e
Wei-Chung Wu and Richard M. Bass
Jerry R. Yeargan
School of Electrical and Computer Engineering
Georgia Institute of Technology
Atlanta, Georgia 30332-0250, USA
Department of Electrical Engineering
University of Arkansas
Fayetteville, AR 72701
Abstract -This paper presents techniques which can
eliminate or reduce the effects of the positive or RHP zero
which appears in the small-signal duty-cycle to output
transfer function of certain dc-dc converters. Several
techniques are presented along with the theory, simulation
11. TECHNIQUES FOR ELIMINATION OF THE
RHP ZERO
This paper investigates three more techniques. Two are based
on the state-space averaging model of Cuk and Middlebrook [2]
and the third is based on the Injected-Absorbed-Current (IAC)
model described in the book by Kislovski, Red1 and Sokal [ 3 ] .
results and experimental results supporting each technique.
Consider the example of the boost converter shown in Fig. 1.
Neglecting capacitor ESR, the state-space averaging model yields
I. INTRODUCTION
There are a number of ways to eliminate the positive or right-
the following small-signal model
E
half plane (RHP) zero in the loop gain equation for the boost
r.
f
1 \2L1
converter. Some methods involve complex control schemes to
eliminate the RHP zero, but the various techniques presented in
this paper involve changing parameters in a conventional pulsewidth modulator ( P W ) controlled boost converter to eliminate
the RHP zero. A paper published in APEC 1990 by Sable, Cho
and Ridley [1] presented different ways to eliminate the RHP
E
zero. The three methods suggested in that paper were
1) use leading-edge modulation
2) select component values to satisfy R ,
L
. C > R (1 - 0 )
~
3) do not integrate the equivalent series resistance (ESR)
generated output voltage switching ripple in the control loop
All three techniques are based on knowing a fixed value of
Fig. 1. Boost converter model considering only PWM feedback control
the capacitor ESR. This paper will focus on eliminating the RHP
zero in converters with low or negligible ESRs.
A well known technique for eliminating the RHP zero is to
operate the converter in the discontinuous conduction mode
(DCM). Once the system enters this mode of operation, the
0-7803-4489-8/98/$10.00 0 1998 IEEE
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conventional system model reduces to a single left-half plane
(LHP) pole, proper system [2].The condition which guarantees
averaging model and the: IAC models converge to the same
model.
the boost converter is operating in DCM is
In summary, the three techniques are
TS
A second technique
2L
'RD( 1 - D)*
(2)
1) operate in the discontinuous conduction mode
2) reduce the inductor value while maintaining CCM operation
continuous conduction mode
(CCM)
3) reduce the frequency while maintaining CCM operation
TS
2L
RD( 1 - D ) 2
111. VERIFKATION TECHNIQUE
(3)
while decreasing the inductor value. With a small inductor and at
Stability of a linear system with feedback can be determined
a high switching frequency, from (l), the effect of the RHP zero
from the phase and gain margins of the loop gain transfer
can be minimized. The RHP zero is shifted towards positive
function using Bode plots. A negative feedback system consisting
infinity, thus reducing it's effect. This also holds true for
of only two poles strictly in the LHP and no zeros is always
increasing the resistor value while maintaining CCM operation.
stable. It can easily be seen that for realistic values for the duty
By operating the converter with large inductor ripple, i.e. near the
cycle, the poles of (1) and (4) are strictly in the LHP. The RHP
DCWCCM boundary, the effect of the RHP zero is reduced.
zero allows the possibility of instability. Using Bode plots and
A third technique also maintains CCM (3) while operating
the system at a lower switching frequency. From the IAC model,
observing the phase and gain margins, the stability of the system,
which depends on the RHE' zero, can be determined.
Figures 2 and 3 demonstrate how stability properties vary
the small-signal model for the boost converter is:
using Bode plots from MA'IZAB[4] for two techniques discussed
in this paper. The Bode pbot for the first technique is not shown
H(s) =
(4)
because it is a standard one pole system. In Fig. 2., the plot with
the resonant peak on the left is the Bode plot of the loopgain of a
From this equation, it can be seen that the RHP zero can become
a LHP zero if the switching frequency is low enough. The
frequency operating range constraint for this third approach is
2L
,>Ts>
R
(
2
D
)( 1 -D ) 2
RD( 1 - D )
2L
(5)
0
-50
-100
The upper limit insures CCM while the lower limit keeps the zero
in the LHP. Unlike the IAC model (4), the state-space averaged
-150
-200
-250
model (1) does not include a T , term, and therefore does not
predict this regime of stable operation. As the frequency
approaches infinity or T , approaches zero, the state-space
-300
I o3
1o5
1oe
10'
1O8
Frequency(radlsec)
Fig. 2. Bode plots from Matlaib using the IAC model demonstratingthe
effects of changing the inductor value.
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boost converter with an inductor value of 60pF. From the phase
minimum component feedback control was implemented
margin, it is clearly unstable. The plot with the resonant peak on
consisting of a comparator and a resistor network as shown in the
the right has an inductor value of 15pH and is stable.In Fig. 3.,
same figure. This simple feedback scheme clearly isolates the
the lower plot shows a boost converter operating at a switching
right-half plane (RHP) zero instability phenomenon which
frequency of 1 MHz, which is unstable. The upper plot shows the
appears only when the output voltage is included in the feedback
same system, but operating at 77 kHz, which is stable.
loop (known as voltage mode control).
The input current and output voltage waveforms for stable
100
and unstable operation are shown in Figures 5, 6, and 7 for the
50
s
D
three techniques: discontinuous mode, lower inductance value,
0
and lower switching frequency, respectively. Fig. 5. shows the
-50
operating mode changing from unstable CCM (25 SZ) to stable
0
g
-100
$
0
z
DCM (loo0 Sz). In this case, the instability manifests itself as an
unwanted, low frequency oscillation. The RHP zero can be
-50
c
-100
attributed as the cause of t h ~ instability.
s
In Fig. 6., the stable and
-150
-200
-250
300
1o3
I o4
1o6
I 0'
I 0'
1O8
Frequency (radlsec)
Fig. 3. Bode plots from Matlab using the IAC model demonstrating the
effects of changing the frequency
l(111)
06.
~
IV. SIMULATION RESULTS
0.4-
2.
0.0
1WY
lWY
1W"
m u
iW"
600"
7WU
BOO"
Bw"
om1
Fig. 5. Output voltage (top) and inductor current (bottom)
demonstratingthe effects of a discontinuousinductor current
,Wok
I
I
I
&-
Fig. 4. Closed loop boost converter system
The boost converter shown in Fig. 4.was simulated using
Saber[5] with the following parameters: E=3V, L=60 pH, C=68
0.0
.
loo"
.
ZDOY
.
300"
.
IW"
.
SW"
SW"
7W"
,
.
BW"
OW"
0.001
4.1
pF, no ESR, R=25 C& and the peak-to-peak sawtooth waveform
Fig. 6. Output voltage (top) and inductor current (bottom)
demonstrating the effects of a smaller inductor value
in the feedback is 200 mV with a DC offset of 1.25 V. A
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unstable CCM operation for inductance values of 15 pH (stable)
resulted. By keeping all the other parameters the same and
and 60 pH (unstable) is shown. Fig. 7. illustrates the CCM
only changing the 100 p H inductor to a 40 pH inductor, the
operation at switching frequencies of 1 MHz (unstable) and 77
oscillations disappear as shown in Fig. 9. When the switching
kHz(stable). The simulations support the theoretical analysis.
frequency of the original oscillating system was lowered, the
oscillations again disappeared as shown in Fig. 10. These
waveforms show only the inductor current. Oscillations in the
output voltage were difficult to see because the peak-to-peak
voltage of the oscillations was on the order of millivolts and it
was masked by measurement noise and the ESR ripple.
,
,
0.0
IW"
,
-0"
.
.
,
.
.
.
MW
400"
540"
m u
703"
BW"
800"
0.001
4.)
Fig. 7. Output voltage (top) and Inductor current (bottom)
demonstrating the effects of a lower switching frequency
V. EXPERIMENTAL RESULTS
1
The boost converter shown in Fig. 4. was built in the lab using
the following parameters: &2V, L=lOO pH, C=68 pF with low
ESR, R=25 Q diode=lN5817, MOSFET=IRF520, Vsaw=
200mVpp
+ 1.25V and switching frequency = 640 kHz. An
Fig. 9. Inductor current showing stable waveform using the same converter as Fig. 8. except at a lower frequency
unstable, oscillating current waveform shown in Fig. 8.
-
-
Fig. 10. Inductor current showing stable waveform using the same converter as Fig. 8. except with a smaller inductor
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VI. CONCLUSIONS
REFERENCES
This paper investigates three techniques which can eliminate
or reduce the effects of the positive or RHP zero which appears in
the small-signal duty-cycle to output transfer function of certain
dc-dc converters. All of these techniques require operating the
converter near the CCMDCM boundary (i.e. large ripple
condition for the inductor current) by varying the inductor or
switching frequency.The techniques are presented along with the
theory, simulation and experimental results supporting each
technique. A verification technique was also presented. While the
theory does not determine exactly if the system will oscillate, it
[ l ] D. M. Sable, B. H. Cho and R. B. Ridley, “Elimination of
the positive zero in fixed frequency boost and flyback
converters,” in APEC Records, 1990, pp. 205-21I.
[2] R. D. Middlebrook and S. Cuk, “A general unified
approach to modeling switching converter power stages,”
in IEEE PESC Records, 1976, pp. 18-34.
[3] A. S. Kislovslu, R. Redl, and N. 0. Sokal, Dynamic
Analysis of Switching-Mode DC/DC Converters,
Lexington, MA: Design Automation, Inc., 1997
[4] R. C. Dorf and R. H. Bishop, Modern Control Systems,
Reading, MA: Addison-Wesley Publishing Co., 1995.
[5] Analogy Inc., “A Guide to Mixed-Signal Simulation and
Design,” Analogy Inc.,
www.analogy.codpubslguidelguide.htm1
does provide a method of eliminating the oscillations.
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