An Analytical Comparison of Alternative Control Techniques for Powering Next-Generation Microprocessors

An Analytical Comparison of Alternative
          Control Techniques for Powering Next-
              Generation Microprocessors
                                       By Rais Miftakhutdinov

The latest microprocessor roadmaps show not only ever-increasing performance and speed, but also the
demand for higher currents with faster slew rates while maintaining tighter supply-voltage tolerances.
This topic addresses these challenges by reviewing and comparing various control approaches for
single- and multi-phase synchronous buck converters. An optimized hysteretic control algorithm is
shown offering a significantly improved transient response, which is illustrated with a specific design

                                                             core voltage at the battery mode, while keeping
                I. INTRODUCTION                              frequency and voltage higher at performance
                                                             mode. In the automatic mode the processor
     An unprecedented push in semiconductor                  continuously adjusts the clock frequency and
technology, especially microprocessors, set the              voltage according to system demand. That means
new level of requirements for OEMs [1]. By the               the microprocessor voltage regulator must be
year 2005, 3.5-GHz processors having over 190                able to quickly change the output voltage on the
million transistors on chip will consume more                basis of the control signals from the system or
then 160 W. The new 0.1-µM technology will                   microprocessor.
drop the core-voltage of high-performance                        The other power saving technique called
processors down to 1.2-V range, requiring up to              “Intel Mobile Voltage Positioning” or IMVP,
130-A current from the voltage regulator.                    uses the droop-compensation approach usually
Developing an efficient, low-cost power-delivery             associated with the extending of transient
system for that type of load is one of the major             window. It also requires the negative core-
problems to be solved.                                       voltage offset for some sleep-mode stages of the
     Another problem relates to the high slew-rate           microprocessor [18]. This approach extends the
current transients, exceeding 40,000 A/µs                    battery life, because power dissipation of the
through the die when a processor abruptly                    microprocessor is inversely proportional to core
changes its operation state. Obviously, special              voltage square. These new requirements must be
packaging, high-frequency decoupling, and a fast             counted during power-delivery system design.
transient-response regulator must be used to keep                A controller IC is a significant part of the
the core-voltage tolerance within the required               microprocessor power supply: it integrates the
few percent.                                                 described power-saving functions. Still, the main
     Additionally, the latest mobile processors              goal of controller is to provide an accurate output
have     implemented      special    power-saving            voltage at steady-state conditions and fastest
technique called SpeedStep , PowerNowTM, and
                                                             response with minimum voltage tolerance at high
LongRunTM [15-17] trying to prolong battery life. In         slew-rate transients. The topic of this article
accordance       to    this    technology,       the         explains the control-approach influence on
microprocessor has different modes of operation;             transient and optimal power-delivery system
i.e., “performance,” “battery” and “automatic.”              design.
The idea is to decrease the clock frequency and

Review of available literature shows that                     the limitations of actual controllers (such as
usually the authors provide rule-of-thumb                          delays, fixed on- or off-time, compensation
recommendations        for    selecting    different               bandwidth, and error-amplifier saturation) that
components on the basis of transient                               might degrade the transient response. A review
parameters [5-7,13]. This topic discusses the                      and comparison of control techniques shows that
transient in a power-delivery system as a whole,                   a hysteretic regulator is one of the most
thereby suggesting more accurate design                            appropriate        solutions       for         powering
procedure. It includes the model selection, the                    microprocessor-type loads having high slew-rate
derivation of transient equations in the time                      and amplitude transients.
domain, the observation of transient waveforms,                        A hysteretic controller and its modifications
and the effect of different system parameters,                     are described in the next section. Although a
including parasitics and controller characteristics.               hysteretic regulator has been used in power
This approach defines the worst condition of                       electronics for a long time [28], earlier
transient, which is determined by the moment                       publications do not address modern applications
within the switching cycle in which the transient                  and conditions. This analysis includes new
occurs. This condition is not described in the                     equations      for      the    switching-frequency
literature, although it influences component                       calculation, both for the typical hysteretic control
selection. This topic addresses the popular                        and for its modifications [26,30,33,34,40,42]. Finally, a
synchronous buck converter and the multi-phase                     design example and the optimization procedure
(interleaved) topology based on it. The equations                  are provided to implement the analysis results
for a required number of output bulk capacitors                    and to address important practical issues.
are derived and an optimization procedure for the
output filter design is suggested both for one- and                                II. TRANSIENT ANALYSIS
multi-phase topologies.
     The next section of the topic reviews and                     A. Power Distribution Model
compares different control approaches most                             The model selection for an analysis is always
suitable for microprocessor power supplies. On                     a tradeoff between practicality and accuracy of
the basis of the previous transient analysis, it first             the results. Fig. 1 shows the power distribution
formulates a control algorithm for the best                        system considered in the analysis.
transient response. The next step explains how

                                              Lo                                            uP or DSP
                                                         +       vA Rb     Lb      vB   with HF decoupling

                          D                        ESR              Load current
          Vin                  1-D                 ESL
                                     Driver    Hyst.
                                 _             Comp.
                                 Q                           _

                                 Q                           +


Fig. 1. Analyzed power-delivery system at load-current transients.

The model includes the synchronous buck                 B. Analysis Approach
converter, the output inductor LO, the output bulk               There are many publications where dc-to-dc
capacitor with parasitics, and the power supply             buck converters for powering microprocessor are
traces between the bulk capacitor and the                   considered at load-current transients [19-24,26,30-
package or cartridge of the microprocessors. The                               .
output bulk capacitor is presented as a series                   Computer simulation is one of the popular
connection of an equivalent ideal capacitor CO,             technique for the transient analysis [20]. It is a
with equivalent series resistance ESR and                   useful tool for design validation, but it does not
equivalent series inductance ESL. The equivalent            reveal analytical relations among the parameters
resistor RB characterizes the resistive voltage             of a power-delivery system. Therefore, it is
drop through the supply paths and is the                    difficult to predict and find the optimal solution.
summarized resistance of the traces and the                      Small-signal analysis is another technique
connectors. The equivalent inductor LB                              . Some authors optimize the small-signal
characterizes the inductive voltage drop across             frequency characteristics—for example the
the traces and connectors. The high-frequency               output impedance [22]. However, this practice is
decoupling capacitors on the die CDIE and inside            questionable, because it applies the small-signal
the package CHf are not included in the analysis,           analysis to the large-signal transient process. This
because usually the load-current slew-rate is               topic shows that small-signal analysis cannot
specified at the package pins. Nevertheless, in             explain the dependence of transient upon the
situations when LB and ESL are too high, it is              moment within a switching cycle when it occurs.
important to decrease the slew-rate of current                   Assuming that a load-current transient is a
through the bulk capacitor CO by adding the high-           linear function of time, the equations are derived
frequency decoupling capacitors around the                  for the voltages and currents of all components of
package. This procedure is also described in the            power-delivery system as a function of time
transient analysis section.The analyzed model is            before, during, and after the load-current
the lumped one, while in an actual power-                   transient. These equations are included in a
delivery system the output capacitors and                   MATHCAD-based software program to view the
parasitics are distributed over the PCB board               transient voltage and current waveforms. The
area. But results of the analysis using this model          same equations also characterize peak values of
are confirmed by the measurements and                       the transient. On the basis of these equations, the
sufficiently accurate for most applications. The            curves for selecting the optimal output filter and
same model of the power-delivery system is                  the minimum number of bulk capacitors selection
suggested in Intel's Power Distribution                     are built [26,30,33,34,40,42].
Guidelines [5-7]. The controller and the drive
circuitry are assumed not to have any delays.               C. Transient Waveforms and Experimental
They are able to maintain the on- or off-state of           Verification
the related power switch during the transients as               Fig. 2 shows the waveforms across the
long as necessary to return the output voltage              different elements of the model (Fig. 1) at the
back to the steady-state level as soon as possible.         load-current step-down.
In this case, the controller does not degrade the               The output current changes between IO(max)
transient response, which is determined by                  and IO(min) (Fig. 2a) with a constant slew-rate. The
passive components only. The hysteretic                     step-up transient waveforms have the same stages
controller in Fig.1 includes the comparator with a          and are qualitatively similar to a step-down,
hysteresis window, the reference voltage, and the           although they are described by different
drive circuitry with a complementary control of             equations.
high- and low-side FETs. Fig. 1 is an example
only, but it will be shown later that it is a good
approximation of the ideal controller.

                                                     with decoupling
                                                        uP or DSP


                                                                                                   Load Current







                                                                                             VIN  +










                              iO, iL

                                                                                                                                                             iC, vCO

                                                                                                                             (e) vESR+vRb

                                                                                                  (d) vESL+vLB



(a) Load-current and output inductor current transient waveforms.
(b) Transient-voltage waveform on the output of the dc-dc converter (point A in Fig. 1).
(c) Transient-voltage waveform on the microprocessor package pins (point B in Fig. 1).
(d) Transient-voltage waveform at the inductive components of the model ESL and LB.
(e) Transient-voltage waveform at the resistive components of the model ESR and RB.
(f) Transient-voltage waveform and current on the-capacitor CO.
Fig. 2. Waveforms through different elements of the model during load-current step-down transition.
To verify the derived equations, the                                                 on an ideal controller without delays, and
MATHCAD transient waveforms are compared                                                 measured waveforms, with the switching
with the measured ones under the same                                                    regulator based on the hysteretic controller
conditions. As shown in Fig. 3, the theoretical                                          TPS5210, shows that the hysteretic control,
and measured waveforms are very close for the                                            despite typical 250 ns delays does not degrade
load-current step-down and step-up conditions.                                           the transient characteristics significantly.
The comparison of analyzed waveforms, based
     VA(50mV/div), IO(14.5A/div)

                                   0   10      20        30        40    50   60
                                             Time (microseconds)

(a) Theory                                                                          ( b) Measurement
   VA(50mV/div), IO(14.5A/div)

                                   0    10          20        30        40    50
                                             Time (microseconds)

(c) Theory                                                                          (d) Measurement
Fig. 3. Theoretical (a), (c) and measured (b), (d) waveforms at load-current step-down (a), (b) and step-
up (c), (d)transitions. [The theoretical waveforms show the output voltage (top) and load-current
(bottom) transients.The measured waveforms include VDS voltage of the low-side FET (Ch1: 20V/div),
output voltage (Ch4: 50mV/div) and load current (Ch3: 14.5A/div)].

D. The Two Extreme Values of a Transient
     Fig. 4 shows typical load-current transient

                                                                                                                 VA(50mV/div), IO(14.5A/div)
waveforms. The output-voltage waveform has
two extreme values, VM1 and VM2.
     For most applications the transient slew rate
of the load current is much higher than the slew
rate of the output-inductor current. Therefore, the                                                                                                   VM1
first peak, VM1, depends mainly on the parasitics                                                                                                                                   VM2
of the output capacitor and supply path. The
controller    transient-response      characteristics
(Fig.5a) do not affect it significantly. The second
peak, VM2, depends on the resistive components                                                          Fig. 4. Typical load-current transient waveforms.
ESR and RB, the capacitive component CO, the
inductor value LO and the converter
characteristics, including the switching frequency
and the type of control (Fig.5b).

                                                                                                                                               V M1
             VM1                                                                                                                                                           V M2

                                  Lo                                            uP or DSP                                                                            Lo
                                                               Lb                                                                                                                                 Lb
                                                                                                                                                                                                                    uP or DSP
                                              +       vA Rb            vB   with HF decoupling
                                                                                                                                                                                +       vA Rb             vB   with HF decoupling

              D                        ESR              Load current                                                                                                      ESR
                                                                                                                                                D                                          Load current
      Vin          1-D                 ESL
                                                                                           Cdie                                                       1-D                                                       Chf            Cdie
                                                               or                                        Vin                                                              ESL                                         Die
                                       Co                                                                                                                                 Co
                                              _                                                                                                                                 _
                         Driver     Hyst.                                                                                                                   Driver    Hyst .
                    _              Comp.                                                                                                               _
                                                  _                                                                                                                   Comp.         _
                    Q                                                                                                                                  Q
                    Q                             +                                                                                                    Q                            +

                                                        Vref                                                                                                                               Vref

(a)                                                                                                            (b)
Fig. 5. Different parameters and their effect on peaks VM1 and VM2.


                                                                   VA(20mV/div), IO and IL(10A/div)
A. Output Inductor Value
    It could be thought, that the lower output
inductor value enables better transient-response
characteristics because of a faster inductor-
current change to the new level after the load-
current transient occurs.
    In reality, the example in Fig. 6 shows that,
after some optimal value (Fig. 6b), further
decreasing of the inductor value increases the
                                                                                                            0   5 10 15 20      25 30 35 40 45 50 55 60 65
peak-to-peak transient amplitude because the
                                                                                                                           Time (microseconds)
output ripple rises significantly. As shown later,
the optimal inductor value depends on the                     (a) LO = 1.6 µH VO(max) = 79 mV TRECOV = 34 µs
switching frequency and the characteristics of the
output bulk capacitors.

                                                                      VA(20mV/div), IO and IL(10A/div)
B. Dependence on Switching Cycle Position
    The output-voltage transient response
depends on where the load-current transient
occurs within the switching cycle. If the load
current steps down, the excessive energy stored
in an output inductor is delivered to the output
capacitor. The worst case for the step-down
transition occurs if the transient takes place at the
end of a conduction time of high-side FET,
because the inductor current is at its maximum.                                                             0   5    10   15    20    25    30    35    40   45   50   55
At that moment the inductor stores the maximum                                                                            Time (microseconds)
energy, while the output ripple voltage also is at
its maximum. So the effect of transient is most               (b) LO = 0.8 µH VO(max) = 62 mV TRECOV = 19 µs
significant at that moment, causing greater output
voltage spikes than at any other moment (Fig. 7).
                                                                         VA(20mV/div), IO and IL(10A/div)

    In contrast, the worst case for the step-up
transition occurs if the transient happens at the
end of the switching cycle, because the inductor
current and output voltage ripple are at their
minimum at that moment. Only the output
capacitor supplies the load during the step-up
transient, while the inductor must restore its
energy and current to the new load-current level.
This fact cannot be described with small-signal
analysis, although the effect must be considered
for reliable output filter selection.                                                                       0    5   10    15    20        25    30    35    40   45   50

                                                                                                                          Time (microseconds)

                                                              (c) LO = 0.4 µH VO(max) = 72 mV TRECOV = 12.5 µs
                                                              Fig. 6. Transient waveforms for different
                                                              inductor values LO.

       VA(20mV/div), IO and IL(10A/div)
                                                                                                                   OF CAPACITORS

                                                                                                           The peak values VM1 and VM2 for the worst-
                                                                                                      case step-down and step-up transients can be
                                                                                                      found         with     the       derived equations.
                                                                                                      (See Appendix 1.) Assume that the output filter
                                                                                                      capacitors are connected in parallel and that each
                                                                                                      capacitor has the characteristics CO1, ESR1 and
                                                                                                      ESL1. The number 1 after a parameter means that
                                                                                                      the parameter relates to one of many capacitors
                                          0   5   10    15   20   25   30   35   40   45   50         connected in parallel. The literature shows
                                                       Time (microseconds)                                       that if ∆VREQ is the maximum allowable
                                                                                                      peak-to-peak transient tolerance, then the
(a) Worst case: Transient occurs at the end of the upper
FETs conduction time.                                                                                 required number of output bulk capacitors N1
                                                                                                      and N2, to meet the conditions VM1 = ∆VREQ and
                                                                                                      VM2 = ∆VREQ, respectively, can be found by
       VA(20mV/div), IO and IL(10A/div)

                                                                                                      equations (1) and (2):
                                                                                                      where KL = ∆ILEQV/∆IO                           (3)
                                                                                                                 VOUT ⋅ (1 – D ⋅ n )
                                                                                                       KL =                                           (4)
                                                                                                               L OEQV ⋅ ∆I O ⋅ f S ⋅ n
                                                                                                              The parameter m depends on the type of
                                                                                                      transient. It is equal to:
                                                                                                       m =1– n⋅D                                     (5)
                                                                                                      for the worst-case step-down transient and to:
                                                                                                            D ⋅ (1 – n ⋅ D )
                                                                                                       m=                                            (6)
                                          0   5   10    15   20   25   30   35   40   45   50

                                                       Time (microseconds)                                   n ⋅ (1 – D )
(b) Best case: Transient occurs at the end of the switching

Fig. 7. Output voltage (bottom curve) and
inductor current (dashed) waveforms for the
different instants when the load-current (top,
solid) step-down transition occurs.

     ESR1          T      æ        TO ö æ TO ⋅ f S ⋅ n ö
          + ESR1 + O + çç ESR1 +            ÷ ⋅ ç1 –   ÷ ⋅ KL
      TO          2C O 1 è       2 ⋅ C O 1 ÷ø è      m ø
N1 =                                                                                                                                                 (1)
                        ∆VREQ L B
                              –      – RB
                         ∆I O   TO
    1é       m           TO æç        ESR12 ⋅ C O 1 ⋅ f S ⋅ n          m             ö
                                                                                     ÷ ⋅ KL +       m          1 ù
      ê                –     + ESR1 +                         +                                              ⋅   ú
    2 ê C O 1 ⋅ f S ⋅ n C O 1 çè              m                 4 ⋅ C O 1 ⋅ f S ⋅ n ÷ø        C O 1 ⋅ f S ⋅ N KL ú
N2 = ë                                                                                                           û (2)
                                                         – RB
                                              ∆I O

for the worst-case step-up transient. The                      Assume that RB = 1.5 mΩ; LB = 1.0 nH;
∆ILEQV is the peak-to-peak ripple portion of the                ∆IO = 23.8 A; vIO = 20 A/µs and TO = ∆IO/vIO =
output inductor current, ∆IO is the load-current                1.2 µs in accordance with the VRM 8.4
step, D = VOUT/VIN is the duty cycle, fS = 1/TS is              requirements [10]. Then the voltage drop through
the switching frequency, and TS is the switching                the supply path is:
cycle. The number of channels in an interleaved                 VB = 35.7 mV+20 mV = 55.7 mV                (11)
(multi-phase) regulator is n. For the one-channel                    For the 1.65-V output power supply, the
converter n = 1. Later discussion shows that, with              voltage drop is almost 3.8%. This example shows
some assumptions, the same transient analysis                   why it is important to keep the output filter
applies to the one- and multi-channel interleaved               capacitors as close as possible to the
converters.                                                     microprocessor package to avoid a significant
      The second peak VM2 exists only if the                    voltage drop due to supply path parasitics.
following condition is fulfilled:                                    Let us consider the other example. The
    m æ 1       1ö                                              expression:
         ⋅ç   + ÷ – ESR ⋅ C O > TO                 (7)
 f S ⋅ n è KL 2 ø                                               æ ∆VREQ ö æ L B 1 ö
                                                                ç           ÷ − çç     ÷÷ − R B 1
      where TO is the load-current transition time.             ç ∆I        ÷      T
                                                                è      O ø è O ø
If this condition is not fulfilled, only the first              is the denominator of the Equation 1.
spike and Equation 1 must be considered. The                         Obviously, the minimum number of bulk
other assumption of the analysis is that TO <                   capacitors N1 can be calculated only if the
D·TS. This assumption does not restrict the                     following inequality is met:
analysis for most applications. Equations 1 and 2                ∆VREQ L B1
can be used for the optimal output-filter design                           >        + R B1
based on the optimization curves N1 = N1(fS,                       ∆IO        TO
LO(eqv), n) and N2 = N2(fS, LO(eqv), n). But first, let         Assume that ∆VC = 96 mV for the previous
us consider some simple but useful relations from               example.
these equations.                                                     In this case ∆VREQ / ∆IO = 4 mΩ. This is an
                                                                important transient characteristic, let’s call it
A. Influence of Supply-Path Parasitics                          Equivalent Transient Resistance (ETR). LB/TO =
    The voltage-transient waveforms on the                      0.84 mΩ. That means that the supply path
converter output pins (point vA in Fig.1) and on                parasitics LB and RB give us 4 mΩ/(4 – 0.84 –
the microprocessor package supply pins (point                   1.5) mΩ. = 4/1.66 = 2.4 times number of
vB in Fig.1) are different because the supply-path              capacitors increase.
resistance RB and inductance LB cause an
additional voltage drop. If the output current step             B. Bulk Capacitor Parameters
∆IO and slew-rate vI are defined as:                                                ESL
 ∆IO = IO(max ) – IO(min )                      (8)                 The expression  To          2 .Co
                                                                                                    is part of the
       ∆I O                                                     numerator of the Equation 1 for N1. It
vI O =                                      (9)                 characterizes       the    selected     capacitor
       TO                                                       characteristics. For example, the aluminum
   then the additional voltage drop VB of the                   electrolytic capacitor 6.3ZA1000 from Rubycon
supply paths is:                                                has ESL = 4.8 nH, ESR = 24 mΩ and CO =
VB = VR B + VL B = ∆I O ⋅ R B + vI O ⋅ L B (10)                 1000 µF. That gives us 4.8 nH/1.2 µs +24 mΩ
                                                                +1.2 µs /(2•1000 µF) = (4+24+0.6) mΩ =
                                                                28.6 mΩ. After substituting 1.66 mΩ for
                                                                denominator from the previous example:
                                                                28.6 mΩ /1.66 mΩ = 17.2

Roughly at least 17 capacitors must be used                    There are two different approaches for a
to meet the requirements. It will be shown later              practical implementation of this idea. Fig. 9a
how this number can be reduced by additional                  shows the simplest way called "passive" droop
high-frequency decoupling.                                    compensation. The droop resistor RD = ESR is
                                                              inserted between the output capacitor and the
C. Active and Passive Droop Compensation                      output voltage sense point VA. Passive droop is
    Equations 1 and 2 show that increasing                    fast and naturally follows the load-current
∆VREQ can lower the required number of bulk                   transient. This approach can be used probably
capacitors. The droop compensation is an                      with any type of control technique. The drawback
effective technique to increase ∆VREQ. Droop                  is that it requires the additional resistor with
compensation means that the dc output voltage of              power losses in it. The other approach, called
the converter is dependent on the load. It is set to          "active" droop compensation, is shown in
the highest level within the specification window             Fig. 9b. It can be implemented by adding an
at no-load condition and to the lowest level at               offset proportional to the output current, which is
full-load. This approach degrades the static load             subtracted from the reference voltage, or by
regulation but increases the output-voltage                   changing the dc gain and the compensation
dynamic tolerance by as much as twofold, thus                 circuitry to get a desirable output impedance
reducing the number of bulk capacitors required.              ZOUT [22,37]. The main challenge here is that it
For the same output filter, this technique allows a           must be accurate and fast enough to follow the
decrease in peak-to-peak output-voltage transient             transients having their own repetition frequency
response. Fig. 31 in the design-example section               up to 100 kHz [8-11].
shows the output voltage budget with droop                        The transient waveforms with and without
compensation.                                                 active droop compensation are shown in Fig. 10.
    The popularity of droop compensation is                   One can see that without droop compensation
confirmed by the fact that it has numerous names              (Fig. 10a) the output-voltage peak-to-peak
such as "Programmable Active DroopTM,”                        amplitude is 146 mV and it exceeds the
"Active Voltage Positioning,” "Adaptive Voltage               requirements, as shown by the cursors. With
Positioning,” "Summing-Mode Control,” "Intel                  droop compensation (Fig. 10b), the peak-to-peak
Mobile Voltage Positioning (IMVP),” etc.                      transient is only 78 mV, keeping the output
                                                              voltage of the same regulator well within the

Passive droop resistor
                                                                   VA RD +            RB            VB

                                                                               ESR Load current
                   VIN               1-D                                      ESL or                     mP


                                       _ Driver      Controller
                                       Q                               _


(a) Passive droop compensation
                                                                           VA +           RB LB     VB

                                      D                                            ESR Load current
                         VIN               1-D                                    ESL or                 mP

                                            Q                                 _
                                            Q             Controller

                                                  Change DC gain and
                                                  compensation to get
                                                     required ZOUT

(b) Active droop compensation
Fig. 9. Passive (a) and active (b) droop compensation technique.

(a) Without droop compensation Vout p-p = 146 mV                                  (b) With droop compensation Vout p-p = 78 mV
Fig. 10. Active droop compensation technique. Channel 2 shows the output voltage (50 mV/div.),
Channel 3 shows the load current (10 A/div.), and the cursors show the required limits for the output

decoupling      capacitors     very     close    to
D. Optimization Curves                                                                                             microprocessor pins.The equations for N1 and
     Equations 1 and 2 for the required numbers of                                                                 N2 can be drawn as a function of TO = ∆IO /vIO
bulk capacitors N1 and N2 can be used to build                                                                     for the selected inductance LO = 2 µH and
optimization curves as a function of the output                                                                    switching frequency fS = 100 kHz (Fig.12).The
inductance LO (Fig.11).                                                                                            graph shows that doubling TO from 1.2 µs to
     These curves are built for a step-down                                                                        2.4 µs will decrease the number of bulk
transient by using aluminum electrolytic                                                                           capacitors from 20 to 15. This transient time
capacitors with:                                                                                                   increase means that the slew-rate of the transient
CO = 1000 µF, ESR = 24 mΩ, ESL = 4.8 nH                                                                            must be decreased from 20 A/µs to 10 A/µs. A
     with the following conditions:                                                                                solution by adding high-frequency ceramic
VIN = 5 V, VOUT = 1.65 V, fS = 100 kHz, ∆VOUT                                                                      decoupling capacitors is illustrated in Fig.13. For
dc = -80 mV / +40 mV, ∆VOUT ac = -130 mV /                                                                         15 electrolytic capacitors the total inductance is
+80 mV, IO(max) = 26 A, IO(min) = 2.2 A, ∆IO =                                                                     ESL1/15 + LB = 4.8 nH/15 + 1 nH = 1.3 nH. The
23.8 A, ∆VREQ = 96 mV,                                                                                             805 1-µF ceramic capacitor has ESL1 = 2.6 nH
vIO = 20 A/µs, TO = 1.19 µs, RB = 1.5 mΩ, LB =                                                                     including inductance of vias and traces. The
1 nH.                                                                                                              equivalent inductance of four capacitors placed
     The higher number of capacitors related to                                                                    very close to the microprocessor is
the curves N1 and N2 must be selected to meet                                                                      2.6 nH/4 = 0.65 nH. Adding four high-frequency
the requirements—in this case for LO = 2 µH, N1                                                                    decoupling capacitors roughly halves the slew-
= 20, while N2 is only 12. These values mean                                                                       rate, in accordance with the following equation:
that the voltage drop associated with ESL and LB                                                                   vIO new = vIO ⋅ (0.65 nH/1.3 nH) = 20 A/µs ⋅ 0.5
is too high. The influence of ESL and LB and the                                                                   = 10 A/µs.
required number of bulk capacitors N1 can be
decreased by placing the high-frequency
                                        24                                                                                                                  30
                                        22                                                                                                                  28
                                                                                                                                                            26               20 caps
                                                                                                                        Number of Capacitors (N1 and (N2)
    Number of Capacitors (N1 and (N2)

                                        18                                                                                                                  22
                                        16                                                                                                                  20                                                N1
                                                                                                                                                                                     15 caps                    N1

                                        14                                             Curve N1
                                                                                (relates to peak Vm1)                                                       16
                                        10                                                                                                                  12
                                                                                 Curve N2
                                        8                                  (relates to peak Vm2)                                                            10
                                        6         Peak Vm2 does not exist in this area at t>TO.
                                                                                                                                                            6                                            N2
                                                                                   1   ∆ Io                                                                 4
                                                 Limitation:   ESR . Co > m . Ts .            To
                                        2                                          2   ∆ IL                                                                 2
                                        0                                                                                                                   0
                                             0        1        2       3        4       5      6        7                                                        0   0.6   1.2 1.8    2.4 3.0   3.6   4.2 4.8        5.4   6.0

                                                                        LO (µ                                                                                                  Transient Duration, TO

Fig. 11. Optimization curves for the required                                                                      Fig. 12. Required number of bulk capacitors N1
number of bulk capacitors based on the equations                                                                   and N2 as a function of the transient duration TO.
for N1 and N2.

This decrease in the slew rate means that                 output filter selection procedure for the
adding four high-frequency ceramic capacitors                 interleaved regulator at high slew-rate load-
close to the microprocessor pins decreases the                current transients. Apparently the analysis and
number of bulk capacitors from 20 to 15. Fig. 12              optimization procedure for the one-channel
shows that further decoupling does not give                   synchronous buck converter can be extended to
significant effect, although four more ceramic                the multi-phase interleaved synchronous buck
capacitors might save roughly two more bulk                   converters.
capacitors. The capacitance and ESR of the                        The power-delivery system considered in the
ceramic capacitors must be big enough to support              analysis appears in Fig. 14. The analyzed model
the load until the bulk capacitors will take over             includes the n-channel interleaved synchronous
the transient.                                                buck converter. Each channel of the model is
                                                              similar to the one-phase solution analyzed earlier
 V. TRANSIENT RESPONSE OF A MULTI-PHASE                       (Fig.1).
      INTERLEAVED BUCK CONVERTER                                  The converter operates in a phase-shifted
                                                              manner at the steady state condition sharing the
     The interleaved synchronous buck converter               current equally between the channels. For the
becomes a popular solution to supply high-                    best response during the transients, all channels
current microprocessors because of the lower                  turn the high-side FETs simultaneously on at
input and output current ripple and the higher                load-current step-up or off at step-down, thus
operating frequency of the input and output                   allowing the fastest recovery and a minimum
capacitors compared to the one-channel solution               dynamic tolerance of the output voltage. The
        . Interleaving also enables spreading of the          control signals do not have delays and the duty
components and the dissipated power over the                  cycle covers all possible ranges from zero to one.
PCB area, but it requires equal current-sharing               When the output voltage returns to the nominal
between the channels. Different control                       level after the transient, the channels resume
approaches to achieve accurate current-sharing                phase-shifted operation with the same sequence
and fast transient response for interleaved                   they had before the transient. This ideal control
microprocessor power supplies are suggested in                algorithm for the best transient response is
the literature [44-46]. Meanwhile, outlining an               illustrated in Figs. 14 and 15.
optimal application area for the one-channel and
interleaved solutions requires an analysis and
                                          10-100A/µ               µs
                                                        100-1000A/µ           µs

                            System                                                     IO
                            Power         I3            I2             I1

                                   Bulk         Decoupling   Packaging          Die
                                Capacitance     Capacitance Capacitance      Capacitance

                                                  æ ∆I ö   æ ∆I ö      L
                                     Roughly:     ç ÷ ≅ç ÷            × N −1
                                                  è ∆t ø N è ∆t ø N −1 LN
Fig. 13. How high-frequency decoupling decreases current slew-rate.

LO                      LB             uP or DSP with
                                                                          + VA RB             VB   HF decoupling

                                     D                              ESR         Load current         Chf        Cdie
                  VIN                     1-D                       ESL                                 Die
                                            _                                                      Controller

                                     2 ... (n-1)
                                                                                                   control signals

                                           1-D                                                     Control signals
                                                                                                   during transient


Fig. 14. Analyzed model of the interleaved synchronous buck converter.

          D x TS             TS                                              The analysis is based on the same approach
                                            (1-D) x TS
                                                                         described earlier for the one-channel synchronous
Control: Ch. 1                                                           buck converter. Assume that (1-D) > n·D, where
                                                                         n - number of channels and D =VOUT/VIN - duty
Control: Ch. 2
                                                                         cycle of each channel. This condition is fulfilled
                                                                         for most microprocessor power supplies, for
      IL: Ch. 1
                                                                         example, the popular 12-V input and 1.6-V
                                  ∆IL                                    output synchronous buck converters with four
      IL: Ch. 2
                                                                         channels have D = 0.13, (1-D) = 0.87, n·D =
                                                 (1-Deq) x TSEQ          0.52, and thus 0.87 > 0.52. It can be shown that
          Deq x TSEQ
      IO,                                                                the summarized current through the inductors and
IL:Ch. 1+Ch. 2                TSEQ                                       output capacitor of an n-channel interleaved
                    ∆ ILEQ                                               converter has the same waveforms as an
             IC                                                          equivalent one-channel converter with the
                                                                         following parameters:
Fig. 15. Control signals, inductor currents                              DEQV = n·D, fS(eqv) = n·fS, TS(eqv) = TS/n, LO(eqv)
through the each channel and summarized                                  = LO/n, VIN(eqv) = VIN/n, ∆ILEQV = ∆IL·(1-
inductor, capacitor and load currents of                                 n·D)/(1-D)
interleaved buck converter at load-current step-
down transient.

where D is duty cycle, fS is switching                          The transient-response dependence on the
frequency, TS is switching period, LO is output                 position of the switching cycle in which the load-
inductor, VIN is input voltage and ∆IL is peak-to-              current transient occurs is the same as that of the
peak inductor-current ripple of each channel of                 one-channel converter. The difference is that the
the interleaved buck converter. DEQV, fS(eqv),                  summarized currents through all the channels and
TS(eqv), LO(eqv), VIN(eqv), and ∆ILEQV are duty cycle,          the output voltage ripple cancellation effects
switching frequency, switching period, output                   must be considered, which is accomplished in
inductor, input voltage and peak-peak inductor-                 this analysis by substituting an equivalent one-
current ripple of the one-channel equivalent buck               channel converter. Further analysis estimates the
converter respectively. The waveforms in Fig.15                 worst-case transient and compares the different
illustrate this statement.                                      type of output capacitors in the interleaved
    During the transients, because all channels                 converter.
turn to the same state simultaneously in                            The example below illustrates how the
accordance with the proposed control algorithm,                 number of channels and capacitor characteristics
the interleaved converter can be considered a                   defines the number of required bulk capacitors
one-channel, which now has the same input                       N1 and N2 [42]. The following requirements are
voltage VIN as the original interleaved one and                 typical for a modern high-end microprocessor:
the output inductor LO/n. The consideration of the
parameters of the equivalent one-channel                        •   VIN = 12 V,
converter helps to explain what kind of                         •   VOUT = 1.5V,
advantages to expect from the interleaved                       •   IO(max) = 50 A,
converter. Interleaving provides the same                       •   IO(min) = 0 A,
properties as the one-channel converter when                    •   ∆IO = 50 A,
operating at higher frequency, having higher duty               •   ∆VREQ = 100 mV,
cycle and lower input voltage and inductor value.               •   vIO = 50 A/µs,
But all this happens only if a good steady state                •   RB = 0.4 mΩ,
and dynamic current sharing are provided                        •   LB = 0.2 nH,
between the channels.                                           •   fS = 200 kHz.
    In an ideal control circuit, the transient                      Optimization curves are built for aluminum
response of the interleaved converter is defined                electrolytic, OS-CON, specialty polymer SP and
by the output-filter characteristics including the              ceramic capacitors. The required number of
slew-rate of the inductor current. It is interesting            capacitors N2 and inductor value LO at different
that for the step-down transient when all low-side              switching frequencies for this application are
switches are turned on, the inductor-current slew               shown in Table 1.
rate equals (VOUT·n)/LO and has the same slew-                      These numbers of capacitors relate to an ideal
rate as during the (1-DEQV)-part of the switching               controller. The practical implementation might
cycle. But for the step-up transient when all high-             require some additional capacitors. But this
side switches are turned on, the slew-rate of the               analysis sets goals to achieve and shows relations
inductor current is (n·(VIN-VOUT))/LO, which is                 between the number of capacitors and number of
much higher than in steady state operation during               channels, switching frequency, inductor value,
the DEQV-part of the switching cycle, where the                 and capacitor and layout parasitics.
slew-rate is only (VIN-VOUT·n)/LO.

                                                     OF CHANNELS

                                         FS     Parameters of each         Number of Capacitors for Different Numbers of
                                         per    capacitor                  Interleaved Channels / LO per Channel
Type           Vendor      Part Number   ch     CO     ESR         ESL     1 one         2 two        3 three       4 four
                                         lkHz   1µF    1mΩ         1nH     Channel       Channel      Channel       Channel
               Rubycon     6.3ZA1000     200    1000   24          4.8     18/0.8µH      16/1.6µH     16/2.4µH      15/3.2µH
OS-CON         Sanyo       4SP820M       200    820    8           4.8     8/0.25µH      7/0.5µH      6/0.75µH      6/1.0µH
               Panasonic   eefcd0d101r   300    100    20          3.2     28/0.1µH      18/0.2µH     15/0.3µH      13/0.4µH
Ceramic,                   grm235y5v
               Murata                    400    22     20          0.5     60/0.05µH     30/0.1µH     20/0.15µH     16/0.2µH
1210                       226z10

   The following is the summary of the
comparison.                                                         VI. CONTROL TECHNIQUES FOR POWERING
• The aluminum electrolytic and OS-CON                               LOW-VOLTAGE, HIGH SLEW-RATE LOAD
   capacitors require significant additional high
   frequency decoupling at slew-rate 50 A/µs                      A. Review of Control Techniques for High-Slew
   because N1 >> N2.                                              Rate Load-Transient Applications
                                                                      Different control approaches are considered
•      Interleaving for aluminum electrolytic and                 in the literature [19-27,30,33,37,40,41,45,46] for the high-
       OS-CON capacitors does not significantly                   slew-rate load-current transient applications.
       decrease the number of required output                     Some solutions require additional circuitry, such
       capacitors. Only the decrease in the input                 as a power amplifier or switch, to regulate the
       filter ripple needs to be considered as the                output voltage if it exceeds some set points at the
       result of interleaving.                                    transients [23,25]. This approach complicates the
•      The 2-channel interleaving is optimal for the              design. Moreover, to avoid the interaction
       specialty-polymer capacitors. They require                 between the main and the additional control
       much lower high-frequency decoupling at                    circuitry at normal operation, the output voltage
       50 A/µs load-current slew-rate in comparison               set points of the additional circuitry have to be at
       with the aluminum electrolytic capacitors.                 least 2-3 percent different from the reference
                                                                  voltage of the main circuitry. This requirement
•      The most significant effect of interleaving                causes some delay until the additional circuitry
       relates to the solution with the ceramic                   starts to control the output voltage when the
       capacitors. The required number of them                    transient occurs. The cost of this delay is the
       drops almost inversely proportional to the                 additional bulk capacitors in the output filter. The
       number of interleaved channels. Ceramic                    other potential problem results from the
       capacitors do not require additional                       occurrence       of     the       transients       in    the
       decoupling at 50 A/µs load-current slew-rate.              microprocessors with high repetitive frequency
                                                                  (up to 100 kHz most of their operation). That
                                                                  means that this additional circuitry could operate
                                                                  frequently and dissipate a significant amount of
                                                                  power, because it usually operates in the linear
                                                                  mode. Although these ideas are interesting and

may find their optimal application niche, this                •   The compensator senses the change of the
topic addresses only the control approaches that                  output voltage when the load-current transient
do not require additional switches or power                       occurs. The tight tolerance requirements for
amplifiers.                                                       the output voltage makes it even more
    Let us review some basics of the control                      difficult to separate the output voltage change
theory before considering and reviewing different                 caused by a load-current transient from noise.
control approaches. In control theory, any
switching regulator with feedback loop includes a             •   The direct sensing of the load current helps to
plant and a compensator. The plant in a switching                 avoid the additional delays. The load current
regulator is its power stage, with at least two                   must be sensed between the output capacitor
different states controlled by the switching                      and the microprocessor because the output
function. The regulated output signal of this                     inductor current or current through the power
control system is the output voltage of the                       switch is not the actual load current.
regulator. The compensator senses the output                  •   The ESR and ESL of the output capacitors
voltage and somehow changes the switching                         and the supply path parasitics help to sense
function to keep the output voltage within the                    the load current and its changes because of
required window. The power stage of the voltage                   the additional voltage drop. But the design
regulator must include a low-pass filter, which is                goal is to avoid this voltage drop by selecting
a second order L-C filter in the applications                     lower ESR and ESL capacitors and
considered.      There    are    many     different               improving the layout. These are also
disturbances, such as input voltage, temperature,                 contradictory requirements.
aging, and load current that change the output
voltage. But for the considered power-delivery                •   The compensator cannot improve the
system with a high-slew-rate transient type of                    dynamic characteristics of the power stage.
load, the main disturbing factors is the load                     Actually the compensator only degrades the
current and the slew-rate of the current transient.               transient response because of the delays and
As discussed in the "Introduction" section above,                 restrictions put by the compensator on the
the load-current transient step and its slew-rate                 power stage.
for the considered type of load is much higher                •   Some think that the compensator has no
than in most generic applications. At the same                    influence and that only the output capacitor
time the required tolerance of the output voltage                 characteristics     define     the     response
of the regulator is very tight. On the basis of this              characteristics because the slew-rate of the
discussion the following main problems can be                     transient is too high. That might be true if the
specified:                                                        output capacitance is so over-killed that any
• The low-pass filter nature of the power stage                   controller is fast enough to handle the
    means that the fast response to the load-                     transient. In reality, an output filter with
    current transients is limited and defined first               minimum size (or cost) can be designed only
    of all by the power stage itself.                             by using a fast and optimal compensator
                                                                  (control approach). In other words, the sooner
•   The power stage with a fast response to the
                                                                  the controller starts to react on the transient,
    load-current transient might require high-slew
                                                                  the better is the response, because the first
    rate current changes through its own
                                                                  spike can be suppressed by high- or mid-
    components. At the same time the power
                                                                  frequency decoupling.
    stage must prevent the transient propagation
    outside the regulator without a significant
    increase of the input filter. These are
    contradictory requirements.

This discussion suggests the desirability of
                                                                       +                                                    +
first finding the optimal power stage, including                     VIN

the low-pass filters. (That attempt was made in                        _                                                    _ RLOAD
the previous section.) After that, on the basis of
                                                                                                                       Error Amp
the transient analysis, the control approach must                                                                          Z1
                                                                                                           COMP            _ Z2
be selected. The control approach must not                                     _Driver    Latch               +
                                                                               Q                                            +
significantly degrade the dynamic characteristics                              Q           Q
                                                                                             R                _
                                                                                             S                                     VREF
of the power stage and must provide some
additional properties that increase the reliability                                                                         OSC
and the robustness of the power-delivery system,
for example, current limit, constant or variable
switching      frequency,     and     pulse-duration           Fig. 16. Voltage mode control.
     There is a large variety of control techniques
in power electronics. The control techniques                           +                                                    +
                                                                                         D 1-D                               VOUT
reviewed and considered in the topic are selected                   VIN
                                                                           _                                                _RLOAD
on the basis of their current popularity in this
application area or of their promising                                                                               Error Amp
characteristics. These control approaches are                                  _ Driver       Latch        Comp            _ R
voltage mode (Fig. 16), peak current mode (Fig.                                Q                  R
                                                                                                               _+          +
17), average current mode (Fig. 18), V2-mode                                   Q              Q
                                                                                                  S                                      VREF
                                                                                                                    Sloop Comp
(Fig. 20), and hysteretic (ripple) mode (Fig. 19).                                                                   for D>0.5
They can be divided in the following two groups:                                                                                  OSC

• Control techniques that do not sense the load
     current or its changes directly, and so do not            Fig. 17. Peak current mode control.
     feed-forward or feed back this signal. This is
     group-control with a “slow” feedback loop,
     because it senses the disturbance caused by                      +
                                                                    VIN                   1-D
     the load current indirectly and uses this signal                  _                                             _RLOAD
     in the main feedback loop. This group                                                              R2 Current Amp
                                                                                                                       Error Amp
     improves the dynamic response by increasing                                                                          Z1
                                                                                                      Comp         _
     the unity-gain-frequency bandwidth of the                             _ Driver Latch                  _ Z2            _ R
                                                                           Q        _S                             +
     voltage mode [20] or of different types of                            Q        QR                     +               +
     current-mode [21,36-39] control. Some authors
     have suggested an optimization of the small-
     signal characteristics for a minimum output
     impedance [22].                                           Fig. 18. Average current mode control.
•   The control techniques that does sense the
    output current transient directly or through               B. Control with Slow Feedback Loop
    the related change of the output voltage and                   This     approach   requires     complicated
    uses this signal in a fast feedback loop or                feedback-loop-compensation circuitry for a stable
    feed-forwards it to improve the transient                  converter operation at all conditions, including
    response. The main examples of this group                  the wide range of the output capacitance that is
    are hysteretic-mode [26, 29-33] and V2 -mode               usually specified by the requirements [11]. The
            control techniques. This is group control          unity-gain bandwidth must be at least a few times
    with a “fast” feedback loop.                               lower than the switching frequency, so the
                                                               dynamic characteristics of the regulator are
                                                               relatively slow. Using a small-signal model to

interpret and improve the large-signal transient
response characteristics, as described in [20,22], is                  +
                                                                      VIN                   1-D
questionable. Although these controllers are very                      _                                                  _RLOAD
popular in most generic applications, their
transient characteristics are worse than the                                                    _ Driver Hyst. Comp
controllers having direct load-current sensing and                                              Q
                                                                                                Q                +
a fast load-current feedback loop.
C. Control with Fast Feedback Loop
    The most popular implementation of the                     Fig. 19. Hysteretic control.
second approach is the hysteretic control and V2
mode control. The hysteretic control (or two-
state, bang-bang, ripple, free-running regulator,                      +
                                                                                       D 1-D
etc.) is the simplest control approach, which has                      _                                              _ RLOAD
been used for a long time [28]. This is probably the
earliest controller or regulator. The hysteretic
controller became the solution in the new                                   _ Driver    Latch                  Error Amp
applications that require the fast load-current                             Q            Q
                                                                                         _R               _
                                                                            Q                                       +
transient response power supplies. This is a very                                        QS
simple solution that does not require the
compensation circuitry and has the excellent
                                                                                Auxilliary Logic, that sets:
dynamic characteristics. The examples of                                          TOFF MAX, TOFF ext,
integrated circuit implementation of this                                         TOFF norm, TON MAX
approach for modern applications are available                 Fig. 20. One of the implementations of V2 mode
now from a few companies, including Texas                      control.
Instruments (Fig. 19).
    Unlike other control approaches, this                          The V2 mode control also has very fast
controller does not have a slow feedback loop: it              transient-response characteristics, but it has
reacts on the load-current transient in the                    drawbacks in its original implementation
switching cycle where the transient occurs. Its                (Fig. 20). This controller uses the output-voltage
transient response time depends only on delays in              ripple as the ramp signal of the modulator. It is
the hysteretic comparator and the drive circuitry.             assumed that the voltage ripple of the output
The high-frequency noise filter in the input of the            capacitor depends mostly on the ESR, while the
comparator also adds some additional delay.                    part of ripple caused by the ESL and CO is
Those delays depend mostly on the level of the                 negligible. In this case the ripple is proportional
selected technology, and so the hysteretic control             to the inductor current ripple. At the transients,
is theoretically the fastest solution. The other               this voltage carries the information about the
advantage of the hysteretic controller is that its             load-current change directly to the comparator,
duty cycle covers the entire range from zero to                bypassing the slow main feedback loop. The V2
one. It does not have the restrictions on the                  mode control actually can be defined as the sort
conduction interval of the power switches that                 of hysteretic control having the additional error
most of the other control approaches have. This                amplifier. This amplifier enables an increase of
capability is very important to decrease the                   the hysteresis window of the comparator without
recovery time of the output voltage after a load-              degrading the accuracy of the output voltage.
current transient.

On the other hand, the use of output ripple-                  TABLE 2. COMPARISON OF CONTROL
voltage as the ramp signal causes the stability to            APPROACHES FOR HIGH SLEW-RATE TRANSIENT
depend greatly on the output capacitor parasitics.                                               APPLICATIONS
This dependence is especially problematic when                                   Hysteretic                              Voltage mode      Current mode
using many high-frequency ceramic or film                                     (ripple) control
                                                                                                  V2 mode control
                                                                                                                             control          control

capacitors in parallel as the output filter. In this                                       150ns - 200ns                5 - 10 µs in best case, because
case, the equivalent output capacitor is almost               Reaction time
                                                                                    (limited only by technology)             of slow feedback loop

ideal, because its parasitics are negligible and an           ON/OFF time                     Constant OFF time in       Limited by switching cycle of
                                                                                            practical implementation,    internal oscillator in constant
output ripple has a parabolic waveform instead of               duration
                                                                               limitation    that degrade transient           switching frequency

a linear ramp. This ripple is also with an angle of
                                                                                                    response                    implementation

                                                               Duty cycle
π/2 out of phase with the output inductor current.                                          No limitation               Yes/No Depends
                                                               limitation                                                                       Yes
                                                                                                                            on part

Providing a stable operation in this situation                    Apparently        any       control      allows
might be a problem. Another possible problem is               implementation of the passive or active droop-
getting a reliable and predictable startup                    compensation techniques discussed in the
characteristic of the V2 mode controller. The                 transient analysis section [Fig. 9]. But the cost of
controller may require additional circuitry for the           the implementation might be different. For
startup control only, as in the original                      example the compensation circuitry of controllers
implementation [Fig. 20]. To avoid these                      with slow feedback loop could be designed with
problems, the modifications of V2 mode control                low load regulation for this purpose. The
superimpose the inductor-current ramp signal                  question is still how fast the active droop
onto the input of a comparator [46].                          compensation is. Usually the droop compensation
    There is another problem related to                       needs to complete its own transient within 1-2 µs
controllers that sense information about load                 to be ready for the next transient. The hysteretic
current by using output capacitor’s ESR: their                controller requires an additional current sense
frequency dependence from the output filter                   amplifier for the active droop implementation.
parasitics and its variation related to tolerances
and variations of the capacitor characteristics.              VII. EFFECT OF CONTROLLER LIMITATIONS ON
This problem could be solved by using the                              THE TRANSIENT RESPONSE
frequency synchronization or again, by
superimposing an additional ramp signal to avoid                   The previous analysis assumed that the
dependence from the output filter [29,40,41]. It is           controller is ideal. The ideal controller does not
useful to mention that the compensation of                    have delays. It is able to maintain the on or off
controllers with the slow feedback loops also                 state of the respective power switches during a
depends on the output capacitor parasitic                     transient as long as necessary. As shown above,
variations. Table 2 shows pros and cons of the                the controller parameters influence the second
discussed controllers for high-slew-rate load-                peak of the transient but have no effect on the
current transient applications.                               first one (Fig. 5b). The real controllers do have
                                                              delays and might have some restrictions on the
                                                              duty cycle. For example, the constant on-time
                                                              controller must complete its on-cycle for the
                                                              high-side FET all the time, even when a transient

The same issue relates to the constant off-
time control, but here the off-time cycle must be                                                      ∆Vc = ∆ Qc/Co
completed. The controllers with a trigger latch                  VCO
working at a constant frequency usually have
restrictions on the maximum duty-cycle. Control
approaches that do not have a fast feedback loop
(which senses the load-current transient directly)                IO
                                                                                               ∆ Qc
require at least a few switching cycles until they                IL
start to change the duty cycle and to respond to a
transient. This delay is caused by their low unity-
gain bandwidth of the compensation circuitry
compared to the switching frequency. Fig. 21
shows how the actual controller degrades the                    VG
transient waveforms at a load-current step-down.
The moment when the transient occurs is selected             (a) “Ideal" controller. The moment of the transient is
to get the worst-case conditions for each kind of            selected to get the worst-case condition.
    The ideal controller in Fig. 21a does not have
delays and duty-cycle restrictions. The capacitive
portion of the output voltage vCO gets the                                                  ∆ Qextra
                                                                                               ∆ QC
additional rise ∆VC because the charge ∆QC is
delivered to the output capacitor CO (Fig. 1) by
the inductor LO. The delay tDEL that any real
controller has adds the extra charge ∆QEXTRA
(Fig. 21b). This additional charge increases the
second peak of the output-voltage transient                      VG
waveform (Fig. 4) by -∆QEXTRA /CO. The
constant on-time controller (Fig. 21c) gets the
                                                             (b) Controller with delays. The moment of the transient is
extra-charge related to the delay and to the fixed           shifted to the left to get the worst-case condition.
on-time. The controller with a slow feedback
loop (Fig. 21d) has even more extra charge and                                        tON
related voltage-soar than the previous controllers.
This extra charge can have significant effect on                                            Qextra
the transient waveforms if the output filter has                                              QC
relatively low output-capacitance like, for                            IL
instance, the ceramic capacitors in a high-
frequency regulator. Actually the extra charge
becomes more important in a low-size high-
frequency solution. There the controller
parameters could significantly affect the regulator
cost and size.
    The control considerations in this section               (c) Constant on-time controller with delays. The moment of
suggest that transient response makes the                    the transient is shifted to the left because of delays and
hysteretic controller is one of the best solutions           fixed on-time.
for powering high slew-rate transient loads. The
next section is dedicated to further analysis of
this controller and some of its modifications.

tDEL                                                  Fast transient-response regulators are needed
                                                                    for powering devices such as modern, high slew-
                              ∆ Qextra                              rate transient microprocessors, DSPs, and
                                                                    memory ICs. That need, and achievements in
       IL                                                           analog IC design technology, have made the
                                                                    hysteretic control interesting again. Modern
                             ∆ QC                                   hysteretic comparators have delays of only tens
                                                                    of nanoseconds, with a hysteresis window around
       VG                                                           10-20 mV. This voltage is far from that of the old
                                                                    Schmidt triggers with hundreds of millivolts of
(d) Controller with delays and slow feedback loop.The               hysteresis in a frequency range of a few kHz. The
additional on-cycle appears because of slow feedback loop.          theoretical analysis of hysteretic regulators also
Fig. 21. Impact of actual controller limitations on                 must now take into account component and
transient.                                                          layout parasitics that cannot be avoided at or
                                                                    above a hundred kHz switching frequency and
                                                                    with extremely high slew-rate transients.
     CONTROLLER AND ITS MODIFICATIONS                               B. Switching Frequency of Hysteretic Regulator
                                                                        Switching-frequency predictability of a
A. Requirements for a Hysteretic Comparator                         hysteretic controller is important for a power-
    There are many examples of new                                  supply design. A simple and accurate method of
requirements and technological achievements                         determining the switching frequency is described
bringing new life to some old ideas and forgotten                   below [26]. Assume that the input and output
solutions. The hysteretic regulator is one of these                 voltage ripple magnitudes are relatively
examples. This regulator, based on a Schmidt                        negligible in comparison with the dc component.
trigger, was very popular in the 60s and early                      Also suppose that the time constant
70s. Unfortunately the simple scheme does not                       L/(RDS(on) + RL) that includes the output inductor,
always mean easy analysis and design. Later,                        L, the on-state resistance of the FET, RDS(on), and
more sophisticated control approaches took over                     the inductor resistance, RL, is high in comparison
most applications by allowing fixed-frequency                       to the switching period. Assume the body-diode
operation and more predictable design                               conduction time and switching-transition time are
procedures. Nevertheless, the hysteretic control,                   much shorter than the switching period. These
which by control-theory definition belongs to the                   assumptions are reasonable for high-efficiency
slide-mode control or a relay system, always has                    regulators with low output- and input-voltage
natural advantages such as fast transient-                          ripple. In such a case the output inductor current
response. The engine of a modern hysteretic                         can be modeled as the sum of the dc component
regulator is its comparator, which has a hysteresis                 (equal to the output current IO) and the ac linear
window. The comparator is intended to keep the                      ramp component, which flows through the output
output voltage and ripple within the hysteresis                     capacitor (Fig.22). The peak-to-peak value of the
window. In reality the ripple is always higher                      inductor current ∆I is equal to the following
than the hysteresis window because of the delays.                   equation:
Another problem is that the switching frequency
depends significantly on the power stage
characteristics and the operation conditions.

∆I =
                  (                 )
       VIN – IO ⋅ R DS(on ) + R L – VOUT
                                               ⋅ D ⋅ TS                                                           (12)

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