L-24(DK&SSG)(PE) ((EE)NPTEL) (4 files merged).pdf

Module
3
DC to DC Converters
Version 2 EE IIT, Kharagpur 1
Lesson
24
and Sepic Converter
n
C uK

Instructional objective
On completion the student will be able to
• Compare the advantages and disadvantages of l
CuK and Sepic converters with those of
three basic converters.
• Draw the circuit diagrams and identify the operating modes of l
CuK and Sepic
converters.
• Draw the waveforms of the circuit variables associated with l
CuK and Sepic converters.
• Calculate the capacitor voltage ripples and inductor current ripples in l
CuK converter.
24.1 Introduction
Switch Mode Power Supply topologies follow a set of rules. A very large number of converters
have been proposed, which however can be seen to be minor variations of a group of basic DC-
DC converters – built on a set of rules. Many consider the basic group to consist of the three:
BUCK, BOOST and BUCK-BOOST converters. The CUK, essentially a BOOST-BUCK
converter, may not be considered as basic converter along with its variations: the SEPIC and the
zeta converters.
The Canonical Cell forms the basis of analyzing switching circuits, but the energy
transport mechanism forms the foundation of the building blocks of such converters. The Buck
converter may consequently be seen as a Voltage to Current converter, the Boost as a Current to
Voltage converter, the Buck-Boost as a Voltage-Current-Voltage and the CUK as a Current-
Voltage-Current converter. All other switching converter MUST fall into one of these
configurations if it does not increase the switching stages further for example into a V-I-V-I
converter which is difficult to realize through a single controlled switch. It does not require an
explanation that a current source must be made to deliver its energy into a voltage sink and vice-
versa. A voltage source cannot discharge into a voltage sink and neither can a current source
discharge into a current sink. The first would cause current stresses while the latter results in
voltage surges. This rule is analogous to the energy exchange between a source of Potential
Energy (Voltage of a Capacitor) and a sink of Kinetic Energy (Current in an Inductor) and vice-
versa. Both can however discharge into a dissipative load, without causing any voltage or current
amplification. The resonant converters also have to agree to some of these basic rules.
24.2 Analysis of l
C uK converter
The advantages and disadvantages of three basic non-isolated converters can be summerised as
given below.
(i) Buck converter
1
S1
L
C
iB
Vin
Fig. 24.1: Circuit schematic of a buck converter
2
Features of a buck converter are
• Pulsed input current, requires input filter.
• Continuous output current results in lower output voltage ripple.
• Output voltage is always less than input voltage.

(ii) Boost converter
1
S1
L
C
R
Vin
Fig. 24.2: Circuit schematic of a boost converter
2
Features of a boost converter are
• Continuous input current, eliminates input filter.
• Pulsed output current increases output voltage ripple.
• Output voltage is always greater than input voltage.
(iii) Buck - Boost converter
Features of a buck - boost converter are
• Pulsed input current, requires input filter.
• Pulsed output current increases output voltage ripple
• Output voltage can be either greater or smaller than input voltage.
It will be desirable to combine the advantages of these basic converters into one converter.
l
CuK converter is one such converter. It has the following advantages.
• Continuous input current.
• Continuous output current.
• Output voltage can be either greater or less than input voltage.
l
CuK converter is actually the cascade combination of a boost and a buck converter.
1
S1
L
C
R
Vin
Fig. 24.3: Circuit schematic of a buck boost converter
2
1
S1
L1
C2
R
Vin
Fig. 24.4: Circuit schematic of a boost-buck converter
2
C
L2
2'
1'
S2
+
-
+
-
S1 and S2 operate synchronously with same duty ratio. Therefore there are only two switching
states.
(i) 0 < t ≤ DT
S1 to (1)
& S2 to (1')
The circuit configuration is given below
L L L1
R
C2
1
C2
R
Vin C1
2
(ii) DT < t < T; S1 to (2) & S2 to (2')
These two topologies can also be obtained from the following circuit which is the so called
l
CuK converter.
(a)
C1
1
C2
R
Vin
Fig. 24.5: Circuit topology of a boost-buck converter during different
switching intervals
(a) 0 ≤ t < DT, (b) DT ≤ t < T
(b)
L L1
R
C2
L2

S 2
1
+
-
+
-
Fig. 24.6: Schematic and Circuit representation of
24.2.1 Expression for average output voltage and inductor
currents
ĈuK converter.
(a) Schematic diagram, (b) Circuit diagram
(a)
L1
C2
R
Vin
L2
C1
2
1
(b)
L1
C2
R
Vin
L2
C1
iL1 iL2
iB
ic2 i0
V0
vc1
L
+ -
+
-
Fig. 24.7: Equivalent Circuit of a ĈuK converter during different
conduction modes.
(a) 0 < t ≤ DT (b) DT < t ≤ T
(a)
1
C2
R
Vin
L2
C1
(b)
iL1 iL2
ic2 i0
V0
+
-
ic1
+
-
+
-
L1
C2
R
Vin
L2
C1
iL1
iL2
ic2 i0
V0
+
ic1
0 < t ≤ DT
L2
VC1
-
DT < t ≤ T
Applying Volt-sec balance across L1
( )( )
1 1
in in C
V DT V V D T
+ − − = 0 (24.1)
∴ ( ) 1
1 0
in C
V D V
− =
or 1
1
in
C
V
V
D
=
−
(24.2)
Applying Volt-sec balance across L2
( ) ( )
0 1 0 1
C
V V DT V D T
+ + − 0
= (24.3)
or 0 1 0
C
V DV
+ = (24.4)
or 0 1
1
in
C
DV
V DV
D
= − = −
−
(24.5)
Expression for average inductor current can be obtained from charge balance of C2
2 0 0
L
I I
+ = (24.6)
∴ 0
2 0
1
in
L
V V
D
I I
R D
= − = − =
− R
(24.7)
From power balance
( )
2 2
2
0
1 0 0 2
1
in
in L
v V
D
V I V I
R R
D
= = =
−
(24.8)
∴
( )
2
1 2
1
in
L
V
D
I
R
D
=
−
(24.9)
24.2.2 Current ripple and voltage ripple calculations
The waveforms of different circuit variables of Fig. 24.7 are given in Fig. 24.8.

iB
DT T
t
t
t
t
t
t
t
Vc2
Vc2
t2
t1
ˆ
L2 p-p
-1/2 I
ˆL2 p-p
1/2 I
ic2
vc1
VC1 MAX
VC1 MIN
VC1
- IL1 MIN
- IL1 MAX
IL2 MIN
IL2 MAX
t2
t1
ic1
IL1 MIN
IL1 MAX
IL1
iL1
IL2 MIN
IL2 MAX
IL2
iL2
Fig. 24.8: Waveforms of circuit variables in a ĈuK converter.
From the waveforms of Fig. 24.8
1 1
1
in
L MAX L MIN
DV T
I I
L
= + (24.10)
1 1 1
1
ˆ in
L L MAX L MIN
p p
V DT
I I I
L
−
= − = (24.11)
From equation 24.9
2
1 1 1 2
2
2
(1 )
in
L MAX L MIN L
V
D
I I I
R
D
+ = =
−
(24.12)
∴ 1 2
1
2
(1 )
in
L MAX
DV
D RT
I
L R
D
⎡
= +
⎢ −
⎣ ⎦
⎤
⎥ (24.13)
1 2
1
2
(1 )
in
L MIN
DV
D RT
I
L R
D
⎡
= −
⎢ −
⎣ ⎦
⎤
⎥ (24.14)
0
2 2 2
2 2
(1 ) in
L MAX L MIN L MIN
V V
I I D T I
L
= − − = + DT
L
(24.15)
∴ 2 2 2
2
ˆ in
L L MAX L MIN
p p
V DT
I I I
L
−
= − = (24.16)
From equation 24.7
2 2 0
2
2
1
in
L MAX L MIN
V
D
I I I
D R
+ = − =
−
(24.17)
∴ 2
2
1
1 2
in
L MAX
DV
RT
I
D L R
⎡ ⎤
= +
⎢ ⎥
−
⎣ ⎦
(24.18)
2
2
1
1 2
in
L MIN
DV
RT
I
D L R
⎡ ⎤
= −
⎢ ⎥
−
⎣ ⎦
(24.19)
For calculating voltage ripples it is noted that
1 0
1
1
DT
c
v i
c
= ∫ 1
c dt (24.20)
but for 0 < t ≤ DT ic1 = iL2 (24.21)
1
0 0
1 1
1 1
DT DT
c
i dt i dt
c c
=
∫ ∫ 2
L (24.22)

or 2 2 0
2
1
2
ˆ
1 2 2 1
L MAX L MIN L
c
1
I I DTI
DT RT
v
C L c
+
⎡ ⎤
= + =
⎢ ⎥
⎣ ⎦
I DT
c
= (24.23)
or
2
1
ˆ
1(1 )
in
c
D V T
v
RC D
=
−
(24.24)
2
2 1
1
ˆ
2
c c
t
v i
t
c
= ∫ 2 dt which is the hatched area under ic2 waveform in Fig. 24.8
∴
2
2 2
1 1
ˆ
1 2 2 2 8
in in
c
V DT V DT
T
v
c L
= × × × =
2
L C
(24.25)
Equations 24.11, 24.16, 24.24 and 24.25 can be utilized to design a l
CuK converter of given
specification
The SEPIC Converter
The previous chapter discussed the single stage conversion Buck and Boost converters
along with the two-stage Buck-Boost converter. This chapter offers a few additional topologies.
Fig. 24.9(a): A basic converter: BUCK converter
Fig. 24.9(a) is that of a basic Buck converter. From the voltage source C1, the converter charges
the current sink constituted by the inductor-diode (L-D). The current is further converted into
voltage without a switching stage (amplification) at C2. The canonical switching cell is
approached if the capacitors C1 and C2 are combined to be represented by a single capacitor C.
The cell includes T-C-L-D, the basic building block of DC-DC converters. The Boost converter
is realized if the positions of D and T are interchanged in Fig.24.9 (a). Now power flows in from
the right. Here, the energy stored in the inductor during each ON period of switch T is
transferred to the Capacitor during its OFF period.
The CUK converter as the dual of the Buck-Boost converter has current input and current
output stages. The basic SEPIC is a modification of the basic Boost and the CuK topologies.
Consider the Boost converter in Fig 24.9(b). At steady state, the average voltage across the input
inductor is zero. Equating the inductor voltages for the period when the switch T is ON with that
when it is OFF,
( )
in ON out in OFF
out in
V .T V V .T
1
or,V ( ).V
1
= −
=
−∂
(24.26)
where, ∂ is the duty ratio of the switch.
Fig. 24.9(b): BOOST converter
Fig. 24.10 Modified Boost with load across Diode for Boost-Buck
Operation. (left) without output filter, (right) with filter.
In the path, Vin-L-D-Vout, in Fig. 24.9(b), the average voltages across all the elements are
known. Thus, that appearing across the diode D is Vout – Vin. This voltage from Eqn 1 is:

in
in
D
V
V
V
)
1
(
.
]
1
)
1
1
[(
∂
−
∂
=
−
∂
−
=
A Boost-Buck converter is thus realized. This is the voltage that would appear in an
unfiltered form at the load in Fig. 24.10 (left). Now, since the source is a current source, the
output stage must be capacitive (voltage sink) which is taken care of by C2-D. The voltage across
D has high ripples, which can be filtered much like the Buck converter with an L (and a C3). The
CUK converter is thus realized. It is a I-V-I converter.
A glaring drawback of this derived converter topology is that the polarity of the output is
reversed. This is not acceptable for various reasons.
Now it is the turn of the Diode to be interchanged with the filter inductor. The inductor is
thus converted to be part of the switching circuit and it not just a filter. The SEPIC results – not
an entirely different one - but easily derivable from the previous topologies.
The SEPIC officially stands for “Single-Ended Primary Inductance Converter”. However,
the unofficial interpretation is more descriptive: “Secondary Polarity Inverted Cuk”.
Fig. 24.11(a): The basic SEPIC topology
Again, the basic input–output relation can be derived by considering the two inductors to
have average null voltage across themselves.
If the link capacitor has a voltage Vc across itself (consider it to be reasonably constant),
then for the input inductor, the volt-secs during the ON and OFF periods of the switch are:
in ON C out in OFF
C out in
OFF
V .T (V V V )T
1
or, V V V (. )
T
= − −
= −
(24.27)
For the output inductor,
C ON out OFF
V .T V .T
= (24.28)
Eliminating, Vc and writing TON = ∂ . T,
out in
V ( )V
1
∂
=
− ∂
(24.29)
Thus the SEPIC is also basically a BOOST-BUCK converter akin to the CUK converter.
(The Boost stage comes first followed by the Buck stage and it is also I-V-I converter)
In the practical SEPIC converter, the two inductors are coupled with the polarities as
indicated by dots in Fig. 24.11(a). The turns ratio is and the coupling is very tight. For such a
coupled-transformer SEPIC, equating the positive and negative volt-secs for the two inductors,
( . . ). ( . ).
= + − −
in C ON out C in out OFF
V K V T V V V K V T (24.30)
for the input inductor, and
' '
( . ). [ ( )].
− = − + −
C in ON out out C in OFF
V K V T V K V V V T (24.31)
Equations (24.28) and (24.29) can be obtained from the above two by substituting both K and K’
to zero to have no coupling between the two coils.
Fig. 24.11(b) The practical SEPIC topology with coupled inductors
The above two equations result in an identity to indicate that such a system cannot work.
This can be explained by examining the operation of the circuit. Initially when the
transistor is OFF, the capacitor C2 charges to the supply voltage Vin. When the transistor is
switched ON, the resulting active circuit is shown in Fig 24.12.
Fig. 24.12: Active part of the circuit when transistor is switched
with C2 charged toVin
The circuits to the left and right of the transistor are identical and both the windings are induced
with the supply voltages, resulting in null emfs on either side, which explains why the ideal
circuit will not work. However, neither the coupling between the inductors nor the effective turns
ratio can be unity. This results in a circuit with the features of the uncoupled circuit and the
circuit performs.
The second voltage source, VC, induces N.VC into the primary, where N is the turns ratio. For the
interesting case, Vin = VC = V1, if the turns ratio, n, is increased slightly from unity, by 1/k (where
k < 1 is the coupling coefficient between windings), then the voltage induced by Vin will increase
the voltage at the Drain of the transistor to N. V1, thereby "bootstrapping" the leakage inductance
of the input inductor. Because the voltage at each end of this leakage inductance is the same, its
inductance is effectively infinite. Consequently, all variations in magnetizing current, (through
M) due to a varying V1 is supplied from the secondary winding source. By symmetry, setting
n = k causes the secondary-winding current to become constant while the primary source
supplies the magnetizing-current variations.
This effect can be desirable because, for n = 1/k, it results in constant (DC) primary current.
Noisy switching current does not appear at the converter input but is diverted instead to the
secondary winding. However, typical values of k are slightly less than one, and turns ratios of
nearly 1:1 may not be easy to wind. One simplification is to use a 1:1 transformer, such as a low-
cost, commodity, common-mode power-line input-filter choke, and add a small additional
inductance in series with the primary winding. This effectively increases the leakage inductance
so that the same secondary-winding dominance of magnetizing current is obtained with n = 1.

The circuit is an alternative to the Boost converter and outputs an range which includes
the input range also being a Boost-Buck converter. It is superior to the other converters both in
terms of the input current purity and efficiency.
Fig. 24.13: Drain voltages of FLYBACK and SEPIC converters
The waveforms in Fig. 24.13 show the voltage at the transistor Drain present on the fly
back (Boost) and SEPIC circuits. The fly back transformer leakage inductance produces a
voltage spike that adds an additional level to the "flat-top" voltage. This level is about 1.5 times
the supply voltage for inputs around 20 V. In comparison, the SEPIC FET switching waveform
is clamped, and shows very little overshoot, or ringing. This clamping results in less switching-
loss, output voltage noise and a power stage that can be operated at a much higher frequency
than that of the fly back.
Again, the fly back transformer leakage inductance also produces a significant voltage
spike relative to the SEPIC at the output diode. A relatively high voltage (~200V) output diode is
required for the fly back to handle the large negative ringing compared to the SEPIC’s 60V
Schottky diode. The 0.5 volt forward drop of the SEPIC’s Schottky diode relative to the one volt
forward drop of the flyback's ultra-fast diode, results in significant power savings for the SEPIC.
Module
3
DC to DC Converters

Lesson
25
Design of Transformer for
Switched Mode Power
Supply (SMPS) Circuits
After completion of this lesson the reader will be able to:
(i) Explain the underlying principles behind the design of a high frequency transformer
and inductor.
(ii) Do a preliminary design of a high frequency transformer for some popular
configurations of SMPS circuits.
(iii) Do a preliminary design of a high frequency inductor.
(iv) Estimate the size of an SMPS transformer of some given VA rating.
Transformers are required for galvanic isolation between input and output voltages and for
voltage and current scaling. It also helps in optimizing the device voltage and current ratings.
The switches, diodes and other circuit elements on the high voltage side of the transformer are
subjected to higher voltages but only lower currents. Similarly the devices put on the low voltage
side are subjected to less voltage stress but higher current stress. The dc-to-dc buck converter
shown in Fig. 25.1, which is used to get a low voltage output from a high input dc voltage
illustrates this point clearly. The circuit in Fig. 25.1(a) uses a step down transformer with proper
turns ratio and has the advantages discussed above. On the other hand the switch and diode and
the filter inductor in Fig. 25.1(b) need to withstand both input side voltage and output side
current. Also, the switch in case (b) will be constrained to operate in a narrow range, which may
cause lesser accuracy in output voltage control.
NP: NS
D1
D2
S
L
C
Edc
Load
Edc
+
_
L
O
A
D
S
L
C
D
(a) (b)
Fig. 25.1: DC to DC buck converters: (a) Isolated type (b) Non-isolated type
Transformers used in switched mode power supply circuits are significantly different from the
power transformers that are used in utility ac supply system. Following are the important
differences:
(i) The input and output voltages and currents of a SMPS transformer are mostly
non-sinusoidal, whereas the transformers connected to utility ac supply are almost
always subjected to sinusoidal voltages and currents.
(ii) The currents and voltages of SMPS transformer are of very high frequency where
as utility type transformers are subjected to low frequency supply voltages.
(iii) SMPS transformers generally handle much smaller power than the utility
transformer.

SMPS transformer-core, because of high frequency operation, is generally made of hard
magnetic material like ferrites whereas the low frequency power transformers mostly use soft
magnetic material like silicon steel. Ferrites have very high ohmic resistance and the area
enclosed under the hysteresis loop of their B-H magnetization curve is significantly lower than
that of silicon steel. As a result, even at very high frequency operation, the hysteresis and eddy
current losses are low. [Low hysteresis loss is due to less B-H loop area and low eddy current
loss is due to very high resistivity of the core material.] The ferrites have low magnetic
permeability (typical value of relative permeability is around 100) and low saturating value of
flux density (typical value is around 0.4 Tesla) that are considerably less than that of silicon
steel. Ferrites are also brittle and fragile. The efforts are on to search for alternatives to ferrites
that may have higher permeability, may handle higher flux density and may be more rugged.
The fundamental principles concerning emf generation etc. in SMPS-transformers and power
transformers are identical and hence, in this lesson, many concepts of conventional transformer
design have been borrowed.
25.1 Recapitulation of Governing Equations for Utility Transformer
In case of sinusoidal flux of peak magnitude ‘ m
φ ’ and frequency ‘f’ linking the transformer
windings, the emf generated per turn of the winding will have a rms magnitude ‘Et’ given by:
Et = 4.44 f m
φ -----------------------------------------------(25.1)
The peak flux through the core is the product of peak flux density (Bm) and the core area (Ac),
i.e.,
m
φ = Bm Ac ----------------------------------------------- (25.2)
The windings are placed around the core and are accommodated in the window of the
transformer. The transformer window area (Aw) is related with the winding’s current rating and
the number of turns. For a single-phase transformer the relation between them is given by:
Aw kw δ = 2 N I ----------------------------------------------- (25.3)
,where kw is the window utilization factor and δ is the current density through the cross-
sectional area of the transformer windings. Window utilization factor, roughly varies between
0.35 to 0.6 and is dependent on the insulation requirements of the windings. A typical figure for
the current density through copper conductors of naturally cooled transformers is 3X106
amps
per square meter. If the current density through primary and secondary windings is taken
identical, they occupy equal window-space of the transformer. Some times the current densities
through the two windings may differ depending on their physical ability to dissipate heat. The
VA rating of a single phase transformer (= N Et I) can now be found from the above equations
as:
VA rating = 2.22 f Bm δ kw Ac Aw ----------------------------(25.4)
For the given operating frequency (f) the product ‘Ac Aw’, known as area product is roughly
proportional to the VA rating of the transformer as other parameters have nearly fixed
magnitudes.
25.2 Derivation of Design Equations for SMPS Transformer
The nature of voltage and flux waveforms in SMPS transformers is different from that of utility
transformer. Moreover SMPS circuits of different topologies generate different kinds of winding
voltages (and hence the flux-linked waveforms) and need to be considered separately.
In this section some representative voltage and flux waveforms have been taken up and through
them the transformer design procedure has been illustrated.
25.2.1 Transformer with Square-Wave Voltage and Bipolar Flux
Fig. 25.2 shows the typical winding voltage and core-flux waveform produced by one of the
popular SMPS topologies that utilizes a H-bridge converter to get high frequency ac voltage
from the dc input. The primary side of the SMPS transformer is connected to the H-bridge output
and the secondary side voltage is rectified and filtered to get regulated dc output voltage of
desired magnitude. The transformer windings carry bi-direction current and the flux linking the
windings is also bipolar. The input dc bus voltage is unregulated and often varies over a large
range. The duty ratio ‘D’ of the switches is controlled within 0<D<0.5 to regulate the output
voltage. The mean of the rectified secondary side voltage, under steady state and after accounting
for voltage drops in the rectifier diode and filter inductor, equals the desired load voltage and can
be assumed fixed to the output voltage ‘Vo’. However under dynamic condition, which may arise
due to sudden change in load or supply voltage, the mean (dc) output voltage on the secondary
side may be significantly higher than its steady state magnitude. For calculation of peak flux in
the core, the worst-case condition will correspond to maximum duty ratio (D=0.5) and maximum
magnitude of input voltage. The worst-case current through the windings will correspond to
maximum duty ratio (D=0.5) and peak magnitude of output (load) current. Now the transformer
may be designed as per the design steps given below:
(i) Determination of primary to secondary turns ratio (NP/ NS):- This can be found from
the knowledge of operating range over which the input dc voltage may vary. Let the
input voltage vary from Vmin to Vmax. With minimum input voltage ‘Vmin’ and duty
ratio ‘D’ = 0.5, the magnitude of square-shaped secondary side voltage should equal
(Vo + VR), where VR is the estimated voltage drop in the transformer winding, output
rectifier and filter circuit under maximum load condition. The transformer turns ratio
can thus be estimated to NP/ NS = Vmin /(Vo + VR). The actual number of turns in the
windings will be found as shown below in step (v).
(ii) Determination of peak magnitude of flux in the transformer core: As per above
discussion, the maximum flux in the core will correspond to a square wave voltage of
magnitude Vmax across the primary winding (refer to Fig. 25.2 with D=0.5). The
frequency of voltage waveform ‘f’(=1/T) is same as the frequency at which the
converter switches are turned on and is fixed beforehand. Now by simple integration
of the square wave voltage waveform, the peak flux ‘ m
φ ’ is related to the input
voltage as, Vmax = 4.0 f m
φ NP = 4.0 f Bm Ac NP ---------------------------(25.5)

(iii) Determination of winding current rating and requirement of window area: Let ‘Iom’
be the peak expected load current. The secondary winding of the transformer should
be rated to supply this current. Most SMPS circuits, with low magnitude of output
voltage, have a center-tapped secondary winding followed by a mid-point rectifier
circuit realized using two diodes (instead of bridge rectifier having four diodes). This
results in only one diode voltage drop during rectification, unlike two diode drops for
the bridge rectifier circuit. For SMPS with low output voltage, saving one diode drop
can result in significant increase in the efficiency. For this same reason, the diodes
used on the secondary side are Schottky diodes having low on-state voltage drop.
Each half of the center-tapped secondary winding requires NS turns as determined in
(i) above and they carry the load (dc) current only in alternate half cycles. Thus the
rms current rating of each half equals om
I
2
and the net copper cross-sectional area
required for the secondary winding is S om
2N I
δ
, where δ is the current density (as
described in relation to Eqn.25.3). If the secondary was not center-tapped, the
rectifier used would be bridge type and the copper area for the secondary would have
been just S om
N I
δ
. The primary side carries the reflected secondary current and the
required copper area for primary would equal S om
N I
δ
. The total window area
requirement for the transformer can now be given as:
(
S om
w w
N I
A k 1 2
= +
δ
) ------------ (25.6),
where Aw is the window area and kw is the window utilization factor (as discussed in
Sec.25.1).
(iv) Expression for VA rating of the transformer: Combining Eqns. (25.5) and (25.6) one
gets,
( )
S
max om m w c w
P
N
V I 1 2 4fB k A
N
+ = δ A --------------------------(25.7)
Using relations derived in (i) above, Eqn.25.7 may be rewritten as:
( )
o om 1 2 m w c w
V I K K 1 2 4fB k A A
+ = δ --------------------------(25.8)
where max
1
min
V
K
V
= , a factor allowing for input voltage variation and 0 R
2
o
V V
K
V
+
= ,
a factor coming due to voltage drop in rectifier diode, filter inductor etc. Vo Iom is the
peak output power from the SMPS. The factor ( )
1 2
+ on the L.H.S. of Eqn.27.8
will become 2.0 if the secondary winding is not center-tapped.
(v) Selection of transformer core and determination of number of turns in the windings:
Knowing the area product ‘Ac Aw’, as given by Eqn.25.8, the appropriate
transformer core is to be selected from the core-manufacturer’s catalog. Once the
area product matches, the details of other dimensions of the transformer core are
found from the catalog. Knowing window area (Aw) and core area (Ac), the number
of turns in the windings can be decided using Eqns. Like (25.5) or (25.6).
25.2.2 Transformer with Unipolar Flux
Many switched mode power supply circuits use only one controlled switch (like the forward
converter discussed in Lesson-23). The winding current and core-flux for most of these
transformers are unidirectional. Fig. 25.3 shows the typical winding voltage along with the
corresponding core-flux waveform for a forward converter. As shown in Lesson-23, when the
forward converter switch is turned on the primary winding is subjected to input dc voltage. As
soon as the primary winding is turned-off, the tertiary winding starts conducting and the voltage
across primary goes negative with a magnitude that equals the product of input voltage and the
turns ratio between the primary and tertiary windings. The maximum duty ratio (Dmax) of the
switch is also limited by the turns ratio between the primary and tertiary winding to allow
resetting of the transformer flux (as given in Sec.23.4 of Lesson-23). The maximum input
voltage (Vmax), switching frequency ‘f’(=1/T) and the maximum duty ratio (Dmax) are related
with the peak magnitude of core-flux is calculated as
Vmax Dmax = f m
φ NP = f Bm Ac NP -----------------------------------------------(25.9)
Eqn.25.9 may be compared with Eqn.25.5 for a typical value of Dmax = 0.5 (which corresponds
to the case when primary and tertiary windings have identical number of turns). Because of
unipolar nature of flux the utilization of core (in terms of emf generation) is poorer here.
The primary to secondary turns ratio (NP/ NS ) for the forward converter can be estimated as
done previously for the H-bridge converter. Accordingly, NP/ NS = Vmin Dmax /(Vo + VR), where
Vo is the required output voltage and VR denotes the voltage drop in output rectifier and filter
circuit.
The maximum rms current through the secondary winding can be equated to om max
I D and the
window area (Aw) requirement is given by S om max
w w
2N I D
A k =
δ
-----------------(25.10)
From Eqn.25.9 and 25.10, the VA rating of the transformer is given as:
( )1.5
S
max om max m w c w
P
N
V I D 0.5fB k A
N
= δ A , which may be rewritten as
o om 1 2 max m w c w
V I K K D 0.5fB k A A
= δ ------------------------------------------(25.11)
Eqn.25.11 is similar to Eqn.25.8 above. The symbols used also denote the same. Knowing the
window area, the transformer core selection and other designs are done as described above in
connection with the H-bridge topology. The extra tertiary winding of a forward converter
transformer carries only magnetization current, which is a quite small and even a thin gauge wire
will serve the purpose. However, with the addition of tertiary winding the insulation requirement

of the transformer increases significantly and hence the window utilization factor (kw) becomes
low.
0
0
Voltage
Flux
+V
-V
+φm
- φm
time
time
T
T/2 3T/2
DT
DT
Fig. 25.2: Winding voltage and core-flux waveforms for a H-bridge type SMPS supply
0
0
Voltage
Flux
+VF
-VR
+φmax
time
time
T
Fig. 25.3: Winding voltage and core-flux waveforms for a forward type SMPS supply
DT
25.2.3 Design of Inductor-Transformer
The fly-back type SMPS circuits use a different kind of transformer, which as indicated in
Lesson-22, may be more appropriately called as inductor-transformer. Such a transformer is
more like two coupled inductors. These two coupled-inductors don’t conduct simultaneously,
unlike the two coupled-windings of a normal transformer. Also, the inductance needs to have a
finite magnitude so that current can build through it during each high frequency cycle and the
inductor may store the desired magnitude of energy. The windings of an inductor-transformer
facilitate energy storage in the magnetic field whereas the windings of an ideal transformer
(having infinitely large permeability ‘μ’ of the core) cannot be used for storing energy as energy
density equals
2
B
0.5
μ
. For finite magnitude of flux density ‘B’, the magnitude of ‘μ’ should be
small to have higher energy per unit volume. ‘μ’ and magnetic reluctance have inverse relation,
as ‘μ’ decreases the reluctance increases. For a practical inductor the reluctance of its flux-path
should not be zero.
For an inductor, working in the linear region of the core’s magnetization, the following relation
holds good between inductance (L), reluctance (R) and the number of turns (N) of the inductor:
2
N
L
R
= . However a practical inductor still requires a good core with high permeability to
increase (i) coupling between the windings, (ii) to guide the flux path and hence decrease the
stray magnetic field lines and (iii) to keep the inductor size small. However to keep the
reluctance of the flux-path at the desired value, an appropriate length of air-gap is introduced in
the flux path. Fig.25.4 shows a double ‘E’ core with windings put around the central limb. After
the windings are placed in position, a non-magnetic material (like, paper) is inserted between the
faces of the core and the two ‘E’s of the core are clamped together. The non-magnetic material
acts like air-gap in the core. A preferred way of creating air-gap may be to grind some length
from only the central limb of the core. If ‘lg’ is the length of air-gap in the core, the inductance
(L) can be expressed as:
2
c 0
g
N A
L
l
μ
= ------------------------------------------------------------------------ (25.12)
where Ac is the area of the core’s limb on which the windings have been placed and μ0 is
the permeability of air-gap. In the above expression for inductance, the fringing effect of the flux
and the reluctance of the flux path through magnetic core have been neglected.
The core material should not saturate with the peak expected current (Ip) in the inductor. The
peak flux density in the core (Bm) can be related with the peak magnitude of current as
p c
LI NA B
= m ------------------------------------------------------------------------ (25.13)
Knowing the current shape through the inductor, one calculates its rms magnitude (Ip,rms) and
determines the window area required as
p,rms
w w
NI
A k =
δ
------------------------------(25.14)
Combining Eqns.25.13 and 25.14, one gets
p p,rms m w c w
LI I B k A A
= δ ------------------------------------------------------------- (25.15)
Eqn.25.15, gives the area product from which rest of the design can be proceeded as in the case
of transformer design shown above. LHS of Eqn.25.15 is indicative of the energy holding
capacity of the inductor (some what like VA rating of the transformer discussed above). Should
there be a couple winding (an inductor-transformer) the area product expression needs to be
modified to include the window space requirement of the secondary winding as well.

H
A
L
F
P
R
I
H
A
L
F
P
R
I
F
U
L
L
S
E
C
F
U
L
L
S
E
C
H
A
L
F
P
R
I
H
A
L
F
P
R
I
Fig. 25.4: A typical SMPS transformer with a double ‘E’ type ferrite core and
interleaved primary and secondary winding
25.3 Transformer Winding
Often sandwiched type windings (as shown in Fig.25.4, where the secondary winding is
sandwiched between two halves of the primary) are used to reduce leakage inductance of the
windings. Sandwiching increases the insulation requirement between the windings.
For very high frequency applications, it may be preferred to use ribbon-conductor or copper foil
in place of solid circular conductors. This helps in better utilization of winding’s copper as high
frequency current is effectively limited to the surface of the conductor.
Many applications require grounded shields around the windings to reduce electro-magnetic
interference (EMI) caused by the SMPS transformers. As discussed in this lesson the SMPS
transformers often carry very high frequency ripples. These shields are essentially 3/4th
turn of a
metallic foil put around the windings. There should be proper insulation between the shield and
Quiz Problems
(1) For a high frequency transformer the relation between the transformer size and frequency
of voltage waveform can be given as:
(a) Size increases with frequency
(b) Size decreases with frequency
(c) Core size reduces but copper weight increases with increase in frequency
(d) Size is independent of frequency
(2) The assembly of fly-back and forward type transformer cores may differ in the following
sense:
(a) Air-gap is inserted in fly-back type but it is undesirable for forward type.
(b) Air-gap in the flux path is undesirable for both types
(c) Only forward type must have a suitably length of air-gap
(d) Little air-gap is deliberately put for both transformers
(3) Transformers of forward type and H-bridge type SMPS circuits of identical VA rating
and frequency differ in the following sense:
(a) The forward type transformer will be bigger
(b) The H-bridge circuit will require bigger transformer
(c) They will be of identical size
(d) Only the window area of H-bridge transformer will be bigger
(4) The size of SMPS transformers operating over large input voltage range will compare
with similar rated transformer operating over a narrower input voltage range in the
following manner:
(a) Larger input voltage range will require larger transformer
(b) Larger voltage range requires smaller transformer
(c) Size remains independent of voltage range
(Answers: 1-b, 2-a, 3-a, 4-a)

Module
3
DC to DC Converters
Lesson
23
Forward Type Switched
Mode Power Supply

(i) Identify the topology of a forward type switched mode power supply circuit.
(ii) Explain the principle of operation of a forward dc-to-dc power supply.
(iii) Calculate the ratings of devices, components, transformer turns ratio for the given
input and output voltages and the required output power.
(iv) Design a simple forward type switched mode power supply circuit.
23.1 Introduction
Forward converter is another popular switched mode power supply (SMPS) circuit that is used
for producing isolated and controlled dc voltage from the unregulated dc input supply. As in
the case of fly-back converter (lesson-22) the input dc supply is often derived after rectifying
(and little filtering) of the utility ac voltage. The forward converter, when compared with the
fly-back circuit, is generally more energy efficient and is used for applications requiring little
higher power output (in the range of 100 watts to 200 watts). However the circuit topology,
especially the output filtering circuit is not as simple as in the fly-back converter.
Fig. 23.1 shows the basic topology of the forward converter. It consists of a fast switching
device ‘S’ along with its control circuitry, a transformer with its primary winding connected in
series with switch ‘S’ to the input supply and a rectification and filtering circuit for the
transformer secondary winding. The load is connected across the rectified output of the
transformer-secondary.
V (o/p)
Switch S
NP : NS
Fig. 23.1: Basic Topology of a Forward Converter
Edc
Load
D1
D2
L
C
Control
Circuit
The transformer used in the forward converter is desired to be an ideal transformer with no
leakage fluxes, zero magnetizing current and no losses. The basic operation of the circuit is
explained here assuming ideal circuit elements and later the non-ideal characteristics of the
devices are taken care of by suitable modification in the circuit design. In fact, due to the
presence of finite magnetizing current in a practical transformer, a tertiary winding
needs to be introduced in the transformer and the circuit topology changes slightly. A
more practical type forward converter circuit is discussed in later sections.
23.2 Principle of Operation
The circuit of Fig. 23.1 is basically a dc-to-dc buck converter with the addition of a transformer
for output voltage isolation and scaling. When switch ‘S’ is turned on, input dc gets applied to the
primary winding and simultaneously a scaled voltage appears across the transformer secondary.
Dotted sides of both the windings are now having positive polarity. Diode ‘D1’, connected in
series with the secondary winding gets forward biased and the scaled input voltage is applied to
the low pass filter circuit preceding the load. The primary winding current enters through its
dotted end while the secondary current comes out of the dotted side and their magnitudes are
inversely proportional to their turns-ratio. Thus, as per the assumption of an ideal transformer, the
net magnetizing ampere-turns of the transformer is zero and there is no energy stored in the
transformer core. When switch ‘S’ is turned off, the primary as well as the secondary winding
currents are suddenly brought down to zero. Current through the filter inductor and the load
continues without any abrupt change. Diode ‘D2’ provides the freewheeling path for this current.
The required emf to maintain continuity in filter-inductor current and to maintain the forward bias
voltage across D2 comes from the filter inductor ‘L’ itself. During freewheeling the filter inductor
current will be decaying as it flows against the output voltage (Vop), but the presence of relatively
large filter capacitor ‘C’ still maintains the output voltage nearly constant. The ripple in the output
voltage must be within the acceptable limits. The supply switching frequency is generally kept
sufficiently high such that the next turn-on of the switch takes place before the filter inductor
current decays significantly. Needless to say, that the magnitudes of filter inductor and capacitor
are to be chosen appropriately.
The idea behind keeping filter inductor current nearly constant is to relieve the output capacitor
from supplying large ripple current. [As per the circuit topology of Fig.23.1, the inductor and
the capacitor together share the load-current drawn from the output. Under steady state
condition, mean dc current supplied by the capacitor is zero but capacitor still supplies
ripple current. For maintaining constant load current, the inductor and capacitor current-
ripples must be equal in magnitude but opposite in sense. Capacitors with higher ripple
current rating are required to have much less equivalent series resistor (ESR) and
equivalent series inductor (ESL) and as such they are bulkier and costlier. Also, the ESR
and ESL of a practical capacitor causes ripple in its dc output voltage due to flow of ripple
current through these series impedances. Since the output voltage is drawn from capacitor
terminal the ripple in output voltage will be less if the capacitor is made to carry less ripple
current.]
For better understanding of the steady-state behavior of the converter, the circuit’s operation is
divided in two different modes, mode-1 and mode-2. Mode-1 corresponds to the ‘on’ duration of
the switch and mode-2 corresponds to its ‘off’ duration.
The following simplifying assumptions are made before proceeding to the detailed mode-
wise analysis of the circuit:
• ON state voltage drops of switches and diodes are neglected. Similarly, leakage currents
through the off state devices is assumed zero. The switching-on and switching-off times
of the switch and diodes are neglected.

• The transformer used in the circuit is assumed to be ideal requiring no magnetizing
current, having no leakage inductance and no losses.
• The filter circuit elements like, inductors and capacitors are assumed loss-less.
• For the simplified steady-state analysis of the circuit the switch duty ratio (δ), as defined
in the previous chapters is assumed constant.
• The input and output dc voltages are assumed to be constant and ripple-free. Current
through the filter inductor (L) is assumed to be continuous.
Mode-1 of Circuit Operation
Mode-1 of circuit starts after switch ‘S’ (as shown in Fig.23.1) is turned ON. This connects the
input voltage, Edc, to the primary winding. Both primary and secondary windings start
conducting simultaneously with the turning on of the switch. The primary and secondary
winding currents and voltages are related to their turns-ratio (NP / NS), as in an ideal transformer.
Fig.23.2 (a) shows, in bold lines, the current carrying path of the circuit and Fig.23.2 (b) shows
the functional equivalent circuit of mode-1. As switch ‘S’ closes, diode D1 in the secondary circuit
gets forward biased and the input voltage, scaled by the transformer turns ratio, gets applied to the
secondary circuit. Diode D2 does not conduct during mode-1, as it remains reverse biased.
Fig.23.2(b): Equivalent circuit in
Mode-1
NP: NS
D1
D2
S
L
C
Edc
Fig. 23.2(a): Current path during Mode-1
Load
L
C Load
S
dc
P
N
E
N
P
N
Switch ‘S’
and D1 ON,
D2 Off.
VO
As can be seen, the output circuit consisting of L-C filter and the load gets a voltage equal
to S
dc
P
N
E
N
during mode-1. This voltage is shown across points ‘P’ and ‘N’ in Fig. 23.2(b) and it is
the maximum achievable dc voltage across the load, corresponding to δ = 1. Mode-1 can be called
as powering mode during which input power is transferred to the load. Mode-2, to be called as
freewheeling mode, starts with turning off of the switch ‘S’.
Mode-2 of Circuit Operation
As soon as switch ‘S’ is turned off, the primary and the secondary winding currents of the
transformer fall to zero. However, the secondary side filter inductor maintains a continuous
current through the freewheeling diode ‘D2’. Diode ‘D1’ remains off during this mode and isolates
the output section of the circuit from the transformer and the input.
Fig.23.2(b): Equivalent circuit in
Mode-2
NP: NS
D1
D2
S
L
C
Edc
Fig. 23.3(a): Current path during Mode-2
Load
L
C Load
P
N
Switch ‘S’
and D1 Off,
D2 ON.
VO
Fig. 23.3(a) shows the current carrying portion of the circuit in bold line and Fig. 23.3(b) shows
the equivalent circuit active during mode-2. Points ‘P’ and ‘N’ of the equivalent circuit are
effectively shorted due to conduction of diode ‘D2’. The inductor current continues to flow
through the parallel combination of the load and the output capacitor. During mode-2, there is no
power flow from source to load but still the load voltage is maintained nearly constant by the large
output capacitor ‘C’. The charged capacitor and the inductor provide continuity in load voltage.
However since there is no input power during mode-2, the stored energy of the filter inductor and
capacitor will be slowly dissipating in the load and hence during this mode the magnitudes of
inductor current and the capacitor voltage will be falling slightly. In order to keep the load voltage
magnitude within required tolerance band, the converter-switch ‘S’ is turned on again to end the
freewheeling mode and start the next powering mode (mode-1). Under steady state, loss in
inductor current and capacitor voltage in mode-2 is exactly made up in mode-1. It may not be
difficult to see that to maintain load voltage within the desired tolerance band the filter inductor
and capacitor magnitudes should be sufficiently large. However, in order to keep the filter cost
and its physical size small these elements should not be unnecessarily too large. Also, for faster
dynamic control over the output voltage the filter elements should not be too large. [It may be
pointed out here that the filter inductor, capacitor, transformer and the heat sinks for the
switching devices together account for nearly 90% of the power supply weight and volume.]
One important factor that directly influences the size of the filter circuit elements and the
transformer is the converter’s switching frequency. High frequency operation of switch ‘S’ will
help in keeping the filter and transformer size small. The switching frequency of a typical forward
converter may thus be in the range of 100 kHz or more. The higher end limit on the switching
frequency comes mainly due to the finite switching time and finite switching losses of a practical
switch.
Switch limitations have been ignored in the simplified analysis presented here. As mentioned
earlier, the switch and the diodes have been assumed to be ideal, with no losses and zero
switching time. Control over switch duty ratio, which is the ratio of ON time to (ON + OFF)
time, provides the control over the output voltage ‘VO’.
Relation Between Input and Output Voltage
The equivalent circuits of mode-1 and mode-2 can be used to derive a steady state relation
between the input voltage, switch duty ratio (δ) and the output voltage. With the assumption of

constant input and output voltage, the instantaneous value of inductor voltage (eL) during mode-1
can be written as:
eL (t) = S
dc
P
N
E
N
- VO ; for 0 ≤ t ≤ δT, -------------------------------- (23.1)
Where t = 0 is the time instant when mode-1 of any steady state switching cycle starts, T is the
switching time period that may be assumed to be constant and δ is the duty ratio of the switch. It
can be seen that δT is the time duration of mode-1 and (1-δ) T is the time duration of mode-2. The
inductor voltage during mode-2 may similarly be written as:
eL (t) = - VO ; for δT ≤ t ≤ T, -------------------------------- (23.2)
Now since voltage across an inductor, averaged over a steady state cycle time, must always be
zero, one gets:
[ S
dc
P
N
E
N
- VO ] δ + [- VO ] (1-δ) = 0,
Or, VO = δ S
dc
P
N
E
N
------------------------------- (23.3)
Thus according to Eqn. (23.3), the forward converter output voltage is directly proportional to the
switch duty ratio. It may be noticed that except for transformer scaling factor the output voltage
relation is same as in a simple dc-to-dc buck converter. It is to be noted that the output voltage
relation given by Eqn. (23.3) is valid only under the assumption of continuous inductor
current. For an improperly designed circuit or for very light load at the converter output, the
inductor current may decay to zero in the midst of mode-2 resulting into discontinuous inductor
current. Once the inductor current becomes zero, diode ‘D2’ in Fig. 23.3(a) no longer conducts
and the points ‘P’ and ‘N’ of the equivalent circuit in Fig. 23.3(b) are no longer shorted. In fact,
the output voltage ‘VO’ will appear across ‘P’ and ‘N’. Thus equation (23.2) remains valid only
for a part of (1-δ) T period. In case of discontinuous inductor current, the output voltage, which is
the average of voltage across points ‘P’ and ‘N’ will have a higher magnitude than the one given
by Eqn. (23.3). Under discontinuous inductor current the relation between output voltage
and switch duty ratio becomes non-linear and is load dependent. For better control over
output voltage discontinuous inductor current mode is generally avoided. With prior knowledge of
the load-range and for the desired switching frequency the filter inductor may be suitably chosen
to keep the inductor current continuous and preferably with less ripple.
23.3 Practical Topology of A Forward Converter Circuit
Fig. 23.4 shows the circuit topology of a practical forward converter. It takes into account the non-
ideal nature of a practical transformer. Other non-idealities of the circuit elements like that of
switch, diodes, inductor and capacitor are taken care of by modifying the circuit parameters
chosen on the basis of ideal circuit assumption. Most common consequence of non-ideal nature of
circuit elements is increase in looses and hence reduction in efficiency of the power supply. A
practical way to get around the consequence of circuit losses is to over-design the power supply.
The design should aim to achieve an output power of o
P
η
, where ‘ o
P ’ is the required output power
and ‘η ’ is the efficiency of the converter. As a first order approximation, a typical efficiency
figure of around 80% may be assumed for the forward converter. Once the efficiency figure has
been considered the circuit may still be designed based on the simplified analysis presented here,
which neglects many of the non-idealities. Another common non-ideality is the low frequency
ripple and fluctuation in input dc supply voltage. In the simplified analysis input supply has been
assumed to be of constant magnitude. In a practical circuit, the variation in input supply is taken
care of by modulating the switch duty ratio in such a manner that it offsets the effect of supply
voltage fluctuation and continues to give the required quality of output voltage.
The non-ideality of the transformer, however, cannot simply be overcome by changing the circuit
parameters of the simplified circuit shown in Fig. 23.1. A practical transformer will have finite
magnetization current and finite energy associated with this magnetization current. Similarly there
will be some leakage inductance of the windings. However, windings of the forward-converter
transformer will have much smaller leakage inductances than those of fly-back converter
transformer. In fly-back transformer’s flux path some air-gap is deliberately introduced by
creating a gap in the transformer core (refer to lesson-22). Introduction of air gap in the mutual
flux path increases the magnitude of leakage inductances. Transformer of a forward converter
should have no air-gap in its flux path.
The forward-converter transformer works like a normal power transformer where both primary
and secondary windings conduct simultaneously with opposing magneto motive force (mmf)
along the mutual flux path. The difference of the mmfs is responsible for maintaining the
magnetizing flux in the core. When primary winding current is interrupted by switching off ‘S’,
the dotted ends of the windings develop negative potential to oppose the interruption of current (in
accordance with Lenz’s law). Negative potential of the dotted end of secondary winding makes
diode ‘D1’ reverse biased and hence it also stops conducting. This results in simultaneous opening
of both primary and secondary windings of the transformer. In case the basic circuit of Fig. 23.1 is
used along with a practical transformer, turning off of switch ‘S’ will result in sudden
demagnetization of the core from its previously magnetized state. As discussed in Lesson-22, a
practical circuit cannot support sudden change in flux. Any attempt to change flux suddenly
results in generation of infinitely large magnitude of voltage (in accordance with Lenz’s law).
Such a large voltage in the circuit will have a destructive effect and that should be avoided. Thus,
after switch ‘S’ is turned off, there must exist a convenient path for the trapped energy in the
primary due to magnetizing current. One solution could be a snubber circuit across the primary
winding, similar to the one shown in Fig.22.6 for a fly-back circuit (refer to lesson-22). Each time
the switch ‘S’ is turned off the snubber circuit will dissipate the energy associated with the
magnetizing flux. This, as has been seen in connection with fly-back converter, reduces the
power-supply efficiency considerably. A more preferred solution is to recover this energy. For this
reason the practical forward converter uses an extra tertiary winding with a series diode, as shown
in Fig. 23.4. When both switch ‘S’ and ‘D1’ turn-off together, as discussed above, the
magnetization energy will cause a current flow through the closely coupled tertiary winding and
the diode ‘D3’. The dot markings on the windings are to be observed. Current entering the dot
through any of the magnetically coupled windings will produce magnetic flux in the same sense.
As soon as switch ‘S’ is turned off, the dotted end voltages of the windings will become negative
in accordance with Lenz’s law. The sudden rise in magnitude of negative potential across the
windings is checked only by the conduction of current through the tertiary winding. As discussed
earlier unless the continuity in transformer flux is maintained the voltages in the windings will
theoretically reach infinite value. Thus turning off of switch ‘S’ and turn-on of diode ‘D3’ need to
be simultaneous. Similarly fall in magnetizing current through primary winding must be coupled
with simultaneous rise of magnetization current through the tertiary winding. In order that the
entire flux linking the primary winding gets transferred to the tertiary, the magnetic coupling
between these two windings must be very good. For this the primary and tertiary winding turns

are wound together, known as bifilar windings. The wires used for bifilar windings of the primary
and the tertiary need to withstand large electrical voltage stress and are costlier than ordinary
transformer wires.
Fig. 23.4: Circuit topology of a practical forward converter
V (o/p)
Switch S
NT : NP : NS
Edc
Load
D3
D1
D2
L
C
Fig. 23.5 shows some of the typical current and voltage waveforms of the forward converter
shown in Fig. 23.4. For these waveforms, once again, many of the ideal circuit assumptions have
been made.
In Fig. 23.5, Vload is the converter output voltage that is maintained constant at VO. ‘IL’ is the
current through filter inductor ‘L’. The inductor current rises linearly during mode-1 as its voltage
is maintained constant as per Eqn.23.1. Similarly the inductor current decays at a constant rate in
mode-2 as it flows against the constant output voltage. Average magnitude of inductor current
equals the load current. ISW and VSW are respectively the switch current and switch voltage. VD3 is
the voltage across diode ‘D3’. Switch conducts only during mode-1 and carries the primary
winding current (IPr) of the transformer. The transformer magnetization current is assumed to be
negligibly small and hence the primary winding essentially carries the reflected inductor current.
As switch ‘S’ turns on, primary winding gets input dc voltage (with its dotted end positive). The
induced voltages in other windings are in proportion to their turns ratios. Diode ‘D3’ of the tertiary
winding is reverse biased and is subjected to a voltage (1 )
T
dc
P
N
E
N
− + .
As soon as switch ‘S’ is turned-off, primary and secondary winding currents fall to zero but diode
‘D3’ gets forward biased and the tertiary winding starts conducting to maintain a path for the
magnetizing current. While ‘D3’ conducts the tertiary winding voltage is clamped to input dc
voltage with its dotted end negative. Primary and secondary windings have induced voltages due
to transformer action. Primary winding voltage equals to P
dc
T
N
E
N
, with dotted end at negative
potential.
In Fig. 23.5, ‘VPr’ denotes the primary winding voltage. The net volt-time area of the primary
winding voltage must be zero under steady state. Voltage across switch ‘S’ can be seen to be the
sum of primary winding voltage and the input voltage and equals (1 )
P
dc
T
N
E
N
+ .
As the tertiary winding current flows against the input dc supply, the magnetization current decays
linearly given by the following relation:
m
T
d
N
dt
Φ
= − dc
E -------------------------------- (23.4)
Where, = flux through the transformer core.
m
Φ
When the transformer is completely demagnetized, diode ‘D3’ turns off and voltage across
transformer windings fall to zero. The transformer remains de-magnetized for the remaining
duration of Mode-2.
When switch ‘S’ is again turned on, in the next switching cycle, the transformer flux builds up
linearly given by the relation: m
P dc
d
N
dt
E
Φ
= , ----------------------------- (23.5)
Under steady-state the increase in flux during conduction of switch ‘S’ must be equal to fall in
flux during conduction of tertiary winding and hence Eqns. 23.4 and 23.5 may be combined to
show that T
T P
t t
N N
= P
, where and are the time durations for which tertiary and primary
windings conduct during each switching cycle. Now,
T
t P
t
P
t = δT = on-duration of switch ‘S’ and the
tertiary winding conducts only during off duration of switch (during mode-2). Hence, (1-δ) T ≥
. As a result,
T
t
1
P
T
N
N
δ
δ
≤
−
, or
( )
P
P T
N
N N
δ ≤
+
------------------------------- (23.6)
Thus if P
N = , the duty ratio must be less than or equal to 50% or else the transformer
magnetic circuit will not get time to reset fully during mode-2 and will saturate. Less duty ratio
means less duration of powering mode (mode-1) and hence less transfer of power to the output
circuit. On the other hand, as described above, if
T
N
P
T
N
N
is increased for higher duty ratio, the switch
voltage stress increases.

V Pr
Time
Time
V load
I Pr
I L
Imax
Imax (NS / NP)
0
0
tON
tON
T
T
Time
0 VO
Imin
V sw
Fig.23.5: Some Typical waveforms of a practical Forward converter circuit
Time
Time
MODE-1 MODE-2
0
tON = δ T T
dc
E
(1 )
P
dc
T
N
E
N
+
MODE-1
0
V D3
(1 )
T
dc
P
N
E
N
− +
dc
E
−
Time
0
dc
E
P
dc
T
N
E
N
−
23.4 Selection of Transformer Turns Ratio
The transformer-winding turns ratio is a crucial design factor. The primary to secondary turns
ratio of the transformer is decided in accordance with Eqn. 23.3. For the required output voltage
(VO), the turns ratio S
P
N
N
is found after considering the minimum magnitude of input supply
voltage ( ) and the maximum allowable duty ratio (δ). The maximum duty ratio of the
converter, as discussed above, is constrained by the primary to tertiary winding turns ratio (given
by Eqn. 23.6) but the choice of primary to tertiary winding turns ratio is often governed by the
voltage stress that the switch must withstand. Higher voltage stress will mean higher cost of
switch. If the tertiary winding turns is kept very high, the switch voltage stress reduces but
allowable duty ratio of switch and the power output of the converter becomes low and diode ‘D
dc
E
3’
voltage rating increases. Thus an optimum design needs to be arrived at to maximize the
performance of the converter.
23.5 Selection of Filter Circuit Inductor and Capacitor
The transformer’s secondary voltage is rectified and filtered suitably to get the desired quality of
output voltage waveform. The filter inductor and capacitor values need to be chosen optimally to
arrive at a cost-effective, less bulky power supply. In this section, some simple guidelines have
been developed to arrive at the required filter size.
Inductor current waveform during a typical switching cycle has been shown in Fig. 23.5. As
described earlier, mean (dc) value of inductor current equals the load current. The filter capacitor
merely supplies the ripple (ac) current of switching frequency. It has also been mentioned earlier
that for linear relation between the output voltage and the switch duty ratio, the inductor current is
desired to be continuous (refer to Eqn.23.3). In case the inductor current becomes discontinuous
the linearity between switch duty ratio and output voltage is lost and the output-voltage controller
circuit, which is often designed using linear control theory, is not able to maintain the desired
quality of output voltage. Hence filter inductor should be chosen to be sufficiently large such that
under expected range of load current variation, the inductor current remains continuous. In many
cases the minimum value of load current may not be specified or may be too low. If the load
connected to the output is very light or if there is no load, the inductor current will not remain
continuous. Hence, as a thumb rule, the filter inductor size may be chosen such that the inductor
current remains continuous for more than 10% of the rated load current. At 10% of the load, the
inductor current may be assumed to be just continuous. This gives a basis for choosing the
inductor value as detailed below:
With reference to the waveforms in Fig. 23.5, under just continuous inductor current, Imin = 0 and
Iload = 0.5 (Imin + Imax) = 0.5 Imax = 0.1 Irated, where Irated is the rated load current.
Again, using Eqn.23.1, (Imax - Imin) = δT ( S
dc
P
N
E
N
- VO)/L ------------------- (23.7)
Thus for Imin = 0 and Imax = 0.2 Irated,
L = δT ( S
dc
P
N
E
N
- VO)/ (0.2 Irated) -------------------------- (23.8)
where VO, the output voltage, is assumed to have a fixed magnitude. Input supply voltage, ,
may itself be varying and the duty ratio is adjusted to keep V
dc
E
O constant in accordance with
Eqn.23.3. Thus even though ‘ ’ and ‘δ’ are varying, their product (δ
dc
E S
dc
P
N
E
N
) will be constant

and equal to VO. [As mentioned earlier, only low frequency variation in supply voltage has
been considered. Switching frequency and the switch control dynamics are assumed to be
much faster.]
Hence, the inductor ‘L’ magnitude should correspond to minimum value of duty ratio and may be
written as min
5
(1 )
O
rated SW
V
L
I f
δ
= − -------------------------------- (23.9)
,where min
δ is the minimum magnitude of duty ratio and SW
f is the constant switching
frequency of the converter switch. Now in accordance with Eqn.(23.6) the maximum value of
duty ratio may be taken as max
( )
P
P T
N
N N
δ =
+
. Again to maintain constant output voltage
,max
max
min ,min
dc
dc
E
E
δ
δ
= . -------------------------------- (23.10)
, where and are maximum and minimum magnitudes of input dc voltage
respectively.
,max
dc
E ,min
dc
E
Thus ,min
min
,max ( )
dc P
dc P T
E N
E N N
δ =
+
and
,min
,max
5
1
( )
dc
O P
rated SW dc P T
E
V N
L
I f E N N
⎡ ⎤
= −
⎢
+
⎢ ⎥
⎣ ⎦
⎥ -------------------------------- (23.11)
The inductor magnitude given by Eqn.(23.11) will limit the worst case peak to peak current ripple
in the filter inductor (= Imax - Imin) to 20% of rated current. [refer to Eqns.(23.7) and (23.8). It
may be noted here that as long as inductor current is continuous the peak-to-peak ripple in
the inductor current is not affected by the dc value of load current. For constant output
voltage and constant current through load, the inductor current ripple depends only on the
duty ratio, which in turn depends on the magnitude of input dc voltage]
Once inductor magnitude is chosen in accordance with Eqn.(23.11), peak to peak ripple in the
capacitor current will also be 20% of the rated current. This is so because the load, under steady
state, has been assumed to draw a constant magnitude of current.
Even though the output capacitor voltage has been assumed constant in our analysis so far,
there will be a minor ripple in capacitor voltage too which however will have only negligible
effect on the analysis carried out earlier. The worst case, peak to peak ripple in capacitor voltage
( ) can be given as:
,
O p p
v − ,
20
rated
O p p
SW
I
v
Cf
− = -------------------------------- (23.12)
, where ‘C’ is the output capacitance in farad. Capacitance value should be chosen, in
accordance with the above equation, based on the allowed ripple in the output voltage.
Quiz
1). If the turns ratio of the primary and tertiary windings of the forward transformer are in the
ratio of 1:2, what is the maximum duty ratio at which the converter can be operated?
Corresponding to this duty ratio, what should be the minimum ratio of secondary to primary
turns if the input dc supply is 400 volts and the required output voltage is 15 volts? Neglect
switch and diode conduction voltage drops.
[Answer: 1/3 and 9/80]
2) Find maximum voltage stress of the switch in the primary winding and diode in the tertiary
winding if the converter-transformer has 10 primary turns and 15 tertiary turns and the maximum
input dc voltage is 300 volts.
[Answer: Switch voltage stress = 500V, diode voltage stress = 750V]
3) Calculate the filter inductor and capacitor values for the forward converter described below:
Maximum duty ratio = 0.5, Input dc remains constant at 200 volts, output dc (under steady state)
= 10 volts ± 0.1 volt, primary to secondary turns = 10:1. The load current is expected to vary
between 0.5 and 5 amps. Assume just continuous conduction of inductor current at 0.5 amp load
current. Take switching frequency = 100 kHz.
[Answer: L = 50 micro Henry and C = 12.5 micro Farad]
(4) What function does the diode ‘D1’ of circuit in Fig.(23.4) have?
(i) rectifies secondary voltage
(ii) blocks back propagation of secondary voltage to transformer
(iii) both (i) and (ii)
(iv) protects diode ‘D2’ from excessive reverse voltage
[Answer: (iii)]

Module
3
DC to DC Converters
Lesson
22
Fly-Back Type Switched
Mode Power Supply

(i) Identify the topology of a fly-back type switched mode power supply circuit.
(ii) Explain the principle of operation of fly-back SMPS circuit.
(iii) Calculate the ratings of devices and components used in fly-back converter for the
specified input and output voltages and for the required output power.
(iv) Design a simple fly-back converter circuit.
22.1 Introduction
Fly-back converter is the most commonly used SMPS circuit for low output power applications
where the output voltage needs to be isolated from the input main supply. The output power of
fly-back type SMPS circuits may vary from few watts to less than 100 watts. The overall circuit
topology of this converter is considerably simpler than other SMPS circuits. Input to the circuit
is generally unregulated dc voltage obtained by rectifying the utility ac voltage followed by a
simple capacitor filter. The circuit can offer single or multiple isolated output voltages and can
operate over wide range of input voltage variation. In respect of energy-efficiency, fly-back
power supplies are inferior to many other SMPS circuits but its simple topology and low cost
makes it popular in low output power range.
The commonly used fly-back converter requires a single controllable switch like,
MOSFET and the usual switching frequency is in the range of 100 kHz. A two-
switch topology exists that offers better energy efficiency and less voltage stress
across the switches but costs more and the circuit complexity also increases
slightly. The present lesson is limited to the study of fly-back circuit of single
switch topology.
22.2 Basic Topology of Fly-Back Converter
Fig.22.1 shows the basic topology of a fly-back circuit. Input to the circuit may be unregulated
dc voltage derived from the utility ac supply after rectification and some filtering. The ripple in
dc voltage waveform is generally of low frequency and the overall ripple voltage waveform
repeats at twice the ac mains frequency. Since the SMPS circuit is operated at much higher
frequency (in the range of 100 kHz) the input voltage, in spite of being unregulated, may be
considered to have a constant magnitude during any high frequency cycle. A fast switching
device (‘S’), like a MOSFET, is used with fast dynamic control over switch duty ratio (ratio of
ON time to switching time-period) to maintain the desired output voltage. The transformer, in
Fig.22.1, is used for voltage isolation as well as for better matching between input and output
voltage and current requirements. Primary and secondary windings of the transformer are wound
to have good coupling so that they are linked by nearly same magnetic flux. As will be shown in
the next section the primary and secondary windings of the fly-back transformer don’t carry
current simultaneously and in this sense fly-back transformer works differently from a
normal transformer. In a normal transformer, under load, primary and secondary windings
conduct simultaneously such that the ampere turns of primary winding is nearly balanced by the
opposing ampere-turns of the secondary winding (the small difference in ampere-turns is
required to establish flux in the non-ideal core). Since primary and
VO
C
Gate pulses
Edc
Switch S
D
N1:N2
Fig. 22.1 Fly Back Converter
Load
Primary Side
secondary windings of the fly-back transformer don’t conduct simultaneously they are more like
two magnetically coupled inductors and it may be more appropriate to call the fly-back
transformer as inductor-transformer. Accordingly the magnetic circuit design of a fly-back
transformer is done like that for an inductor. The details of the inductor-transformer design are
dealt with separately in some later lesson. The output section of the fly-back transformer, which
consists of voltage rectification and filtering, is considerably simpler than in most other switched
mode power supply circuits. As can be seen from the circuit (Fig.22.1), the secondary winding
voltage is rectified and filtered using just a diode and a capacitor. Voltage across this filter
capacitor is the SMPS output voltage.
It may be noted here that the circuit shown in Fig.22.1 is rather schematic in nature. A more
practical circuit will have provisions for output voltage and current feedback and a controller for
modulating the duty ratio of the switch. It is quite common to have multiple secondary windings
for generating multiple isolated voltages. One of the secondary outputs may be dedicated for
estimating the load voltage as well as for supplying the control power to the circuit. Further, as
will be discussed later, a snubber circuit will be required to dissipate the energy stored in the
leakage inductance of the primary winding when switch ‘S’ is turned off.
Under this lesson, for ease of understanding, some simplifying assumptions are made. The
magnetic circuit is assumed to be linear and coupling between primary and secondary windings
is assumed to be ideal. Thus the circuit operation is explained without consideration of winding
leakage inductances. ON state voltage drops of switches and diodes are neglected. The windings,
the transformer core, capacitors etc. are assumed loss-less. The input dc supply is also assumed
to be ripple-free.
[A brief idea of a more practical fly-back converter will be given towards the end of this
lesson.]

22.3 Principle of Operation
During its operation fly-back converter assumes different circuit-configurations. Each of these
circuit configurations have been referred here as modes of circuit operation. The complete
operation of the power supply circuit is explained with the help of functionally equivalent
circuits in these different modes.
As may be seen from the circuit diagram of Fig.22.1, when switch ‘S’ is on, the primary winding
of the transformer gets connected to the input supply with its dotted end connected to the positive
side. At this time the diode ‘D’ connected in series with the secondary winding gets reverse
biased due to the induced voltage in the secondary (dotted end potential being higher). Thus with
the turning on of switch ‘S’, primary winding is able to carry current but current in the secondary
winding is blocked due to the reverse biased diode. The flux established in the transformer core
and linking the windings is entirely due to the primary winding current. This mode of circuit has
been described here as Mode-1 of circuit operation. Fig. 22.2(a) shows (in bold line) the current
carrying part of the circuit and Fig. 22.2(b) shows the circuit that is functionally equivalent to the
fly-back circuit during mode-1. In the equivalent circuit shown, the conducting switch or diode is
taken as a shorted switch and the device that is not conducting is taken as an open switch. This
representation of switch is in line with our assumption where the switches and diodes are
assumed to have ideal nature, having zero voltage drop during conduction and zero leakage
current during off state.
Vpri = Edc , Vsec = Edc*N2/N1
Fig.22.2(b): Equivalent circuit in Mode-1
+
Edc
Fig.22.2(a): Current path during Mode-1 of
circuit operation
+
Edc Vpri Vsec
N1 : N2
VO VO
Under Mode-1, the input supply voltage appears across the primary winding inductance and the
primary current rises linearly. The following mathematical relation gives an expression for
current rise through the primary winding:
Pr Pr
DC i
d
i
E L
dt
= × i ------------------------------------------------------------(22.1),
where is the input dc voltage, is inductance of the primary winding and i is
the instantaneous current through primary winding.
DC
E Pri
L Pri
Linear rise of primary winding current during mode-1 is shown in Fig.22.5(a) and Fig.22.5(b).
As described later, the fly-back circuit may have continuous flux operation or discontinuous flux
operation. The waveforms in Fig.22.5(a) and Fig.22.5(b) correspond to circuit operations in
continuous and discontinuous flux respectively. In case the circuit works in continuous flux
mode, the magnetic flux in the transformer core is not reset to zero before the next cyclic turning
ON of switch ‘S’. Since some flux is already present before ‘S’ is turned on, the primary winding
current in Fig. 22.3(a) abruptly rises to a finite value as the switch is turned on. Magnitude of the
current-step corresponds to the primary winding current required to maintain the previous flux in
the core.
At the end of switch-conduction (i.e., end of Mode-1), the energy stored in the magnetic field of
the fly back inductor-transformer is equal to 2
Pr 2
i P
L I , where denotes the magnitude of
primary current at the end of conduction period. Even though the secondary winding does not
conduct during this mode, the load connected to the output capacitor gets uninterrupted current
due to the previously stored charge on the capacitor. During mode-1, assuming a large capacitor,
the secondary winding voltage remains almost constant and equals to .
During mode-1, dotted end of secondary winding remains at higher potential than the other end.
Under this condition, voltage stress across the diode connected to secondary winding (which is
now reverse biased) is the sum of the induced voltage in secondary and the output voltage
(
P
I
2 1
/
Sec DC
V E N N
= ×
2 1
/
diode O DC
V V E N
= + × N ).
Mode-2 of circuit operation starts when switch ‘S’ is turned off after conducting for some time.
The primary winding current path is broken and according to laws of magnetic induction, the
voltage polarities across the windings reverse. Reversal of voltage polarities makes the diode in
the secondary circuit forward biased. Fig. 22.3(a) shows the current path (in bold line) during
mode-2 of circuit operation while Fig. 22.3(b) shows the functional equivalent of the circuit
during this mode.
In mode-2, though primary winding current is interrupted due to turning off of the switch ‘S’, the
secondary winding immediately starts conducting such that the net mmf produced by the
windings do not change abruptly. (mmf is magneto motive force that is responsible for flux
production in the core. Mmf, in this case, is the algebraic sum of the ampere-turns of the
two windings. Current entering the dotted ends of the windings may be assumed to
produce positive mmf and accordingly current entering the opposite end will produce
negative mmf.) Continuity of mmf, in magnitude and direction, is automatically ensured as
sudden change in mmf is not supported by a practical circuit for reasons briefly given below.
[mmf is proportional to the flux produced and flux, in turn, decides the energy stored in
the magnetic field (energy per unit volume being equal to 2
2
B μ , B being flux per unit
area and μ is the permeability of the medium). Sudden change in flux will mean sudden
Vpri = VO*N1/N2 , Vsec= VO
Vpri Vsec
N1: N2
+
Edc
+
Edc
VO
VO
Fig:22.3(a) : Current path during Mode-2 of
circuit operation Fig.22.3(b): Equivalent circuit in Mode-2

change in the magnetic field energy and this in turn will mean infinite magnitude of
instantaneous power, some thing that a practical system cannot support.]
For the idealized circuit considered here, the secondary winding current abruptly rises from zero
to 1 2
P
I N N as soon as the switch ‘S’ turns off. and denote the number of turns in the
primary and secondary windings respectively. The sudden rise of secondary winding current is
shown in Fig. 22.5(a) and Fig. 22.5(b). The diode connected in the secondary circuit, as shown in
Fig.22.1, allows only the current that enters through the dotted end. It can be seen that the
magnitude and current direction in the secondary winding is such that the mmf produced by the
two windings does not have any abrupt change. The secondary winding current charges the
output capacitor. The + marked end of the capacitor will have positive voltage. The output
capacitor is usually sufficiently large such that its voltage doesn’t change appreciably in a single
switching cycle but over a period of several cycles the capacitor voltage builds up to its steady
state value.
1
N 2
N
The steady-state magnitude of output capacitor voltage depends on various factors, like,
input dc supply, fly-back transformer parameters, switching frequency, switch duty ratio
and the load at the output. Capacitor voltage magnitude will stabilize if during each switching
cycle, the energy output by the secondary winding equals the energy delivered to the load.
As can be seen from the steady state waveforms of Figs.22.5(a) and 22.5(b), the secondary
winding current decays linearly as it flows against the constant output voltage (VO). The linear
decay of the secondary current can be expressed as follows: Sec Sec O
d
L i
dt
V
× = − ---------- (22.2),
Where, and are secondary winding inductance and current respectively.
is the stabilized magnitude of output voltage.
Sec
L Sec
i
O
V
Under steady-state and under the assumption of zero on-state voltage drop across diode, the
secondary winding voltage during this mode equals VO and the primary winding voltage =
VON1/N2 (dotted ends of both windings being at lower potential). Under this condition, voltage
stress across switch ‘S’ is the sum total of the induced emf in the primary winding and the dc
supply voltage (Vswitch = + V
DC
E ON1/N2).
The secondary winding, while charging the output capacitor (and feeding the load), starts
transferring energy from the magnetic field of the fly back transformer to the power supply
output in electrical form. If the off period of the switch is kept large, the secondary current gets
sufficient time to decay to zero and magnetic field energy is completely transferred to the output
capacitor and load. Flux linked by the windings remain zero until the next turn-on of the switch,
and the circuit is under discontinuous flux mode of operation. Alternately, if the off period of the
switch is small, the next turn on takes place before the secondary current decays to zero. The
circuit is then under continuous flux mode of operation.
During discontinuous mode, after complete transfer of the magnetic field energy to the output,
the secondary winding emf as well as current fall to zero and the diode in series with the winding
stops conducting. The output capacitor however continues to supply uninterrupted voltage to the
load. This part of the circuit operation has been referred to as Mode-3 of the circuit operation.
Mode-3 ends with turn ON of switch ‘S’ and then the circuit again goes to Mode-1 and the
sequence repeats.
+ +
Edc
E
Figs.22.4(a) and 22.4(b) respectively show the current path and the equivalent circuit during
mode-3 of circuit operation. Figs.22.5(a) and 22.5(b) show, the voltage and current waveforms of
the winding over a complete cycle. It may be noted here that even though the two windings of
the fly-back transformer don’t conduct simultaneously they are still coupled magnetically
(linking the same flux) and hence the induced voltages across the windings are proportional to
their number of turns.
dc
Fig.22.4(b): Equivalent circuit in Mode-3
VO
VO
Fig:22.4 (a) : Current flow path during Mode-
3 of circuit operation

Time
Time
Time
Time
V load
V pri
I sec
I pri
MODE-1
MODE-2
IP
IP X N1 / N2
T
T
T
tON
tON
tON
0
0
0
Fig.22.5(a): Fly-back circuit waveforms under continuous magnetic flux
EDC
VO X N1 / N2
VO
MODE-1
Io
Io X N1 / N2
22.4 Circuit Equations Under Continuous-Flux Operation
The waveforms in Fig.22.5(a) correspond to steady state operation under continuous magnetic
flux. ‘tON’ denotes the time for which the fly-back switch is ON during each switching cycle. ‘T’
stands for the time period of the switching cycle. The ratio ( tON /T) is known as the duty cycle
(δ) of the switch. As can be seen from Fig.22.5(a), the primary winding current rises from I0 to IP
in ‘δT’ time. In terms of input supply voltage (Edc) and the primary winding inductance ( )
the following relation holds:
Pri
L
(IP - I0) = (Edc / ) δT ------------------------------------------(22.3),
Pri
L
Under steady state the energy input to primary winding during each ON duration equals: 0.5Edc
(IP + I0) δT and similarly the output energy in each cycle equals V0 ILoad T, where V0 is the
output voltage magnitude and ILoad denotes the load current. Equating energy input and energy
output of the converter (the converter was assumed loss-less) in each supply cycle, one gets:
0.5Edc (IP + I0) δ = V0 ILoad ------------------------------------------(22.4),
The mean (dc) voltage across both primary and secondary windings must be zero under every
steady state. When the switch is ON, the primary winding voltage equals input supply voltage
and when the switch is OFF the reflected secondary voltage appears across the primary winding.
Under the assumption of ideal switch and diode,
Edc δ = (N1 / N2) V0 (1-δ) ------------------------------------------(22.5),
where N1 and N2 are the number of turns in primary and secondary windings and (N1/N2)V0 is
the reflected secondary voltage across the primary winding (dotted end of the windings at lower
potential) during mode-2 of circuit operation.
One needs to know the required ratings for the switch and the diode used in the converter. When
the switch is OFF, it has to block a voltage (Vswitch) that equals to the sum of input voltage and
the reflected secondary voltage during mode-2.
Thus, Vswitch = Edc + (N1 / N2) V0 -----------------------------------(22.6),
When the switch in ON, the diode has to block a voltage (Vdiode) that equals to the sum of output
voltage and reflected primary voltage during mode-1, i.e.,
Vdiode = V0 + Edc (N2 / N1) -----------------------------------(22.6a)
Since the intended switching frequency for SMPS circuits is generally in the range of 100kHz,
the switch and the diode used in the fly-back circuit must be capable of operating at high
frequency. The switch and the transformer primary winding must be rated to carry a repetitive
peak current equal to IP (related to maximum output power as given by Eqns. 22.3 to 22.5).
Similarly the secondary winding and the diode put in the secondary circuit must be rated to carry
a repetitive peak current equal to the maximum expected load current. The magnetic core of the
high frequency inductor-transformer must be chosen properly such that the core does not saturate
even when the primary winding carries the maximum expected current. Also, the transformer

core (made of ferrite material) must have low hysteresis loss even at high frequency operation.
Since the ferrite cores have very low conductivity, the eddy current related loss in the core is
generally insignificant.
22.5 Circuit Equations Under Discontinuous-Flux Mode
Fig. 22.5(b) shows some of the important voltage and current waveforms of the fly-back circuit
when it is operating in the discontinuous flux mode. During mode-3 of the circuit operation,
primary and secondary winding currents as well as voltages are zero. The load, however,
continues to get a reasonably steady voltage due to the relatively large output filter capacitor.
With the turning ON of the switch, the primary winding current starts building up linearly from
zero and at the end of mode-1 the magnetic field energy due to primary winding current rises to
2
1
2
pri P
L I . This entire energy is transferred to the output at the end of mode-2 of circuit operation.
Under the assumption of loss-less operation the output power (Po) can be expressed as:
Po = 2
1
2
pri P
L I fswitch -----------------------------------(22.7),
Time
Time
Time
Time
V load
V pri
I sec
I pri
MODE-1
MODE-2 MODE-3
IP
IP X N1 / N2
0
0
0
0
tON
tON
tON
T
T
T
Fig.22.5(b): Fly-back circuit waveforms under discontinuous flux
EDC
VO X N1 / N2
MODE-1
VO
where fswitch (=1/T) is the switching frequency of the converter.
It may be noted that output power Po is same as ‘V0 ILoad’ used in Eqn.(22.4). The volt-time area
equation as given in Eqn.(22.5) gets modified under discontinuous flux mode of operation as
follows:
Edc δ ≤ (N1 / N2) V0 (1-δ) ------------------------------------------(22.8)
Average voltage across windings over a switching cycle is still zero. The inequality sign of
Eqn.22.8 is due to the fact that during part of the OFF period of the switch [= (1-δ)T], the
winding voltages are zero. This zero voltage duration had been identified earlier as mode-3 of the
circuit operation. The equality sign in Eqn.(22.8) will correspond to just-continuous case, which
is the boundary between continuous and discontinuous mode of operation. The expression for
Vswitch and Vdiode, as given in Eqns.(22.6) and (22.6a), will hold good in discontinuous mode also.
22.6 Continuous Versus Discontinuous Flux Mode of Operation
A practical fly-back type SMPS circuit will have a closed loop control circuit for output voltage
regulation. The controller modulates the duty ratio of the switch to maintain the output voltage
within a small tolerable ripple voltage band around the desired output value. If the load is very
light, very small amount of energy needs to be input to the circuit in each switching cycle. This is
achieved by keeping the ON duration of the switch low, resulting in low duty ratio (δ). Within
this small ON time only a small amount of current builds up in the primary winding. The off
duration of the switch, which is (1-δ) fraction of the switching time period, is relatively large.
Mode-2 duration of the circuit operation is also small as the magnetic field energy is quickly
discharged into the output capacitor. Thus, at light load, the circuit is in mode-3 for significant
duration. As the load increases the mode-3 duration, during which there is zero winding currents
and zero flux through the core, reduces and the circuit is driven towards continuous flux mode.
The circuit operation changes from discontinuous to continuous flux mode if the output power
from the circuit increases beyond certain value. Similarly if the applied input voltage decreases,
keeping the load power and switching frequency constant, the circuit tends to go in continuous
flux mode of operation.
For better control over output voltage, discontinuous flux mode of operation is preferred.
However, for the given transformer and switch ratings etc., more output power can be transferred
during continuous flux mode. A common design thumb rule is to design the circuit for operation
at just-continuous flux mode at the minimum expected input voltage and at the maximum (rated)
output power.
22.7 A Practical Fly-Back Converter
The fly-back converter discussed in the previous sections neglects some of the practical aspects
of the circuit. The simplified and idealized circuit considered above essentially conveys the basic
idea behind the converter. However a practical converter will have device voltage drops and
losses, the transformer shown will also have some losses. The coupling between the primary and
secondary windings will not be ideal. The loss part of the circuit is to be kept in mind while
designing for rated power. The designed input power (Pin) should be equal to Po/η, where Po is
the required output power and η is the efficiency of the circuit. A typical figure for η may be

taken close to 0.6 for first design iteration. Similarly one needs to counter the effects of the non-
ideal coupling between the windings. Due to the non-ideal coupling between the primary and
secondary windings when the primary side switch is turned-off some energy is trapped in the
leakage inductance of the winding. The flux associated with the primary winding leakage
inductance will not link the secondary winding and hence the energy associated with the leakage
flux needs to be dissipated in an external circuit (known as snubber). Unless this energy finds a
path, there will be a large voltage spike across the windings which may destroy the circuit.
Fig.22.6 shows a practical fly-back converter. The snubber circuit consists of a fast recovery
diode in series with a parallel combination of a snubber capacitor and a resistor. The leakage-
inductance current of the primary winding finds a low impedance path through the snubber diode
to the snubber capacitor. It can be seen that the diode end of the snubber capacitor will be at
higher potential. To check the excessive voltage build up across the snubber capacitor a resistor
is put across it. Under steady state this resistor is meant to dissipate the leakage flux energy. The
power lost in the snubber circuit reduces the overall efficiency of the fly-back type SMPS circuit.
A typical figure for efficiency of a fly-back circuit is around 65% to 75%. In order that snubber
capacitor does not take away any portion of energy stored in the mutual flux of the windings, the
minimum steady state snubber capacitor voltage should be greater than the reflected secondary
voltage on the primary side. This can be achieved by proper choice of the snubber-resistor and
by keeping the RC time constant of the snubber circuit significantly higher than the switching
time period. Since the snubber capacitor voltage is kept higher than the reflected secondary
voltage, the worst-case switch voltage stress will be the sum of input voltage and the peak
magnitude of the snubber capacitor voltage.
V (o/p)
Edc
D
N1:N2
Fig. 22.6 A Practical Fly Back Converter
Load
C
PWM
Control
Block
S
N
U
B
B
E
R
N3
RS
Current Feedback
The circuit in Fig.22.6 also shows, in block diagram, a Pulse Width Modulation (PWM) control
circuit to control the duty ratio of the switch. In practical fly-back circuits, for closed loop output
voltage regulation, one needs to feed output voltage magnitude to the PWM controller. In order
to maintain ohmic isolation between the output voltage and the input switching circuit the output
voltage signal needs to be isolated before feeding back. A popular way of feeding the isolated
voltage information is to use a tertiary winding. The tertiary winding voltage is rectified in a way
similar to the rectification done for the secondary winding. The rectified tertiary voltage will be
nearly proportional to the secondary voltage multiplied by the turns-ratio between the windings.
The rectified tertiary winding voltage also doubles up as control power supply for the PWM
controller. For initial powering up of the circuit the control power is drawn directly from the
input supply through a resistor (shown as RS in Fig.22.6) connected between the input supply
and the capacitor of the tertiary circuit rectifier. The resistor ‘RS’ is of high magnitude and causes
only small continuous power loss.
In case, multiple isolated output voltages are required, the fly-back transformer will need to have
multiple secondary windings. Each of these secondary winding voltages are rectified and filtered
separately. Each rectifier and filter circuit uses the simple diode and capacitor as shown earlier
for a single secondary winding. In the practical circuit shown above, where a tertiary winding is
used for voltage feedback, it may not be possible to compensate exactly for the secondary
winding resistance drop as the tertiary winding is unaware of the actual load supplied by the
secondary winding. However for most applications the small voltage drop in the winding
resistance may be tolerable. Else, one needs to improve the voltage regulation by adding a linear
regulator stage in tandem (as mentioned in Chapter-21) or by giving a direct output voltage
feedback to the control circuit.
Quiz Problems
(i) What kind of output rectifier and filter circuit is used in a fly back converter?
(a) a four-diode bridge rectifier followed by a capacitor
(b) a single diode followed by an inductor-capacitor filter
(c) a single diode followed by a capacitor
(d) will require a center-tapped secondary winding followed by a full wave rectifier and
a output filter capacitor.
(ii) A fly-back converter operates in discontinuous conduction mode with fixed ON duration of
the switch in each switching cycle. Assuming input voltage and the resistive load at the
output to remain constant, how will the output voltage change with change in switching
frequency? (Assume discontinuous conduction through out and neglect circuit losses.)
(a) Output voltage varies directly with switching frequency.
(b) Output voltage varies inversely with switching frequency.
(c) Output voltage varies directly with square root of switching frequency.
(d) Output voltage is independent of switching frequency.
(iii) A fly-back converter has primary to secondary turns ratio of 15:1. The input voltage is
constant at 200 volts and the output voltage is maintained at 18 volts. What should be the
snubber capacitor voltage under steady state?
(a) More than 270 volts.
(b) More than 200 volts but less than 270 volts.
(c) Less than 18 volts.
(d) Not related to input or output voltage.

(iv)A fly-back converter is to be designed to operate in just-continuous conduction mode when
the input dc is at its minimum expected voltage of 200 volts and when the load draws
maximum power. The load voltage is regulated at 16 volts. What should be the primary to
secondary turns ratio of the transformer if the switch duty ratio is limited to 80%. Neglect
ON-state voltage drop across switch and diodes.
(a) 20 :1
(b) 30 :1
(c) 25 :2
(d) 50 :1
Answers to quiz problems: (i-c), (ii-c), (iii-a), (iv-d).

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