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Design How-To

Switch-Mode Power Supplies---SPICE Simulations and PracticalDesigns

3.2 Stability CriteriaAmong stability tools (Nyquist, Nichols, . . .), Bode's approach isprobably the most popular owing to its simplicity. When othermethods require manipulating data in the complex plane, the Bodediagram offers an immediate insight as the transfer functionamplitude appears in the frequency domain.

We know that a feedback system takes a portion of the outputvariable and compares it to a stable reference. It then further"amplifies" the error between these signals, via the loop gain, togenerate a corrective action. In other words, if the output voltagedeviates from its target—let us assume it increases—the errorsignal must reduce to instruct the converter to diminish its output.On the contrary, if the output voltage stays below the target, theerror voltage will increase to let the converter know that there is ademand for more output voltage. The control action consists ofopposing the variation observed on the regulated output, hencethe term negative feedback. As the frequency increases, theconverter output stage H(s) introduces further delay (we say it"lags") and its gain drops. Combined with the correction loop H(s),a case might quickly appear where the total phase differencebetween the control signal and the output signal vanishes to 0°.Theory thus shows that if, for any reason, both output and errorsignals arrive in phase while the gain loop reaches unity (or 0 dBin a log scale), we have built a positive feedback oscillator,delivering a sinusoidal signal at a frequency fixed by the 0 dBcrossover point.

When we compensate a power supply, the idea is not to build anoscillator! The design work will thus consist of shaping thecorrection circuit G(s) to make sure that (1) when the loop gaincrosses the 0 dB axis, there exists sufficient phase differencebetween the error and the output signal and (2) G(s) offers a high

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gain value in the dc portion to reduce the static error and theoutput impedance and to improve the input line rejection. Thisphase difference is called the phase margin (PM). How muchphase margin must be selected? Usually, 45° represents theabsolute minimum, but rock-solid designs aim for around 70° to80° phase margin, offering good stability and a fast nonringingtransient response.

Figure 3-7 represents the loop gain of a compensated CCM buckvoltage-mode converter and highlights the phase margin. We canread a PM greater than 50° and a 0 dB crossover frequency (or abandwidth) of 4.2 kHz. Please note that PM on this drawing isread as the distance between the phase curve and the 0° line.Sometimes, in textbooks, PM is assessed as the distance betweenthe phase curve and the -180° line. It leads to the sameinterpretation either way.

(Click on Image to Enlarge)

Figure 3-7: The compressed loop gain of a compensated CCMvoltage-mode buck converter.

Note that a null phase margin at above or below the 0 dB pointoffers what is called conditional stability. That is, if the gain movesup or down (the phase shape remaining the same), the unity gaincrossover point can coincide with a 0 phase margin, engenderingoscillations. What matters is the distance between the 0 dB axisand the point at which danger can occur. This situation can beseen on the right side of Fig. 3-7 where both error and outputsignals are in phase (0°). If the gain increased by 20 dB, we wouldbe in trouble. . . The gain increase (or decrease in some cases)necessary to reach the 0 dB axis is called the gain margin (GM).Good designs ensure at least a 10 to 15 dB margin to cope withany gain variations, due to loading conditions, componentdispersions, ambient temperature, and so on.

(Click on Image to Enlarge)

Figure 3-8: In this configuration, the bandwidth did not

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change, but the phase margin did.

Figure 3-8 represents the same CCM buck but now featuring areduced phase margin of 25° at the crossover frequency. This istoo low. Furthermore, the phase almost hits 0° around 2 kHz. If thegain reduces by 20 dB and crosses 0 dB at that particular point,oscillations will occur, this is the conditional stability describedabove.

3.3 Phase Margin and Transient ResponseA relationship exists between the phase margin of a second-orderclosed-loop system and the quality coefficient Q of its transferfunction [1]. If the phase margin is too small, the peaking induceshigh output ringing, exactly as in an RLC circuit. On the contrary, ifthe phase margin becomes too large, it slows down the system:the overshoot goes away but to the detriment of response andrecovery speed. An equivalent quality coefficient of 0.5 brings atheoretical phase margin of 76° as highlighted in [1]. It leads to acritically-damped converter, combining response speed and lack ofovershoot. Based on this statement, the converter phase margintarget must be set to 70°, with a worst case of 45°.

The CCM buck featuring a 4.3 kHz bandwidth was simulated in atransient load step together with various phase margins imposedby the compensation network. Figure 3-9 has collected alltransient responses. We can see that a weak phase margin givesbirth to oscillations and large overshoots: the system becomesundamped. This is obviously not an acceptable design. As thephase margin increases, the response time slows down a little, butthe overshoot fades away. For a 76° phase margin, the overshootkeeps within a 0.5 percent window.

(Click on Image to Enlarge)

Figure 3-9: Various step load responses versus phasemargin. The undershoot depends on different parameters,

including the output capacitor, but the recovery time links tothe phase margin.

3.4 Choosing the Crossover FrequencyThe crossover frequency is chosen depending on various designfactors and constraints. In a power converter, it is possible toapproximate its closed-loop output impedance by the outputcapacitor impedance at the crossover frequency fC. Therefore, theoutput voltage undershoot level Vp occurring during an outputtransient step ΔIout can be approximated by the following formula[2](Eq. 3-3)

where Cout is the output capacitor and fc the crossover frequency.Note that this equation holds as long as the output capacitor ESRis less than the reactance of Cout at the crossover frequency,implying that the capacitor is solely held responsible for the

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undershoot. This condition can be expressed by (Eq. 3-4)

As Fig. 3-9 shows, the undershoot depends on Eq. (3-3), but therecovery time mostly depends on the phase margin at thecrossover frequency.

Equation (3-3) can help you make the decision on the crossovervalue once the output capacitor has been selected based on theneeded ripple performance and its rms current capability, forinstance. However, there are other limiting factors that you needto consider. For instance, if your converter features a RHP zero,like in CCM boost, buck-boost or flyback converters, then thecrossover frequency fc cannot be higher than 30 percent of itsworse-case lowest position. It quickly closes the debate! Involtage-mode operated converters, the peaking of the LC network(L or Le) also bounds the crossover frequency: trying to fix fc tooclose to the resonant frequency f0 of the LC network will bringobvious stability troubles given the phase lag at resonance. Makesure to select f0 at least equal to three times f0 in worse-caseconditions.

In absence of RHP zero, however, one-tenth to one-fifth of theswitching frequency (10 percent to 20 percent of FSW) looks like apossible target. Extending the crossover frequency can bringadditional problems such as noise pick-up: a theoretical designmight show adequate PM and GM at the chosen cutoff frequency,but the reduction to practice can show instability because of noisesusceptibility brought by the wide loop bandwidth. Do not push thecutoff frequency beyond what you really need to avoid thisproblem: there is no need for a 15 kHz crossover if a 1 kHz onecan do the job transient wise!

3.5 Shaping the Compensation LoopThe stability exercise requires shaping the compensation circuitG(s) in order to provide adequate phase margin at the selectedcrossover point, together with a high gain in dc. To do so, severalcompensation circuits can be used, assembling poles and zeros.What we usually need is a phase boost at the crossover frequencyto provide the right phase margin. This is done by forcing the loopto cross over with a -1 slope, or -20 dB/decade in the vicinity ofthe crossover frequency. However, the needed boost issometimes so large that you cannot reach the crossoverfrequency you have in mind. You must revise your goal and adopta more humble target. Let us review the basics about passivefilters first, quickly followed by operational amplifier-based circuits.

3.5.1 The Passive PoleFigure 3-10a represents an RC circuit producing a so-calledpassive single-pole response. Also known as a low-pass filter, itintroduces phase lag (or delay) as the frequency increases. ItsLaplace transfer function has the following form (Eq. 3-5):

(Click on Image to Enlarge)

The cutoff pulsation of this passive filter, that is, when the dc"gain" reduces by -3 dB, is given by the classical formula (Eq. 3-6)

ωo = 1/RC

Figure 3-10a represents the electrical construction of such a low-pass filter whereas Fig. 3-10b shows its Bode plot.

(Click on Image to Enlarge)

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Figure 3-10a, b: A single-pole RC network and its frequencyresponse.

A single pole is often inserted in compensation circuits to roll offthe gain at a certain point. The rate at which the amplitude goesdown is -20 dB by decade. That is, after the cutoff frequency, theamplitude difference at frequencies f1 and f2, where f2 = 10 f1, willbe -20 dB. On the Bode plot, this is shown as a "-1" slope,whereas a -40 dB by decade circuit (typically a second-ordernetwork) would be designated as a -2 slope. A pole correspondsto a root in the transfer function denominator D(s). Solving for theroots gives an indication of the system stability (please see App.2B for more details).

3.5.2 The Passive ZeroIf the transfer function contains a zero, it appears in the numeratorN(s). At the zero frequency, the numerator cancels and nulls thetransfer function. Equation (3-7) describes the generalized form ofthe zero (Eq. 3-7):

Such an expression describes a 0 dB "gain" in dc, followed by a+20 dB/decade slope (a +1 slope) occurring at the zero location.The phase is now positive, as seen in Fig. 3-11a. This is theproperty of a zero that actually "boosts" the phase, compared to apole that "lags" the phase. Zeros are thus introduced in G(s) tocompensate for excessive phase lag occurring in the power stageresponse.

Back to the passive circuits, Fig. 3-11b represents a high-passfilter. The transfer function of such a simple RC circuit alsocontains one pole and one zero, but placed at the origin. It lookslike (Eq. 3-8)

(Click on Image to Enlarge)

where the cutoff pulsation is the same as Eq. (3-6).

(Click on Image to Enlarge)

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Figure 3-11a: A single zero network and its frequencyresponse.

(Click on Image to Enlarge)

Figure 3-11b, c: A high-pass RC network and its frequencyresponse.

This filter features a low-frequency asymptote of a +20 dB/decadeslope (a +1 slope) and high-frequency gain of 1 or 0 dB. Here wehave a zero located at the origin nulling the transfer function in dc(s = 0) as Fig. 3-11c portrays.

In all the above equations, a negative numerator root sign signifiesa left half-plane zero (LHPZ) position. In some converters, a righthalf-plane zero (RHPZ) can exist and stability is jeopardized.

3.5.3 Right Half-Plane ZeroThe RHPZ is not part of the loop shaping toolbox. You actuallyundergo a RHPZ rather than create it for stability purposes! Itsgeneral form looks pretty much like Eq. (3-7) except that anegative sign appears (Eq. 3-9a):

G(s) = 1 - s/ωo

A RHPZ can be formed by using the circuit in Fig. 3-12a where wecan see an active high-pass filter whose inverting output (thenegative sign) is summed with the input. The transfer function iseasy to derive (Eq. 3-9b):

(Click on Image to Enlarge)

with Eq. (3-6) again ruling the cutoff frequency.

(Click on Image to Enlarge)

Figure 3-12a: A RHPZ artificially created via an active high-pass filter and an adder.

(Click on Image to Enlarge)

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Figure 3-12b: The Bode plot looks like a zero, but the phaselags.

As Fig. 3-12b shows, the gain output looks like a traditional zero:a +1 slope of 20 dB/decade with a cutoff frequency imposed by R1and C1. The difference lies in the phase diagram. Instead of aphase boost, the perfidious RHPZ gives you a phase lag andfurther degrades the phase margin you strived to save. Right half-plane zeros usually exist in indirect energy transfer converterswhere energy is first stored (on time) prior to being dumped in theoutput capacitor (off time). If we take the example of the boostconverter, the average diode current equals the load dc current.This diode current Id actually equals the inductor current IL duringthe off time, or d'Tsw. Its average value can thus be written (Eq. 3-10):

(Click on Image to Enlarge)

Suppose we have a 40 percent duty cycle in a CCM-operatedboost converter. A sudden load step occurs which, via thefeedback loop, pushes the duty cycle to 50 percent. The current inthe inductor increases accordingly, but what about the averageoutput current in the diode? It drops because as d' = 1 - d, if dincreases, d' shrinks and the output capacitor first dischargesinstead of increasing! The change goes in the wrong direction untilthe current builds up in the inductor, bringing the average diodecurrent to its right value and finally lifting up the output voltage.Since it is a closed-loop system, the converter becomes unstable.There is nothing you can do about it, except to severely roll off thebandwidth in order to not undergo the RHPZ additional phase lag.

The RHPZ frequency position unfortunately changes with the dutycycle. The typical rule of thumb recommends that you select acrossover frequency to be around one-third of the lowest RHPZposition. If you try to increase the bandwidth closer to the RHPZlocation, you might encounter a problem as the phase lagbecomes too large. The RHPZ occurs in CCM operatedconverters such as buck-boost, boost, or flybacks. The RHPZdisappears in DCM although some academic studies state thepresence of one in DCM, but relegated to higher frequencies.

Figure 3-13 plots a converter featuring a RHPZ. In the presence ofa load step, the duty cycle suddenly increases. As a result, theinductor current increases as the switch stays closed for a longertime. But as d' has diminished, the diode average current nowgoes down. This situation translates into a decreasing outputvoltage, the opposite of what the loop is asking for. Then theaverage current eventually catches up with the inductor current,and the output voltage rises.

The above lines showed how passive poles and zeros help toshape the loop gain. Unfortunately, used on their own, they do notprovide any dc gain, which is badly needed for a low static error,good input rejection, and so on. Associated with operationalamplifiers (op amps), these so-called active filters provide thenecessary transfer function together with the required

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amplification. Three different types have been identified.

(Click on Image to Enlarge)

Figure 3-13: A RHPZ effect in a boost converter operating inCCM.

p. 2, Switch-Mode Power Supplies, Copyright McGraw-Hill,2008