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
3.6 An easy stabilization tool—The K Factor
How can we easily position the pole(s) and zero(s) to cross over at a selected frequency with a specified phase margin? In the 1980s a gentleman named Dean Venable introduced the concept of the k factor [3]. It consists of deriving a number k based on observation of the open-loop Bode plot of the converter we want to stabilize. This k factor indicates the necessary separation (the distance) between the frequency position of the pole(s) and zero(s) brought by the compensation network (see Fig. 3-19a, b, and c). Then, by selecting the desired crossover frequency fc and the amount of phase margin you need at fc, the k factor automatically places the poles and zeros to make fc the geometric mean between their respective locations: this is the place where the highest phase boost occurs. Depending on the value of k, different phase boosts are thus brought at the crossover frequency.
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Figure 3-19a, b, and c: The k factor adjusts the distance between the positions of the poles and zeros to get the desire phase boost at the crossover frequency.
Once k is calculated together with different data values coming from the open-loop Bode plot of the converter to stabilize, the derivation of all compensation elements for types 1, 2, and 3 is straightforward. Follow the steps.
3.6.1 Type 1 Derivation
The type 1 amplifier, with its pole at the origin (pure integrator), always features a k factor of 1 and introduces a permanent phase delay of -270°. It implies that both zeros and poles occupy the same position on the frequency axis. Here G is an indication of the gain needed to cross the 0 dB axis at the select crossover frequency. It must compensate (in both ways, either by amplifying or by attenuating) the gain developed by the power stage at the considered crossover frequency. Suppose you need a crossover frequency fc of 1 kHz. Looking at the open-loop Bode plot, you read at 1 kHz a gain Gfc of +18 dB. Therefore, the capacitor C must be calculated so that the type 1 amplifier delivers a gain GG of +18 dB at 1 kHz (Eqs. 3-27):
3.6.2 Type 2 Derivation
A complex number a + jb features an argument equal to (Eq. 3-28):
If we consider a transfer function featuring one pole and one zero, we can also calculate the phase boost that the function introduces (Eq. 3-29):
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Yes, if both the zero and the pole are coincident, there is no phase boost at all since arg(T(f)) = 0. But let us assume we place a zero at frequency and a pole at a frequency Equation (3-29) can then be updated (Eq. 3-30):
We know from the past trigonometric classes that the following identity is true (Eq. 3-31):
Now, if we extract from arctan(1/x) Eq. (3-31) and inject it into Eq. (3-30), we obtain (Eq. 3-32)
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Rearranging leads to (Eq. 3-33)
Solving for k gives (Eq. 3-34)
Equation 3-34 links the value of k to the amount of needed phase boost occurring at the crossover frequency. The needed phase boost is obtained from reading the phase shift information (PS) on the open-loop Bode plot of the converter you have to stabilize and the phase margin (PM) you finally want. The first phase delay is given by the phase shift brought by the converter power stage. Then, you add the -90° phase shift brought by the integrator (origin pole). After the addition, you have to calculate how much positive phase you need to add (the boost) to obtain the desired phase margin PM which keeps you away from the -180° limit (Eq. 3-35a).
solving for boost gives: (Eq. 3-35b):
where: PM is the phase margin you want at fc and PS is the negative phase shift brought by the converter, also read at fc.
Now, based on these numbers [boost, G—via Eq. (3-27a)—fc and k], Dean Venable linked the pole and zero locations via the following formulas, where component labels correspond to Fig. 3-15a (Eqs. 3-36, 3-37, 3-38):
To show the effect of varying k, that is, changing the distance between the pole and zero, we can update Fig. 3-15b with a new plot (Fig. 3-20a), where k moves from 1 to 10.
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Figure 3-20a: Adjusting the value of k allows the designer to modulate the phase boost. Here, G has been selected to be 1 (0 dB), and fc to be 1 kHz. Note the decrease of dc gain as k increases.
For k = 1, the pole and zero are coincident, and the phase boost is null [see Eq. (3-29)]. You can recognize the type 1 response. As k increases, so does the distance between the pole and zero locations, offering a greater boost at fc. Unfortunately, increasing the phase boost comes at the price of
reducing the dc gain. The k factor can thus be considered as a gain loss penalty that you pay for a greater phase boost. This statement also holds for the type 3 amplifier.
We will see soon the successive steps to successfully stabilize a converter using a type 2
amplifier.
3.6.3 Type 3 Derivation
Dean Venable also considered the type 3 amplifier and derived pole-zero equations based on his k factor. As with type 2, the k factor adjusts the distance between the pole-zero pairs and defines the phase boost it brings. Venable defined a zero at frequency f/√k and a pole at frequency √kf. Capitalizing on results obtained by Eq. (3-30) gives (Eq. 3-39)
However, we are going to place a double zero and a double pole. Therefore, the phase boost given by Eq. (3-39) must be multiplied by 2 if the double zeros and poles are, respectively, coincident. Therefore, the boost given by a type 3 amplifier is (Eq 3-40)
Using Eq. (3-31), we can write (Eq. 3-41)
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Solving for k gives (Eq. 3-42):
Based on the definition of the phase boost in Eq. (3-35b), we can define all compensation values, where labels are coming from Fig. 3-18a (Eqs. 3-43, 3-44, 3-45, 3-46, 3-47):
Here fc is the crossover frequency and G the needed gain at fc.
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Figure 3-20b: Adjusting the value of k allows the designer to modulate the phase boost. The phase boost, as in type 2, depends on the distance between the pole-zero pair. Note the decrease of dc gain as k increases.
If we enter these definitions into the simulator and sweep k, we generate the Fig. 3-20b plot, showing the effect of varying the k factor. A phase boost up to 180° can theoretically be attained.
3.6.4 Stabilizing a Voltage-Mode Buck Converter with the k Factor
As you will see, stabilizing a converter can be quite easy if you use the k factor technique. Let us follow the steps:
1. Generate an open-loop Bode plot. This open-loop plot can come from a laboratory sweep obtained via a network analyzer, or you can make it with an averaged SPICE model. If we take the example of a buck converter, Fig. 3-21 shows a typical simulation sketch. This is a 100 kHz CCM buck converter in voltage mode. The PWM block uses a 2 V peak-to-peak sawtooth Vpeak, hence the -6 dB attenuation brought by the XPWM subcircuit (1/VPEAK). The
input voltage varies between 10 V and 20 V. The maximum output current is 2 A, and the
minimum is 100 mA, implying a load variation between 2.5 ohms and 50 ohms. We will plot the Bode plot for a low line input, maximum current. This is what Fig. 3-22 represents.
2. Select a crossover frequency and a phase margin. As we operate to 100 kHz, experience shows that we could select a crossover frequency of one-fourth of the switching frequency, or 25 kHz. Let us be humble and select 5 kHz for the first step. The phase margin we want for this example is 45°.
3. Read the Bode plot at the crossover frequency. From Fig. 3-22, we can see a power stage phase shift of -146&#deg; and attenuation Gfc of -9.2 dB, both measured at 5 kHz.
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Figure 3-21: A buck converter where the calculation of the compensation network has been automated.
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Figure 3-22: The open-loop Bode plot of the CCM buck converter. We select a 5 kHz crossover frequency.
4. Select the amplifier type. From the previous lines, we can see a power stage phase shift down to -180°, implying a type 3 amplifier. The LC network resonates at 1.2 kHz.
5. Apply formulas. Use Eqs. (3-35) and (3-42) to (3-47) to calculate the compensation elements:
Phase boost = 101°, k = 7.76, G = 2.88, C1 = 7.5 nF, C2 = 1.1 nF, C3 = 7.72 nF, R2 = 11.9 kilohms; R3 = 1.5 kilohms. From calculations, the k factor places a double zero at 1.8 kHz and a double pole at 14 kHz.
6. Sweep the open-loop gain with the above values. The result appears in Fig. 3-23a and b
as Rload = 2.5 ohms and Rload 50 ohms at the two input voltage extremes.
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Figure 3-23a: The compensated gain, at both input voltages, highest load.
7. Check that phase margin and gain margin are within safe limits in all cases.
8. Vary the output capacitor(s) ESRs. The ESR varies with the capacitor age but also its internal temperature. Make sure the associated zero variations do not jeopardize point 7.
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Figure 3-23b: The compensated gain, at both input voltages, highest load. The buck has toggled in the DCM mode thanks to the automatic mode transition of the model.
9. Step load the output. Replace the fixed load by a signal-controlled switch and step load it between two operating points at both input voltage extremes to check the stability. This is done in Fig. 3-24. A controlled current source can also perform the task. Following Eq. (3-3), we should have the following approximate undershoots:
Figure 3-24 confirms these numbers.
VP = 275 mV, hence a drop to 5 - 0.275 = 4.72 V @fc = 5 kHz
VP = 171 mV, or a drop down to 5 - 0.171 = 4.82 V @fc = 8 kHz.
As shown in various screen shots (Fig. 3-21, for example), the schematic capture SNET from Intusoft offers the ability of automating the k factor compensation elements. This is made through a text window on which the keyword parameters appears. The equations following the keyword then describe how the element values must be evaluated before the simulation is begun. You can thus quickly change the crossover frequency or the phase margin and see how it affects the transient response. OrCAD can also do that, and examples on the CD-ROM make use of it. For those who still would like to separately calculate the element values, a simple Excel spreadsheet lets you do so. It also includes the formulas described below for the manual pole and zero placements. The spreadsheet is described in App. 3A.
p. 1, Switch-Mode Power Supplies, Copyright McGraw-Hill, 2008
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