Introduction This application note details how to calculate a type III compensation network and investigates the relationship between phase margin and load transient response for the Skyworks family of voltage mode control step-down converters. This family includes the AAT84, AAT85, AAT89, AAT687, AAT688, and AAT689. Voltage mode control has become a very popular topology for DC to DC converters, especially with low noise output systems including DSL and cable modems, notebook computers, satellite set-top boxes, and wireless LAN systems. Background In order to reduce the DC-DC converter s output voltage ripple, the equivalent series resistance (ESR) of the output capacitor needs to be reduced. Ceramic output capacitors have a very small equivalent series resistance (ESR), low cost, and small size, making them the ideal output filter solution for DC-to-DC converters. However, the use of low ESR ceramic capacitors significantly affects the design of the error amplifier in the feedback loop. The power stage consists of a double pole due to the L C OUT filter and an ESR zero. The ESR zero is pushed far away from the double pole frequency which results in inadequate phase margin at the cross-over frequency. Therefore, type III compensation is used to stabilize the loop and optimize the output transient response to dynamic load changes. Voltage Mode Control Loop As illustrated in Figure, a typical voltage mode control loop has three main stages: step-down power stage, compensation network, and PWM modulator. The Type III compensation network generates two zeros and two poles. The two zeros are placed from 60% to 50% of double pole frequency to counter the 80 phase lag due to the L C OUT output filter. The two poles are set at the switching frequency of the converter to nullify the ESR zero and attenuate the high frequency noise. Driver V IN STEP-DOWN POWER STAGE R DCR L C OUT R ESR R OUT V OUT PWM MODULATOR COMPENSATION NETWORK COMP V reff R fbh RAMP COMPARATOR Error Amp R C Cff R ff C R fbl Figure : Closed Loop Step-Down Converter with Type III Network Compensation. Skyworks Solutions, Inc. Phone [78] 376-3000 Fax [78] 376-300 sales@skyworksinc.com www.skyworksinc.com 0376A Skyworks Proprietary Information Products and Product Information are Subject to Change Without Notice. September, 0
Step-Down Power Stage Transfer Function The transfer function of the power stage of the step-down converter can be determined by the voltage division: Eq. : V OUT V IN = Z OUT Z L + Z OUT Where Z L and Z OUT are the inductor impedance and output impedance of the power stage. The R DCR includes the DC winding resistance, the turn-on resistance of the MOSFET, and the trace resistance. R ESR is the equivalent series resistor of the output capacitor. Z L and Z OUT are calculated using Equations and 3. Eq. : s C OUT R LOAD R ESR + R Z LOAD OUT = R LOAD // R ESR + = s C OUT (R LOAD + R ESR ) + Eq. 3: Z L = s L + R DCR sc OUT Where the complex variable s = j w and - j = Driver V IN STEP-DOWN POWER STAGE L Z L R DCR C OUT R ESR Z OUT Figure : Step-Down Converter Power Stage. The step-down power stage open loop gain is given by substituting Equations and 3 into Equation. Algebraic manipulation yields the following expression for the open-loop transfer function of the power stage: Eq. 4: V OUT R LOAD (s C OUT R ESR + ) G P = = V IN s L C OUT (R LOAD + R ESR ) + s{l + C OUT [R DCR (R LOAD + R ESR ) + R LOAD R ESR ]} + R LOAD + R DCR R LOAD V OUT A typical Bode plot of the step-down converter power stage is illustrated in Figure 3. A double pole at the cut-off frequency causes the gain to roll off with a -40dB/decade slope (blue) and the phase to exhibit a very sharp slope downward from 0 degree to -80 degree phase lag (red). The ESR zero is observed at a very high frequency due to the ceramic output capacitor. Skyworks Solutions, Inc. Phone [78] 376-3000 Fax [78] 376-300 sales@skyworksinc.com www.skyworksinc.com 0376A Skyworks Proprietary Information Products and Product Information are Subject to Change Without Notice. September, 0
Output Double Pole at cut-off frequency Gain (db) and Phase (degree) -80 degree phase lag due to double pole Frequency (Hz) -40dB/dec ESR zero at very high frequency -0dB/dec Figure 3: The Bode Plot of the Output Stage. Error Amplifier Transfer Function Calculation The error amplifier transfer function with type III compensation as shown in Figure 4 is calculated from Equation 5: Eq. 5: G E = V COMP V OUT // R + s C s C = R fbh // R ff + s C ff V COMP Error Amp R C V REF C ff R fbh Z P R ff V OUT Z C R fbl P Figure 4: Error Amplifier With Type III Compensation Network. Skyworks Solutions, Inc. Phone [78] 376-3000 Fax [78] 376-300 sales@skyworksinc.com www.skyworksinc.com 0376A Skyworks Proprietary Information Products and Product Information are Subject to Change Without Notice. September, 0 3
By algebraic manipulation, G E can be explicitly expressed in terms of zeros and poles in Equation 6. R Eq. 6: fbh + R G ff E = R fbh R ff C s + s + R C (R fbh + R ff ) C ff (C + C ) s s + s + R C C R ff C ff Equation 6 gives two zeroes at frequencies F Z and F Z and two poles at frequencies F P and F P in the following expressions: F Z = and π (R fbh + R ff ) C ff F P = π R ff C ff F Z = π R C and F P = C C π R C + C Gain (db) and Phase (degree) Placing the two zeros close to the output double pole frequency 80 degree phase boost Placing the two poles at cross-over frequency Figure 5: Error Amplifier With Type III Compensation Bode Plot. Type III compensation provides two zeros and two poles which push the cross-over frequency as high as possible and boosts the phase margin greater than 45 degree. A higher bandwidth yields a faster load transient response. The faster transient response results in a smaller output voltage spike. PWM Modulator Stage The PWM modulator gain is inversely proportional to the peak-to-peak input ramp voltage of the oscillator and is derived via Equation 7. Eq. 7: G M = V IN V RAMP 4 Skyworks Solutions, Inc. Phone [78] 376-3000 Fax [78] 376-300 sales@skyworksinc.com www.skyworksinc.com 0376A Skyworks Proprietary Information Products and Product Information are Subject to Change Without Notice. September, 0
Step-Down Converter Loop Gain with Type III Compensation The loop gain of the system is expressed in terms of G M, G E, and G P factors as shown in Equation 8. Eq. 8: G LOOP = G M G E G P The magnitude in db and the phase in degree of the converter loop gain are derived from Equations 9 and 0. Eq. 9: G LOOP (db) = 0.log (G LOOP ) = 0.log (G M G E G P ) Eq. 0: P LOOP = arg(g LOOP ) 80 π The magnitude and phase Bode plots of the converter loop gain with type III compensation are shown in Figure 5. By placing the two zeros close to the output double pole and the two poles at switching frequency, the crossover frequency is pushed to 0% to 60% of switching frequency and in the vicinity of maximum phase boost in order to achieve an optimum phase margin Φ M. Gain (db) and Phase (degree) Output Double Pole Placing the two zeros close to the output double pole frequency Φ M Cross-Over Frequency at /0 Switching Frequency Figure 6: Step-Down Converter Loop Gain With Type III Compensation Bode Plot. Skyworks Solutions, Inc. Phone [78] 376-3000 Fax [78] 376-300 sales@skyworksinc.com www.skyworksinc.com 0376A Skyworks Proprietary Information Products and Product Information are Subject to Change Without Notice. September, 0 5
Type III Compensation Design Process For Voltage Mode Control Step-Down Converter: For example, assume the voltage mode step-down converter has the following specifications: V IN = 6V to 4V V OUT = 3.3V V OUT - V REF 3.3V - 0.6V R FBL = 6.04KΩ, R FBH = R FBL = 6.04K = 7.4KΩ V REF 0.6V V V IN RAMP = L = 4.7µH C OUT = xµf, ESR = mω I OUT =.5A F SW = 490KHz. Set the crossover frequency in the range of /6 to /0 of switching frequency to avoid the Niquist pole: Eq. : F C = F SW 0 = 49KHz. Place the first zero from 60% to 50% of the double pole frequency of the L C OUT filter: Eq. : C ff = L C OUT 4.7µH 44µF = = 48pF K R fbh. 7.4KΩ Where the value of factor K is within the range of 0.6 to.5. 3. Set the first pole at switching frequency and calculate R ff from: Eq. 3: R ff = = = 675Ω π C ff F SW π 48pF 490KHz 4. At cross-over frequency (F C ) the loop gain is unity. Setting G LOOP = at s = jw c, the value of R is given by Equation 4. (π F Eq. 4: C ) L C OUT + V (π 49KHz) 4.7µH 44µF R RAMP = = =.6KΩ π F C C ff π 49KHz 48pF 5. Set the second zero to coincide with the first zero, and solve for C: V IN Eq. 5: C = L C OUT 4.7µH 44µF = = pf K R..6KΩ 6. Place the second pole from switching frequency to one decay higher for adequate phase margin, and solve for C: Eq. 6: C = = = 8pF π R F SW π.6kω 490KHz 6 Skyworks Solutions, Inc. Phone [78] 376-3000 Fax [78] 376-300 sales@skyworksinc.com www.skyworksinc.com 0376A Skyworks Proprietary Information Products and Product Information are Subject to Change Without Notice. September, 0
The Relationship between Frequency Domain and Time Domain in a Step-Down Converter Knowing the relationship between the phase margin in the frequency domain and load transient response in time domain is beneficial to achieving the best results. In this way, we can select either a slow output transient response but without any overshoot or, a faster output transient response with a small amount of overshoot. Let s concentrate on the small area in the vicinity of the cross over frequency (see Figure 7). The curve has two different slopes (-0dB/ decade and -40dB/ decade) due to the location of the original pole w 0 and the high frequency pole w. Assuming the other compensation pole w and the ESR zero are cancelled out. The open loop transfer function in this region can be approximated by Equation 7: Eq. 7: T(s) s ω 0 + s ω The close loop transfer function can derive from T(s): Eq. 8: G LOOP (s) = + T(s) = = s s s s + + + ω 0 ω ω 0 ω r ω r Q + Where the quality coefficient Q and the resonant frequency w r are defined using Equations 9 and 0. Eq. 9: Q = ω 0 ω Eq. 0: ω r = ω 0 ω The cross-over frequency w c can be solved by equating Equation 8 to unity at the crossover frequency: ω 0 ω + 4 - Eq. : ω c = ω = ω + 4(Q) - Gain (db) and Phase (degree) Output Double Pole Placing the two zeros close to the output double pole frequency Φ M Cross-Over Frequency at /0 Switching Frequency Figure 7: The Gain Curve Has Two Different Slopes (-0dB/decade and -40dB/decade) at Crossover Frequency due to the Location of the Original Pole ω 0 and the High Frequency Pole ω. Skyworks Solutions, Inc. Phone [78] 376-3000 Fax [78] 376-300 sales@skyworksinc.com www.skyworksinc.com 0376A Skyworks Proprietary Information Products and Product Information are Subject to Change Without Notice. September, 0 7
ω C - ω 0 ω C ω - - C Eq. : arg T(ω C ) = - tan + tan = -tan - 0 ω ω π Eq. 3: ϕ M = π + arg T(ω C ) = tan - ω - = tan ω C + 4Q 4 - The relationship between the phase margin and the quality coefficient can be derived from Equation 3: + tan(ϕ M ) Eq. 4: Q = = tan(ϕ M ) cos(ϕ M ) sin(ϕ M ) The percent overshoot and quality factor in the second order system are given by Equation 5. Eq. 5: %OS = 00 e -π 4Q - = 00 e -π 4cosϕ M - sin ϕm Figure 8 plots the percent overshoot versus phase margin of a typical second order system. Pecent Overshoot (%) 80 70 60 50 40 30 0 0 Percent Overshoot vs. Phase Margin 0 0 0 0 30 40 50 60 70 80 Phase Margin (degree) Figure 8: Percent Overshoot vs. Phase Margin for Second Order System. The output transient response of a 3.3V output step-down converter with different phase margin is measured in Figure 9. The step load is generated from 00mA to.5a with A/µs slew rate. The red curve corresponding to 68 phase margin has 60µs recovery time without overshoot and a transient voltage spike of 404mV. The black and green curves experience very fast recovery time (40µs) with very small overshoot and a small transient voltage spike of 80mV. Finally, the blue and pink curves reveal an unstable system due to the phase margin of less than 45. 8 Skyworks Solutions, Inc. Phone [78] 376-3000 Fax [78] 376-300 sales@skyworksinc.com www.skyworksinc.com 0376A Skyworks Proprietary Information Products and Product Information are Subject to Change Without Notice. September, 0
3.6V Output Voltage (00mV/div) 3.5V 3.4V 3.3V 3.V 3.V PM=68º PM=58º PM=48º PM=33º PM=6º 3.0V Time (40µs/div) Figure 9: The Relationship Between Phase Margin, Overshoot and Recovery Time of the Output Transient Response of a 3.3V Output Buck Converter. Phase Margin and Transient Response vs. DC Gain (F C = 50KHz) Based on the discussion above of the frequency domain and time domain, the recovery time can be adjusted faster to reduce the peak-to-peak output transient response of a step-down converter. This can be done by pushing the zeros a bit above the double poles frequency (K =.) in order to boost the DC gain from 65dB to 75dB. Figure 0 illustrates the relationship between the phase margin and load transient response for K = 0.6 and K =. at the same crossover frequency of 50KHz. A higher DC gain along with a smaller phase margin of 58 yields a faster recovery time of 60µs, which results in a smaller peak-to-peak output transient response (80mV) for a 00mA to.5a dynamic load. K=0.6 Frequency Domain DC Gain = 65dB F co =50KHz Frequency (Hz) PM=68º Time Domain t r =40 us V PP =404mV Time (40μs/div).5A 00mA Cff = nf Rff = 365Ω R = 3.34kΩ C = 3300pF C = 47pF K=. DC Gain = 75dB PM=58º tr=60us V PP =80mV.5A Cff = 470pF Rff = 68Ω R =.5kΩ C = nf C = 7pF F co =50KHz 00 ma Frequency (Hz) Time (40μs/div) Figure 0: Phase Margin and Transient Response For Differing K Factors (K = 0.6 and K =.). Skyworks Solutions, Inc. Phone [78] 376-3000 Fax [78] 376-300 sales@skyworksinc.com www.skyworksinc.com 0376A Skyworks Proprietary Information Products and Product Information are Subject to Change Without Notice. September, 0 9
Phase Margin and Transient Response vs. Bandwidth (K =.) As illustrated in Figure, the output voltage spike can be further improved by pushing the crossover frequency (F C ) to 80KHz if a small amount of overshoot is acceptable. However, further increasing the bandwidth reduces the phase margin below 45, resulting in an unstable system. In addition, increasing the bandwidth to exceed the effective control bandwidth no longer reduces the output voltage spike due to the voltage drop across the ESR of the output capacitor which dominates the transient voltage spike. For a 3.3V output voltage buck converter using a 4.7µH inductor during a load transient step from 00mA to.5a, the effective control bandwidth is derived from Equation 6. Eq. 6: F CE = V O 4 I O L 3.3V = = 76KHz 4.5A 4.7µH K=. K=. Frequency Domain F co =50KHz Frequency (Hz) F co =80KHz Frequency (Hz) PM=58º PM=55º K=. V PP =80mV.5A 00mA K=. V PP = 66mV.5A Time Domain Time (40μs/div) 00mA Time (40μs/div) Cff = 470pF Rff = 68Ω R =.5kΩ C = nf C = 7pF Cff = 470pF Rff = 68Ω R = 8.7kΩ C = 680pF C = 8pF Figure : Frequency Domain vs. Time Domain For Different Bandwidth (F CO = 50KHz and F CO = 80KHz). 0 Skyworks Solutions, Inc. Phone [78] 376-3000 Fax [78] 376-300 sales@skyworksinc.com www.skyworksinc.com 0376A Skyworks Proprietary Information Products and Product Information are Subject to Change Without Notice. September, 0
Loop Gain Measurement The following guidelines show the method used to measure the loop gain of a DC-DC converter:. Break the feedback loop and insert a 50Ω resistor between the broken original connection. Insert the secondary winding terminal of the one-to-one isolation transformer between the 50Ω resistor. Configure the specified test equipment as shown in Figure.. Inject a sinusoidal signal from SOURCE OUT of the network analyzer to the loop through the primary winding terminal of the transformer while monitoring the ratio of CHA and CHB on the network analyzer. 3. Set the converter output current to heavy load while monitoring the LX node of the converter on the oscilloscope (to obtain a good result the converter must be in continuous PWM mode). 4. Sweep the frequency from SOURCE OUT of the network analyzer from 0Hz to MHz and adjust the magnitude of the injected signal (around 0mV to 00mV) in order to have a clean PWM waveform at the LX node. Power Supply V VIN 5A C IN Oscilloscope VIN LX L 4.7µH Buck Converter FB PGND Analog Network Analyzer SOURCE OUT CHA CHB Isolation Transformer Rfbh R fbl 50 Broken Original Connection R ff Cff VOUT COUT LOAD A Figure : Loop Gain Measurement Set-up. Skyworks Solutions, Inc. Phone [78] 376-3000 Fax [78] 376-300 sales@skyworksinc.com www.skyworksinc.com 0376A Skyworks Proprietary Information Products and Product Information are Subject to Change Without Notice. September, 0
Conclusion Using low ESR ceramic output capacitors for voltage mode controlled buck converters yields very low output voltage ripple, but requires type III compensation for adequate phase margin. The type III compensation network provides two zeros and two poles that push the crossover frequency to a possible maximum value with adequate phase margin for the control loop. The trade-off between the stability and output transient response can be adjusted by using the factor K, which represents the position of zeros in the vicinity frequency of the output double poles. In applications which require no overshoot, the two zeros are placed at 60% (K = 0.6) of the output double poles frequency to achieve approximately 70 degrees of phase margin. However, if the transient output voltage spike is critical, the two zeros can be placed up to 50% (K =.5) of the output double pole frequency if a small amount of overshoot is acceptable. In addition, a higher bandwidth yields a faster transient response. However, a bandwidth higher than the critical bandwidth can no longer reduce the transient output voltage spike. A typical bandwidth for type III compensation is in the range of 0% to 60% of switching frequency. Copyright 0 Skyworks Solutions, Inc. All Rights Reserved. Information in this document is provided in connection with Skyworks Solutions, Inc. ( Skyworks ) products or services. These materials, including the information contained herein, are provided by Skyworks as a service to its customers and may be used for informational purposes only by the customer. Skyworks assumes no responsibility for errors or omissions in these materials or the information contained herein. Skyworks may change its documentation, products, services, specifications or product descriptions at any time, without notice. Skyworks makes no commitment to update the materials or information and shall have no responsibility whatsoever for conflicts, incompatibilities, or other difficulties arising from any future changes. No license, whether express, implied, by estoppel or otherwise, is granted to any intellectual property rights by this document. Skyworks assumes no liability for any materials, products or information provided hereunder, including the sale, distribution, reproduction or use of Skyworks products, information or materials, except as may be provided in Skyworks Terms and Conditions of Sale. THE MATERIALS, PRODUCTS AND INFORMATION ARE PROVIDED AS IS WITHOUT WARRANTY OF ANY KIND, WHETHER EXPRESS, IMPLIED, STATUTORY, OR OTHERWISE, INCLUDING FITNESS FOR A PARTICULAR PURPOSE OR USE, MERCHANTABILITY, PERFORMANCE, QUALITY OR NON-INFRINGEMENT OF ANY INTELLECTUAL PROPERTY RIGHT; ALL SUCH WARRANTIES ARE HEREBY EXPRESSLY DISCLAIMED. SKYWORKS DOES NOT WARRANT THE ACCURACY OR COMPLETENESS OF THE INFORMATION, TEXT, GRAPHICS OR OTHER ITEMS CONTAINED WITHIN THESE MATERIALS. SKYWORKS SHALL NOT BE LIABLE FOR ANY DAMAGES, IN- CLUDING BUT NOT LIMITED TO ANY SPECIAL, INDIRECT, INCIDENTAL, STATUTORY, OR CONSEQUENTIAL DAMAGES, INCLUDING WITHOUT LIMITATION, LOST REVENUES OR LOST PROFITS THAT MAY RESULT FROM THE USE OF THE MATERIALS OR INFORMATION, WHETHER OR NOT THE RECIPIENT OF MATERIALS HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. Skyworks products are not intended for use in medical, lifesaving or life-sustaining applications, or other equipment in which the failure of the Skyworks products could lead to personal injury, death, physical or environmental damage. Skyworks customers using or selling Skyworks products for use in such applications do so at their own risk and agree to fully indemnify Skyworks for any damages resulting from such improper use or sale. Customers are responsible for their products and applications using Skyworks products, which may deviate from published specifications as a result of design defects, errors, or operation of products outside of published parameters or design specifications. Customers should include design and operating safeguards to minimize these and other risks. Skyworks assumes no liability for applications assistance, customer product design, or damage to any equipment resulting from the use of Skyworks products outside of stated published specifications or parameters. Skyworks, the Skyworks symbol, and Breakthrough Simplicity are trademarks or registered trademarks of Skyworks Solutions, Inc., in the United States and other countries. Third-party brands and names are for identification purposes only, and are the property of their respective owners. Additional information, including relevant terms and conditions, posted at www.skyworksinc.com, are incorporated by reference. Skyworks Solutions, Inc. Phone [78] 376-3000 Fax [78] 376-300 sales@skyworksinc.com www.skyworksinc.com 0376A Skyworks Proprietary Information Products and Product Information are Subject to Change Without Notice. September, 0