## Problem

**Oscillation**. The addition of an input filter to a
switching-mode power supply can cause the combination to become unstable and
oscillate. Prior to oscillation, underdamped ringing may occur. In
current-programmed converters, the system may go unstable with no degradation
of voltage-loop gain or output impedance -- conventional indicators of the
onset of instability. The oscillations or underdamped ringing are due to the
negative resistance input characteristic of the regulator.

**Degradation**. In addition to sustained oscillations or
underdamped ringing, other performance parameters of the regulator can be
degraded, especially the output impedance of the regulator.

**Source**. Certain source and line impedances can cause
similar problems. For example, the inductance of a long input power cable,
interacting with a capacitance on the input of the switching mode power supply
can cause ringing and oscillation.

**Load**. Adding capacitance to the load can also cause the
problem.

**Sustained Oscillation Characteristics**

**Amplitude**. The sustained output voltage oscillations due to
input-filter interactions can have an amplitude of up to twice the input
voltage of the converter, but is usually less.

**Frequency**. The frequency is near the resonant frequency of
the input filter.

**Example**. For example, a 5 Volt regulator operating from a
28 V battery with an LC input filter whose resonant frequency is 1 kHz, could
have a 56 Volt peak-to-peak one kHz sinusoidal oscillations on the 5-volt
output, however it is usually less, e.g. 5 volts peak-to-peak (0 to 10 V
peak).

**Cause**. The cause of the oscillation is the negative input
resistance characteristic of switching-mode regulators or other constant-power
devices.

**Negative Resistance Input Characteristic**

**Transformer Model**. A simple model of a switching-mode power
supply is to consider it a dc transformer with the turns ratio u. Input voltage
Vin is u*Vout and the input current Iin is (1/u)*Iout. However, the input
resistance Rin is not the output resistance u^2*Rout but is -u^2*Rout. The
negative sign comes from the constant power characteristics of a switching-mode
regulator as is shown in the derivation:

**Derivation**.

dVin d P P Vin Vout Rin = ---- = ---- --- = - ------- = - --- = -u^2 ---- dIin dIin Iin Iin^2 Iin Iout = -u^2*Rout

More sophisticated derivations in the references take into account efficiency and other regulator characteristics such as the frequency range over which the input impedance is negative.

## Relevance

**System Effects**. The stability and performance of a
switching-mode power supply installed in a system may be radically different
than the stability and performance of the power supply measured by itself.

**Cause**. Problems are usually caused by the addition of an
input EMI filter, but can also be caused by long cables to the input power
source, and in some cases, by added load capacitance.

**Programming**. Duty-ratio (voltage) programmed power supplies
usually show degradation in the loop-gain and output impedance before going
unstable. This is not necessarily true of current-programmed power supplies
which may show no degradation of the open voltage-loop gain or phase or the
closed loop output impedance before going unstable.

**Worst Case**. The worst-case condition is at low line and
full load. For current-programmed converters operating conditions near the
discontinuous conduction mode boundary may also be a problem.

## Solvability

**Analysis**. For optimum design, the switching-mode power
supply and the input filter or EMI filter should be designed together. Analysis
techniques have been developed that allow the stability solution of a
switching-mode power supply with an input filter. These are usually complex
expressions.

**Graphs**. To simplify the design problems, graphical
solutions have been developed that, if passed, assure there is no stability
problem or performance degradation. If failed, there may or may not be a
problem and the more complex analytical expressions must be solved to determine
stability and performance. The graphical solutions are dependent on the control
mode of the converter.

**Criteria**. For duty-ratio-programmed control (voltage-mode
control) converters, the graphical solution is called the Middlebrook criteria
and is simple to construct. For current-programmed converters (current-mode
control) one more criteria associated with the controller feedback loops must
be added to ensure stability.

**Basic Papers**. The basic design papers for input filter
interaction with switching-mode power supplies are Middlebrook's paper for duty-ratio
(voltage) programmed converters and the Jang
and Erickson modification for current-programmed converters, which contains
the information for both voltage-programmed and current-programmed converters
on the same graphs.

A useful paper on the methods of damping input filters is the Phelps and Tate paper.

**Shunt-Damped**. One method of damping a single section LC
filter is to place a dc-blocked damping resistor across the input-filter output
capacitor. The analysis for this simple circuit is surprisingly complex and if
this popular damping method is used, the Middlebrook paper, which also reviews his
earlier work, is highly recommended.

**Timeline**. The timeline of key
papers can be used to locate other papers for this design problem.

## Solution

### Analytical Techniques

Analytical techniques are beyond the scope of this presentation. However, some discussion is warranted as an introduction to the graphical techniques.

**Modeling.** The most common analytical approach to the
problem of input filter interaction with switching-mode supplies is based on
the state-space-averaged canonical model. From this analysis, the loop gain
T(s), input-to-output transfer function F(s), and the output impedance Zo(s)
can be calculated for the regulator without an input filter. By use of the
extra-element theorem the effect of the input filter can then be determined in
terms of the original analysis.

**Stability.** For duty-ratio (voltage) programmed control,
which normally do not contain right-half-plane poles, adequate gain and phase
margins in the open-loop voltage control loop Bode plots can be used to
determine stability. Degradation of the closed loop output impedance also
serves as an indicator of the onset of instability. For current-programmed
converters, the onset of instability may not show in either the voltage-loop
Bode plots or the output impedance. The best monitoring point for observing the
onset of instability in current-mode converters is the regulator input voltage
or current (the node/branch between the input filter and the regulator.) Since
the cause of instability in current-mode converters is often the appearance of
right-half-plane poles, the full Nyquist criteria is usually used to analyze
stability. This criteria must be applied to the full loop, not just the voltage
or current loops. A good description of the loops that have to be considered is
in the Jang and Erickson paper.

**Once Again**. It is important to re-emphasize the fact that
in current-programmed converters, the often measured external voltage-loop gain
and closed loop output impedance used to indicate stability may show no
indication of the onset of instability caused by adding input filters to these
converters.

### Graphical Techniques

**Middlebrook
Criterion** The Middlebrook Criterion is a graphical method for
determining if the input filter of a switching mode power supply will cause
instability or degrade performance parameters of a duty-ratio (voltage)
programmed dc-to-dc converter switching-mode power supply. As usually applied,
the output impedance of the input filter is overlaid on the open-loop input
impedance of the switching-mode power supply at the worse-case conditions of
low-line and full-load and low-line with shorted output.

Error: In the above diagram, Lo should extend into region P1 as a thick solid line and Ro between regions P1 and P2 should be a dotted line whose main purpose is the separate regions P1 and P2.

This graph is a simplification for this discussion only and does not contain any criteria on the current-programmed case. If possible, it is recommended that the reader view the curves in Jang and Erickson (both voltage-programmed and current-programmed controllers) or one of the Middlebrook papers (voltage-programmed controllers) while reading this.

**Instability.** If the output impedance of the filter is
greater than the open-loop input impedance of the power supply at any
frequency, sustained oscillations may be possible at that frequency and further
analysis is necessary.

**Degradation.** If the output impedance of the input filter is
greater than the short-circuit open-loop input impedance of the power supply,
performance degradation, especially in output impedance, is possible.

**More Information.** More information is found in the Timeline of key papers, especially the Middlebrook '76 and Middlebrook '78 papers, and the Jang and Erickson paper.

### Explanation of Middlebrook Criteria Plot

**Assumptions.** The circuit is the canonical model of a buck,
boost, or buck-boost converter operating in the continuous current mode with
duty-ratio (voltage-mode) control. The input filter is a damped LC filter.

**Input Impedance.** The heavy blue line is the criterion used
to test for stability. Ro, Co, and Lo are the equivalent load resistance,
filter capacitor, and inductor reflected through the regulator (these are not
necessarily the physical values of these components, but are those derived in
the canonical models). The low frequency input impedance is Ro, which then
breaks with the output capacitor, Co, which then breaks with the output
inductor, Lo. Near the LC break point, the impedance is modified by the damping of the output filter.

**Short Circuit.** The light blue line is the reflected input
impedance of the output filter with the load shorted. Rd is the reflected
series damping resistance of the output filter.

**Input Filter.** The heavy red line is the output impedance of
the input filter. Ls and Cs are the inductor and capacitor impedance. The LC
peaking is controlled by filter damping.

### Placement of Input Filter

**No Problem in the Old Days.** When 20 kHz switching-mode power
supplies began to be used in the mid 1960's usually the only EMI
specification invoked was for conducted emissions above 150 kHz. This placed the
input filter out to the right of location P1, beyond the negative input
resistance frequency of the power supply, and adding a filter never caused
instability.

**Now a Problem.** Later when MIL-STD-461 limits CE01 and CE03
were invoked and filters had to meet emission requirements down to 30 Hz, the
added filter often resonated at location P3, the absolute worst placement. At
first, filters where often place at location P2, but to get sufficient
attenuation at the switching frequency forced it close to Lo and adding
inductance (from long input lines or added system filters) caused Ls to migrate
left into the region of instability.

**Placement.** Placing the input filter resonance at P4 insures
stability and no degradation, but may result in a filter larger than the
regulator. P5 is usually the most practical placement.

### Practical Placement of Input Filter

**Placement.** P5 is the most practical placement of the input
filter.

**Input L.** In this location, added input inductance from a
system EMI filter or long leads to the power source will not cause the power
supply to go unstable since Ls will migrate to the left, which is safe, and
will not increase the Q.

**Output C.** However, added output capacitance may cause Co to
migrate causing instability. The input filter capacitance Cs should be greater
than the reflected output capacitance Co, including any added load
capacitance.

**Degradation.** Penetrating the Rd line will degrade regulator
output impedance. This is often acceptable if the output impedance degradation
is less than the maximum regulator output impedance. This maximum usually
occurs near where the loop gain passes through unity.

**Damping.** A well-damped input filter simplifies the design
and there are other compelling reasons to damp the input filter.

### Filter Damping

**Input Filter.** Feedforward techniques have been proposed to
prevent the peaking of the input filter from causing instability and
performance degradation. See Kelkar and
Lee for an example. This technique is controversial and not recommended by
some investigators. However, the primary reason for damping the input filter is
usually to control the amplification of input voltage modulation caused by the
resonances of the input filter and feedforward should not be necessary.

**Example.** For example, MIL-STD-461, CS01, applies an 8 Vp-p
signal (50 W maximum input) on a 28V input in the likely range of the input
filter peak (20 Hz to 9 kHz). Other modulations invoked by specification or
actual environment are similar. Very little gain from the input filter Q is
tolerable, and a well-damped filter is called for. This necessarily well-damped
filter usually aids in meeting the various criteria without the use of
controversial feedforward techniques.

**Output Filter.** Explicit damping of the output filter is
normally not required. However damping of the output filter may be desirably to
control the minimum of the open-loop input impedance and the Rd degradation
line.

### Current-Programmed Criteria

**Y-Parameters.** Using y parameters, both the analytical and
graphical criteria for determining stability and degradation with the addition
of an input filter have been worked out by Erich-and-Polivka, and by Kohut. Erich-and-Polivka use impedance for
their graphical criteria and Kohut uses admittance.

**Modified.** The above work has probably been overtaken by the
work of Jang and Erickson on
current-programmed control. They show that if the original Middlebrook criteria
is passed, then one more criteria associated with the feed-forward loop of the
current-programmed controller must be applied. This criteria is important near
the boundary between continuous and discontinuous modes of conduction and at
high frequency.

**Important!** A key difference between voltage-programmed and
current-programmed converters is that current-programmed converters may show no
degradation in gain or phase margins on Bode plots of the voltage loop just
before going unstable. Either the current-programmed criteria or application of
the full Nyquist criterion to the total loop gain of the filter and converter
combination must be used to assure stability.

### Cascaded Converters and Multiple Converters on Common Bus

Cascaded converters are used in distributed power systems where an ac-dc converter or a dc-dc source converter provides a regulated bus for several dc-dc load converters operating in parallel. Lewis et. al. showed that the negative input impedance of the load converters can introduce right-half-plane poles in the source converter and affect its performance. They develop a criteria for preventing this and for introducing a filter between the source and load converters. This is expanded into a simple design procedure by Choi and Cho in 1995. Martin Florez-Lizarraga and Arthur Witulski further develop these concepts in techniques in a 1993 paper and an updated 1996 paper.

### Input Filter Interaction - Summary

**Negative R.** Switching-mode power supplies have an
incremental negative input resistance. Adding an LC filter on their input can
cause them to go unstable or suffer performance degradation. The problem and
solution has been discussed in the current literature starting in 1971. Bode
plots of the voltage-gain loop may give no warning of the impending
oscillations in current-programmed control, and applying the full Nyquist
criterion to the total loop may be necessary.

**Voltage-Mode.** A fairly simple criteria to insure stability
was worked out in 1976 by R. D. Middlebrook and is generally considered
correct for duty-ratio programmed converters in the continuous-current mode. It
was found not to be appropriate for current-programmed control.

**Current-Mode.** After initial work by Erich-and-Polivka, and by Kohut, Jang
and Erickson worked out an addition to the Middlebrook Criteria in 1991 for
current-programmed converters.

**Cascaded Converters** Criteria for cascaded converters with
intermediate filters has been developed by Choi and Cho and by Florez-Lizarraga and Witulski.

**Bottom Line**. Start with the Jang and Erickson paper and refer to other
referenced papers as required to improve understanding or to apply to multiple
converters on the same bus.

## Personal Anecdote

Familiar with the 1971 Yu and Biess papers and Nathan Sokal's 1973 paper, I set out to make a simulation using CSMP software on a PDP 11-40 computer of how adding an EMI filter to a switching-mode power supply caused the combination to go unstable. This was in 1974 at the Naval Ocean Systems Center (NOSC)

What I found was the published criteria didn't really work. The criteria predicted oscillations in stable systems and did not predict oscillations in unstable systems.

At Powercon I (Beverly Hills, March 20-22, 1975) I brought up the question from the floor to a panel of chief engineers of power supply companies. None of them had heard of the problem and generally did not believe it. However, several others in the audience had experienced it and we met at break time. There was a enough experience in this group to report after the break that it was a real problem and should be considered by designers.

Later, I tried to get Thomas Wilson at Duke University to look at the problem, but Duke was swamped with NASA work. What I wanted were two procedures for MIL-HDBK-241. One procedure would let a designer design an EMI filter and switching-mode power supply combination that would not oscillate. The other procedure would allow a filter to be designed having only "black box" measurements on the power supply.

At PESC'75 (Culver City, June 9-11, 1975) I talked to R. David Middlebrook at Caltech, who had just given a paper in which he discussed the problem. He thought he could do what I asked using a new canonical model of switching-mode power supplies developed at Caltech. I funded the work. The result was both a section for MIL-HDBK-241 and Middlebrook's landmark IEEE paper.

This was also the first contract I issued with the provision that if the author published the work in a conference likely to be well attended and read by American power supply designers, the government report could simply be a cover letter including a copy of the paper. This concept proved very successful and resulted in government sponsored work being published in the open literature rather than buried in obscure government reports never read by those most needing the information. I had a lot to be proud of during my eight years at NOSC, but this policy, combined with getting funding for needed research, is what I feel best about. I was sorry to see it abandoned when I left NOSC. Almost nothing was published in the open literature on the Navy 100 W per Cubic Inch program.

## On the Web

I have not found any information on the Web.