power-supply devices. Leman holds a BSEE and MSEE from Santa Clara
University (Santa Clara, CA) and is a member of Tau Beta Pi, the EE
honor society. His leisure activities include soccer and skiing.

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