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The following text, which focuses on feedback and control loops, is excerpted from Chapter 3 of the book Switch-Mode Power Supplies—SPICE Simulations and Practical Designs, by Christophe P. Basso. Reprinted with permission from McGraw-Hill, copyright 2008. McGraw-Hill offers our readers a 20 percent discount on this book. Click here for more information.—VJB
For nearly 100 percent of the applications, a switch-mode converter delivers a parameter—a voltage or a current—whose value must remain constant, independent of various operating conditions, such as the input voltage, the output loading, the ambient temperature. To perform such a task, a portion of the circuit must be insensitive to any of the above variations. This portion is called the reference, usually a voltage source, Vref, which is precise and well stable over temperature. A fraction (α) of the converter output variable (for instance, the output voltage Vout) is permanently compared to this reference. Thanks to a loop that feeds the information back to it, hence the term feedback loop, the controller strives to maintain the theoretical equality between these two levels (Eq. 3-1):
If you go through some power electronics books, you often see the converter modeled using the classical feedback representation. This approach can sometimes confuse the reader as the network divider featuring the α ratio appears in the chain. Unfortunately, if the ratio surely plays a role in setting the dc output, because of the op amp and its virtual ground, its action disappears in the ac analysis. This is discussed in greater detail in App. 3D.
Figure 3-1a portrays the simplified switch-mode converter as it appears on your bench. As we said, thanks to the error amplifier, the whole loop strives to satisfy Eq. (3-1).
Based on Fig. 3-1a, we can draw Fig. 3-1b as a simplified static representation where we
purposely reduce the whole chain in a unity-return configuration. From this new drawing, we can write a few equations (Eqs. 3-2):
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Figure 3-1a: A simplified schematic of the switch-mode power converter operating in closed-loop mode.
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Figure 3-1b: A simplified static representation of the switch-mode converter.
Here T(0) represents the static loop gain linking Vout and Vref. The term T(0)/(1 + T(0)) illustrates the static error between the theoretical output value you want and the final measurement you read with a voltmeter once the loop is closed. This is something already seen in Chap. 1. Thus, using a large open-loop gain op amp is key to reducing the static error, but it also helps to provide enough low-frequency gain to fight the rectification ripple in off-line supplies.
Figure 3-2 illustrates the small-signal representation of the power supply. As we explained, the op amp keeps its noninverting pin to zero in small-signal conditions. Therefore, Rlower naturally fades away, and the loop gain is solely fixed by Rupper and Zf. Changing the divider network ratio (α) has no effect on the loop gain, as demonstrated in App. 3D. This is what Fig. 3-2 suggests, around a familiar buck converter. The output signal feeds the inverting input of the operational amplifier (op amp) whose frequency response is affected by a compensation network made around Zf and Rupper. The purpose of this compensation network is to tailor the converter frequency response to make it stable once operated in closed-loop conditions. We will find more complex arrangements later, however. The output of this op amp, Verr(s), flows through the PWM gain block to finally generate the control variable, the duty cycle of the power stage. The power stage is affected by a transfer function H(s). In this configuration, the loop gain is simply T(s) = H(s)G(s)GPWM.
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Figure 3-2: A small-signal representation of the switch-mode converter.
The power converter output, unfortunately, does not solely depend on the control variable d. Some external perturbations contribute to its deviation from the imposed target: they are the input voltage Vin and the output current Iout. We have seen in Chap. 1 how negative feedback reduces these effects. Loop analysis will consist of studying the open-loop gain/phase response of the total chain (the transfer function), most of the time through a Bode plot, and shaping it
via the compensation network to stabilize the power supply over the various input/output conditions the converter will encounter in its lifetime.
3.1 Observation Points
Various methods exist to reveal the transfer function of a converter. The simplest one consists of opening the loop—we say breaking the loop—to inject an input signal and observe what is obtained on the other side of the opened path. Capitalizing on the previous figure, we can propose a simple way that the next analysis sketches will adopt (Fig. 3-3).
In this example, a dc source fixes the operating point, and the ac modulation comes in via a coupling capacitor. A network analyzer monitors Vstim and Verr and computes the gain by 20 log10(Verr/Vstim). If this method works great in a SPICE environment, it suffers from a major problem linked to bias point runaway in the presence of high open-loop gains. We therefore do not recommend it for practical experiments.
Observing the voltage on the power stage output will reveal a dc gain and a phase starting from 0°, then becoming negative down to -180° in the case of a second order system (e.g., our voltage-mode CCM buck). If we now look at the voltage delivered by the op amp output, given its inverting configuration, we will add another -180° phase shift to the inverted output stage signal. In classical technical literature, the phase representation of an open-loop system is plotted between 0° and -360°. When it comes to phase margin study, very often, authors purposely omit the -180° inversion brought by the op amp and display the response between 0° and -180°. However, unlike the classical representations, SPICE often bounds its phase display between -180° and +180°. It considers a complete phase rotation when the phase hits 0°.
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Figure 3-3: Operating the loop before the modulating point using an external bias source.
This modulo 2π representation explains why, later on in the text, the phase margin is read as the distance between the open-loop phase trace and the 0° axis. After all, if you observe two waveforms W1 and W2 on an oscilloscope,
stating that W2 lags W1 by -270° is similar to say that W2 leads W1 by +90°!
Figure 3-4 shows a typical Bode plot for the CCM buck operated in voltage mode. The upper section depicts the power stage only, H(s). The lower trace represents the total loop gain T(s) after compensation. As indicated in the above paragraph, the observed loop phase starts from 180° in the low frequencies and heads towards 0° if no proper compensation is provided.
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Figure 3-4: A typical Bode plot of a CCM buck power stage, followed by the op amp circuit response (no compensation, no origin pole).
Exploring the ac behavior of Fig. 3-3 can be performed by SPICE via the familiar network LoL and CoL, as already studied before. Figure 3-5 shows it again for reference. Sometimes, a kind of noise appears in ac sweeps using this technique. Adding a series resistor with either CoL or LoL will cure the problem (typically 100 milliohms).
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Figure 3-5: The inductor closes the loop in dc (during bias point calculation) andopens it in ac, blocking all modulation signals coming from the error chain.
The important improvement brought by Fig. 3-5 is the closed-loop dc point you will not have in Fig. 3-3. Any change in the load or the input voltage will automatically adjust the duty cycle to keep the output constant (within the converter capabilities, of course).
As we explained in the previous lines, if you take a power supply and follow Fig. 3-3 opening recommendations, in other words, you physically open the loop and fix a bias point via an external dc source (that you tweak to obtain the right output voltage), then you might encounter difficulties in maintaining the right operating point in the presence of a high dc gain (feedback with an op amp, for instance). A shift of a few millivolts on the external supply due to temperature variations, and the error amplifier hits one of its stops (if it is the upper stop, a loud noise is usually heard!). The transformer method, already tackled in Figs. 2-45 and 2-46, is actually the best and recommended as best practical measurement practice.
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Figure 3-6a and b: Inserting the ac source in series represents another viable solution.
In SPICE, this solution consists of inserting the ac source in series with either the error signal or the output signal. Figure 3-6a and b portrays these possibilities; please note the ac source polarity which appears in series with the preceding signal. The source connections must have satisfied some impedance requirements, however. The - shall connect to a low-impedance point, and the + must go to a high-impedance point. Based on the author's experience, using the LoL/CoL method offers the easiest way with SPICE to probe the transfer function at any point. This is so because the excitation source refers to the ground and does not float. To obtain the loop gain in Fig. 3-5 using IntuScope (IsSpice graphical tool) or Probe (Pspice graphical tool), you simply need to type the following commands:
Gain:
IsSpice: click on dB V(vout)
Probe: type dB(V(vout))
Phase:
IsSpice: click on phase V (vout)
Probe: type Vp (vout)
No further signal manipulations are required. The same applies if you want to probe another node, for instance, the output stage signal, before the op amp divider.
On the contrary, if you use the approaches of Fig. 3-6a and b, since the source floats, you need to apply imaginary signal algebra:
Gain:
IsSpice: click on dB V(vout), it gives waveform 1 (W1); click on dB V(vin), it gives waveform 2 (W2). Now plotting W1-W2, you obtain the transfer function.
Probe: type dB(V(vout)/V(vin))
Phase:
IsSpice: click on phase V(vout), it gives waveform 1 (W1), click on phase V(vin), it gives waveform 2 (W2). Now plot W1-W2, you obtain the phase transfer function.
Probe: type Vp(vout) - Vp(vin)
Note: under IntuScope, pressing the keyboard letter b directly plots the Bode diagram on the screen.
As you can see, the floating source requires some manipulations compared to the LoL/CoL method. However, to run a transient analysis, you must absolutely reduce LoL/CoL to 1 pH and 1 pF in order to not disturb the loop. Hence, toggling between ac and transient analysis can quickly become a tedious task. Something you really do not care about with the floating source method as a transient analysis automatically puts this ac source to 0. It thus shields you from toggling between ac and transient schematics. For the sake of simplicity, we will use the LoL/CoL method in this book, but the floating source can be applied the same way.
p. 1, Switch-Mode Power Supplies, Copyright McGraw-Hill, 2008
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