ISSUE: April Fig. 1. Simplified block diagram of power supply voltage loop.

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1 ISSUE: April 200 Why Struggle with Loop ompensation? by Michael O Loughlin, Texas Instruments, Dallas, TX In the power supply design industry, engineers sometimes have trouble compensating the control loop or their power supply. They try to get the loop to cross over at a very high switching requency in an attempt to improve large-signal transient response, only to end up struggling with stability issues. One o the most popular control methods in power supply design is peak current-mode control. Even though this method is supposed to be easier to compensate than voltage-mode control, some power supply designers still struggle with compensating the voltage loop. The purpose o this article is to give some pointers that will, hopeully, make compensating the voltage loop in peak current-mode control easier. ontrol Blocks In A Power Supply When we studied control theory back in university, all control systems could be simpliied by transer unction blocks. The voltage control loop in a peak current-mode-controlled power converter is no dierent. The voltage loop (T ()) can be simpliied and represented as the product o two dierent transer blocks (Figure ). The irst transer block is the power-stage control-to-output transer unction (G O ()), which can be described as the ratio o the change ( ) in output voltage over the change in control voltage ( ). Note that this block is actually the combination o the pulse width modulation (PWM) modulator gain (K) and the power supply output ilter gain (G F ()). The second transer block is generally the output-to-control transer unction (G ()), sometimes reerred to as the compensation transer unction, which can be described as ratio over the change in. I an optoisolator is used, it will have a transer unction block as well G OPTO () that would be ound on the line between blocks K and G () blocks. Fig.. Simpliied block diagram o power supply voltage loop. Figure 2 shows the unctional schematic o a peak current-mode-control orward converter represented by the block diagram in Figure. The control blocks are separated by dashed lines. 200 How2Power. All rights reserved. Page o

2 Figure 2. Simpliied block diagram o power supply voltage loop. Originally, the idea behind peak current-mode control was to control the average current through the inductor o the power stage, making it look like a current source by removing the double pole that occurs between the interaction o the output capacitor ( ) and the power stage s inductor (L ). A control block diagram o this model is presented in Figure How2Power. All rights reserved. Page 2 o

3 Figure 3. Modeling the inductor as a current source in a peak current-mode-controlled converter. A simpliied control-to-output transer unction (G O ()) or Figure 2 is presented below. In this equation, a is the transormer s turns ratio, LOAD is the converter s output load impedance, is the converter s output ilter capacitance, and ES is s equivalent series resistance. Another term, S(), is the angular velocity as a unction o requency. S ( ) 2π j From the control-to-output transer unction (G O ()), you will see that there is a zero ( ZO ) that occurs between the interaction o and ES and a pole ( PO ) that occurs between the interaction o LOAD and. G O ( ) Δ Δ a LOAD SENSE ( S( ) ( S( ) ES LOAD + ) + ) ZO PO 2π 2π ES LOAD Over time, engineers using peak current-mode control discovered there was a double pole ( PP ) in G O () that occurs at roughly hal the switching requency ( S ). The ollowing equations describe the G O () o a peak 200 How2Power. All rights reserved. Page 3 o

4 current-mode-controlled orward converter that includes the eects o PP. Note that i you analyze the orward converter with a network analyzer, you will ind this transer unction does not exactly match what the model describes. The zero (F ZO ) that occurs because o the interaction o ES and moves with load. The PP happens at slightly more than hal the switching requency. How will you ever compensate the voltage loop without an accurate model? You can do what engineers have been doing or years: ompensate the voltage loop by using a network analyzer to measure G O () and then ollow a ew simple rules or stability that will be presented in this article. PP S 2 Δ Gco() Δ a LOAD SENSE (S() (S() ES LOAD + ) + ) S() + 2π pp S() + 2π pp 2 Slope ompensation It was discovered that in a peak current-mode-control converter there could be subharmonic oscillations caused by sudden changes in duty cycle. This is because changes in duty cycle cause errors in the average output current (I, I 2 ) due to the control voltage ( ) not being able to correct or the changes in duty cycle ast enough. To correct or this error a technique called slope compensation was developed. This technique adds a triangular voltage waveorm to the current-sense signal (2 SLOPE + SENSE ) that orces the average output current to not change drastically with sudden changes in duty cycle. eer to Figure 4 or details. 200 How2Power. All rights reserved. Page 4 o

5 SLOPE ( )( ) AMP B + SENSE SENSE 2 SENSE + SLOPE Figure 4. Slope compensation is needed or stability. One o the most important steps in setting up the control loop or peak current-mode control is correctly adding slope compensation to the current-sense signal ( SENSE ). I you do not use slope compensation, you will be ighting subharmonic oscillation, even though your network analyzer is telling you that the loop should be stable. I you add too much slope compensation, the converter will operate in voltage-mode control and not operate correctly, and may be unpredictable. As a general rule to help ensure stability, the amount o slope compensation ( SLOPE ) added to the current-sense signal should be equal to hal the down slope o the output inductor current (di L ). The ollowing equations calculate the slope compensation ( SLOPE ) or the peak current-mode-controlled orward converter presented in Figure 2. In these equations, di L is the change in inductor ripple current and is the output voltage. L is the output ilter inductance and D is the converter s duty cycle. ariable s is the converter s switching requency. di L L dt L ( D) s I a transormer is used in your design, the magnetizing current (di LM ) in the primary o the transormer caused by the magnetizing inductance o the primary (L M ) will add some slope compensation and needs to be accounted or when adding slope compensation. To ensure that the converter is not operating in voltage-mode 200 How2Power. All rights reserved. Page 5 o

6 control, it is recommended that you choose a transormer or your design with a di LM that is less than hal the relected down slope (di L ) o the output inductor current. The ollowing equations can be used or selecting the correct amount o slope compensation or the orward converter presented in Figures and 2. di LM IN dt L M L IN M ( D) s di LM dil 0. 5 a dil 2 a SLOPE dilm SENSE General ules For Stability In the power supply control loop (T ()), when the loop is 80 degrees out o phase, this is equivalent to swapping the polarities o the inputs o an operational ampliier used in the eedback network (G ()). I this occurs at the voltage-loop crossover when the eedback loop has a gain o one, the loop could become unstable and break into oscillations. To ensure this does not occur, we generally design T () or 45 degrees o phase margin (PM) at voltage-loop crossover. In most switch-mode power supplies, the control loop will eventually approach a 80-degree phase shit. To ensure this does not cause loop instability, we generally design or greater than 6 db o gain margin (GM) to ensure the control signal is attenuated when T () is 80 degrees out o phase. When evaluating the control loop (T ()) the phase margin can be read as the magnitude o phase during crossover. The gain margin is calculated in the traditional way, 0 db minus the gain in db when the loop is 80 degrees out o phase. The gain and phase margin rule is a staple in good control-loop design. This rule is summarized below.. Set PM 45 degrees at voltage-loop crossover, which is deined as the requency where the magnitude o the loop gain (T ()) (i.e. 0 db). 2. Set GM > 6 db, where GM is deined as 0 db - gain 80 DEGEE PHASE SHIFT. Where Should My oltage-loop rossover Be For T ()? According to Nyquist, or voltage-loop stability, the crossover requency ( c ) needs to be less than hal the converter s switching requency ( s ). 3. < S 2 In peak current-mode control, the voltage loop should be crossed over a decade beore the double pole that occurs in G O (). Depending on the topology used, this double pole may occur at less than hal the switching requency. Using a network analyzer allows the designer to know exactly where the double pole occurs. 4. < PP 0 Measure G O () With A Network Analyzer Even i you have a good model o your control-to-output transer unction, you will end up modiying the control loop based on measured results rom a network analyzer. It is easier to compensate the voltage by initially using the voltage ampliier network (G ()) as an integrator and measure the actual G O () characteristics. This can be accomplished by setting capacitor P in Figures and 2 to µf to measure G O () and not populating F and Z. The loop will not be optimized and the input voltage and load currents should be adjusted slowly to avoid oscillations. The ollowing two plots (Figures 5 and 6) show the measured gain and phase o a 600-W peak current-mode controlled, phase-shited, ull-bridge converter using TI s U28950 secondary-side controller. This controller 200 How2Power. All rights reserved. Page 6 o

7 does not require an optoisolator and a standalone voltage eedback ampliier (TL43), making the voltage loop easier to compensate. Figure 5. ontrol-to-output gain GO() in db.. Figure 6. ontrol-to-output phase G O () in degrees. The G O () is more complicated than what is deined above and you could spend hours deriving a transer unction that closely models the measured results. However, it is not necessary to compensate the loop once the actual requency response data is obtained with a network analyzer. From the plots below it can be observed that the low-requency pole ( PO ) rom the interaction o and LOAD changes requency with changes in output power. The zero in G O () that is caused by the interaction o and ES also moves with load. The PP o G O () or this converter occurs at roughly 60 khz. Note that G O () should be set up to cross the voltage loop (T ()) a decade beore this double pole at roughly 6 khz. 200 How2Power. All rights reserved. Page 7 o

8 PP 0 60kHz 0 < 6 khz To set up G () requires knowing the highest gain o G O ( ) at crossover. From the measured G O () this occurs at the 60-W load and is roughly -0 db. G ( ) db O 0 Setting Up The oltage Ampliier (G ()) One o the more popular compensation techniques or peak current-mode control is a type-two compensator that is presented in Figures 2 and 3. The ollowing equation describes the transer unction. It has a pole that occurs at the origin. The type-two ampliier also has a zero ( Z ) that can be programmed by selecting F and Z values. Additionally, the type-two compensation network has a pole ( P ) that can be programmed by selecting F and P. Δ Gc() Δ Z P S 2π F Z P 2π F + ()( + ) Z Z P I (S() F Z + ) P Z S() + F Z + P Z P esistor I and A are selected based on dc output voltage and resistor F is set at loop crossover to correct or the gain at G O (c). In this power converter, I was selected to be 9.09 kω. To cross over the voltage loop at roughly 6 khz required an F resistor o 28.7 kω. G ( 0dB) k 0 F I O ( ) 28.7k apacitor Z is set to give added phase margin at crossover and can be set a decade below the crossover requency ( ). z c 2π 0 9.2nF For this design a standard capacitor value o 0 nf was used or Z. z 0nF This leaves one pole in the G () eedback that is used to cancel the phase gain caused by the ES o the output capacitor in G O () ater. This helps maintain stability, ensuring the gain continues to roll o ater voltageloop crossover. > < P PP To ensure the gain roles o beore the double-pole requency, set the pole requency o the compensator at twice the crossover requency. To compensate this voltage loop, a standard 680-pF capacitor or P is used. 200 How2Power. All rights reserved. Page 8 o

9 P 2 2kHz P 460 pf 2 π P A standard capacitor o 470 pf was used or P. P 470 pf Ater selecting the compensation components or G (), double check the voltage loop with a network analyzer and adjust it i needed. The plots in Figures 7 and 8 were taken with a network analyzer to measure the voltage loop T () at 60 W and 600 W. These plots show that the voltage loops crossed over ( ) at roughly 3.8 khz at 600-W load with a phase margin at crossover o 0 degrees. At a 60-W load T () crossed over at roughly 5 khz with greater than 45 degrees o phase margin at. The voltage loop at 0 percent load crossed over at khz less than the design target. However, loop compensation is not an exact science and being within to 2 khz is quite acceptable. Note that the gain was less than -30 db as the phase o T () approached 80 degrees. This yields a gain margin o greater than 60 db. The network analyzer always has problems measuring -80 degrees. It cannot determine whether the phase is ±80 degrees. Figure 7. T() loop gain in db. 200 How2Power. All rights reserved. Page 9 o

10 Figure 8. T() loop phase in degrees. Misconception Speeding up the small-signal voltage loop T () reduces the output capacitor bank. emember there is an inductance in most switched-mode power supplies somewhere that will resist sudden changes in current. Largesignal current transients pass through and the s ES. To meet large-signal transient speciications requires selecting and ES to hold up and suppress large-current load transients. The ollowing equations should be useul in selecting the output ilter capacitance required or the design. ariable I TANSIENT is the large-signal instantaneous current load step, and variable dt is the amount o time the output capacitance is expected to suppress the large-signal transients. ariable I AEAGE is the average current ater the load step. Worst case would be stepping rom no-load to ull-load conditions. These equations place 90 percent o the load-transient burden on the ES and 0 percent on. ES TANSIENT, I TANSIENT 0.9 I AEAGE TANSIENT dt 0. onclusion Over the years I have compensated many peak current-mode control voltage loops in power supplies. In the beginning, I struggled to get the control loops to cross over at much higher switching requencies than necessary, only to have the loop become unstable due to the converter s double-pole requency. The techniques presented in this paper or compensating these voltage loops saved me a lot o time and eort. I hope these techniques will provide the same beneits to you as well. eerence: Unitrode SEM 300, Topic, urrent-mode ontrol o Switching Power Supplies, Lloyd H. Dixon, Jr, How2Power. All rights reserved. Page 0 o

11 About the Author Michael O'Loughlin is an applications engineer with the Power Supply ontrol Products group at Texas Instruments. He specializes in oline and isolated power supply design and has authored numerous articles on power actor correction and power supply design related topics. Michael received his Bachelor o Science degree rom the University o Massachusetts. Michael can be reached at ti_mikeoloughlin@list.ti.com. For urther reading on loop compensation, see the How2Power Design Guide, search the Design Area category and select ontrol Methods as the subcategory. To narrow your search, try entering keywords such as control loop, stability, or compensation. 200 How2Power. All rights reserved. Page o