(Invited Lecture) - Boost DC/DC Converter Nonlinearity and RHP-Zero: Survey of the Control-to-Output Transfer Function Linearization Methods
Abstract
This invited paper is dedicated to both the nonlinear behavior of a step-up DC/DC converter, and to its linearization. It explains the nonlinear DC and AC control-to-output transfer function and its RHP-zero. The explanation is based on linearized models developed in the paper. These models are suitable for SPICE-like environment, and allow to obtain accurate symbolic equations in Matlab format. The linearization methods based on tri-state PWM and predistortion are also described. The description is followed by a Matlab algorithm allowing fast computation of the voltage-mode PID controller for pre-distorted PWM modulator. As a result, a linearized converter operating in wide duty-cycle range is designed with voltage mode feedback loop.
ISBN 978-80-261-0602-9, © University of West Bohemia, 2016
Boost DC/DC Converter Nonlinearity and
RHP-Zero: Survey of the Control-to-Output
Transfer Function Linearization Methods
Vratislav Michal*, Denis Cottin†, Patrik Arno†
*†STMicroelectronics, 12 rue Jules Horowitz, 38000 Grenoble − France
*Infineon Technologies Austria AG, Siemensstraße 2, 95000 Villach − Austria
vratislav.michal@infineon.com
Abstract−This invited paper is dedicated to both the
nonlinear behavior of a step-up DC/DC converter,
and to its linearization. It explains the nonlinear
DC and AC control-to-output transfer function
and its RHP-zero. The explanation is based on
linearized models developed in the paper. These
models are suitable for SPICE-like environment,
and allow to obtain accurate symbolic equations in
Matlab format. The linearization methods based on
tri-state PWM and predistortion are also
described. The description is followed by a Matlab
algorithm allowing fast computation of the voltage-
mode PID controller for pre-distorted PWM
modulator. As a result, a linearized converter
operating in wide duty-cycle range is designed with
voltage mode feedback loop.
I. INTRODUCTION: IDEAL CHARACTERISTICS
A boost converter is a non-isolated power converter
that may be used when a higher output voltage than
the one provided by the input source is required. It
contains two power switches, one inductor and an
output capacitor. Boost converter, including dominant
parasitic resistances, is shown in Fig. 1.
Fig. 1. Basic topology of the boost DC-DC converter.
The converter operates in two phases: phase D, where
SWL is closed during D-portion of the clock period
and the inductor current increase, and (1−D) phase
where the inductor energy is released to the load via
the high-side switch SWH. For an ideal converter
(RCOIL, RLOW, and RHIGH = 0), the output voltage can be
determined by the volt-second balance as a function of
the input voltage VIN and duty-cycle D [1]:
1
IN
OUT
V
V
D
(1)
The inductor (input) current is given by dividing the
load current by high-side switch conduction time:
2
11
OUT IN
COIL
L
I V
ID
R D
(2)
Fig. 2. Conversion ratio and small-signal gain GC of the ideal boost
converter [14].
Although these basic characteristics of the boost
converter are well known, its modeling can be
complicated due to the strong nonlinear behavior. This
concerns namely the nonlinearity of DC transfer
function (1), and the nonlinear dynamic response with
the well-known right-half plane (RHP) zero. Unlike
the (fairly linear) buck converter, boost converter
belongs to a class of nonlinear and time-variant
systems. Consequently, its transient response depends
on the steady-state operating point [2], [3].
A. Static Control Gain GC
Most significant non-linearity of the boost converter is
the static control-to-output transfer function (1), and
its static (DC) gain GC. This gain directly impacts the
stability of the feedback control loop. It can be
obtained by derivation of (1):
2
1
OUT IN
C
V V
GD
D
(3)
The control gain GC is shown together with the output
voltage (1) as a function of the duty-cycle in Fig. 2.
Here, high variation (up to 30dB) of GC with duty-
cycle can be seen.
B. Nonlinear Dynamic (transient) Response
The dynamic response is governed by the operating
point given by VIN, VOUT, IOUT and D. As an example,
the Bode-plot of the control-to-output transfer
function for three values of D is shown in Fig. 3.
Here, each duty-cycle exhibits a different frequency
characteristic, and thus a different transient response.
Fig. 3. Control-to-output transfer functions GC(s) for three values of
the duty-cycle D.
This paper is organized as follows: averaged switches
model allowing to obtain mathematical description of
both DC and AC transfer function is presented in
section II. Explanation of the RHP zero is provided in
section III. Methods for the transfer function
linearization based on the tri-state PWM and
predistortion PWM generator are described in sections
IV, V, and VI, whereas synthesis of the voltage-mode
feedback control loop is presented in section VII.
II. LINEAR DC AND AC MODEL OF THE BOOST
CONVERTER
The switching between low and high side switch
results in a periodical switching between two sub-
circuits of the boost converter. Consequently, direct
DC and AC analysis cannot be used.
For frequencies below fSW, AC and DC modeling can
be performed using averaging techniques. The
averaging aims to replace the pulsating currents and
voltages of the switches by their respective averaged
values (averaged within one conduction cycle). In
particular, state-space averaging SSA [4],[5], and
modeling by averaged switches [6],[7],[8] are
frequently used. Circuit’s parameters obtained by
averaging are usually referred as periodic steady-state
values, and are labeled by 〈-〉 bracelets.
This section provides an “user friendly” model with
averaged switches. This model includes all the
parasitic resistances shown in Fig. 1. Additionally,
accurate symbolic DC and AC transfer functions are
also generated. This facilitates both electrical
simulation and accurate symbolic manipulation with
(e.g.) Matlab or Simulink.
The modeling with averaged switches is based on the
substitution of SWL and SWH by a couple of linear
voltage and current controlled sources VCVS and
CCCS [7],[8]. Generally, available models [6],[7],[8]
consider only the inductor parasitic resistance RCOIL.
However, resistances RLOW and RHIGH can significantly
influence the parameters of the model. On this
account, these resistances have been included in the
averaged model Fig. 4 developed in the following.
Main advantage of this technique is the possibility to
directly use accurate SPICE models of the
semiconductors switches (MOSFET or diodes) in the
linearized model. As demonstrated in Biolek 2008 [11]
(on an example of the boost converter model), the
accuracy of the simulation is greatly improved.
A. Switch Averaging Model of a Boost Converter
While the low-side switch SWL is ON during D-
portion of the switching period TSW, its average
current is 〈iSW(L)〉 = D∙ICOIL [1]. As shown in Fig. 4,
current 〈iSW(L)〉 is then realized by CCCS with current
gain D.
The VCVS output 〈vB〉 is defined by the volt-second
balance. In periodic steady-state, the volt-second
balance corresponds to a zero average voltage across
the ideal inductor during [9]:
0
( ) 0
SW
T
L L
v v t dt
(4)
This rule can be used to obtain the DC transfer
function of the boost converter from Fig. 1 as follows:
during the low-side conduction phase D, inductor
voltage is VIN(L) − VA, where VA = ICOILRLOW. Similarly,
inductor voltage during the high-side switch
conduction phase (1−D) is VIN(L) – (VOUT + VRHIGH).
Here, VRHIGH = RHIGH∙ICOIL. The volt-second balance
can be then written as:
1 0
A OUT RHIGH
IN L IN L
V V D V V V D
(5)
In order to satisfy this condition by linearized model, a
voltage controlled voltage source vB is added between
the inductor and RHIGH.
Fig. 4. Nonlinear averaged-switch model of converter from Fig. 1.
Controlled sources vB and iSW(L) are defined as follows:
v
B =
v(D)*v(out)−v(a), and
i
SW(L) = i(L)*v(D). Term relating to the ESR
in vB as described in [6] is neglected. This simplification assumes
the use of a low-ESR (ceramic) output capacitor COUT.
Condition 〈vL〉 = 0 from (4) allows to assume
〈vIN(L)〉 = 〈vLX〉. Average output voltage can be then
written as:
OUT B
IN L RHIGH
v vv v (6)
Since 〈iout〉 = (1−D)∙ICOIL, average voltage 〈vRHIGH〉
from (6) can be written as 〈vRHIGH〉 = (1−D)∙VRHIGH.
Moreover, non-pulsating terms from (5) can be
replaced by their average values: VIN(L) = 〈vIN(L)〉, and
VOUT = 〈vOUT〉. Solving (5) for 〈vIN(L)〉, and applying it
in (6) result in:
B OUT A
v D v v
(7)
here 〈vA〉 = D∙VA is the voltage available in the circuit
shown in Fig. 4. As already mentioned, 〈vB〉 and
〈iSW(L)〉 can be implemented by controlled sources
VCVS and CCCS, available in SPICE environments.
For simulations with variable duty-cycle (e.g.
feedback loop transient, AC control-to-output transfer
function), the controlled sources should enable insert
an expressions D∙ICOIL and (7), that refers to variable
signal D(t) or AC = 1. An example of comparison
between switched-mode simulation and simulation
with linearized model is shown in Fig. 6.
One limitation of the averaged model from Fig. 4 is
the estimation of power efficiency η. The inaccuracy
concerns the simulation of the power dissipated by
resistances RLOW and RHIGH. The RMS value of the
PWM pulsating current with duty-cycle D is given as
= √ [1]. Related dissipated power is
. While average current 〈〉= , the
power measured on Fig. 4 model resistances RLOW and
RHIGH is
and
(1 − ), respectively.
As a result, the power dissipated on resistances RLOW
and RHIGH must be divided by their respective
conduction ratios, i.e. PLOW =RLOW∙〈()
〉/D, and
PHIGH = RHIGH∙〈()
〉/(1−D).
B. DC Transfer Function
Compared to ideal output voltage VOUT (1), real boost
converter from Fig. 1 generates lower output voltage
for the same duty-cycle. Output voltage for such real
boost converter can be obtained by analysis of Fig. 4
model as:
2
1
2
L IN
OUT
L L HIGH LOW L HIGH COIL
R D V
VD R D R R R R R R
(8)
The decrease of VOUT is shown in Fig.5. The
conversion characteristics contains three significant
areas [10]:
1) D < DCRIT: normal (positive gain) operations. The
VOUT is close to ideal value given by Eq.(1),
(curve RL = ∞),
2) D = DCRIT: providing maximum output voltage
VMAX and maximal possible output power [10].
3) D > DCRIT: negative gain area, where the output
voltage decreases when duty-cycle increases.
The value of critical duty cycle DCRIT can be obtained
by setting the 1st derivative d(VOUT)/dD = 0 as:
1
CRIT COIL LOW L
D R R R
(9)
where DCRIT is independent on RHIGH [10]. More
accurate equations of inductor current (2) and output
resistance (being ideally zero) can also be derived by
the help of Fig. 4 model as:
22
IN
COIL
L L HIGH L OW L HIGH COIL
V
ID R D R R R R R R
(10)
2
1
1
COIL HIGH LOW
OUT
R D R DR
R
D
(11)
Both values increase with D. The ROUT given by (11)
allows also to obtain an alternative expression (8) as:
( )
OUT IN IDEAL OUT OUT
V V R I
(12)
where VIN(IDEAL) is given by (1). Improved control-to-
output static gain given previously by ideal Eq.(3) can
be obtained by derivative of (8) as:
Duty-cycle
Output voltage (Eq. 2)
RL=10Ω
RL=20Ω
∞
RL=40Ω
∞
RL=
negative
gain area
positive
gain area
VMAX(1)
VMAX(2)
VMAX(3)
Critical duty-
cycles DCRIT
DCRIT
RCOIL = 0.8Ω
RLOW = 0.3Ω
RHIGH = 0.5Ω
VIN = 1V
(Eq.1)
DMAX
Fig. 5. VOUT/VIN DC voltage transfer function plotted with Eq.(8)
[10].
2
2
2
1
1
L COIL LOW L IN
C
L LOW HIGH COIL HIGH
R D R R R V
G
R D D R R R R
(13)
A more accurate method allowing to obtain DC output
voltage of real boost convertor is to use the power
efficiency concept presented in [1]. This method
assumes that power PIN delivered by the input source
is equal to the sum of output power POUT and power
PLOSS dissipated in the converter:
–
OUT IN LOSS
P P P
(14)
This equation can be developed as the sum:
21
OUT IN LOSS
OUT OUT IN COIL COIL COIL LOW HIGH
P P P
V I V I I R DR D R
(15)
By expanding this expression with = /,
and = /(1 − ), the output voltage from (8)
can be obtained. Advantageously, term PLOSS can
include other power-loss factors, which are not
considered in Fig. 4 model. By this way, accuracy of
VOUT can be improved e.g. by including ESR of COUT,
MOSFETs switching power, high-side rectification
diode nonlinear forward voltage drop, inductor
frequency-dependent power loss, or power dissipated
due to the inductor triangular (RMS) current.
C. AC transfer Function
The linearization of the averaged model allows also to
build a small-signal model for the converter. Mainly,
we can deliver the control-to-output transfer function
GC(s) = OUT/
shown in Fig. 3, and the output
impedance ZOUT(s) = O UT/̂OUT [6], [7].
The linearization is performed for a given steady-state
DC operating point ICOIL, VOUT and duty-cycle D.
These values can be obtained either from expressions
(1) and (2), or by more accurate (8) and (10). The AC
signals produced by the controlled sources vB, and
isw(L) can be written as:
( ) ˆ
ˆ ˆ
,
ˆ
ˆ ˆ ˆ
sw L COIL COIL
B OUT OUT A
i d I D i
v d V D v v
(16)
Here, ^ corresponds to AC voltages, and
is the
AC excitation allowing to calculate the transfer
function GC(s) = OUT/
. The AC analysis of the
Fig. 4 model with controlled sources defined by
Eqs.(16) allow to obtain the transfer function, which is
usually presented in the form [12], [14]:
1 2
2 2
0
0
2
0
1 1
ˆ
( ) ˆ1
OUT
C C
s z s z
v
G s G
ds s
Q
(17)
This control-to-output transfer function (17) contains
two poles and two zeros: one in the left and one in the
right-half-plane (RHP). Parameters of (17) are
collected in Tab. 1 (Ω0 and Q calculated for RLOW =
RHIGH = 0). Whereas, GC is the static gain given by (3)
or (13). Typically, the transfer function (17) is
delivered for zero resistances RLOW and RHIGH [6], [12].
Although this achieves sufficient accuracy, a non-
negligible gain error can occur for high duty-cycle
values. In order to improve the accuracy, complete DC
and AC transfer functions corresponding to Fig. 4
model are presented in Tab. II.
The AC analysis of Fig. 4 model output node allows
to extract the output impedance ZOUT. This impedance
allows to determine the shape of the load transient
response, as shown in the example in section VI. D,
Eq.(32). Calculation of ZOUT does not require steady-
state values ICOIL and VOUT. Simplified form for
RLOW = RHIGH = 0 can be obtained as (see also Tab. II):
2
1
1 1
COIL OUT
OUT
OUT COIL OUT
R ESR C s
Z s
D ESR C s R C s
(18)
TABLE I.
PARAMETERS OF CONTROL-TO-OUTPUT TRANSFER FUNCTION (17)
Linear PWM generator Modulated ramp PWM
generator
z1
1
ESR OUT
R C 1
ESR OUT
R C
z2
2
(1 )
L COIL LO W
D R R R
L
2
2
1L
COIL LOW
CON
RR R
L I
Ω0
2
1
COIL L
OUT ESR L
R R D
LC R R
2 2
L CON COIL
OUT L ESR
R I R
LC R R
Q
0
2
1
OUT L ESR
OUT L ESR COIL
LC R R
C R R D R L
0
2
OUT ESR L
OUT L ESR COIL
CON
LC R R
C R R R L
I
III. RIGHT-HALF PLANE (RHP) ZERO AND DYNAMIC
RESPONSE
The positive 90° phase lag of the Right-Half Plane
zero of (17) constrains significantly the feedback
control of the boost converter. It increases the high
frequency phase lag of (17) to −180° (Fig. 3). The first
reaction to a positive incremental step of D is then a
negative drop of the output voltage [13]. This is
shown by transient simulation in Fig. 6. Here, a duty-
cycle increment of ∆D = +0.1 was applied to a steady-
state operating converter with D = 0.6 and
IOUT = 100mA. Although the final output voltage
increases by +500mV, the immediate response of VOUT
is negative. Obviously, the response in wrong
direction “confuses” the feedback controller and slows
Fig. 6. Response of the boost converter to the increase of duty-cycle
from D = 0.6
D = 0.7. Comparison of switched-mode and
averaged models.
TABLE II.
FULL MATLAB-COMPATIBLE EXPRESSIONS OBTAINED BY THE
ANALYSIS OF THE FIG. 4 AVERAGED-SWITCH MODEL.
Eqs. DC characteristics
(8) Vout = Rl*(1-D)*Vin/(Rl*D^2-D*(2*Rl+
Rhigh-Rlow)+Rhigh+Rcoil+Rl)
(10) Icoil = Vin/(Rl*D^2-D*(2*Rl+Rhigh-
Rlow)+Rhigh+Rcoil+Rl)
(13) Gc = Vin*Rl*(Rl*(1-D)^2-Rcoil-Rlow)/
(Rl*(1-D)^2+D*(Rlow-Rhigh)+Rhigh
+Rcoil)^2
Full-accuracy AC characteristics
(17) Vout = (1+ESR*Cout*s)*Rl*Vin*(-s*L-Rcoil-
2.*D*Rl+D^2*Rl+Rl-Rlow)*d/
((D^2*Rl+D^2*Rl*ESR*Cout*s-D*Rhigh-
D*Rhigh*Cout*s*Rl-
D*Rhigh*ESR*Cout*s+D*Rlow+D*Rlow*Co
ut*s*Rl+D*Rlow*ESR*Cout*s-2.*D*Rl-
2.*D*Rl*ESR*Cout*s+Rhigh+Rhigh*Cout*s*
Rl+Rhigh*ESR*Cout*s+s*L+s^2*L*Cout*Rl+
s^2*L*ESR*Cout+Rcoil+Rcoil*Cout*s*Rl+Rc
oil*ESR*Cout*s+Rl+ESR*Cout*Rl*s)*(Rl-
2*D*Rl+D^2*Rl+Rcoil+D*Rlow+ Rhigh-
D*Rhigh))
(18) Zout = (D*(Rlow-Rhigh)+Rhigh+Rcoil+
(D*(Rlow-Rhigh)+Rhigh+Rcoil)*ESR*Cout
*s)/((1-D)^2+(D*(Rlow-Rhigh)+Rhigh+
Rcoil+ESR*(1-D)^2)*Cout*s)
(26) Vout = RL*alfa*Vin*Icon/(Rcoil*Icon^2*
(Rcoil+Rlow)+(Rhigh-Rlow)*Icon*alfa
+RL*alfa^2)
(27)
Gc_ICON = Rl*alfa*Vin*(Rl*alfa^2-
(Rcoil+Rlow)*Icon^2)/((Rcoil+Rlow)*Icon^2+
(Rhigh-Rlow)*Icon*alfa+Rl*alfa^2)^2
down the recovery of the output voltage.
Unfortunately, RHP zero cannot be eliminated by
control methods like current-mode control [17].
The understanding of the RHP zero can be made
easier writing down the inductor current transfer
function ICOIL(s)/D(s), obtained from the averaged
model Fig. 4:
2 2
0 0
2
1
L OUT
IN
COIL
s R C
V
I s L D s Q
(19)
Interestingly, ICOIL(s) does not contain the RHP zero.
This signifies that a positive increment ΔD produce an
immediate (but slow) increase of ICOIL(t). Indeed, the
inductor current in Fig. 6 increases, in ~100μs, from
an initial value of 250mA up to the final steady-state
value of 333mA (IOUT = 100mA, Eq.(2)). Compared to
this, duty-cycle increment ΔD = 0.1 reduces
instantaneously the high-side switch conduction time.
As a result, ISW_HIGH drops by 25% at the transition
time. The output capacitor is therefore discharged by a
current equal to ISW-HIGH − IOUT. This discharge
continues until the high-side switch deliver full load
current, i.e. ISW_HIGH = IOUT = 100mA.
Recovery time τ due to RHP zero can be reduced by
maximizing its frequency (z2 in Tab. I). In particular,
inductor value L, RCOIL and RLOW are to be reduced.
Similarly, it is preferable to make the converter
operating at low duty-cycle. On the other hand, COUT
does not have an impact on τ.
It is interesting to mention that Ω0 of ICOIL(s) and
VOUT(s) given by Eqs.(19) and (17) are identical.
However, quality factor Q of (17) and (19) are
different. Particularly ICOIL(s) reaches a higher value:
() = /( + ). It results
that inductor current can reach a very-high
(dangerous) value during the transient event.
Generally speaking, the poor transient-response of the
boost converter operating at high duty-cycle results
from Eq.(2), and it is due to a too short high-side
switch conduction time. An increase of the inductor
current ∆ICOIL during e.g. load transient is
considerably larger, when compared to buck
converter. For instance, a 50mA load transient step,
with D = 0.7, requires the inductor current to be
increased by ∆ICOIL = 250mA. Obviously, reaching
∆ICOIL = 250mA takes a long time, resulting in large
output voltage undershoot.
IV. LINEARIZATION OF THE BOOST CONVERTER
Several approaches allowing either a partial or a
complete linearization of the boost converter have
been proposed in the past. Two of these techniques are
commonly used:
o DC linearization by predistortion [14],[15],
enabling to linearize the static DC transfer function
(1), whereas the AC nonlinearity and RHP zero of
transfer function (17) are not modified.
o Structural modification [16] allowing to obtain
both linear static and linear dynamic transfer
functions. Furthermore, this method removes the
RHP zero from the control-to-output transfer
function (17).
In the following sections, it will be shown that the
linearization enable either a simple open-loop control
of the boost converter, or provide stable closed-loop
operations by using voltage-mode control scheme.
Advantageously, the voltage-mode control does not
require expensive current sensing, typically used by
the current-mode control scheme [17]. This
simplification is important for cost reduction, and it
also allows obtaining good performances even in case
of critical conditions (e.g. for low supply voltage).
V. TRI-STATE SWITCHING BOOST CONVERTER
As shown in section III, RHP zero and its initial
undershoot occurs due to the high-side switch SWH. It
has been shown, that while duty cycle increment ∆D
generates a slow increase of the inductor current, the
reduction of the high-side switch SWH conduction
time TON(H) produces an immediate drop of the current
delivered to COUT and to the load. Accordingly, RHP
zero can be eliminated by keeping constant conduction
time TON(H). Consequently, a control of the inductor
current independent on TON(H) has to be implemented.
An obvious approach to control the inductor current is
through the regulation of the input voltage VIN, while
duty-cycle D (i.e. TON(H)) is maintained constant. This
can be implemented by regulation of the input voltage
VIN by an LDO regulator, or buck power stage. By
doing so, converter then operates at high (constant)
duty cycle D, but also with inductor current higher
than (2) – see below.
Alternatively, RHP zero elimination and DC
linearization can be obtained for low output current by
maintaining DCM operations [18].
A. Tri-state Boost Converter
A more straightforward way to control of the inductor
current and maintain constant TON(H) is to use an
additional switch SWAUX. This configuration is
labeled tri-state boost converter, and its basic
schematic and related waveforms are shown in Fig. 7
a, b) [16]. The voltage VLX has three phases:
Fig. 7 a) Tri-state boost converter allowing to provide linear transfer
function and eliminate RHP zero, b) main waveforms of the
converter in CCM.
1) ∙ : SWL is closed and VLX = 0V. ICOIL
increases linearly with slope VIN/L.
2) ( − ): SWAUX is closed and VLX = VIN. In
CCM, the inductor current remains constant due
to ∫ = 0.
3) ( − ): SWH is closed and VLX = VOUT.
Inductor current ICOIL linearly decreases with
slope of ( − )/.
Considering k = 1, SWAUX is always open and
converter behaves as standard boost converter from
Fig. 1. The value of constant duty-cycle D value can
be then obtained from highest required conversion
VOUT(MAX)/VIN given by Eqs. (1) or (8). While lower
VOUT is required, k < 1 and SWAUX is closed during
time (1 − k)D·TSW. Ratio k < 1 is then used to control
the inductor current without modifying the conduction
time of high-side switch SWH.
B. Linear DC Transfer Function
The output voltage VOUT can be obtained by applying
the volt-second balance (4). The inductor voltage can
be written as:
(1 ) 0 1 0
IN IN OUT
k D V k D D V V
(20)
and the output voltage can be written in the form:
11
OUT IN
D
V k V
D
(21)
Here, we can see that VOUT is a linear function of k.
The inductor current can be obtained as a function of
the load as:
1
1 1 1
OUT IN
COIL
L
I V D
I k
D D R D
(22)
The behavior of the tri-state boost converter is
demonstrated in Fig. 8 by sweeping the input control
signal k in the range 0 to 1, with fixed D = 0.8. Fig. 8
shows the DC characteristics obtained with ideal
(RCOIL, RAUX, RLOW, RHIGH = 0) and real model of the
boost converter. We can notice that very high linearity
is provided even for real converter simulation case.
Fig. 8. DC transfer characteristics of VOUT and IOUT for k varying
from 0 to 1 of real and ideal (RC OIL, RAUX, RLOW, RHIGH = 0 ) tri-state
boost converter. Comparison of switched-model Fig. 7, and
averaged-switches model Fig. 9.
C. Averaged Switch Model
Similarly as presented in section II. A, the volt-second
balance was applied to the nonideal tri-state converter
shown in Fig. 7. The obtained average switch model
shown in Fig. 9 allows to perform both AC and DC
simulations. The additional switching state of SWAUX
is represented in Fig. 9 by 〈iAUX〉. Second current
source 〈iAUX〉 is used to provide 〈vAUX〉 referred to
GND. Different inductor and input source currents
highlighted in Fig. 7 a) does not allow to include DC
source resistance RIN into RCOIL. On this account, RIN
was added to the circuits in Figs. 7 and 9.
Fig. 9. Averaged switches model with VCVS and CCCS of the tri-
state boost converter from Fig. 7.
A comparison between linear and switched model of
the tri-state boost converter is shown in Fig. 8. The
simulation of AC transfer function can be obtained by
considering = +
, whereas
is the AC signal
input AC = 1.
Load transient behavior of the tri-state boost converter
is shown in next section in Fig. 16. Here, we notice
that undershot and global shape of the transient
response is almost independent of the conversion
ratio k. Note: for zero switches and inductor
resistances, the transient characteristics will be exactly
identical for all duty-cycles. It result, that control-to-
output transfer function is no more dependent on D
and RL, and belongs to the LTI (linear-time invariant)
class of systems.
D. Power-Efficiency Consideration
Important benefit of the DC linearization and RHP
zero elimination is paid by higher inductor current,
given by (22). As an example, tri-state converter
operating at D = 0.8 and k = 0.5 result in three-time
higher ICOIL than boost converter from Fig. 1 with
identical VOUT. Consequently,
leads to
approximately nine-time higher power loss when
compared to the standard boost converter.
VI. RAMP-MODULATED PWM GENERATOR WITH
BATTERY VOLTAGE FEEDFORWARD
Second class of the linearization method allowing to
achieve constant DC gain (GC) is the PWM pre-
distortion. Predistortion aims to provide nonlinear
duty-cycle generation, that compensate the original
DC nonlinearity of the boost converter (1). On the
contrary, nonlinear dynamic of the boost converter
remains unchanged. This means, that predistortion
preserves both poles and RHP zero of the control-to-
output transfer function mowing with D and RL (as
mentioned in Tab. I. and shown in Fig. 3). This
situation is demonstrated by frequency characteristics
shown in Fig. 10 [14].
Fig. 10. Control-to-Output transfer functions of ramp-modulated
boost converter [14]. Simulation parameters are: VIN = 10V, L =
5µH, C = 22µF, RCOIL = 150mΩ, ESR = 20mΩ, RL = 100Ω Vb =
0.5V, C = 1pF, T = 0.3125µs (α = 1.6µA), RLOW = RHIGH = 0.
Main benefit of the predistortion is the compatibility
with existing boost converter power stage. Moreover,
predistortion partially linearize the converter without
power efficiency loss as discussed in section V. D.
The predistortion is usually implemented inside the
PWM modulator. The predistortion is either based on
an inner modulator feedback loop [15], or on direct
predistortion described further below. The advantage
of direct predistortion is a faster response of the
modulator, which does not alter the feedback loop by
adding extra poles. As mentioned in [14], the
predistortion can be advantageously used also for
other architectures such as buck-boost or flyback.
A. Pre-distorted PWM Modulator Circuit
The PWM predistortion described in this section
allows to obtain a linear VOUT/VIN conversion
characteristic, and also to provide battery voltage
feedforward. This means that, for given input control
signal (e.g. VERROR), the boost converter output voltage
VOUT is ideally independent of input (battery) voltage
VIN. Advantageously, this improves line-transients
regulation.
The technique of predistortion was originally
presented by Arbetter and Maksimovic [19] in 1995.
Then, the battery voltage feedforward was reinvented
by Kazimierczuk 1997 [20], PWM predistortion by
Egawa 2010 [21], and finally also by the author of this
paper in 2012 [14].
Main concept of the ramp-modulated PWM generator
is shown in Fig. 11. Here, the input control signal
(VERROR) is applied to the current source ICON. For
instance, ICON is directly proportional to the PID
controller output voltage VERROR. As a result, the ramp
amplitude is proportional to the control signal. The
variable ramp amplitude VC is shown in Fig. 12.
The duty-cycle D generated by the modulator can be
determined from the time analysis of capacitor voltage
VC(t). As indicated in Fig. 12, current ICON (considered
constant during one clock period T) generates the
voltage ()= (/) . The capacitor voltage is
then compared with an arbitrary reference voltage Vb.
While the capacitor is periodically discharged with
period T, the duty-cycle D is defined within the
condition () > ∙ / as:
Fig. 11. Nonlinear modulated-ramp PWM generator [14],[19].
Fig. 12. Capacitor voltage VC(t) for two values of ICON1 and ICON2
[14]. ICON is considered constant during one clock period..
1 1
b
CON CON
V C
DI T I
(23)
B. DC Output Voltage with Feedforward
The relationship (23) between duty-cycle D and
control current ICON can be substituted in the
conversion characteristic (1). As a result, the output
voltage is a linear function of ICON:
CON
OUT IN
b
I T
V V
V C
(24)
Moreover, if the term Vb in (24) is proportional to
input voltage VIN (e.g. = ∙ ), the term VIN can
be eliminated from (24). This realizes the so-called
“battery-voltage feedforward”:
CON
OUT IN CON
IN
I T T
V V I
k V C k C
(25)
The output voltage is then only determined by the
clock frequency, the ramp capacitor and the control
1
SW
IN
CON
Cf
D k V I
Fig. 13. Example of comparison between boost converter with fixed
duty-cycle D = 0.6, and battery voltage feedforward with
parameters: C = 15pF, k = 0.25, fSW = 1MHz, ICO N = 9.375μA, RCOIL
= 500mΩ, RHIGH = 300mΩ, RLOW = 200mΩ, and RL = 40Ω.
current ICON. On the other hand, VOUT is independent
of VIN. An example of comparison between boost
converter with fixed duty-cycle, and battery voltage
feedforward is shown in Fig. 13.
For a real boost converter with predistortion, the
output voltage can be obtained by inserting the
equation of D (23) into (8). This results in:
2 2
CON L
OUT
CON COIL LOW CON HIGH LO W L
I R
VI R R I R R R
(26)
The derivative of (26) for RHIGH = RLOW = 0 allows to
obtain DC gain (GC_ICON):
2
2
_
2
2
2
1
1
COIL CON
OUT IN LOAD
C ICON
CON COIL CON
LOAD
R I
V V R
GIR I
R
(27)
More accurate equations (26) and (27) including RHIGH
and RLOW are available in Tab. II.
An example of VOUT/VIN conversion characteristics
for both ideal and real boost converters with ramp-
modulated PWM generator are shown in Fig. 14.
Here, we can notice a small decrease of the output
voltage for high values of ICON. Advantageously, when
compared to the abrupt decrease of VOUT above the
DCRIT in Fig. 5, smooth decrease of VOUT with ramp-
modulated PWM generator crates a natural duty-cycle
limitation preventing the inversion of the conversion
gain [10].
Fig. 14. Comparison of ideal (RL = 0) and real (RL = 150 mΩ) boost
converter conversion characteristics with modulated-ramp PWM
generator (VIN = 10V Vb = 0.5V, RL = 150 mΩ, R= 100 Ω, C = 1pF,
T = 0.3125µs or α = 1.6µA.
C. AC Characteristics
As already mentioned, the predistortion technique
achieved by the modulated-ramp generator does not
modify the nonlinear dynamic behavior. This means
that the transfer function (17) remains unchanged,
except for the DC gain GC_ICON (27). Both values of
poles and zeros can be obtained as a function of ICON
replacing the duty cycle (23) into the values from
Tab. I. (left). Resulting values are mentioned for
RLOW = RHIGH = 0 in Tab I (right). Final control-to-
output transfer function can be then obtained by i)
substituting values from Tab. I. right, and ii) gain
GC_ICON (27) (Tab. II.) into (17). An example of
control-to-output transfer functions bode plots for
three values of duty-cycle is shown in Fig. 10.
D. Voltage-mode Feedback Control Loop: Example.
An example of the voltage-mode feedback control
loop is shown in Fig. 15. Here, the output voltage of
PID controller VERROR [22] is connected to a V/I
converter with conversion gain 1/Rv2i (Ω-1). This PID
controller exhibits a transfer function:
2
1 1 1 2 2 1 2 1 2
2 1 1
1 2
0
1
1 ( ) ( )
( ) 1
1 1
1
B B
PID
B
R C C R C R s R R C C R s
F s sR C C R s
s z s z
Gs s p
(28)
As shown in Fig. 15, the pre-distorted PWM generator
then drives a standard boost converter from Fig. 1. For
convenience, this scheme also contains the tri-state
simulation case used in the demonstration in next
section.
Fig. 15. Simulation scheme of the Voltage mode feedback loop of
the tri-state and pre-distorted boost converter.
During the design phase of the PID controller, the
double pole of the transfer function (17) can be
eliminated by double zero of the controller (28). On
the other hand, RHP zero z2 is eliminated by the
controller pole. Obviously, this cancellation is not
mathematically possible, but allows to obtain
sufficient starting point for the final PID controller
parameter adjusting. The Open-loop transfer function
()/() is then:
_1
0
2
1
1PID
OL
v i
s z
F s G
R
_2 1
PID
s z
_1 1
PID
s s p
2
0 2
_
1
PID
C ICON
F s
s z
G
0
1 /s
2
BOOST
F
Here, the quality factor Q of (17) was considered ½,
and the zero coming from ESR was neglected. The
resulting transfer function is of a first order:
2
OL
F s BW s
(30)
where BW is the open loop bandwidth of the loop.
This bandwidth is used as a starting-point parameter
for the feedback loop transfer function synthesis. By
equaling (29) and (30), the PID controller DC gain G0
can be obtained as a function of the bandwidth,
GC_ICON and Rv2i: as:
0 2 _
2
v i C ICON
G BW R G
(31)
By several iterations of BW, suitable PID controller
transfer function can be obtained. As previously
mentioned, PID controller zeros are zPID-1 = zPID-2 = Ω0,
and pPID-1 = z2 (Tab. 1). The values of passive
components can be computed by comparing the
coefficients of FPID(s) with the transfer function (28)
see ref. [22], or Tab. III.
A Matlab algorithm allowing a fast computation of the
PID controller values for a given duty-cycle (e.g. D =
0.6), the visualization of gain/phase margins and the
load and VREF transients, is presented in Tab III. The
load transient response used in the algorithm is
obtained by the help of output impedance (18) [23]:
_ 2
( )
( ) ( ) 1 ( )
OUT OUT
LOAD
LOAD C ICON PID v i
V s Z
F s
I s G s G s R
(32)
VII. COMPARISON OF THE LINEARIZATION METHODS
The voltage model control loop was implemented for
both boost converters with tri-state PWM and pre-
distorted modulated ramp PWM generator. The
simulation scheme is shown in Fig. 15. In order to
provide a visible response for a 100mA load step, the
PID controller was designed for unreasonably low
control-to-output transfer function bandwidth (30)
BW = 500Hz (Tab. III, D = 0.6, Fig. 15.).
Fig. 16. Comparison of 100mA load transient response for tri-state
converter and converter with predistortion. Parameters of
simulations are identical with parameters in Tab. III. except BW =
500Hz, and RAUX = 0.1Ω. Duty-cycle of tri-state converter D = 0.8,
i.e. TSW(H) = 100ns).
Parameters of the PID controller for fast regulation
scheme (BW = 30kHz) are shown in Fig. 15 in italic.
One simulation results for BW = 30kHz is shown for
VREF = 0.4V in Fig. 16.
From the transient responses shown in Fig. 16, we can
notice that:
Tri-state boost converter provide transient
response almost independent of the duty-cycle D,
Modulated ramp predistortion provide
improving response with decreasing duty-cycle
value (decreasing VOUT).
This different behavior corresponds to the different
level of linearization offered by each method. The tri-
state converter provides linearization of the complete
transfer function, whereas the predistortion technique
maintains the resonant frequency and other parameters
of GC(s) listed in Tab. 1 depending (sometimes
favorably) on the duty-cycle D.
While the Tri-state boost converter operates with high
(constant) duty-cycle D = 0.8, the equivalent resonant
frequency Ω0 is low. Therefore, despite the absence of
RHP zero and higher inductor current, the transient
TABLE. III: MATLAB ALGORITHM FOR PID CONTROLLER
DESIGN FOR THE PRE-DISTORTED BOOST CONVERTER.
close all; clear all; s = tf('s');
L=input('L (2e-6)='); Cout=input('C (10e-6)=');
Rcoil=input('Coil+Battery resistance (0.3)=');
Rlow=input('Rlow (0.1)='); Rhigh=input('Rhigh (0.2)=');
RL=input('RL (40)='); ESR=input('ESR (0.02)=');
Rv2i=input('V/I converter resistance (200e3)=');
Vin=input('Vin (1)=');
BW=input('Open-loop bandwidt (10000)=');
k=input('feedforward ratio (0.25)=');
Cramp=input('pwm ram capacitor (10e-12)=');
Fsw=input('switching frequency (2e6)=');
D=input('steady-stte duty-cycle D (0.6)=');
% PID controller gain G0
Vb=k*Vin;Icon = -Vb*Cramp*Fsw/(D-1);
alfa=k*Vin*Cramp*Fsw;
Gc=RL*alfa*Vin*(RL*alfa^2-
(Rcoil+Rlow)*Icon^2)/((Rcoil+Rlow)*Icon^2+(Rhigh-
Rlow)*Icon*alfa+RL*alfa^2)^2;
G0=2*pi*BW*Rv2i/Gc
omega=sqrt(((1-D)^2*RL+Rcoil)/RL)/sqrt(L*Cout);
ppid=((1-D)^2*RL-Rcoil)/L; ('PID transfer function');
Fpid= zpk(G0*(1+s/omega)^2/(s*(1+s/ppid)))
% GM, PM for D 0.1 to 0.9
for n=1:8;
D=0.1*n; Out(n,1)=D;
Out(n,2)=-Vb*Cramp*Fsw/(D-1); % Icon for D
Out(n,3)= RL*alfa*Vin*(RL*alfa^2-
(Rcoil+Rlow)*Out(n, 2)^2)/((Rcoil+Rlow)*Out(n,
2)^2+(Rhigh-Rlow)*Out(n, 2)*alfa+RL*alfa^2)^2;%
gain Gc
% Transient, bode and gain and phase margins plots
for D = 0.1 to 0.8
Fboost = Gc*(1+s*ESR*Cout)*(1-s*L/((1-D)^2*RL-
Rcoil))/(1+s*L*Cout*RL*(Rcoil/L+1/(Cout*(RL+ESR))
)/(Rcoil+(1-D)^2*RL)+s^2*L*Cout*RL/(Rcoil+(1-
D)^2*RL));
Fol=Fpid*Fboost/Rv2i; Fcl=Fol/(1+Fol);
Zout=(D*(Rlow-Rhigh)+Rhigh+Rcoil+ (D*(Rlow-
Rhigh)+Rhigh+Rcoil)*ESR*Cout *s)/((1-
D)^2+(D*(Rlow-Rhigh)+Rhigh+ Rcoil+ESR*(1-
D)^2)*Cout*s);
FLoad_tran=Zout/(1+Fol);
[Out(n,4),Out(n,5)]=margin(Fol); %Out(n,4) - gain
margin Out(n,5) - phase margin
Out(n,4)=20*log10(Out(n,4));
figure(1); step(Fcl); grid on; hold on; title('control to
output transient');
figure(2); bode(Fol); grid on; hold on; title('Open-loop
transfer function');
figure(3); step(-FLoad_tran); grid on; hold on;
title('1A Load Transient (with linearized TF)');
n=n+1;
end
figure(4); plot(Out(:,1),Out(:,4)); grid on
title('Gain margin (dB) vs. Duty-cycle')
figure(5); plot(Out(:,1),Out(:,5)); grid on
title('Phase margin (degre) vs. Duty-cycle')
figure(6); bode(Fpid); grid on;
title('PID bode plot - check the gain at Fsw (<20dB)')
('computation of passive components');
RA = input('choose Ra (e.g. 250000)=');Vref =
input('Choose Vref (e.g. 0.4)=');
Vout = input('Choose Vout (e.g. 2)=');
RA = RA, RB = RA*(Vout-Vref)/Vref, R1 = -
RA*omega*(Vout-Vref)/((omega-ppid)*Vref),
R2 = (Vout-Vref)*RA*G0/(Vref*omega), C1 = -(omega-
ppid)*Vref/(RA*omega*ppid*(Vout-Vref)),
C2 = Vref/(RA*G0*(Vout-Vref))
response is not significantly improved in Fig. 16,
when compared to the predistortion (for similar open-
loop BW). However, the Tri-state boost converter is
better in some configurations, (high value of L, that
could create very low-frequency RHP zero), where a
stable boost converter can be difficult to design in a
wide duty-cycle range with only the predistortion
technique.
CONCLUSION
This paper presents a tutorial on the nonlinear
behavior, modeling, and linearization of the boost
converter. It aims to provide a compact introduction to
the topic with insight on the mathematical description
and models, being useful with help of standard CAD
environment. It also presents an application example
of the boost converter in pure voltage mode, which
allows to provide a stable behavior and good quality
of the regulation in a wide range of operation
parameters.
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- [Show abstract] [Hide abstract] ABSTRACT: A generalization of the model of PWM (Pulse Width Modulation) switch according to Dijk et al. (1) for averaged modeling of switched DC-DC converters is proposed. The original model by Dijk consists of controlled current and voltage sources for modeling the ideal active and passive switch. A new model, proposed in this paper, is based on the same strategy, but it takes into account the nonideal V/I characteristics of semiconductor switches. The rules for determining the parameters of controlled sources from converter state variables remain the same as in (1), but a special procedure provides for the extraction of these parameters directly from PSpice models of concrete transistors or diodes, representing the active and passive switches. The way this model is compiled allows for the influence of parasitic ESRs of the capacitors in the circuit such that the resulting equations are equivalent to the averaged state space equations. The procedure of PSpice modeling and the results of computer simulation of boost converter are shown.
- [Show abstract] [Hide abstract] ABSTRACT: The control-to-output and the duty ratio-to-inductor current transfer functions of peak current-mode controlled PWM dc-dc buck converter in CCM are derived and illustrated. The closed inner-current loop and the power stage inductor current-to-output voltage transfer functions are used to derive the control-to-output transfer function of the peak current-mode controlled buck converter in CCM, which is essential for the outer-voltage loop controller design. A small-signal model including parasitic components and delay is used to derive the power stage transfer function, necessary for proper analysis of the power stage. The loop gain of the inner-current loop is derived and is shown that it is independent of converter topology.
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