# Designing Stable Compensation Networks for Single Phase Voltage Mode Buck Regulators

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1 Designing Stable Compensation Networks for Single Phase Voltage Mode Buck Regulators Technical Brief December 3 TB47. Author: Doug Mattingly Assumptions This Technical Brief makes the following assumptions:. The power supply designer has already designed the power stage of the single phase buck converter. The last step to the design is the compensation network.. The designer has at least a basic understanding of control systems theory. 3. The designer has a basic understanding of Bode plots. Introduction Synchronous and nonsynchronous buck regulators have three basic blocks that contribute to the closed loop system. These blocks consist of the modulator, the output filter, and the compensation network which closes the loop and stabilizes the system. ERROR AMPLIFIER _ REFERENCE Modulator MODULATOR NETWORK OUTPUT FILTER OUTPUT FIGURE. BASIC BLOCKS OF THE BUCK REGULATOR The modulator is shown in Figure. The input to the modulator is the output of the error amplifier, which is used to compare the output to the reference. V OSC OUTPUT OF ERROR AMPLIFIER OSC PWM COMPARATOR FIGURE. THE MODULATOR V IN Output Filter The output filter consists of the output inductor and all of the output capacitance. It is important to include the DC resistance (DCR) of the output inductor and the total Equivalent Series Resistance (ESR) of the output capacitor bank. The input to the output filter is the node and the output is the regulator output. Figure 3 shows the equivalent circuit of the output filter and its transfer function. L O The transfer function for the output filter shows the well known double pole of an LC filter. It is important to note that the ESR of the capacitor bank and the DCR of the inductor both influence the damping of this resonant circuit. It is also important to notice the single zero that is a function of the output capacitance and its ESR. Open Loop System DCR ESR Figure 4 illustrates the open loop system and presents the transfer function. C O s ESR C OUT = FILTER s ( ESR DCR) C s L C OUT OUT OUT E/A OUTPUT FIGURE 3. THE OUTPUT FILTER L O DCR C O ESR The output of the modulator is the node. The gain of the modulator is simply the input voltage to the regulator, V IN, divided by the peaktopeak voltage of the oscillator, V OSC, or: V s ESR C IN OUT OPENLOOP = V OSC s ( ESR DCR) C s L C OUT OUT OUT FIGURE 4. THE OPEN LOOP SYSTEM MODULATOR = V IN V OSC The peak to peak voltage of the oscillator can be obtained from the data sheet for the controller IC. CAUTION: These devices are sensitive to electrostatic discharge; follow proper IC Handling Procedures. 888INTERSIL or Intersil (and design) is a registered trademark of Intersil Americas Inc. Copyright Intersil Americas Inc. 3. All Rights Reserved All other trademarks mentioned are the property of their respective owners.

2 Technical Brief 47 Figure 5 shows the asymptotic Bode plot of the open loop system gain. V IN log V pp (db) F LC 4dB/DEC Figure 5 represents a generic open loop system. Specific systems will have different double pole and ESR zero frequencies. For systems with very low DCR and ESR parameters, the phase will experience a very sharp slope downward at the double pole while the gain will have a rather high peak at the double pole. Systems that have such resonant output filters will be more difficult to compensate since the phase will need an extra boost to provide the necessary phase margin for stability. Systems such as this will typically need a Type III compensation, which will be discussed later in this brief. Closing The Loop The Compensation Network Closing the control loop allows the regulator to adjust to load perturbations or changes in the input voltage which may adversely affect the output. Proper compensation of the system will allow for a predictable bandwidth with unconditional stability. In most cases, a Type II or Type III compensation network will properly compensate the system. The ideal Bode plot for the compensated system would be a gain that rolls off at a slope of db/decade, crossing db at the desired bandwidth and a phase margin greater than 45 o for all frequencies below the db crossing. For synchronous and nonsynchronous buck converters, the bandwidth should be between to 3% of the switching frequency. Type II Compensation F ESR db/dec FREQUENCY (Hz) FIGURE 5. OPEN LOOP SYSTEM Figure 6 shows a generic Type II compensation, its transfer function and asymptotic Bode plot. The Type II network helps to shape the profile of the gain with respect to frequency and also gives a 9 o boost to the phase. This boost is necessary to counteract the effects of the resonant output filter at the double pole. Figure 7 shows the closed loop system with a Type II compensation network and presents the closed loop transfer function. The following guidelines will help calculate the poles and zeroes, and from those the component values, for a Type II network.. Choose a value for R, usually between k and 5kΩ.. Pick a gain ( /R ) that will shift the Open Loop Gain up to give the desired bandwidth. This will allow the db crossover to occur in the frequency range where the Type II network has a flat gain. The following equation will calculate an that will accomplish this given the system parameters and a chosen R. F ESR DBW V OSC = R F LC F ESR V IN 3. Calculate by placing the zero a decade below the output filter double pole frequency: = π F LC 4. Calculate C by placing the second pole at half the switching frequency: C = π F sw Figure 8 shows the asymptotic Bode gain plot and the actual gain and phase equations for the Type II compensated system. It is recommended that the actual gain and phase plots be generated through the use of commercially available analytical software. Some examples of software that can be used are Mathcad, Maple, and Excel. The asymptotic plot of the gain and phase does not portray all the necessary information that is needed to determine stability and bandwidth. The compensation gain must be compared to the open loop gain of the error amplifier. The compensation gain should not exceed the error amplifier open loop gain because this is the limiting factor of the compensation. Once the gain and phase plots are generated and analyzed, the system may need to be changed somewhat in order adjust the bandwidth or phase margin. Adjust the location of the pole and/or zero to modify the profile of the plots. If the phase margin proves too difficult to correct, then a Type III system may be needed. If the output voltage of the regulator is not the reference voltage then a voltage programming resistor will be connected between the inverting input to the error amplifier and ground. This resistor is used to offset the output voltage to a level higher than the reference. This resistor, if present, has no effect on the compensation and can be ignored.

3 Technical Brief 47. C R s R C = TYPEII R C C C s s R C REFERENCE V COMP (db) db/dec π C π R C C π R C log R db/dec FREQUENCY (Hz) FREQUENCY (Hz) o BOOST 9 FIGURE 6. GENERIC TYPE II NETWORK 3

4 Technical Brief 47. V IN V OSC OSC PWM COMPARATOR L O DCR C O C ESR R V COMP REFERENCE s C V IN s ESR C OUT = SYSTEM R C C C s s V OSC s ( ESR DCR) C s L C OUT OUT OUT R C FIGURE 7. CLOSED LOOP SYSTEM WITH TYPE II NETWORK ERROR AMP DC OPEN LOOP ERROR AMP (db) BANDWIDTH PRODUCT.F LC.5 F SW MODULATOR & FILTER F LC F ESR CONVERTER BANDWIDTH db/dec FREQUENCY db () f = MODULATOR FILTER TYPEII () f = MODULATOR FILTER TYPEII V IN Where: = MODULATOR log V OSC FILTER log ( πf ESR C OUT ) ( πf) L OUT C = log OUT ( πf ( ESR DCR) C OUT ) FILTER = atan[ πf ESR C OUT ] atan πf ESR DCR C OUT πf L OUT C OUT ( πf R C ) C C log πf R C TYPEII = log[ ( C )] log πf C C 9 o C C = TYPEII atan[ πf R ] atan πf R C C FIGURE 8. TYPE II COMPENSATED NETWORK 4

5 Technical Brief 47 Type III Compensation Figure 9 shows a generic Type III compensation, its transfer function and asymptotic Bode plot. The Type III network shapes the profile of the gain with respect to frequency in a similar fashion to the Type II network. The Type III network, however, utilizes two zeroes to give a phase boost of 8 o. This boost is necessary to counteract the effects of an under damped resonance of the output filter at the double pole. Figure shows the closed loop system with a Type III compensation network and presents the closed loop transfer function. The guidelines for positioning the poles and zeroes and for calculating the component values are similar to the guidelines for the Type II network.. Choose a value for R, usually between k and 5kΩ.. Pick a gain ( /R ) that will shift the Open Loop Gain up to give the desired bandwidth. This will allow the db crossover to occur in the frequency range where the Type III network has its second flat gain. The following equation will calculate an that will accomplish this given the system parameters and a chosen R. Figure shows the asymptotic Bode gain plot for the Type III compensated system and the gain and phase equations for the compensated system. As with the Type II compensation network, it is recommended that the actual gain and phase plots be generated through the use of a commercially available analytical software package that has the capability to plot. The compensation gain must be compared to the open loop gain of the error amplifier. The compensation gain should not exceed the error amplifier open loop gain because this is the limiting factor of the compensation. Once the gain and phase plots are generated the system may need to be changed after it is analyzed. Adjust the poles and/or zeroes in order to shape the gain profile and insure that the phase margin is greater than 45 o. = DBW F LC V OSC R V IN 3. Calculate by placing the zero at 5% of the output filter double pole frequency: = π F LC 4. Calculate C by placing the first pole at the ESR zero frequency: C = π F ESR 5. Set the second pole at half the switching frequency and also set the second zero at the output filter double pole. This combination will yield the following component calculations: R R 3 = F SW F LC C 3 = π R 3 F SW 5

6 Technical Brief 47 C R 3 C 3 R REFERENCE V COMP s s R R R 3 ( R R 3 ) C 3 TYPEIII = R R 3 C C s s R s C R C 3 3 R R 3 C π R R 3 π ( R R ) C 3 3 C C π R C C (db) log R π R π R C 3 3 FREQUENCY (Hz) o BOOST FREQUENCY (Hz) 9 FIGURE 9. GENERIC TYPE III NETWORK 6

7 Technical Brief 47 V OSC OSC PWM COMPARATOR V IN L O DCR C O C ESR C 3 R 3 R V COMP REFERENCE. s R R 3 C s ( R R 3 ) C V 3 IN s ESR C OUT = SYSTEM R R 3 C C s s V s OSC s ( ESR DCR) C OUT s L OUT C OUT C R 3 C 3 FIGURE. CLOSED LOOP SYSTEM WITH TYPE III NETWORK F Z =.5F LC F P =F ESR F Z =F LC F P =.5F SW ERROR AMP DC (db) CONVERTER OPEN LOOP ERROR AMP BANDWIDTH PRODUCT MODULATOR & FILTER db/dec F LC F ESR BANDWIDTH FREQUENCY db () f = MODULATOR FILTER TYPEIII () f = MODULATOR FILTER TYPEIII V IN Where: MODULATOR = log V OSC FILTER log ( πf ESR C OUT ) ( πf) L OUT C = log ( πf ( ESR DCR) C OUT OUT ) πf ESR DCR C OUT FILTER = atan[ πf ESR C OUT ] atan πf L C OUT OUT log ( πf R C ) C C log[ πf R ( C C TYPEIII = )] log πf R C C log ( πf ( R R ) C ) log ( πf R C ) o C C = TYPEIII atan[ πf R ] atan πf R atan[ πf ( R R C C 3 ) C 3 ] atan[ πf R C 3 3 ] FIGURE. TYPE III COMPENSATED NETWORK 7

8 Technical Brief 47 Example The following example will illustrate the entire process of compensation design for a synchronous buck converter. Converter Parameters Input Voltage: V IN 5V Output Voltage: 3.3V Controller IC: IC ISL65A Osc. Voltage: V OSC.5V Switching Frequency: f SW 3kHz Total Output Capacitance: C OUT 99µF Total ESR: ESR 5mΩ Output Inductance: L OUT 9nH Inductor DCR: DCR 3mΩ Desired Bandwidth: DBW 9kHz First, a Type II compensation network will be attempted. The low ESR of the output capacitance and the low DCR of the output inductor may make the implementation of a Type II network difficult. The guidelines given for designing a Type II network were followed in order to calculate the following component values: R = 4.kΩ (chosen as the feedback component) = 5.8kΩ C = 8.464pF =.373nF These calculated values need to be replaced by standard resistor values before the gain and phase plots can be plotted and examined. R = 4.kΩ = 4kΩ C = 8.pF =.nf Upon analysis of the bode plots in Figure, it can be seen that the system does not meet the stability criteria previously set. The bode plot for the gain is acceptable. The gain rolls off at db/decade with a perturbation at the resonant point of the LC filter. After the perturbation, the gain again begins to roll off about db/decade until it crosses db right around 9kHz. The phase plot shows the problem with this Type II system. The low ESR and DCR values create a very sharp slope downward at the double pole of the LC filter. The dive in the phase is so sharp that the 9 o phase boost of the Type II network does not compensate the phase enough to have sufficient phase margin. At approximately 6kHz, the phase margin goes below 45 o and never recovers. There is nothing more that the Type II system can do to improve the phase. The Phase of the compensation is at it s peak when the phase of the filter is at it s minimum. Another problem with the Type II compensation network in this example is that the compensation gain intersects and then exceeds the gain of the error amplifier open loop gain. As the open loop gain of the error amplifier is the limiting factor to the compensation gain, the actual gain and phase is affected by the limit and will not exceed it. Due to these issues, a Type III network will need to be implemented to compensate for the phase properly. The guidelines for the Type III network were then followed to produce the following component values: R = 4.kΩ (chosen as the feedback component) =.863kΩ R 3 = 5.85Ω C =.587nF =.86nF C 3 = 6.987nF Again, these calculated values need to be replaced by standard resistor values before the gain and phase plots can be plotted and examined. R = 4.kΩ =.5kΩ R 3 = 5Ω C =.nf =.7nF C 3 = 6.8nF The gain plot of the Type III compensated system in Figure 3 looks very good. The gain rolls off at db/decade from low frequency all the way to the db crossover with a small perturbation from the LC filter double pole resonant point. The phase plot shows a system that is unconditionally stable. All Intersil U.S. products are manufactured, assembled and tested utilizing ISO9 quality systems. Intersil Corporation s quality certifications can be viewed at Intersil products are sold by description only. Intersil Corporation reserves the right to make changes in circuit design, software and/or specifications at any time without notice. Accordingly, the reader is cautioned to verify that data sheets are current before placing orders. Information furnished by Intersil is believed to be accurate and reliable. However, no responsibility is assumed by Intersil or its subsidiaries for its use; nor for any infringements of patents or other rights of third parties which may result from its use. No license is granted by implication or otherwise under any patent or patent rights of Intersil or its subsidiaries. For information regarding Intersil Corporation and its products, see 8

9 Technical Brief Gain [db] 4 MODULATOR & FILTER ERROR AMP OPEN LOOP ENCROACHING ON ERROR AMP OPEN LOOP 4 SYSTEM BANDWIDTH 6 Frequency 4 SYSTEM 6 Phase [degrees] 8 MODULATOR & FILTER MARGIN 8 Frequency FIGURE. BODE PLOT OF THE TYPE II SYSTEM EXAMPLE 9

10 Technical Brief 47 ERROR AMP 8 OPEN LOOP 6 4 BANDWIDTH 4 MODULATOR & FILTER SYSTEM 6 7 MODULATOR & FILTER 3 SYSTEM MARGIN 8 FIGURE 3. BODE PLOT OF THE TYPE III SYSTEM EXAMPLE

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