Designing Stable Control Loops

5-1 Designing Stable Control Loops By Dan Mitchell and Bob Mammano ABSTRACT The objective of this topic is to provide the designer with a practical review of loop compensation techniques applied to switching power supply feedback control. A top-down system approach is taken starting with basic feedback control concepts and leading to step-by-step design procedures, initially applied to a simple buck regulator and then expanded to other topologies and control algorithms. Sample designs are demonstrated with Mathcad simulations to illustrate gain and phase margins and their impact on performance analysis. I. INTRODUCTION Insuring stability of a proposed power supply solution is often one of the more challenging aspects of the design process. Nothing is more disconcerting than to have your lovingly crafted breadboard break into wild oscillations just as it is being demonstrated to the boss or customer, but insuring against this unfortunate event takes some analysis which many designers view as formidable. Paths taken by design engineers often emphasize either cut-and-try empirical testing in the laboratory or computer simulations looking for numerical solutions based on complex mathematical models. While both of these approaches have a place in circuit design, a basic understanding of feedback theory will usually allow the definition of an acceptable compensation network with a minimum of computational effort. II. STABILITY DEFINED Fig. 1 gives a quick illustration of at least one definition of stability. In its simplest terms, a system is stable if, when subjected to a perturbation from some source, its response to that perturbation eventually dies out. Note that in any practical system, instability cannot result in a completely unbounded response as the system will either reach a saturation level – or fail. Oscillation in a switching regulator can, at most, vary the duty cycle between zero and 100% and while that may not prevent failure, it will ultimate limit the response of an unstable system. Perturbation Perturbation Stable System Unstable System t t Response Response Fig. 1. Definition of stability. Another way of visualizing stability is shown in Fig. 2. While this graphically illustrates the concept of system stability, it also points out that we must make a further distinction between large-signal and small-signal stability. While small-signal stability is an important and necessary criterion, a system could satisfy this requirement and yet still become unstable with a large-signal perturbation. It is important that designers remember that all the gain and phase calculations we might perform are only to insure small-signal stability. These calculations are based upon – and only applicable to - linear systems, and a switching regulator is – by definition – a non-linear system. We solve this conundrum by performing our analysis using small-signal perturbations around a large-signal operating point, a distinction which will be further clarified in our design procedure discussion. 5-2 Unconditionally Stable Unstable Small-Signal Stable Large-Signal Unstable Fig. 2. Large-signal vs. small-signal stability. III. FEEDBACK CONTROL PRINCIPLES The basic regulator is shown in Fig. 3 where an uncontrolled source of voltage (or current, or power) is applied to the input of our system with the expectation that the voltage (or current, or power) at the output will be very well controlled. The basis of our control is some form of reference, and any deviation between the output and the reference becomes an error. In a feedback-controlled system, negative feedback is used to reduce this error to an acceptable value – as close to zero as we want to spend the effort to achieve. Typically, however, we also want to reduce the error quickly, but inherent with feedback control is the tradeoff between system response and system stability. The more responsive the feedback network is, the greater becomes the risk of instability. Output Feedforward Feedback System Input Reference Fig. 3. The basic regulator. At this point we should also mention that there is another method of control – feedforward. With feedforward control, a control signal is developed directly in response to an input variation or perturbation. Feedforward is less accurate than feedback since output sensing is not involved, however, there is no delay waiting for an output error signal to be developed, and feedforward control cannot cause instability. It should be clear that feedforward control will typically not be adequate as the only control method for a voltage regulator, but it is often used together with feedback to improve a regulator’s response to dynamic input variations. The basis for feedback control is illustrated with the flow diagram of Fig. 4 where the goal is for the output to follow the reference predictably and for the effects of external perturbations, such as input voltage variations, to be reduced to tolerable levels at the output. G H ����u y + _ Negative Feedback Reference Output Inputs GH1 G u y GuGH)1y( GHyGuy + = =+ −= Fig. 4. Flow graph of feedback control. Without feedback, the reference-to-output transfer function y/u is equal to G, and we can express the output as Guy = With the addition of feedback (actually the subtraction of the feedback signal) yHGGuy −= and the reference-to-output transfer function becomes GH1 G u y + = If we assume that 1GH �� , then the overall transfer function simplifies to H 1 u y = 5-3 Not only is this result now independent of G, it is also independent of all the parameters of the system which might impact G (supply voltage, temperature, component tolerances, etc.) and is determined instead solely by the feedback network H (and, of course, by the reference). Note that the accuracy of H (usually resistor tolerances) and in the summing circuit (error amplifier offset voltage) will still contribute to an output error. In practice, the feedback control system, as modeled in Fig. 4, is designed so that HG �� and 1GH �� over as wide a frequency range as possible without incurring instability. We can make a further refinement to our generalized power regulator with the block diagram shown in Fig. 5. Here we have separated the power system into two blocks – the power section and the control circuitry. The power section handles the load current and is typically large, heavy, and subject to wide temperature fluctuations. Its switching functions are by definition, large-signal phenomenon, normally simulated in most stability analyses as just a two- state switch with a duty cycle. The output filter is also considered as a part of the power section but can be considered as a linear block. The control circuitry will normally be made up of a gain block – the error amplifier – and the pulse-width modulator, used to define the duty cycle for the power switches. Source PowerCircuitry Load Control Circuitry Reference Feedforward Feedback Control Power System Fig. 5. The general power regulator. IV. THE BUCK CONVERTER The simplest form of the above general power regulator is the buck – or stepdown – topology whose power stage is shown in Fig. 6. In this configuration, a DC input voltage is switched at some repetitive rate as it is applied to an output filter. The filter averages the duty cycle modulation of the input voltage to establish an output DC voltage lower than the input value. The transfer function for this stage is defined by :where,dVV T tV iiONO =⋅� � � � � � = timeonswitchtON −= periodrepetitiveT = (1/fs) cycledutyd = Vi VO VS2 t tON 1/fS dVftVVV iSONiO)DC(O === R L C S1 S2 VS2 + _ VO + _ Vi + _ PWM Control Fig. 6. The buck converter. Since we assume that the switch and the filter components are lossless, the ideal efficiency of this conversion process is 100%, and regulation of the output voltage level is achieved by controlling the duty cycle. The waveforms of Fig. 6 assume a continuous conduction mode (CCM) meaning that current is always flowing through the inductor – from the switch when it is closed, and from the diode when the switch is open. The analysis presented in this topic will emphasize CCM operation because it is in this mode that small-signal stability is generally more difficult to achieve. In the discontinuous conduction mode (DCM), there is a third switch condition in which the inductor, switch, and diode currents are all 5-4 zero. Each switching period starts from the same state (with zero inductor current), thus effectively reducing the system order by one and making small-signal stable performance much easier to achieve. Although beyond the scope of this topic, there may be specialized instances where the large-signal stability of a DCM system is of greater concern than small-signal stability. There are several forms of PWM control for the buck regulator including, • Fixed frequency (fS) with variable tON and variable tOFF • Fixed tON with variable tOFF and variable fS • Fixed tOFF with variable tON and variable fS • Hysteretic (or “bang-bang”) with tON, tOFF, and fS all variable Each of these forms have their own set of advantages and limitations and all have been successfully used, but since all switch mode regulators generate a switching frequency component and its associated harmonics as well as the intended DC output, electromagnetic interference and noise considerations have made fixed frequency operation by far the most popular. With the exception of hysteretic, all other forms of PWM control have essentially the same small-signal behavior. Thus, without much loss in generality, fixed fS will be the basis for our discussion of classical, small-signal stability. Hysteretic control is fundamentally different in that the duty factor is not controlled, per se. Switch turn-off occurs when the output ripple voltage reaches an upper trip point and turn-on occurs at a lower threshold. By definition, this is a large-signal controller to which small-signal stability considerations do not apply. In a small- signal sense, it is already unstable and, in a mathematical sense, its fast response is due more to feedforward than feedback. V. CONTROLLING PULSE-WIDTH MODULATION A typical implementation for PWM control (in a form which we now call “voltage-mode control”) is illustrated in Fig. 7. From the block diagram it can be seen that the “width” of the PWM signal is determined by the point in time where the sawtooth, or ramp waveform (VR) crosses the voltage level at the output of the error amplifier (VE). Since VR traverses from zero to VP within a switching period, it follows that when VE = zero, the width of the output pulse will be zero, and it will increase linearly reaching 100% when VE = VP. Therefore the duty cycle of the modulator will be VE / VP and since, in a buck converter, the duty cycle has already been determined to be VO /Vi, the control gain of the modulator is: P i E O V V V V = Sawtooth Reference Amplified Error Voltage, VE Comp + _ DC Reference Feedback Signal Error Amp + _ PWM Control d = tONfS = VE/VP VE PWM VP t t tON T = 1/fS Fig. 7. Typical PWM control implementation. Note that if VP is made proportional to VI, a feature which can be accomplished with feedforward, then the duty cycle will vary inversely proportional to the input voltage and input-to-output voltage regulation can ideally be achieved with no change in VE. In this analysis we have assumed complete linearity and that the output of the error amplifier, VE, is a DC voltage. If, in addition to the intended DC component, VE contains excessive “ripple”, due to error amplifier gain at the switching frequency, then those switching frequency components can mix with the sawtooth frequency components causing the regulator to exhibit large-signal “switching instability”, even if it has excellent small-signal stability. This type of instability can cause the regulator to produce even more ripple, usually at a subharmonic of the switching frequency, although it may still regulate at the proper output voltage. 5-5 VI. CHARACTERISTICS OF A LOADED L-C FILTER The schematic of Fig. 8 shows an L-C filter with a load, R, where the components have been converted to impedances in the frequency domain through the use of Laplace transforms. The overall transfer function of this network is described by the use of Kirchhoff’s law as )ps)(ps( LC 1 LC 1 RC ss LC 1 LsCs 1R Cs 1R )s(V )s(V 21 2i O −− = ++ = � � � � � � + = LS R VO(s) + _ Vi(s) + _ 1 CS Fig. 8. Frequency response of a loaded LC filter. By setting the transfer function numerator and denominator each equal to zero, we can derive the roots of both the numerator, which are the zeros of the system (none in this equation), and the denominator which gives us the poles. This second-order expression contains two poles, d2d1 jpandjp ω−α−=ω+α−= where: 1jand, LC 1, RC2 1 2 d −=α−=ω=α For lightly damped filters (typical of switching regulators), we can often use the approximation of: LC 1 d =ω≈ω With ω= js , we see that transfer functions are complex numbers containing a real part and an imaginary part. The amplitude of a complex number is the square root of the sum of the squares of the real and imaginary parts. The phase is the inverse tangent (arctan) of the ratio of the imaginary part to the real part. By evaluating the transfer functions as a function of frequency, we can determine the point where both the magnitude and the phase make transitions. The most common way of doing this is to plot the gain in dB (20 times the log of gain), and the phase in degrees, against the log of frequency. These are called Bode plots and allow easy visualization of the characteristics that we will use to help define system stability. From the transfer function equations for Fig. 8, we can determine that: • The gain = 1 and the phase = 0 for LC 1 ��ω . • The gain = L CR and the phase = �90 for LC 1 =ω . • The gain slope = LC 1 2ω − and the phase = �180 for LC 1 ��ω . For this example of a loaded L-C filter, Mathcad was used to draw the plots shown in Fig. 9, with the assumption of an arbitrarily assigned set of numerical values (which we will later use for our buck converter example): Ω=µ=µ= 5.0Rand,F540C,H16L from which 31085.1 ⋅=α and 41006.1 ⋅=ω . 5-6 From Fig. 9 we substantiate that the gain of this filter is unity at low frequencies, experiences a resonant peak (determined by R) at the second- order pole frequency, and then falls with a slope of 12 dB/octive (20 dB/decade) at higher frequencies; while the phase goes through a shift from zero to a 180o lag. This higher frequency slope is sometimes called a –2 slope since, in this region, the function is proportional to 21 ω , or 2−ω . 180 135 90 45 0 -45 -90 -135 -180 Ph a s e in De gr ee s 10 100 103 104 105 Frequency 40 30 20 10 0 -10 -20 -30 -40 G ai n in d B 10 100 103 104 105 Frequency 4 d 3 1006.11085.1 5.0RF540CH16L •=ω•=α Ω=µ=µ= Fig. 9. Bode plot of a sample loaded LC filter. It is worth reinforcing that a system must be linear before frequency-domain techniques, such as Laplace transforms and Bode plots, apply. A switching regulator is not even continuous, let alone linear. Therefore, approximations will have to be made – first, to average the switching effects so that we have a continuous system, and secondly, to apply a small-signal approximation in order to assume linearity. And, of course, all this is done under the additional assumptions of linear passive components and ideal switches. VII. LINEARIZING THE BUCK CONVERTER With ideal components, a switching regulator is a linear circuit for any given switch condition. The concept of averaging can be applied when the switching rate is fast with respect to the rate of change of any other parameters of interest. To quantify this, we can say that the accuracy of the approximations is excellent up to one-tenth the switching frequency, “pretty good” for one-third, and one-half is the theoretical limit based on the Nyquist sampling criterion. Fig. 10 shows the process of averaging the operation of the circuit by combining the condition when the active switch is closed with that when it is open. The relationship between these two conditions is the duty cycle of the switching and its effect is accounted for through the use of a hypothetical DC transformer with a turns ratio of the duty cycle, d. With this model, the primary current is now ILd and the secondary voltage is Vid. This DC transformer is an artifact from Dave Middlebrook at Caltech. It is not a real component but it is valid for your Spice library. We now have a continuous system but it is nonlinear because the transformer turns-ratio, d, is a variable - namely the control variable - and not a constant. 5-7 Vi VC + + _ _ R Ii L C IL VO + _ VC + _ R L C IL VO + _ Vi VC + + _ _ R Ii = ILd L C IL VO + _ 1:d Vid + _ Switch Condition II [(1-d)th of the time] Switch Condition I [d th of the time] Circuit Model Using DC Transformer Concept. ILd and Vid are nonlinear terms. Fig. 10. Averaged buck converter. With an averaged, continuous model, the next step is to linearize it. We do this exactly the same way we would linearize any nonlinear, continuous system, namely we define the small- signal parameters based on a large-signal operating point. This is shown in Fig. 11. Mathematically, the linearization process involves separating each variable into its DC (in capitals) and signal frequency ac (with a “hat”) components, and neglecting the products of two hat terms. For the example calculation shown in Fig. 11, the product Vid is linearized about the operating point, ViD. Vi VC + + _ _ R Ii = ILd L C IL VO + _ 1:d Vid + _ Vi VC + + _ _ R Ii L C IL VO + _ 1:D ViD + _ dˆIL +_ dˆVi ILD 0dˆVˆwheredˆVdv)dˆD)(vˆV(dv,exampleFor products"hat"NeglectdˆDdiˆIivˆVv iiiiii LLLiii ≈+=++≈ +=+=+= Fig. 11. Linearized buck converter. The advantage of linearization is that Laplace transforms (i.e. impedance concepts) apply so that closed-form algebraic solutions can be found and plotted (e.g. Bode plots). The range over which the linear approximations are valid depends upon the accuracy desired. In general, as long as the signals are small enough so that the duty factor is not clamped either full on or full off for several switching cycles, the linear approximation works very well. And in any case, small-signal stability as evidenced using Bode plots is still a necessary condition for overall stability. VIII. APPLYING THE LINEARIZED MODEL The flow diagram of the closed-loop linearized buck regulator can be derived by applying the generalized control law to the linearized power circuit described above, as shown in Fig. 12. This control law determines how d(s), the Laplace transformed control variable, varies as a function of key circuit parameters. For a second-order system, such as the buck regulator with one input voltage, it can be expressed as: )s(Vˆ)s(Q)s(Vˆ)s(F)s(Iˆ)s(F)s(d i1C2L1 +−−= 5-8 That is, the inductor current variable, the output voltage variable, and the input voltage variable, can each individually contribute to the system control variable. As it pertains to switching regulators, the expression “voltage- mode control” implies that there is no current feedback, i.e. F1(s) = 0. “Current-mode control” means that there is a current loop as well as a voltage loop. In either case, there may or may not be feedforward control, Q1(s). And finally, note that in this case, F2(s) includes the reference and feedback summing components. This generalized control law can then be made specific to our voltage-mode controlled, buck regulator as shown in Fig. 13, where the feedback summing point is the differential input to the Error Amplifi