# AN-1106: An Improved Topology for Creating Split Rails from a Single Input Voltage

### Introduction

Even with the widespread use of rail-to-rail single supply op amps, there is still often the requirement for dual rails (for example, ±15 V) to be generated from a single (positive) input power rail to power different parts of the analog signal chain. These are often low current (such as 10 mA to 500 mA) with relatively well-matched loads on the positive and negative supplies.

One solution to this problem is to use two different converters; one to provide the positive rail and one to provide the negative rail. This can be expensive and, as this application note shows, unnecessary. Another solution is using a flyback; however, the supplies tend not to track each other very well with differential loading, it requires a large and expensive transformer, and it tends to be inefficient.

A better solution is a SEPIC-Ćuk converter. This topology consists of an unregulated Ćuk converter tied to the same switching node as a regulated SEPIC converter. This combination results in two supplies that track each other very well under all but a 100% load mismatch.

An analysis of the converter’s operation and implementation using the Analog Devices, Inc., ADP161x demonstrates the versatility of this topology. In addition, a revolutionary new design tool is introduced, providing a quick path to implementing a SEPIC-Ćuk in user applications.

### Description of Topology

Initially, the SEPIC-Ćuk appears to be a complicated converter with four different inductors and switches. Fortunately, it can be broken down into its two constituent converters, simplifying the analytical problem. For a SEPIC or Ćuk converter, the Q1 and Q2 switches operate in the opposite phase from one another. Figure 2 shows the current flow diagram for the two different switch states in a SEPIC converter.

It is not immediately obvious, but the transfer capacitor
(C1) voltage is approximately constant V_{IN} (with
small ripple).

Figure 4 shows the idealized waveforms for a SEPIC converter.
When Q1 is on, the voltage at SN2 is equal to −V_{IN}. Thus,
during the time that Q1 is on (Q2 is off), the voltage across
both L1a and L1b is V_{IN} and when Q1 is off (Q2 is on), then
the voltage across both L1a and L1b is −V_{OUT}. Applying the
principles of inductor-volt second balance, the equilibrium dc
conversion ratio as shown in Equation 1 can be calculated. D
is the converter’s duty cycle (the fraction of the switching cycle
that Q1 is on).

The Ćuk converter operates in a similar manner to the SEPIC converter, however, in this case, Switch Q2 is connected to ground rather than the output and the Inductor L2b is connected to the output instead of ground. Figure 3 shows a current flow diagram for the Ćuk converter during both switch positions.

The Ćuk is a negative output converter, so current flowing out of the load is actually delivering power to the output.

The idealized waveforms for a Ćuk converter are shown in
Figure 4. Applying the principles of inductor-volt second
balance and capacitor charge balance, the voltage across C2 is
V_{IN} + V_{OUT}. Therefore, the SN2 switch node switches between
GND, when Q2 is closed, and −(V_{IN} + V_{OUT}). The voltage across
both L2a and L2b while Q1 is on (Q2 is off), is V_{IN} and, while
Q1 is off (Q2 is on), the voltage across both L2a and L2b
is −V_{OUT}.

Comparing the waveforms in Figure 4 and Figure 5, note that the voltages across the inductors in a Ćuk are identical to those for the SEPIC. Thus, the duty cycle equation for a Ćuk is simply negative the duty cycle for the SEPIC,` as shown in Equation 2.

The fact that the duty cycles are equal and opposite, the switch node (SN1) voltages are identical, and the inductor currents are identical is what makes it possible to simply attach the two converters together at Node SN1. The combined converter is shown in Figure 1.

Q2 and Q3 have been replaced by diodes because these supplies are generally lower power analog supplies where an asynchronous controller makes good sense. In addition, two inductors (L1a and L2a) are in parallel. The reason for this is that L1a and L1b, and L2a and L2b, are coupled together using two separate coupled inductors. This has multiple advantages.

Coupling the inductors reduces current ripple in the inductors by a factor of two (see the Ćuk-Middlebrook paper cited in the References section). In addition, it significantly reduces the complexity of the small signal model and enables higher bandwidth by eliminating the SEPIC and Ćuk resonances located according to Equation 3 and Equation 4. This enables the use of a wide variety of off-the-shelf parts since there are not many three winding 1:1:1 inductors available.

A six winding part, such as found in Coilcraft’s Hexapath line product line, or a custom three winding transformer could also be used.

#### Limits to the Coupling Coefficient

Even though coupling the inductors has distinct advantages, it
is undesirable for the coupling to be tight enough for there to
be significant energy transfer through the core. To avoid this
situation, the designer must ensure that the magnitude of the
complex impedance of C1 (and C2) at the switching frequency
is less than a tenth that of the impedance of the leakage
inductance (L_{LKG}) plus the DCR of a single winding.

This inequality is designated in Equation 5. The leakage
inductance (L_{l}) can be calculated using Equation 6 and
the coupling coefficient (K) generally found on coupled
inductor data sheets. L_{m} is the measured self-inductance that
appears in the data sheet. Note that in Equation 5, the x in C_{x}
and L_{x} refers to either C1 or C2 or L1 or L2.

#### Differential Load and Output Voltage Tracking

By nature, the Ćuk (negative) output of the SEPIC-Ćuk is unregulated; thus, there is some amount of load variation with changes in output current and, particularly with load mismatch, compared to the SEPIC (positive) output. Note that the tracking is much better than a similarly configured flyback converter, especially in the case of a transient or a load mismatch. This is because the coupling between channels is a direct connection rather than through the transformer with its inherent leakage inductance.

Figure 6 shows a 30 mA transient applied to the Ćuk (−V_{OUT})
output of a SEPIC-Ćuk converter, while a constant 100 mA
remains on the SEPIC output. It shows that both outputs
respond to the transient load. This is the worst-case transient
because the Ćuk output is unregulated. Interestingly, most of
the deviation shown on the −V_{OUT} rail is actually dc regulation
shift caused by the mismatch between the loads applied to the
two rails (I_{OUT}+, I_{OUT}−).

With an identical load on both supplies, at steady state, the most significant error terms are a mismatch in the DCR of the inductors and the forward voltage of the diodes, both of which can be made quite small relative to the output voltage.

With substantial load mismatch, the error grows as shown in Figure 7. Therefore, in some applications it may be necessary to put a small dummy load on one or both of the channels to keep both supplies in their regulation window. Note that, in general, analog chips, like op amps, are largely insensitive to dc changes in their power supplies as long as there is sufficient head room available.

#### Small-signal Analysis and Loop Compensation

A complete small-signal analysis of the SEPIC-Ćuk converter is beyond the scope of this paper; however, the equations provided in this application note should allow the designer to correctly compensate their design. The ADP161x SEPIC-Ćuk design tool uses a more complete model which is more accurate, but much more complicated. The equations shown refer to the ADP161x part in SEPIC-Ćuk and may not be accurate for other parts made by Analog Devices or the company’s competitors.

The small-signal model for a SEPIC-ĆUK looks very similar to a SEPIC converter with no attached Ćuk as long as a few design requirements are met. It is assumed that identical inductors are used on the SEPIC-Ćuk rails. This requirement makes sense because both outputs are designed for the same voltage and current.

In their paper, Ćuk and Middlebrook (see the References
section) show that a coupled inductor, from both a small signal
and a large signal, behaves like an inductor with twice its
single winding inductance value, without the SEPIC or Ćuk
resonances. Therefore, analysis in this application note is shown
using the effective inductance, that is, twice the single winding
inductance value that appears on coupled inductor data sheets.
The analysis assumes identical resistive loads, though the
converter remains stable with significant load imbalance. The
two transfer capacitances (C1 and C2) should be nearly the
same value, erring on the side of having C2 slightly larger than
C1. These are assumed to be ceramic capacitors and, thus, the
designer needs to take into account the differences in their dc
bias value when calculating their effective capacitances.
The first step in compensating a SEPIC Ćuk is to choose an
achievable target crossover frequency. Like most boost and
buck-boost topologies, the SEPIC-Ćuk has a right half plane
zero (RHP) located according to Equation 7. An RHP has the
dual effect of adding gain, like a zero, and subtracting phase,
like a pole. Therefore, the converter must be compensated for
a crossover frequency a maximum of one fifth of the frequency
of the RHP (f_{RHP}).

The SEPIC-Ćuk has an additional resonance caused by the
leakage inductance (L_{lkg} ) and transfer capacitance (C1) that
occurs at F_{res}. This resonance is generally well damped by the
DCR of the inductors, but can introduce significant phase lag;
therefore, it is good to crossover at least a decade before it. In
addition, a current mode controller with standard Type II
compensation is used, thus, the maximum achievable crossover
frequency is approximately one-tenth the switching frequency.
Target fu should, therefore, be chosen as the minimum of these
three constraints, as shown in Equation 9.

The compensation values in Figure 8 can be calculated as
follows. Since it is assumed ceramic output capacitors will be
used, C_{C}2 can be selected as 10 pf.

where:

*f _{p}* is the dominant pole for the current mode converter with
some correction factors to account for ramp compensation and
finite current gain.

*A _{c}* is the magnitude of the open-loop converter gain at the
crossover frequency fu.

*M _{c}* and

*F*are terms derived from Ridley’s thesis (see the References section) on current mode control.

_{m}V_{ramp} and A_{cs} are fixed constants within the chip.

#### Power Component Stress

As is often the case, a 30% ripple in the inductors generally results in a reasonable value (see Equation 19). However, with large step down ratios it can be more optimal to increase this ripple percentage in the input inductor to 50% or 60%.

The currents in the FET Switch Q1 and the two diode switches, Q2 and Q3, are shown in Figure 9. The dc components of the switch current are also shown in Figure 9. Note that Q1 carries the current for both the SEPIC and the Ćuk rails. The peak currents depend on the ripple chosen in Equation 19.

Calculating the switching loss in the primary Switch Q1 is
beyond the scope of this application note. Note that, in many
cases, the switching loss can be quite large since the voltage
swing the switch sees is large (~V_{IN} + V_{OUT}) and so are the
currents (see Figure 9).

The ADP1612/ADP1613 work to reduce this loss by switching
very quickly. The FET chosen must be rated to withstand at
least V_{IN} + V_{OUT} and good engineering allows some margin for
switch node ringing due to stray inductances, in addition to
thermal stress from RDS on loss and switching losses.

The peak-to-peak output voltage ripple on the SEPIC (positive)
output is (ΔV_{ripple SEPIC}) and is approximated by

The value of the current through the capacitor (I_{RMS Cout SEPIC}) is

The peak-to-peak output voltage ripple on the Ćuk (negative)
output (ΔV_{ripple Ćuk}) is approximated by

The I rms value of the current into the C_{OUT} on the Ćuk
(negative) output (ΔV_{rip Ćuk}) is approximated by

The ripple on C1 and C2 should be chosen for around 5% of
V_{IN}. As stated earlier, they should have similar values despite the
difference in dc voltage across them.

It is important to consider I rms ratings when choosing C1 and C2 since the current through them is quite large.

Since Q2 and Q3 are generally diodes, there are several things
to consider when choosing a component. V_{ds max} must be rated
to at least V_{IN} + V_{OUT}. The continuous current should be at
least 1/3 the peak current to be seen. Interestingly, because of
the phase relationship between the output voltage ripple of the
two supplies, the SEPIC diode actually receives the full switch
for some amount of time before the current achieves a more
even split. As expected though, the average current through
both diodes is the same, I_{OUT}. In addition, the package must
be able to handle the I_{OUT} in the thermal environment of the
application.

#### Output Filter

The SEPIC-Ćuk as a dual rail converter is typically used for analog power supplies, which often require very low output ripple. Low output ripple (down to 1 mV) is generally easily achieved on the Ćuk (negative) output rail simply by using ceramic output capacitors because the output current is continuous like the output current of a buck converter.

On the SEPIC (positive) rail, the output current is discontinuous like the input current of a buck converter. This results in a step change in the current into the output capacitors. These switching spikes are not well attenuated even by ceramic capacitors because of their inductance. Therefore, it is often necessary to put a small, damped output pi filter on the output of the SEPIC winding.

Although this filter affects the small-signal model in new and interesting ways, this issue is not fully discussed in this application note. As long as the damping resistor is chosen according to the Equation 31 and Equation 32, and the converter is designed to crossover at a tenth of ωo or less, no instability should be caused by the pi filter.

C_{OUT1} should be chosen for around 2% output ripple and C_{OUT2}
should be chosen to match the output capacitor of the Ćuk
output using the equations in the power components stress
section. A good value for L_{filt} is generally 1 μH, and Q_{o} should
be set to 1.

### ADP161X Design Tool

The ADP161x SEPIC-Ćuk design tool is a fully integrated
Excel^{®}-based designer for the ADP161x chips in a SEPIC-Ćuk
configuration. Once the user has enabled macros (which may
require a change of the security settings in Excel), the **Enter
Inputs** dialog box appears, or can be found by pressing the
**Find Solution** button. In the dialog box, enter the voltages
and currents required for the design and choose whether to
optimize for cost, loss, or size.

If the **View Solution** button is pressed, the design tool outputs
a complete, optimized design. This includes a costed BOM with
compensation values, an accurate, tested efficiency plot across load,
a plot of power loss across load, a full load bode plot, performance
parameters, component stresses, and power dissipation for every
component. In addition, the **Build Your Design** tab provides the
same BOM, but with the components arranged to fit on the blank
demo board (ADP161x-BL3-EVZ) and any extra components
required to configure the demo board.

Additional customization tools are available in the **Advanced
Settings** dialog box. Here the user can select parameter
specifications for output voltage ripple, current, transient
response, optional output filter usage, an external UVLO, and
more. A more in-depth description of the functionality of these
options is provided in the **Program Details** dialog box available
by clicking the **Program Details** button found on the **Enter
Inputs** dialog box.

One of the most powerful features of this tool are the
component buttons found on the **User Interface** tab. This
functionality gives the user the ability to individually change
each component to fully customize the design.

Each of the components in the drop-down list have been preselected from a database of thousands of components to produce a functional design, and sorted according to the optimization chosen in the Enter Inputs dialog box. The components must be selected in order, from top to bottom, since there are dependencies between the different components.

### Lab Results

To demonstrate the efficacy of the design tool, a design was
done using the tool for 5 V_{IN}, ±5 V_{OUT} at 50 mA with the
advanced specifications shown in Figure 11 and Figure 12.
In addition, the diode was changed for slightly lower loss.
The jagged efficiency line at around 10 mA is caused by the
converter going into discontinuous mode. Once both the
switches have turned off, the switch node rings causing zero
voltage switching at specific load currents. A schematic for the
circuit is shown in Figure 14.

### Conclusion

In conclusion, the SEPIC-Ćuk provides an inexpensive and robust way to create dual rails using only one controller. The ADIsimPOWER™ design tool allows complete customization of the design and can be relied on to create robust SEPIC-Ćuk designs quickly.

### References

Ćuk, Slobodan and R.D. Middlebrook. 1983. “CoupledInductor and Other Extensions of a New Optimum Topology
Switching DC-DC Converter.” *Advances in Switched-Mode
Power Conversion*, Volumes I and II. Irvine, CA: Tesla Co.

Ridley, Dr. Ray. 1990. “A New Continuous-Time Model for Current-Mode Control.” Brandenton, FL: Ridley Engineering.