Part Two
A New Way to Model
Current-Mode Control
By Robert Sheehan, Principal Applications Engineer,
National Semiconductor, Santa Clara, Calif.
Unified models using general gain parameters
provide the solution for any peak- or valleyderived current-mode converter.
I
n Part I of this article (Power Electronics Technology, May
2007), the basic operation of current-mode control
was broken down into its component parts, allowing
a greater intuitive understanding for the practical designer. A comparison of the modulator gain was made
to voltage-mode operation, and a simple analogy showed
how the optimal slope-compensation requirement could be
obtained without any complicated equations.
Now unified models using general gain parameters are
introduced, along with simplified design equations, and an
in-depth treatment of the analysis and theory is presented.
This general modeling technique explains how previous
models can complement each other on various aspects of
the current-mode-control theory.
frequency, continuous-conduction-mode (CCM) operation. Reference [1] covers the theoretical background for
this subject, providing an exhaustive analysis of the buck
regulator with its associated models and results. To prevent
duplication, the boost regulator of Fig. 1 forms the basis for
the discussion here. A more rapid approach to using this
information is to bypass reference [1] and follow the general
guidelines for slope compensation described in the first part
of this article. Then the simplified equations can be used to
determine the frequency response.
A current-mode switching regulator is a sampled-data
system, the bandwidth of which is limited by the switching
frequency. Beyond half the switching frequency, the response
of the inductor current to a change in control voltage is
not accurately reproduced. To quantify this effect for linear
modeling, the continuous-time model of reference [2] successfully placed the sampling-gain term in the closed-current
Modeling Continuous-Conduction Mode
This article provides models and solutions for fixed-
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Fig. 1. This switching model of a boost regulator topology provides an example for modeling and simulating continuous-conduction-mode
operation.
Power Electronics Technology June 2007
22
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CURRENT-MODE CONTROL
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To place either sampling-gain term into the linear models
for the buck, boost and buck-boost, the following relationships are applied: FM(s) = FM × HP(s) and GI(s) = GI × H(s).
The accuracy limit for the sampling-gain term is identified
by comparing Q to the modulator voltage gain KM and the
feed-forward term K. Q is directly related to the slope-compensation requirement. The derivation starts with the ideal
steady-state modulator gain, the physical reason being that
at the switching frequency, the relative slopes are fixed with
respect to the period T. A change in control voltage is then
related to a change in average inductor current. Any transfer function that is solely dependent on KM in the forward
dc-gain path will have excellent agreement to the switching
model up to half the switching frequency. However, any
transfer function that includes K in the forward dc-gain path
will show some deviation at half the switching frequency.
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Simplified Transfer Functions
No assumptions for simplification were made during
the derivation of the transfer functions. The only initial
assumptions are the ones generally accepted to be valid in
a first-order analysis. Voltage sources, current sources and
switches are ideal, with no delays in the control circuit.
Amplifier inputs are high impedance, with no significant
loading of the previous stage. Simplification of the results
was made after the complete derivation, which included all
terms. Reference [1] has examples for the buck regulator.
To show the factored form, the simplified transfer functions assume that the poles are well separated by the currentloop gain. Expressions for the low-frequency model do not
show the additional phase shift due to the sampling effect.
The control-to-output transfer function with the samplinggain term accurately represents the circuit’s behavior up to
half the switching frequency. The line-to-output expressions
for audio susceptibility are accurate at dc, but diverge from
the actual response as frequency increases.
The current-sense gain is defined as RI = GI × RS, where
GI is the current-sense amplifier and RS is the sense resistor.
For all transfer functions,
K × RI
1
ωZ =
and ω L = M
.
C OUT
L
OUT × R C
To include the sampling-gain term in the control to output transfer function, the term
Fig. 2. For a buck regulator, sampling gain HP(s) is placed in the forward
path (upper circuit), and sampling gain H(s) is placed in the closedcurrent feedback loop path (lower circuit).
feedback loop. This allows accurate modeling of the controlto-output transfer function using the term HE(s).
To accurately model the current loop, the unified model
of reference [3] placed the sampling-gain term in the forward
path. For peak or valley current modes with a fixed slopecompensation ramp, this also accurately models the controlto-output transfer function using the term FM(s).
To develop the theory for emulated current-mode control,
reference [1] used a fresh approach, deriving general gain
parameters, which are consistent with both models. In addition, a new representation of the sampling-gain term for the
closed-current loop was developed, identifying limitations
of the forward-path sampling-gain term.
The upper circuit in Fig. 2 represents the unified form of
the model, with K being the feed-forward term. In the lower
circuit, KN is the dc audio susceptibility coefficient from the
continuous-time model. The linear model sampling-gain
terms, as shown in Fig. 2, are defined as:
H P ( s) =
1

Q 
1+  s×

 ωN 
, H( s ) = 1 + ( s × K E ) +
s2
π
and ωN = ,
2
T
ωN
s
s2
s
+ 2 in the lowis replaced with 1 +
ωN × Q ωN
ωL
frequency equations. This represents the closed-current-loop
sampling-gain term. Inclusion of this term in the line-tooutput equations will not produce the same accuracy of
results. For peak or valley current mode with a fixed slopecompensating ramp, ω L = Q × ω N .
where T is the switching period. The term KE is new and
emerged from the derivation of the closed-loop expression
for H(s). This derivation used slope-compensation terms
other than the classic fixed ramp for peak or valley current
1
mode. KE can be expressed as
,
ω N × QE
but this serves no purpose, because QE would need a value
of infinity for the condition KE = 0. To date, no method has
been found which successfully incorporates KE into the openloop expression for HP(s). Use of HP(s) is limited to peak or
valley current mode with a fixed slope-compensating ramp,
for which the value of KE = 0.
Power Electronics Technology June 2007
1+
Sampling Gain Q
Using a value of Q = 0.637 will cause any tendency toward
sub-harmonic oscillation to damp in one switching cycle.
With respect to the closed-current-loop control-to-output
24
www.powerelectronics.com
CURRENT-MODE CONTROL
Mode
Peak
current
mode
Peak
current
mode
Slope
compensation
Fixed slope
VSL = SE × T
Proportional
slope
T
K SL = R I ×
L
For Q = 0.637
(single-cycle
damping)
Valley
current
mode
Valley
current
mode
Fixed slope
VSL = SE × T
Proportional
slope
T
K SL = R I ×
L
For Q = 0.637
(single-cycle
damping)
Emulated- Fixed slope
peak
VSL = SE × T
current
mode
Emulated- Proportional
peak
slope
current
T
K SL = R I ×
mode
L
For Q = 0.637
(single-cycle
damping)
SE, SN
mC, Q
SE =
VSL
T
mC = 1 +
SN =
VAP × D′
D × RI
L
Q=
SE =
VAP × D × K SL
T
mC = 1 +
SN =
VAP × D′
D × RI
L
Q=
SE =
VSL
T
mC = 1 +
SN =
VAP × D × R I
L
Q=
SE =
VAP × D′
D × K SL
T
SN =
VAP × D × R I
L
Q=
SE =
VSL
T
mC =
SN =
VAP × R I
L
Q=
SE =
VAP × K SL
T
mC =
SN =
VAP × R I
L
Q=
KM, K
SE
SN
KM =
KE
KE = 0
1
T V
(0.5 − D) × R I × + SL
L VAP
1
T
D
π × (m C × D
D′′ − 0.5)
5 K = 0.5 × R I × × D × D′
L
SE
SN
KM =
1
T
(0.5 − D) × R I × + 2 × K SL × D
L
K E = −K SL × D ×
L
RI
1
T
D + K SL × D2
π × (m C × D
D′′ − 0.5)
5 K = 0.5 × R I × × D × D′
L
SE
SN
KM =
1
π × (m C × D − 0.5)
5
mC = 1 +
SE
SN
1
π × (m C × D − 0.5)
5
SE
SN
SE
SN
1
π × (m C − 0.5)
5
K E = −K SL × D ′ ×
T
×D×D
D′′ − K SL × (D ′)2
L
1
K E = −D × T
T V
(D − 0.5) × R I × + SL
L VAP
K = −0.5 × R I ×
KM =
T
× D × D′
D
L
1
T
(D − 0.5) × R I × + 2 × K SL × D′
D
L
K = −0.5 × R I ×
KM =
1
π × (m C − 0.5)
5
T V
(D − 0.5) × R I × + SL
L VAP
K = −0.5 × R I ×
KM =
KE = 0
1
T
× D × D′
D
L
1
K E = −D × T
T
(D − 0.5) × R I × + K SL
L
K = −0.5 × R I ×
T
× D × D′
D + K SL × D
L
Table 1. Summary of general gain parameters.
function, the effective sampled-gain inductor pole is given
by:
1
f L (Q) =
× ( 1 + (4 × Q2 ) − 1) .
4×T×Q
is Q = 0.5 (δ = 1). Using Q = 1 may make an incremental
difference for the buck, but is inconsequential for the boost
and buck-boost with the associated right-half-plane zero of
ωR. For the peak-current-mode buck with a fixed slope-compensating ramp, the effective sampled-gain inductor pole is
only fixed in frequency with respect to changes in line voltage
when Q = 0.637. Proportional slope-compensation methods
will achieve this for other operating modes.
To determine the effect of reducing the slope compensation to increase the voltage-loop bandwidth, an emulated-
This is the frequency at which a 45-degree phase shift
occurs because of the sampling gain. For Q = 0.637, fL(Q)
occurs at 24% of the switching frequency. For Q = 1, fL(Q)
occurs at 31% of the switching frequency. For second-order
systems, the condition of Q = 1 is normally associated with
best transient response. The criteria for critical damping
Power Electronics Technology June 2007
26
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L
RI
CURRENT-MODE CONTROL
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for a stable voltage loop, at the expense of under-damping
the current loop. With Q = 1, sub-harmonic oscillation is
quite pronounced during transient response, but damps
at steady state. The reader is encouraged to simulate and
observe these effects directly. A simulation example for the
boost is provided after the linear models and transfer functions are presented.
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Linear Models
Simple, accurate and easy-to-use linear models have been
developed for the buck, boost and buck-boost converter
topologies. Each linear model has been verified using results
from its corresponding switching model. In this manner,
validation for any transfer function is possible, identifying
the accuracy limit of the given linear model. General gain
parameters are listed in Table 1. These parameters are independent of topology, and written in terms of the terminal
voltage (VAP) and duty cycle (D).
The coefficients for the linear model of the buck regulator
shown in Fig. 3 are:
V
V − VOUT
VAP = VIN , D = OUT , D ′ =(1 − D)= IN
,
VIN
VIN
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Fig. 3. The low-frequency linear model for this buck regulator was made
using SIMetrix.
peak-current-mode buck with proportional slope-compensation switching circuit was implemented in SIMPLIS.
A standard type-II 10 MHz error amplifier was used for
frequency compensation. With T/L = (5 µs/5 µH) and
RI = (0.1 V/A), the best performance was achieved with
Q = 0.637 for a crossover frequency of 40 kHz and 45-degree
V ×M
K
phase margin. By setting Q = 1, a crossover frequency of
M = D, IC = AP
and FM = M .
R OUT
VAP
50 kHz was achieved, again
with1/4p
45-degree
phase margin
PwrElec-Ventronics
DigiPwr
5/9/07
1:13 PMbutPage 1
The control-to-output simplified transfer function is:
reduced gain margin. This appears to be the practical limit
s
1+
v OUT
R OUT
ωZ
=
×
,
vC
R1 × K D 
s  
s 
 1 + ω  ×  1 + ω 
P
L
NEW! Digi-Power
Multi-Charger
and the line-to-output simplified transfer function is:
s
1+
v OUT R O × D × K N
ωZ
=
×
,
RI × KD

v IN
s  
s 
 1 + ω  ×  1 + ω 
P
L
where
Charges Computers,
Digital Cameras,
and 4 AA or AAA NiMH/NiCD‘s
KD = 1 +
Complete with:
• 5V USB Plug
• 12V Car Plug
• AC Wall Plug
ωP =
R OUT
1
K
, KN =
− and
KM × RI
KM D
1
C OUT
OUT
 1
1 
×
+
.
 R OOUT K M × R I 
The coefficients for the linear model of the current-mode
boost regulator shown in Fig. 4 are:
VOUT − VIINN
V
VAP = VOUT , D = OUT
, D ′ = (1 − D) = IN ,
VOUT
VOUT
M=
The control-to-output simplified transfer function is:

s  
s 
1−
× 1 +


v OUT R OUT × D′
D  ω R   ω Z 
=
×
,
RI × KD

vC
s  
s 
 1 + ω  ×  1 + ω 
P
L
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Power Electronics Technology June 2007
V ×M
K
1
, IC = AP
and FM = M .
D′
R OUT
VAP
28
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CURRENT-MODE CONTROL
and the line-to-output simplified transfer function is:
s
1+
v OUT R OUT × D′
D × KN
ωZ
,
=
×
RI × KD

v IN
s  
s 
 1 + ω  ×  1 + ω 
P
L
where
R
× D′
D2  1
R1
K
1
K D = 2 + OUT
×
+
, KN =
+
,
RI
K M R OUT × D′
D2
 K M D ′ 
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KN =
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Fig. 4. The low-frequency linear model for this boost regulator was
made using SIMetrix.
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
s  
s 
1+
× 1+
R OUT × D × D ′ × K N  ω K   ω Z 
=
×
,
RI × KD

s  
s 
 1 + ω  ×  1 + ω 
P
L
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Fig. 5. The low-frequency linear model for this buck-boost regulator
was made using SIMetrix.
R OUT × D′
D2  1
K
×
+ ,
RI
 K M D′ 
the optimal Q at one input voltage. The control-to-output
gain plots in Fig. 6 show only a slight deviation between
the two models at half the switching frequency, where fSW
= 200 kHz. For the simulation, slope compensation was set
for Q = 0.637.
The choice of simulation program is important, since not
all SPICE programs calculate parameters with the same degree of accuracy. For switching-model simulation, SIMPLIS
is able to produce Bode plots directly from the switching
model. This program was used to produce the switchingmodel simulation results. The low-frequency model was
made with SIMetrix, which is the general-purpose simulator
for the SIMetrix/SIMPLIS program. This simulator only
handles Laplace equations for s in numerical form, where the
numerator order must be equal to or less than the denominator order. PSpice is much better suited for linear models
with Laplace functions in parameter form. It is more accurate
than the SIMetrix/SIMPLIS program but cannot produce
Bode plots directly from the switching model. PSpice or a
program with similar capability may be used to obtain the
simulation results for the linear model.
R OUT × D
D′ 2
RI × D
1
K
− +
, ω R = OUT
,
2
K M D R OUT × D′
D
L×D
2
R OUT
OUT × D ′ × K N
and
L×K
 1 + D D′2  1
1
K 
ωP =
×
+
×
+  .
C OUTT  R OOUT R I  K M D ′  
ωK =
Boost Regulator Simulation Example
For the peak-current-mode boost converter example,
comparisons of results from the switching circuit of Fig. 1
were made to the linear model of Fig. 4 using the samplinggain term HP(s). To use the forward-path sampling-gain
term, slope compensation was implemented with a fixed
ramp. The results will be slightly different if a proportional
ramp is used, as this modifies the modulator gain term KM
and feed-forward term K. For an actual boost-converter
implementation with a fixed ramp, it is only possible to get
Power Electronics Technology June 2007
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and the line-to-output simplified transfer function is:
KD = 1 + D +
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The control-to-output simplified transfer function is:

s  
s 
1−
× 1 +


v OUT R OUT × D′
D  ω R   ω Z 
=
×
,
RI × KD

vC
s  
s 
 1 + ω  ×  1 + ω 
P
L
where
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The coefficients for the linear model of the current-mode
buck-boost regulator shown in Fig. 5 are:
VOUT
VIN
VAP = VIN + VOUT , D =
, D ′ = (1 − D) =
,
VIN + VOUT
VIN + VOUT
V ×M
K
D
M = , IC = AP
and FM = M .
D′
R OUT
VAP
v IN
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R
× D′2
ω R = OUT
and
L
 2
1
D′2  1
K 
ωP =
×
+
×
+  .
C out  R OUT R I  K M D ′  
v OUT
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30
www.powerelectronics.com
CURRENT-MODE CONTROL
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Fig. 6. This comparison of control-to-output transfer functions for a peak-current-mode boost converter using fixed-slope compensation
reveals the switching and linear models behave similarly, with a slight discrepancy at 100 kHz, which is half the switching frequency.
Unified Modulator Modeling
Topology
In Part I of this article, the criteria for current-mode
control was considered. This led to the linear model, with
the gain terms being easily identified. The importance of
the concept of KM as the modulator voltage gain cannot be
overstated. Most linear models for current-mode control
have allowed the math to define the model. In reference 1,
an intuitive understanding of the modulator was used to
drive the math. By algebraic manipulation, both the averaged
iL
Buck
iL = i
Boost
i L = iG
Buckboost
iL = i + iG
LE
LE = L
E(s)
E(s) =
VOFF
D
LE =
L


s×L
E(s) = VOFF ×  1 −

(D′
D ′)2
 D ′ × (VOF
OFF
F / I ON 
LE =
L

VOFF 
s×L
E(s) = OFF
× 1 −

(D′
D ′)2
D
/
I
 D ′ × (VOF
OFF
F
ON 
Notes: VOFF = VAP ; ION = IC
Table 2. Corrections and clarifications for reference [3].
model and continuous-time model were redefined to fit the
form of the unified model. Combining the unified-model
gain blocks with the three-terminal PWM switch resulted in
the linear models used here.
A new closed-current-loop sampling-gain term has been
defined that accommodates any fixed-frequency peak- or
valley-derived operating mode. Limitation of the forwardsampling-gain term has been identified, providing direction
for further development in linear modeling.
PETech
References
1. Sheehan, Robert, “Emulated Current Mode Control for
Buck Regulators Using Sample and Hold Technique,” Power
Electronics Technology Exhibition and Conference, PES02,
October 2006. An updated version of this paper, which
includes complete appendix material, is available from
National Semiconductor Corp.
2. Ridley, R.B., “A New, Continuous-Time Model for Current
Mode Control,” IEEE Transactions on Power Electronics,
Vol. 6, Issue 2, pp. 271-280, 1991.
3. Tan, F.D. and Middlebrook, R.D., “A Unified Model for
Current-Programmed Converters,” IEEE Transactions on
Power Electronics, Vol. 10, Issue 4, pp. 397-408, 1995.
Power Electronics Technology June 2007
32
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