Carlos. J Solis 1
Gabriel A. Rincón-Mora 1
-
(Georgia Institute of Technology, Analog Devices Inc)
Copyright © The Institute of Electronics and Information Engineers(IEIE)
Key words
Design, analysis, stability, high bandwidth, dc–dc power supply, hysteretic current-mode control
I. POWERING WIRELESS MICROSYSTEMS
Emerging wireless microsystems sense, process, and report information that can save
money, energy, and lives [1]. With so much functionality and so little space, however, their onboard batteries
deplete easily. So to conserve energy, they idle when possible and wake only on demand,
and often quickly and to high-power states. Power supplies must therefore be compact,
efficient, and fast.
Switched-inductor dc–dc power supplies are popular in many applications because they
can deliver over 90% of the power they draw [2-4]. When pulse-width-modulated (PWM), they normally require multiple switching cycles
to respond and recover after sudden load dumps [5]. Hysteretic converters are faster because they can slew the inductor's current uninterruptedly
until the current can satisfy the load and replenish the output capacitor [6]. Slewing the inductor this way is the fastest possible way that a switched inductor
can respond to load variations.
Although already analyzed in different ways and from different perspectives in literature
[7-9], stability requirements for these hysteretic power supplies remain largely algebraic,
abstract, and theoretical (without experimental validation). This brief explains and
demonstrates with experiments how those requirements relate to the inductor and the
load dumps it supports. With this insight, a designer can more readily stabilize and
identify under which operating conditions a hysteretic power supply will remain stable.
To explain this, Sections 2–5 describe circuit operation, stability requirements,
validating measurements, and relevant conclusions.
II. HYSTERETIC CURRENT-MODE CONVERTER
1. Operation
The hysteretic current-mode buck dc–dc converter in Fig. 1 closes a feedback loop that, in keeping output $v_O$ near reference $v_R$, supplies
the current that the load requires. Transconductor $G_{OSC}$ is an oscillator that
ripples inductor $L_O$'s current $i_L$ across a window that comparator $CP_{OSC}$'s
hysteretic thresholds set and about a level that amplifier $A_E$ dictates with $v_{ERR}$.
So together, $A_E$ compares $v_O$ and $v_R$ to generate an error $v_{ERR}$ that adjusts
the level about which $G_{OSC}$ oscillates $i_L$ to match and supply the current $i_{LD}$
that the load demands.
Fig. 1. Hysteretic current-mode buck.
Fig. 2. Measured inductor current in continuous conduction mode (CCM).
The system delivers power by energizing and draining $L_O$ from the input $v_{IN}$
into $v_O$ in alternating phases of the oscillating period tOSC. When transistor $M_{IN}$
energizes $L_O$ with energizing voltage $v_E$ or $v_{IN}$ – $v_O$, $i_L$ in Fig. 2 rises across energizing period $t_E$\. $i_L$ similarly falls when switch $M_G$ drains
$L_O$ with drain voltage $v_D$ or –$v_O$ across drain period $t_D$. $i_L$ ripples
this way about the load level the feedback loop sets with $v_{ERR}$
When $L_O$'s and $C_{IL}$'s corner frequencies with $R_{IL}$ and RLESR are well below
the oscillating frequency f$_{OSC}$, $sL_O$ and $R_{IL}$ overwhelm RLESR and 1/$sC_{IL}$
near f$_{OSC}$. So $L_O$'s voltage $v_L$ is $i_L$$sL_O$, $v_C$ is $v_L$/$sR_{IL}$$C_{IL}$
or $i_L$$L_O$/$R_{IL}$$C_{IL}$, and $v_I$ is $i_L$(10$L_O$/$R_{IL}$$C_{IL}$) or $i_L$$A_R$
where current sense gain $A_R$ equals:
$v_I$ therefore rises and falls with $i_L$, and when $v_I$ reaches comparator $CP_{OSC}$'s
upper threshold, $CP_{OSC}$ trips low to open $M_{IN}$ and close $M_G$, and that way,
end $t_E$. The opposite happens when $v_I$ falls to $CP_{OSC}$'s lower threshold
to end $t_D$. $i_L$ oscillates this way across the window that $CP_{OSC}$'s hysteresis
$V_{HYS}$ sets with V$_{HYS}$/$A_R$ and about the level that $v_{ERR}$ sets with $v_{ERR}$/$A_R$.
Together, $R_{IL}$, $C_{IL}$, $CP_{OSC}$, $M_{IN}$, $M_G$, and $L_O$ realize an oscillator
$G_{OSC}$ that oscillates $i_L$ about $i_{LD}$.
Fig. 3. Measured inductor current in discontinuous conduction mode (DCM).
Fig. 4. Voltage-mode hysteretic control for buck converters.
When load current $i_{LD}$ falls below half of $i_L$'s ripple Δ$i_L$, $i_L$ can reverse,
and that way, burn unnecessary power. Comparator $CP_{ZCS}$ keeps this from happening
by sensing and opening $M_G$ when $i_L$ reaches zero. In this way, like Fig. 3 shows, $i_L$ falls to zero and remains at zero in discontinuous-conduction mode (DCM)
until the next cycle. But since Δ$i_L$ is less than prescribed by V$_{HYS}$/$A_R$,
$v_O$ must rise above $v_R$ to trip $CP_{OSC}$. In other words, $v_O$ rises slightly
when $i_{LD}$ falls in DCM.
2. Variants
The voltage-mode counterpart of the hysteretic current-mode buck is the most compact
hysteretic buck because $v_O$ and $v_R$ feed directly into $CP_{OSC}$ like Fig. 4 shows [10], without using $A_E$ or a current sensor. Unfortunately, the circuit requires a resistive
$C_O$, the equivalent series resistance (ESR) of which produces higher ripple in $v_O$.
Adding a current sensor without gain and feeding its output $v_I$ into $CP_{OSC}$
remove this requirement [11,12]. This way, $CP_{OSC}$ keeps $v_O$ near $v_R$ because $v_I$ is $v_O$ + $v_C$. $v_C$'s
$i_L$$L_O$/$R_{IL}$$C_{IL}$, however, appears as an offset in $v_O$. The purpose of
$A_E$ in Fig. 1 is to eliminate this offset. The state of the art adds peripheral blocks to Fig. 1 to keep the oscillating frequency constant [11-13]. These additions, however, do not alter the stability limits and effects of the hysteretic
core.
Fig. 5. System-level block diagram modeling current loop as transconductance G$_{\mathrm{OSC}}$.
Hysteretic current-mode boost and buck–boost configurations are also possible. What
changes in these configurations are the voltages $v_E$ and $v_D$ that energize and
drain $L_O$. $v_E$, for example, is $v_{IN}$ instead of $v_{IN}$ – $v_O$, and $v_D$
in the boost is $v_{IN}$ – $v_O$ instead of just –$v_O$ [14]. So as long as the analysis is with respect to $v_E$ and $v_D$, the mechanics explained
here apply to all configurations. The output diode or switch in boost and buck–boost
topologies, however, produce the effect of an out-of-phase, right-hand-plane (RHP)
zero $z_{RHP}$ that does not appear in buck converters [14]. But as long as the unity-gain frequency $f_{0dB}$ is below $z_{RHP}$, which is a
necessary requirement for these converters, all stability conditions are the same.
III. STABILITY
For the feedback loop that controls $v_{\mathrm{O}}$ to stabilize, the loop gain $A_{\mathrm{LG}}$
must reach unity (at $f_{\mathrm{0dB}}$) with less than 180$^{\circ}$ of phase shift.
Across the loop (in Fig. 5), $G_{\mathrm{OSC}}$'s bandwidth establishes one pole $p_{\mathrm{G}}$ and output
capacitor $C_{\mathrm{O}}$ shunts $v_{\mathrm{O}}$ to produce another pole $p_{\mathrm{O}}$.
$C_{\mathrm{O}}$'s series resistance $R_{\mathrm{CESR}}$ limits $C_{\mathrm{O}}$'s
shunting current to eliminate the effects of $p_{\mathrm{O}}$ past zero $z_{\mathrm{CESR}}$.
But $R_{\mathrm{CESR}}$ for low-ripple applications is so low that $z_{\mathrm{CESR}}$
is negligibly high [15]. And $C_{\mathrm{O}}$ is so high that $p_{\mathrm{O}}$ is low and dominant. A phase
margin of 45$^{\circ}$ balances speed and stability by limiting the number of damped
oscillating rings to three or less [16]. So to maintain 45$^{\circ}$ of phase margin PM, $f_{\mathrm{0dB}}$ should be at
or below $p_{\mathrm{G}}$:
$p_G$ is another way to quantify $G_{OSC}$'s delay time $t_D$ [17]. In the case of hysteretic current-mode converters, $t_D$ is the time $L_O$ requires
to slew $i_L$ across a load dump Δ$i_{LD}$. Because $i_L$ climbs faster with higher
$L_O$ voltage $v_L$ ($v_{SW}$ – $v_O$ in Fig. 1), $t_D$ is longer and $p_G$ is lower when energizing and drain $v_L$ voltages $v_E$
and $v_D$ are lower and Δ$i_{LD}$ is higher. And since $i_L$ requires about four time
constants 4τG to reach 98% of its target, $t_D$ is roughly 4τG or τG is $t_D$/4 and
$p_G$ is no less than the lowest $v_L$ and highest Δ$i_{LD}$ dictate:
At worst case conditions, this yields:
In practice, the application defines $v_{IN}$, $v_O$, and Δ$i_{LD}$ and volume, power-rating,
and conversion-efficiency constraints limit $L_O$. So for the fastest 45° response,
$C_O$ should be just high enough to define a p$_O$ that keeps $f_{0dB}$ near the minimum
$p_G$ or $p_G$($M_{IN}$) that these parameters set. To quantify this, first consider
that $A_{LG}$ is the gain across $A_E$ and $G_{OSC}$ into the output impedance that
$R_O$ and $C_O$ in parallel establish:
where $G_{OSC0}$ is $G_{OSC}$'s low-frequency translation 1/$A_R$:
Since $p_G$'s effect on $A_{LG}$ near $f_{0dB}$ is small and $A_{LG}$ at $f_{0dB}$
is one, $f_{0dB}$ is roughly equivalent to $A_{LG}$'s gain–bandwidth product:
Fig. 6. Loop-gain response.
where $A_{LG0}$ is $A_{LG}$'s low-frequency translation $A_E$$G_{OSC0}$$R_O$. The
Bode plot in Fig. 6 illustrates pole locations and the small-signal dynamics needed for stable conditions.
When worst-case extremes for $v_{IN}$, $v_O$, Δ$i_{LD}$, and $L_O$ in the buck of
Fig. 1 are 1.1 V, 1 V, 150 mA, and 3.3 μH, $v_E$'s $v_{IN}$ – $v_O$ or 100 mV is lower than
$v_D$'s $v_O$ or 1 V. So $p_G$ can be 130 kHz, and for 45° of phase margin, $f_{0dB}$
should not exceed $p_G$'s 130 kHz. $C_O$ should therefore be no less than 15 μF when
$A_E$ is 12 V/V, $R_{IL}$ is 33 kΩ, and $C_{IL}$ is 1 nF, which means $A_R$ is 1 Ω.
Relative to the state of the art, the analysis in [7] requires a model of the system in the form of complex state-space matrices. Although
[8] and [9] decompose the system into transfer functions, their expressions are still complex.
So drawing intuition to understand how each component in the system affects stability
is challenging.
The analysis presented here is much easier to understand and implement. Expressions
are insightful and correspond directly to components in the system. The analysis does
not require state-space matrices. And thanks to its segmentation, the analysis is
scalable with respect to additional gain stages with poles and zeros.
IV. PROTOTYPE
1. Hardware
The 0.6-$mm^2$ 0.18-μm CMOS die in Fig. 7 integrates the error amplifier $A_E$, oscillating comparator $CP_{OSC}$, dead-time
logic, power transistors $M_{IN}$ and $M_G$ and their drivers, and zero-current sensing
comparator $CP_{ZCS}$. The current sensor, inductor $L_O$, and output capacitor $C_O$
are off chip on the two-layer board alongside test circuits used for experiments.
$L_O$ and $C_O$ measure 10.7 × 10 × 5.4 and 1.6 × 0.81 × 0.91 mm3 and incorporate
15 and 4 mΩ, respectively.
Fig. 7. Prototyped 0.18-μm CMOS die and two-layer board.
Fig. 8. Measured output voltage regulation across output load and input voltage.
Fig. 9. Measured response to 40-, 80-, and 180-mA load dumps.
Fig. 8 shows the average output voltage $v_O$ across load level $i_{LD}$ at two input voltage
$v_{IN}$ operating points. $v_O$ decreases with increasing $i_{LD}$ due to its finite
output resistance at a rate, i.e. load regulation, of 77 mV/A. Across the same loading,
the worst-case output voltage difference $\Delta v_O$ is 1.4 mV which results in a
line regulation of 4.7 mV/V.
2. Response
The measured output $v_O$ in Fig. 9 ripples 5 to 10 mV and responds within 5.6 μs to rising 40-, 80-, and 180-mA load
dumps. Response time tR is basically how long $L_O$ requires to slew $i_L$ across
these load steps. The system responds faster (within 3.4 μs) to similar falling load
dumps because $L_O$'s drain voltage $v_D$ is higher at $v_O$'s 1.0 V than $L_O$'s
energizing counterpart $v_E$, which is $v_{IN}$ – $v_O$ or 1.5 – 1.0 V, which is 0.5
V.
Fig. 10. Measured load-dump response when the input is 1.4 and 1.8 V.
Phase margin essentially quantifies the propensity of a feedback loop to oscillate
when perturbed. In this case, $v_O$ in Fig. 5 is more prone to ringing when responding to rising than to falling load dumps. This
is because $L_O$ requires more time to respond to rising loads. In other words, the
bandwidth pole of the oscillator is lower, and as a result, closer to the unity-gain
frequency $f_{0dB}$ of the loop. Similarly, the ringing worsens as the load step increases
from 40 to 180 mA because $L_O$ requires more time to slew across wider load steps.
Ringing also worsens as input voltage falls in Fig. 10 from 1.8 to 1.4 V for the same reason, because with a lower voltage across $L_O$,
$i_L$ slews more slowly.
Using Eqs. (2-8), phase margins for 40-, 80-, and 180-mA load dumps are 75°, 62° and 40°, respectively.
These calculated margins correspond well with Fig. 9. The 40-mA load dump response, for example, is closer to the over-damped response
that 90° produces. The under-damped response to the 180-mA load dump rings 2–3 times
before settling. This corresponds to the response that 45° produces.
3. Efficiency
$A_E$, $CP_{OSC}$, $CP_{ZCS}$, $M_{IN}$, $M_G$, the drivers, and dead-time logic consume
quiescent, ohmic, and switching power P$_Q$, PR, and $P_{SW}$. As load current $i_{LD}$
climbs, output power $P_O$ increases linearly, whereas PR rises quadratically. Up
to at least 200 mA, however, $P_O$ outpaces PR, so power-conversion efficiency ηC
in Fig. 11 generally increases with $i_{LD}$. Although PR is very low below 5 mA, $P_O$ is also
low, so P$_Q$ and $P_{SW}$ become large fractions of $P_O$. This is why ηC drops below
88% abruptly when $i_L$ is less than 5 mA.
Fig. 11. Measured power-conversion efficiency.
V. CONCLUSIONS
Hysteretic current-mode converters are compact, fast, and efficient, but although
widely stable, not immune to instabilities. The bandwidth delay of the oscillating
current loop unfortunately introduces a secondary pole that falls with lower inductor
voltages and higher load-dump currents. This is why the 200-mA, 1-V, 0.18-μm CMOS
buck prototyped here responds more slowly to rising than to falling load dumps and
rings more with lower input voltages and higher load steps. Understanding these stability
limits is critical when designing hysteretic current-mode power supplies, which given
their size, speed, and efficiency benefits, are increasingly prevalent in consumer
electronics.
ACKNOWLEDGMENTS
The authors thank J.D. Morris, B. Legates, T. Bonte, and Analog Devices Inc. for
their sponsorship and support.
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Author
Carlos Solis received the B.S. degree from the University of Puerto Rico at Mayaguez
in 2010, and the M.S. and Ph.D. degree from the Georgia Institute of Technology in
2012 and 2018, all in electrical engineering.
His research interests include switching power supplies, and analog IC design.
He is currently in Analog Devices as an Analog Design Engineer.
Gabriel A. Rincón-Mora has been a Professor at Georgia Tech since 2001, Visiting Professor
at National Cheng Kung University since 2011, and Design Team Leader at Texas Instruments
in 1994–2003.
He is a Fellow of the National Academy of Inventors, Fellow of the Institute of Electrical
and Electronics Engineers, and Fellow of the Institution of Engineering and Technology.
Recognitions include induction into Georgia Tech's Council of Outstanding Young Engineering
Alumni, National Hispanic in Technology Award, Charles E. Perry Visionary Award, Three-Year
Patent Award, Orgullo Hispano Award, Hispanic Heritage Award, and a State of California
Commendation Certificate.
His scholarly output includes 9 books, 2 handbooks, 4 book chapters, 42 licensed patents,
over 180 articles, over 26 commercial power-chip products, and over 150 lectures/keynotes/speeches.