**Source resistance: the efficiency killer
in DC-DC converter circuits **

*The DC-DC converter is very commonly used in battery-operated
equipment and other power-conserving applications. Like a linear regulator, the
DC-DC converter can regulate to a lower voltage. Unlike linear regulators,
however, the DC-DC converter can also boost an input voltage or invert it to
V _{IN-}. As an added bonus, the DC-DC converter boasts efficiencies
greater than 95% under optimum conditions. However, this efficiency is limited
by dissipative components, and the main cause is resistance in the power
source.*

Losses in source resistance can lower the efficiency by 10% or more,
exclusive of loss in the DC-DC converter! If the converter has adequate input
voltage, its output will be normal and there may be no obvious indication that
power is being wasted. Fortunately, testing the input efficiency is a simple
matter (see the *Source* section).

A large source resistance can cause other, less obvious effects. In extreme
cases, the converter's input can become bistable, or its output can decrease
under maximum load conditions. Bistability means that the converter exhibits two
stable input conditions, each with its own efficiency. The converter output is
normal, but system efficiency may be drastically affected (see *How to Avoid
Bistability*).

Should this problem be solved simply by minimizing the source resistance? No, because the practical limits and cost/benefit trade-offs posed by the system may suggest other solutions. A prudent selection of power-supply input voltage, for example, can considerably minimize the need for low source resistance. Higher input voltage for a DC-DC converter limits the input current requirement, which in turn lessens the need for a low source resistance. From a systems standpoint, the conversion of 5V to 2.5V may be far more efficient than the conversion of 3.3V to 2.5V. Each option must be evaluated. The goal of this article is to provide analytic and intuitive tools for simplifying the evaluation task.

**A systems view**

As shown in
**Figure 1**, any regulated power-distribution system can be divided into
three basic sections: source, regulator(s) (a DC-DC converter in this case), and
load(s). The source can be a battery or a DC power supply that is either
regulated or unregulated. Unfortunately, the source also includes all the
dissipative elements between the DC voltage and load: voltage-source output
impedance; wiring resistance; and the resistance of contacts, PC-board lands,
series filters, series switches, hot-swap circuits, etc. These elements can
seriously degrade system efficiency.

*Figure 1. A regulated power-distribution system has three basic
sections.*

Calculation and measurement of the source efficiency is very simple.
EFF_{SOURCE} equals (power delivered to the regulator) / (power provided
by VPS) multiplied by 100%:

Assuming that the regulator draws a negligible amount of current when
unloaded, you can measure source efficiency as the ratio of V_{IN} with
the regulator at full load to V_{IN} with the regulator unloaded.

The regulator (DC-DC converter) consists of a controller IC and associated
discrete components. Its characterization is described in the manufacturer's
data sheet. Efficiency for the DC-DC converter (EFF_{DCDC}) equals
(power delivered by the converter) / (power delivered to the converter)
multiplied by 100%:

As specified by the manufacturer, this efficiency is a function of input voltage, output voltage, and output load current. It's not unusual for the efficiency to vary no more than a few percent over a load current range exceeding two orders of magnitude. Because the output voltage is fixed, we can say the efficiency varies only a few percent over an "output-power range" exceeding two orders of magnitude.

DC-DC converters are most efficient when the input voltage is closest to the output voltage. If the input variation is not extreme with respect to the data sheet specifications, however, the converter's efficiency can usually be approximated as a constant between 75% and 95%:

This discussion treats the DC-DC converter as a two-port black box. For those
interested in the nuances of DC-DC converter design, see **References 1-3**.

The load includes the device to be driven and all dissipative elements in
series with it, such as PC-trace resistance, contact resistance, cable
resistance, etc. Because the DC-DC converter's output resistance is included in
the manufacturer's data sheet, that quantity is specifically excluded. Load
efficiency (EFF_{LOAD}) equals (power delivered to the load) / (power
delivered by the DC-DC converter) multiplied by 100%:

The key to optimum system designs is in analyzing and understanding the interaction between the DC-DC converter and its source. To do this we first define an ideal converter, then calculate the source efficiency, then test our assumptions against measured data from a representative DC-DC converter-in this case, the MAX1626 buck regulator.

**The ideal DC-DC
converter**

An ideal DC-DC converter would have 100% efficiency,
operate over arbitrary input- and output-voltage ranges, and supply arbitrary
currents to the load. It would also be arbitrarily small and available for free!
For this analysis, however, we assume only that the converter's efficiency is
constant, such that input power is proportional to output power:

For a given load, this condition implies that the input current-voltage (I-V)
curve is hyperbolic and exhibits a negative differential-resistance
characteristic over its full range (**Figure 2**). This plot presents I-V
curves for the DC-DC converter as a function of increasing input power. For real
systems with dynamic loads, these curves are also dynamic. That is, the power
curve moves farther from the origin as the load demands more current.

*Figure 2. These hyperbolas represent constant-power input characteristics
for a DC-DC converter.*

Considering a regulator from the input port instead of the output port is an
unusual point of view. After all, regulators are designed to provide a
constant-voltage (sometimes constant-current) output. Their specifications
predominantly describe the output characteristics (output-voltage range,
output-current range, output ripple, transient response, etc.). The input,
however, displays a curious property: within its operating range it acts as a
constant-power load (**Reference 4**). Constant-power loads are useful in the
design of battery testers, among other tasks.

**Calculating source
efficiency**

We now have enough information to calculate the
source's power dissipation and therefore its efficiency. Because the
open-circuit value of source voltage (V_{PS}) is given, we need only
find the DC-DC converter's input voltage (V_{IN}). From equation [5],
solving for I_{IN}:

I_{IN} can also be solved in terms of V_{PS}, V_{IN},
and R_{S}:

Equate the expressions from equations [6] and [7] and solve for
V_{IN}:

To understand their implications, it is very instructive to visualize
equations [6] and [7] graphically (**Figure 3**).

*Figure 3. This plot superimposes a load line for source resistance on the
DC-DC converter's I-V curve.*

The resistor load line is a plot of all possible solutions of equation [7],
and the DC-DC I-V curve is a plot of all possible solutions of equation [6]. The
intersections of these curves, representing solutions to the pair of
simultaneous equations, define stable voltages and currents at the DC-DC
converter's input. Because the DC-DC curve represents constant input power,
(V_{IN+})(I_{IN+}) = (V_{IN-})(I_{IN-}). (The +
and - suffixes refer to the two solutions predicted by equation [8], and
correspond to the ± signs in the numerator.)

The optimum operating point is at V_{IN+} / I_{IN+}, which
minimizes I_{IN2}R_{S} loss by drawing minimum
current from the power supply. The other operating point causes large power
dissipation in any dissipative components between V_{PS} and
V_{IN}. System efficiency drops dramatically. But you can avoid such
problems by keeping R_{S} low enough. The source efficiency
[(V_{IN} / V_{PS}) · 100%] is simply equation [8] divided by
V_{PS}:

It's easy to get lost in the equations, and therein lies the value of the
load-line analysis plot of Figure 3. Note, for example, that if the series
resistance (R_{S}) equals zero, the resistor load-line slope becomes
infinite. The load line would then be a vertical line passing through
V_{PS}. At this point V_{IN+} = V_{PS} and the
efficiency would be 100%. As R_{S} increases from 0W, the load line continues to pass through
V_{PS} but leans more and more to the left. Concurrently,
V_{IN+} and V_{IN-} converge on V_{PS} / 2, which is
also the 50% efficiency point. When the load line is tangent to the I-V curve,
equation [8] has only one solution. For larger R_{S}, the equation has
no real solution and the DC-DC converter no longer functions properly.

**DC-DC converters-theory vs.
practice**

How do these ideal-input curves compare with those of an
actual DC-DC converter? To examine this question, a standard MAX1626 evaluation
kit (**Figure 4**) was configured for an output voltage of 3.3V and a load
resistor of 6.6W. We then measured the
input's I-V curve (**Figure 5**). Several nonideal characteristics were
evident immediately. Note, for example, that for very low input voltages the
input current is zero. A built-in undervoltage lockout (denoted as
V_{L}) ensures that the DC-DC converter is off for all input voltages
below V_{L}. Otherwise, large input currents could be drawn from the
power supply during start-up.

*Figure 4. A standard DC-DC converter circuit illustrates the ideas of
Figure 3.*

*Figure 5. Above V _{MIN}, the MAX1626 input I-V characteristic
closely matches that of a 90%-efficient ideal device.*

When V_{IN} exceeds V_{L}, the input current climbs toward a
maximum that occurs when V_{OUT} first reaches the preset output voltage
(3.3V). The corresponding input voltage (V_{MIN}) is the minimum
required by the DC-DC converter to produce the preset output voltage. For
V_{IN} > V_{MIN}, the constant-power curve for 90% efficiency
closely matches the MAX1626 input curve. Variations from the ideal are caused
primarily by small variations in DC-DC converter efficiency as a function of its
input voltage.

**How to avoid
bistability**

The power-supply designer must also guarantee that the
DC-DC converter never becomes bistable. Bistability is possible in systems for
which the load line intersects the DC-DC converter curve at or below
V_{MIN} / I_{MAX} (**Figure 6**).

*Figure 6. A closer look at the intersection points indicates a possibility
of bistable and even tristable operation.*

Depending on the load line's slope and position, a system can be bistable or
even tristable. Note that a lower V_{PS} value can allow the load line
to intersect at a single point between V_{L} and V_{MIN},
resulting in a system that is stable, but nonfunctional! As a rule, therefore,
the load line must not touch the cusp of the DC-DC converter curve and must not
move below it.

In Figure 6, the load-line resistance (R_{S}, which has a value of
-1/slope) has an upper limit called R_{BISTABLE}:

The source resistance (R_{S}) should always be smaller than
R_{BISTABLE}. If this rule is broken, you risk highly inefficient
operation or a complete shutdown of the DC-DC converter.

**An actual case**

It might be
helpful to plot, for an actual system, the relationship shown in equation [9]
between source efficiency and source resistance (**Figure 7**). Assume the
following conditions:

V_{PS} = 10V Open-circuit power-supply
voltage

V_{MIN} = 2V Minimum input voltage that
ensures proper operation

P_{IN} = 50W Power to the
DC-DC converter's input (P_{OUT} / EFF_{DCDC}).

*Figure 7. This plot of source efficiency vs. source resistance indicates
multiple values of efficiency for a given R _{S}.*

Using equation [12], R_{BISTABLE} can be calculated as 0.320W. Subsequently, a plot of equation [9] shows
that source efficiency drops as R_{S} increases, losing 20% at
R_{S} = R_{BISTABLE}. Note: this result cannot be generalized.
You must perform the calculations for each application. One component of
R_{S} is the finite output resistance found in all power supplies,
determined by the load regulation and usually defined as:

A 5V / 10A power supply with 1% load regulation, for example, would have only 5.0mW of output resistance-not much for a 10A load.

**Source efficiency for common
applications**

It's useful to know how much source resistance
(R_{S}) can be tolerated and how this parameter affects system
efficiency. R_{S} must be less than R_{BISTABLE}, as stated
earlier, but how much lower should it be? To answer this question, solve
equation [9] for R_{S} in terms of EFF_{SOURCE}, for
EFF_{SOURCE} values of 95%, 90%, and 85%. R_{S}95 is the
R_{S} value that yields a 95% source efficiency for the given input and
output conditions. Consider the following four example applications using common
DC-DC converter systems.

**Example 1** derives 3.3V from 5V with a load current of 2A. For 95%
source efficiency, be careful to keep the resistance between the 5V source and
the DC-DC converter's input well under 162mW. Notice that R_{S}90 =
R_{BISTABLE}, by coincidence. This value of R_{S}90 also implies
that the efficiency could as easily be 10% as 90%! Note that system efficiency
(as opposed to source efficiency) is the product of source efficiency, DC-DC
converter efficiency, and load efficiency.

** Example 1. Application Using a MAX797 or MAX1653 DC-DC
Converter (I _{OUT} = 2A) **

**Example 2** is similar to Example 1 except for output-current capability
(20A vs. 2A). Notice that the series-resistance requirement for 95% source
efficiency is 10 times lower (16mW vs.
162mW). To achieve this low resistance,
use 2oz. copper PC traces.

** Example 2. Application Using a MAX797 or MAX1653 DC-DC
Converter (I _{OUT} = 20A) **

**Example 3** derives 1.6V at 5A from a source voltage of 4.5V (i.e.,
5V-10%). The system requirement of 111mW
for R_{S}95 can be met, but not easily.

** Example 3. Application Using a MAX1710 DC-DC Converter
with Separate +5V Supply (V _{PS }= 4.5V) **

**Example 4** is the same as Example 3, but with higher supply voltage
(V_{PS} = 15V instead of 4.5V). Notice the useful trade-off: a
substantial increase in the difference between input and output voltages has
caused an efficiency drop for the DC-DC converter alone, but the overall system
efficiency is improved. R_{S} is no longer an issue because the large
R_{S}95 value (>1W) is easily
met. A system with an input filter and long input lines, for example, can
maintain a source efficiency of 95% or more without special attention to line
widths and connector resistances.

** Example 4. Application Using a MAX1710 DC-DC Converter
with Separate +5V Supply (V _{PS }= 15V) **

**Conclusion**

When looking at
DC-DC converter specifications, it is tempting to maximize efficiency by setting
the supply voltage as close to the output voltage as possible. This strategy,
however, can increase costs by placing unnecessary limitations on elements such
as the wiring, connectors, and trace layout. System efficiency may even suffer.
The analytic tools presented in this article should make such power-system
trade-offs more intuitive and obvious.

**References**

(1) Erickson,
Robert W. *Fundamentals of Power Electronics.*Chapman and Hall,
1997.

(2) Lenk, Ron. *Practical Design of Power Supplies.* IEEE
Press & McGraw Hill, 1998.

(3) Gottlieb, Irving M. *Power Supplies,
Switching Regulators, Inverters and Converters.* Second Edition, TAB Books,
1994.

(4) Wettroth, John. "Controller Provides Constant Power Load."
*EDN,* March 14, 1997.