*Stability of feedback amplifiers and control systems like SMPS are targeted *

*Comprehensive articles and books about feedback loop design and test abound, Bode Plots and Nyquist Diagrams, both highly impractical, are discussed at great length and recommended, while the superior Nichols Chart is hardly ever mentioned. *

*By Dr.- Ing. Artur Seibt*

The purpose of this article is to present the Nichols Chart, invented by the American engineer Nathaniel Nichols in 1947, as the superior practical method. It overcomes the disadvantages of the other methods and can be used without knowledge of Function Theory and Complex Variables. Its application does not require special instruments nor software, the results are obvious, easy to interpret and directly applicable to the design and test of feedback systems.

### Stability, wild oscillations, the necessity of tests

Feedback amplifiers and control systems can oscillate if the time delay between the input and feedback signals becomes too great. For practical applications it is necessary to know the kind and degree of static and dynamic stability. The former is obtained by measuring Nichols Charts (or equivalent, however less practical, plots), the latter by testing the transient response.

A word of caution: by no means will wild oscillations always be apparent! If the output voltages of a SMPS or other voltage regulator seem to be all right on a voltmeter, this is no proof of proper function or acceptable stability. Many SMPS oscillate in the application without that this is detected because the average output voltages remain constant. Also, an audio amplifier may seem to function while oscillating far above the audio band. Examples of other causes of oscillations: the negative input resistance of SMPS can produce oscillations which have nothing to do with the regulation loop; the negative input impedance of circuits like emitter followers may cause oscillations in the hundreds of MHz, active components such as power mosfet’s may also oscillate in the hundred MHz region. In all these cases some parasitic inductance is enough. The use of a scope is required, but not all scopes are capable of detecting hf oscillations, also passive probes exhibit very low input impedances at high frequencies which can damp oscillations so far that they stop when the probe is connected!

In practice, wild oscillations of SMPS will show up on the scope and mostly also acoustically, because the bandwidth of the loops is predominantly in the audio range, the sound will be emitted by the ferrite cores and some ceramic capacitors. The test with the scope is only negative, i.e. if there are oscillations, the circuit is unstable, but if none are visible, this will be no proof of stability; the circuit may be on the verge of instability, and a small change of load, temperature or input voltage can suffice. Especially awkward are oscillations which occur only when a circuit is actually driven. Example: an amplifier may seem stable at rest, but when it is driven e.g. by a sine wave signal, oscillations will appear during portions of the waveform. Transient tests by load steps from electronic loads which apply periodic steps are too mild and can not uncover hidden problems. It is necessary to apply stochastic steps, both in amplitude and time. The decay transients will often superimpose which overdrives the loop amplifier; the result are sudden very high and long excursions of the output voltage, far in excess of the excursions caused by periodic loads! Sensitive loads like microcomputers dislike excessive supply voltage deviations and may malfunction. Another example are “dc” motor loads; dc motors draw high ac currents, much to the surprise of many engineers who only designed for the average current. Not only must the demand by the positive peak currents be met, the negative peak currents must also be accepted by the SMPS, this can lead to the outbreak of oscillations or even a temporary interruption of the regulation loop.

Textbooks generally talk about the design and test of “the” stability plot. In the real world the static and dynamic stability depend on: 1. Component tolerances, temperature and long term behaviour, 2. The load, 3. Often also the input voltage. Consequently, a professional design requires that a set of stability plots is compiled and transient tests performed for several temperature and load combinations - at least!

- The design engineer has to design in and verify a sufficient stability margin, taking into account the tolerances and temperature dependencies of all vital components. Example: electrolytic capacitors are notorious troublemakers: their tolerances of typ. - 20 to + 50 % are only valid when new and at 20 degr. C, the capacitance value depends strongly on temperature and decreases sharply at low temperatures, with time, it will be reduced by loss of electrolyte. The ESR, mostly important for stability, drops with increasing temperature and rises with age. This can result in an amplifier or SMPS being stable at 20 degr. C and starting to oscillate after warming up. The manufacturers guarantee only a maximum ESR. Thus SMPS may suddenly oscillate in production after a batch of “better” capacitors was installed. Due to the exponential dependence of life on the internal temperature, electrolytics constitute the main cause of SMPS failures resp. cause the onset of oscillations with age.
- Between light and high load several parameters of SMPS change, ripple increases which can overdrive the loop amplifier and cause nonlinear behaviour, rarely understood. Inductive components may approach saturation.
- The influence of the input voltage depends on the circuit configuration chosen. Examples: With the simple buck regulator the influence is greatest, with a flyback using current-mode control it is least.

For completeness, it should be mentioned that wild oscillations vary all over the map: from mild oscillations which do not interfere with the delivery of power to destructive ones which, once initiated, destroy the SMPS immediately. Therefore weak oscillations should always be regarded as a warning sign, and the SMPS taken out of service, because they can build up to destruction, and many SMPS components can burn and emit obnoxious gases.

### Theory and Practice

Starting with Strecker 1931, Nyquist 1932 and Dr. Bode’s famous book 1945, the ground work of feedback and linear control systems theory was laid by the 60’s. These texts, typically covering several hundred pages, request the readers’ knowledge of higher mathematics. There are basically two approaches to stability prediction.

Some methods assume that the system response can be described by a set of equations, stability is then analysed by purely mathematical methods. These methods ignore the complexity of the real world, assuming that practical circuits comply with equations. The mathematical treatment ignores amplitudes, dynamic ranges, nonlinearities, overdrive, limiting, the properties of semiconductors, passive components, parasitic components, to name a few. In the case of SMPS ripple and its effects are disregarded, and the fact that all SMPS are sampling systems and its implications are seldomly mentioned.

If transfer functions are not available, Nyquist, Bode or Nichols plots can be obtained and graphed from frequency response data gained from measurements of the real system.

Dr. Bode was a mathematician at the Bell Labs with an astounding understanding of electronics engineering. He realized that real-world electronics systems and their components are far from ideal, also, that complex numbers are a mathematical construct useful for describing electrical phenomena, but that in the real world there are neither imaginary components nor measuring results, hence that all plots in the complex plane are impractical. He was first to introduce a representation in engineering units, the pair of “Bode plots” in which the results of amplitude or gain and phase measurements are entered directly in dB and degrees. His plots can be used without reference to or knowledge of Function Theory and Complex Variables.

Although more practical than Nyquist, the Bode Plots suffer from the division into two separate plots. It is obvious whether a system is stable or not, but it is far from easy to read from the pair of plots which changes to a circuit or system have to be made to stabilize it or to improve the margins, nor are critical frequencies obvious.

### The Nichols Chart

Nichols recognized the shortcomings of the Nyquist and Bode plots and performed the logical step further. It is hard to understand why Bode Plots are strongly advertised in most textbooks while the superior Nichols Chart is rarely mentioned. It is derived from the Nyquist plot by conformal mapping into rectangular real coordinates. It combines the advantages of the Nyquist plot and the Bode plot pair. The frequency is the parameter along the curve from right to left. The list of advantages is impressive:

- The whole information is concentrated in one curve as in the Nyquist plot, but on a real rectangular coordinate system with dB and degree axes as in the Bode plots, so that measurement results can be entered directly; no calculations necessary, no knowledge of higher mathematics.
- Every full revolution of the Nyquist plot is mapped onto another 360 degree wide half plane in parallel to the vertical axis such that it can be extended on the left as needed. Nyquist plots of complicated circuits are almost impossible to interpret, in the Nichols representation the interwoven Nyquist curve is so to speak unfolded into the left half planes and easy to interpret.
- Gain and phase margins can be read directly, and, like in the Nyquist plot, the closest distances to the critical frequencies. Necessary changes to the circuit in order to stabilize or improve stability margins are immediately and clearly obvious.
- Due to the logarithmic vertical scale any gain change just needs a vertical shift.
- Nichols Charts are powerful tools for determining closed loop response from open loop measurements. On the other hand, open loop measurements are often impossible because the gain and the offset drift of the amplifier may be so high that no input signal can be applied without saturating it. In such cases Nichols Charts can be used to convert closed loop measurements to open loop responses. Open loop measurements are entered into the rectangular grid, a second grid of contour lines for gain and phase is overlaid from which the closed loop response can be read; closed loop gain is obtained from the gain contour which the curve touches, and the phase from the phase contour which it intersects.

Example of a Nichols Plot, showing in Figure 1 how the gain and phase margins can be directly read from the diagram: the phase margin is the distance from the intersection of the curve with the 0 dB axis and the critical - 180 degree point, here 35 degrees.

The gain margin is the vertical distance between the - 180 degree - point and the intersection between a vertical line through the - 180 degree - point and the curve, here 15 dB. The frequency is the parameter along the curve from right to left. The minimum distance between the - 180 degr. point and the curve indicates the critical frequency.

Further information about the Nichols Chart can be found in older literature.

### Some remarks on design and test of SMPS

Design of feedback circuits It is not the purpose of this article, but it should be emphasized that the design and test of a SMPS loop is a fairly simple task and requires only a minute fraction of the whole design time, this is also true for nested loops as they are used e.g. in current-mode control. Too many texts exaggerate and inflate the task. The simple mathematical models rarely reflect the real world. The assumption of linear operation, e.g., is bold, disregarding the facts that SMPS are sampling systems and that there are such ugly phenomena as hf ripple, magnetic and capacitive interference, coupling due to poor e.c. board layout, nonlinear, voltage- and temperature-dependent properties of components etc. It does not make sense to even think about loop design before the whole circuit design and layout has been perfected so much that all such influences on the loop have been minimized! Example: the disastrous effects of hf ripple on the loop amplifier are seldomly recognized; most are of the simple lf LM 224 op amp type. Such amplifiers can not digest 100 KHz+ ripple and saturate already at very low levels which results in highly nonlinear behaviour. This is aggravated by any lead compensation because this boasts high frequencies. As the ripple is load-dependent, loop behaviour may be acceptable at low load, but goes out of control at high load. By and large it can be said from decades of experience: if a loop proves difficult to stabilize, something is very wrong in the whole design and layout which should be remedied instead of trying more complicated “loop compensations”.

The gist of it is: there is no way around testing and obtaining the Nichols Charts for all important modes of operation resp. loads and at a minimum of 3 typical temperatures. The proof of the pudding is the transient behaviour for various (stochastic) load steps. As the latter is simpler and takes the least time, it is practical to first design for an optimum transient performance and then measure the Nichols Charts. There is another good reason for this procedure: it is difficult to predict the transient performance from the Nichols (or other) Charts which can cause time-consuming iterations. It should be noted that the static and dynamic performance of a well designed SMPS is excellent and hardly inferior to that of a linear regulator - despite so many statements of the contrary.

### Some measuring problems

Quite often, with amplifiers, it is necessary to measure amplitude and phase far into the MHz range, although the frequency range of interest is only audio, because it is necessary to get the curve in the vicinity of the critical point. SMPS are sampling systems with an optimum operating frequency range between 100 and 250 KHz, the bandwidth of the control loop has to stay well below half the clock frequency.

As mentioned, loop amplifiers are mostly of the lf LM 224 type, the undistorted output amplitude decreases fast with increasing frequency; it is safe to assume that the available swing decreases along with the frequency response. The input test signal must be accordingly reduced, otherwise the amplifier will saturate, the measurement be grossly in error! This a dangerous pitfall. The signal-to-noise ratio will become poor, noise, flicker, SMPS ripple, hf interference will crop up. Standard ac voltage measuring instruments are wideband and will - depending on their measuring principle - measure the interference as well. Filters would improve the amplitude measurement but upset the phase. Gain - Phase - Meters or Network Analyzers will accept signals with a fairly poor S/N ratio, but it is best to always check the signals with a scope!

When the S/N ratio becomes really poor, the scope remains the last resort for performing such measurements, because even if the picture is corrupted by noise, flicker, hf etc., it is still possible to see the signal and to estimate the amplitude. For measuring phase never use XY plots but the two-channel mode with triggering from one channel. Phase is nothing else but time difference and can be measured with high resolution by overdriving the inputs, mark the zero crossings and selecting faster time base settings. And the time difference can still be estimated even if interference is present!

This article was triggered by the article “Three stability assessment methods every engineer should know about” in “Bodo’s Power, July 2014, p. 50. The statements of the author about the shortcomings of the Bode Plots are to the point.