A topic from the SMPS Technology Knowledgebase.
Analytical Techniques
Analytical techniques are beyond the scope of this version of the SMPS Knowledgebase. However, some discussion is warranted as an introduction to the graphical techiques.
Modelling. The most common analytical approach to the problem of input filter interaction with switching-mode supplies is based on the state-space-averaged cannonical 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 stabililty. 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 Ereckson paper.
Once Again. It is important to re-emphasize the 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.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 Ereckson (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 Ereckson 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. Ro then takes control again. 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 20kHz switching-mode power supplies began to be used in the mid 1960's usually the only EMI specification invoked was for conducted emissions above 150kHz. 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 30Hz, 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 (50W maximum input) on a 28V input in the likely range of the input filter peak (20Hz to 9kHz). 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 desireable 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 Ereckson 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 discontinous 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 intitial work by Erich-and-Polivka, and by Kohut, Jang and Ereckson 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 with the Jang and Ereckson paper and refer to other referenced papers as required to improve understanding or to apply to multiple converters on the same bus.
Return to next item, Personal Perspective.
This hypertext contains the personal experience and research of Jerrold Foutz on the design of switching-mode power supplies. Substantial effort has gone into making the information as accurate as possible but experience teaches that it will contain errors. This material is provided on an "as is" basis. Do not use it for design or any other purpose without independent verification of the material. The author shall have neither liability nor responsibility to any person or entity with respect to any loss or damages arising from the information contained in this hypertext.
© Copyright 1996 Jerrold Foutz, All rights reserved.
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