Switch-Mode Power Supplies---SPICE Simulations and Practical Designs _ EE Times3.pdf

Switch-Mode Power Supplies---SPICE Simulations and Practical Designs | EE Times

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Switch -M -Mode ode Power Supplies ----SPIC SP ICE E Simulations Simul ations and Practical Designs

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3.5.4 Type 1 Amplif ier—Active Integrator  3.5.4 Implementing Implem enting the largest dc gain naturally naturally pushes the usage of an op amp as part of the corrective corrective loop. Rather than cascading the passive passiv e networks followed by the high-gain op amp, designe designers rs often combined them to form active filters. filters. This is the case for the pure integrator shown in Fig. 3-14a. Note that all configu configuration rationss are inverting the input signal, but we omitted the minus sign for the sake of clarity. (Click on Image to Enlarge)

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Figure 3-14a: 3-14a: A t ype 1 amplifier. No phase boost, just dc gain. The transfer function of this pure integral compensator is easy to derive (Eq. 3-11):

It features an origin pole, given by R 1 and C 1. In dc mode, when

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Switch-Mode Power Supplies---SPICE Simulations and Practical Designs | EE Times

the capacitor is open, the op amp open-loop gain fixes the gain. We purposely put it to 60 dB in all the op amp models appearing in the following examples. Then, as the frequency rises, the capacitor impedance drops and reduces the gain with a -1 slope, or -20 dB/decade. The phase curve stays flat and does not provide any boost in phase: cumulating the -180° phase reversal due to the inverting op amp configuration plus the -90° brought by the origin pole, the type 1 compensator permanently rotates the input phase by -270° or +90° when displayed modulo 2π in SPICE. Figure 3-14b shows the resulting frequency sweep brought by such a configuration. Note the action of the op amp origin pole on the dashed curves (30 Hz, 60 dB open-loop gain). It further  degrades the phase in higher frequencies and must be accounted for in the final design. Fortunately, SPICE does it naturally as we can pick the op amp model we have selected for the loop design. (Click on Image to Enlarge)

Figure 3-14b: The resulti ng Bo de plot of a type 1 amplifier. Note the action of the op amp origin pole, here active around 30 Hz. Note that R lower  does not play a role in the ac response as long as the op amp ensures a virtual ground. Why? Simply because the op amp maintains 0 V on the inverting pin, thus making R lower  useless for the ac analysis. However, R lower  helps to select the needed dc output voltage together with R 1. See App. 3D for more details. 3.5.5 Type 2 Amplif ier—Zero-Pole Pair  The previous amplifier type did not provide us with any phase boost, which we badly need if the phase margin is too low at the desired crossover frequency. Figure 3-15a depicts such a compensator, referenced as a type 2 amplifier. It produces an integrator together with a zero-pole pair. (Click on Image to Enlarge)

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Switch-Mode Power Supplies---SPICE Simulations and Practical Designs | EE Times

Figure 3-15a: A type 2 amplifier can boost the phase. Its transfer function can be obtained via a few lines of Laplace equations (Eq. 3-12): (Click on Image to Enlarge)

We immediately can see a zero (Eq. 3-13)

a pole at the origin (the integrator) (Eq. 3-14):

and a high-frequency pole (Eq. 3-15)

Figure 3-15b shows how the phase and gain evolve with frequency. We can clearly see the phase increasing between the pole and zero locations. The phase boost depends on the distance between these two points, as we will discover in a few moments. The phase boost peaks right in the geometric mean of  ωz and ωp2  which occurs at a pulsation equal to √ωzωp2 . (Click on Image to Enlarge)

Figure 3-15b: Type 2 amplifier response. The dashed lin e illlustrates the op amp origin pole. 3.5.6 Type 2a—Origin Pole Plus a Zero By suppressing capacitor C 2, it is possible to get rid of the highfrequency pole and change the frequency response of the

Switch-Mode Power Supplies---SPICE Simulations and Practical Designs | EE Times

compensation network. Figure 3-16a shows how the type 2 amplifier transforms. (Click on Image to Enlarge)

Figure 3-16a: Suppressi ng

C 2 gives

a different compensation

network and a diff erent Bode plot shape. The transfer function now becomes (Eq. 3-16)

with a zero described by Eq. (3 -17) and a pole at the origin induced by R 1 and C 1 (Eq. 3-17, 3-18)):

 As the frequency increases, the equation reduces to a gain imposed by the two resistors (Eq. 3-19):

Figure 3-16b plots the resulting frequency sweep, again showing the op amp origin pole effect: (Click on Image to Enlarge)

Figure 3-16b: The modified ty pe 2 amplif ier featuring a singl e high-frequency zero. The op amp origin pole effect clearly appears on the graph. 3.5.7 Type 2b—Proportional Plus a Pole  Another variation of the type 2 amplifier consists of add ing a resistor to make a proportional amplifier and removing the integral term present with the two previous configurations. Figure 3-17a depicts such an arrangement where a capacitor C 1 placed in parallel with the resistor R 1 introduces a high-frequency gain, necessary to roll off the gain.

Switch-Mode Power Supplies---SPICE Simulations and Practical Designs | EE Times

(Click on Image to Enlarge)

Figure 3-17a: A type 2b amplifier where proportional control is necessary. (Click on Image to Enlarge)

Figure 3-17b: The dc gain is flat until the high-frequency pole starts to act and imposes a -1 slope decay. The transient response imposed by this type of amplifier  resembles that of Fig. 1-9b, bringing less overshoot in steep load steps. This type of amplifier offers a flat gain imposed by R 2 and R 1, until the pole imposed by C 1 starts to act. The transfer  function is (Eq. 3-20):

The pole obeys the classical formula (Eq. 3-21)

Figure 3-17b portrays the ac response brought by such a configuration. 3.5.8 Type 3—Origin Pole Plus Two Coinc ident Zero-Pole Pairs The type 3 amplifier is used where a large phase boost is necessary, for instance, in CCM voltage-mode operation of  converters which feature a second-order response. Its transfer  function can be quickly derived, by calculating the impedance made of Z f  = (1/sC 2) || {R 2 + (1/sC 1)}  divided by the input series impedance ZI  = R 1 || {(R 3 + (1/sC3)} . See Fig. 3-18a. (Click on Image to Enlarge)

Switch-Mode Power Supplies---SPICE Simulations and Practical Designs | EE Times

Figure 3-18a: The type 3 amplifier circuitry. Two coincident pole-zero pairs associated with an integrator. We obtain the following expression, highlighting the pole and zero definitions (Eqs. 3-22, 3-23, 3-24, 3-25a, 3-25b, 3-26): (Click on Image to Enlarge)

(Click on Image to Enlarge)

Figure 3-18b: The type 3 amplif ier intr oduces an in tegrator, a double zero, and a double pole. Figure 3-18b plots the frequency response of the Fig. 3-18a amplifier and shows the slope evolution. 3.5.9 Selecting the Right Ampli fier Type Both the converter type and the transient response you need for  your design will guide you through the selection of one particular  compensation type. •Type 1: As it does not offer any phase boost, the type 1 amplifier  can be used in converters where the power stage phase shift is small, e.g., in an application where you would like to roll off the gain far away from the resonant frequency of a second-order filter.  As in any integral type compensation , it brings the largest overshoot in the presence of a sudden load change. This type is widely used in power factor correction (PFC) applications, for  instance, via a transconductance amplifier.

Switch-Mode Power Supplies---SPICE Simulations and Practical Designs | EE Times

•Type 2: This amplifier is the most widely used and works fine for  power stages lagging down to -90° and where the boost brought by the output capacitor ESR must be canceled (to reduce the gain in high frequency). This is the case for current-mode CCM and voltage-mode (direct duty cycle control) converters operated in DCM. •Type 2a: The application field looks the same as for type 2, but when the output capacitor ESR effect can be neglected, e.g., the zero is relegated to the high-frequency domain, then you can use a type 2a. •Type 2b: By adding the proportional term, it can help reduce the under- or overshoots in severe design conditions. We have seen that it prevents the output impedance from being too inductive, therefore offering superior transient response. Nevertheless, you pay for it by a reduction of the dc gain, hence a larger static error. •Type 3: You use this configuration where the phase shift brought by the power stage can reach -180°. This is the case for CCM voltage-mode buck or boost-derived types of converters. p. 3, Switch-Mode Power Supplies, Copyright McGraw-Hill, 2008 EMAIL THIS PRINT COMMENT < PREVIOUS PAGE 3 / 3

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