简单环路稳定性判定

SLVA381A–December2009–Revised September2011

Simplifying Stability Checks John Stevens PMP-DC/DC Controllers

ABSTRACT

This application report explains a method for verifying relative stability of a circuit by showing the relationship between phase margin in an AC loop response and ringing in a load-step analysis.

Contents

1What is Stability and Phase Margin? (1)

2What is Ringing and How Does it Relate to Stability? (2)

2.1Example Test Setup (4)

2.2Examples of Load Step Analysis (4)

2.3TPS5430Laboratory Tests (6)

3Conclusion (7)

List of Figures

1Calculating Phase Margin From a Frequency Response Plot (2)

2Comparison of Load Step Responses for Varying Phase Margin (3)

3Example Test Setup for Finding Step-Response (4)

4Load Step With Phase Margin=45.7°(1Peak) (4)

5Load Step With Phase Margin=40.61°(2Peaks) (5)

6Load Step With Phase Margin=34.72°(3Peaks) (5)

7Loop Response of TPS5430Tested in Laboratory(Phase Margin=55°) (6)

8Load-Step Response of TPS5430Tested in Laboratory(Phase Margin=55°) (6)

List of Tables

1Phase Margin vs Ringing in Load-Step Response (3)

1What is Stability and Phase Margin?

When discussing control systems,stability is a term referring to the ability of a controller to regain a

constant or decaying output after an input or load disturbance,such as a step function.Stability is an

important criterion for control circuits because they have inherent delay in the feedback loop(s).If the

output signal lags one full wavelength,it becomes superimposed on the input signal of the controller and the circuit becomes unstable and oscillates.

Phase margin(PM)is a measure of relative stability,in degrees,that indicates the likelihood of a

closed-loop control system to oscillate when given a disturbance such as a step function.For systems with

a negative angle phase response,the PM is the difference between–180°(due to the inverting nature of

most amplifiers)and the phase angle of the frequency response at the same frequency where the

magnitude response is1or0db.For systems with a positive phase response,the phase margin is merely the phase itself at the frequency where the magnitude response is1or0db.A positive PM technically yields a stable system whereas a negative PM yields an unstable system.However,a system with a3°PM,for example,is not necessarily stable because practical factors make such a small degree of margin unreliable.Thus,a PM of about45°is considered generally acceptable,but this can vary based on the designer’s preference.This classic method of finding the phase margin can be observed in Figure1.

Figure1.Calculating Phase Margin From a Frequency Response Plot

As can be seen in the plot of Figure1,an AC response,(Magnitude on top and Phase underneath),is

given.The Magnitude Response plot is shown to be0db at15.99kHz.The phase at15.99kHz is80.88°.

Thus,the phase margin of this system is80.88°.

Although this method is acceptable for simulated devices,in real systems it creates many problems.For example,many devices with control systems cannot tolerate the wide range of frequencies necessary for such an analysis.Further,to conduct such a test requires an isolation of the feedback loop which in some cases is difficult or impossible.Although tools such as network analyzers can test stability directly,the cost sometimes restricts their availability.Therefore,an alternate method to test stability using a common

laboratory oscilloscope is now considered.

2What is Ringing and How Does it Relate to Stability?

Ringing is the unwanted oscillation seen in a voltage or current signal when the input or load is changed very quickly.If the controller cannot correct the output properly,overshoot and/or undershoot can occur, until it is damped out according to the damping factor of the system.In the extreme case,the system

becomes unstable and the output oscillates indefinitely at the frequency where the gain=1and phase margin is0dB.In a circuit with a positive phase margin,the oscillations decay and are limited in number.

Considering this,it can be stated that fewer bumps(peaks)or ripples in the transient load-step response

correspond to more damping and a higher phase margin.In Figure2,this relationship can be observed.

Figure2.Comparison of Load Step Responses for Varying Phase Margin

As Figure2illustrates,the response with the highest PM,54.08°,(in brown),damps the fastest.As PM decreases,the number of oscillatory peaks increases until eventually,at very low PM,the circuit is in lightly damped oscillation.For a quick reference,see Table1.

Table1.Phase Margin vs Ringing in Load-Step Response

Phase Margin(Degrees)Ringing(Bumps)

80.880

60.750

57.0

54.080

50.161

45.7 1.5

40.612

34.723

27.784

19.436

9.0917

Scope

2.1Example Test Setup

The device under test (DUT)can be set up as seen in Figure 3

Figure 3.Example Test Setup for Finding Step-Response

2.2Examples of Load Step Analysis

For a better definition of what an oscillatory bump or peak is,see Figure 4,Figure 5,and Figure 6.

Figure 4.Load Step With Phase Margin =45.7°(1Peak)

Figure5.Load Step With Phase Margin=40.61°(2Peaks)

Figure6.Load Step With Phase Margin=34.72°(3Peaks)2.3TPS5430Laboratory Tests

To verify that these simulations produce realistic results,the TPS5430was tested in a laboratory.

Figure7.Loop Response of TPS5430Tested in Laboratory(Phase Margin=55°)

NOTE:Uses a50%Load Step at1-10µs Slew Rate

Figure8.Load-Step Response of TPS5430Tested in Laboratory(Phase Margin=55°) In the laboratory test of the load-step response in Figure8,there are no oscillatory peaks.At a phase margin of55°,this result matches what is found in Table1.

www.ti.com Conclusion 3Conclusion

This application report explains the link between the load-step response and the loop response of a given circuit.Based on this relationship,this document provides details to make an estimate at relative stability based on phase margin.The designer must always consider that this measurement requires some

interpretation of load-step response and that these measurements can vary.

Revision History

Changes from Original(December,2009)to A Revision Page •Corrected notation in Figure2and related description for better clarification (2)

NOTE:Page numbers for previous revisions may differ from page numbers in the current version.

7 SLVA381A–December2009–Revised September2011Revision History Submit Documentation Feedback

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