**A New Way to Model**

**Current-Mode Control**

### Unified models using general gain parameters

### provide the solution for any peak- or

### valley-derived current-mode converter.

### By Robert Sheehan

### , Principal Applications Engineer,

### National Semiconductor, Santa Clara, Calif.

## 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 de-signer. 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 uniﬁ ed models using general gain parameters are introduced, along with simpliﬁ ed 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.

### Modeling Continuous-Conduction Mode

This article provides models and solutions for ﬁ

xed-frequency, continuous-conduction-mode (CCM)
opera-tion. 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 ﬁ rst part
of this article. Then the simpliﬁ ed 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] suc-cessfully placed the sampling-gain term in the closed-current

**Part Two**

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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.*

feedback loop. This allows accurate modeling of the
control-to-output transfer function using the term H_{E}(s).

To accurately model the current loop, the uniﬁ ed model
of reference [3] placed the sampling-gain term in the forward
path. For peak or valley current modes with a ﬁ xed
slope-compensation ramp, this also accurately models the
control-to-output transfer function using the term F_{M}(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 addi-tion, 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 uniﬁ ed form of
the model, with K being the feed-forward term. In the lower
circuit, K_{N} is the dc audio susceptibility coefﬁ cient from the
continuous-time model. The linear model sampling-gain
terms, as shown in **Fig. 2**, are deﬁ ned as:

H s
s Q
H s s
T
P
H s_{P}
H s
N
E
N
N
( )
H s( )
H s = H sH s( )( ) (((s Ks K_{E}_{E}))) ,
+ ×s
+ ×s
+ ×
+ ×
+ ×
+ ×_{}
+ ×_{}
+ ×
+ ×_{}
+ ×
+ ×
= + ×((((((s Ks Ks Ks K××× _{E}_{E}_{E}_{E}))))))++++ =
1
1
1
= +1
= + 2_{2}
ω
ω ω
π
, and

where T is the switching period. The term K_{E} 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 ﬁ xed ramp for peak or valley current
mode. K_{E} can be expressed as 1

other than the classic ﬁ xed ramp for peak or valley current 1

other than the classic ﬁ xed ramp for peak or valley current ωNNNNN××QQEEEEE

,

but this serves no purpose, because Q_{E} would need a value
of inﬁ nity for the condition K_{E} = 0. To date, no method has
been found which successfully incorporates K_{E} into the
open-loop expression for H_{P}(s). Use of H_{P}(s) is limited to peak or
valley current mode with a ﬁ xed slope-compensating ramp,
for which the value of K_{E} = 0.

### CURRENT-MODE CONTROL

### CURRENT-MODE CONTROL

To place either sampling-gain term into the linear models
for the buck, boost and buck-boost, the following
relation-ships are applied: F_{M}(s) = F_{M} × H_{P}(s) and G_{I}(s) = G_{I} × H(s).
The accuracy limit for the sampling-gain term is identiﬁ ed
by comparing Q to the modulator voltage gain K_{M} and the
feed-forward term K. Q is directly related to the
slope-com-pensation requirement. The derivation starts with the ideal
steady-state modulator gain, the physical reason being that
at the switching frequency, the relative slopes are ﬁ xed with
respect to the period T. A change in control voltage is then
related to a change in average inductor current. Any
trans-fer function that is solely dependent on K_{M} 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.

### Simpliﬁ ed Transfer Functions

No assumptions for simpliﬁ cation were made during the derivation of the transfer functions. The only initial assumptions are the ones generally accepted to be valid in a ﬁ rst-order analysis. Voltage sources, current sources and switches are ideal, with no delays in the control circuit. Ampliﬁ er inputs are high impedance, with no signiﬁ cant loading of the previous stage. Simpliﬁ cation 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 simpliﬁ ed transfer func-tions assume that the poles are well separated by the current-loop 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 sampling-gain 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 deﬁ ned as R_{I} = G_{I} × R_{S}, where
G_{I} is the current-sense ampliﬁ er and R_{S} is the sense resistor.
For all transfer functions,

ω ω ωZ ω ωZ ω ω ω OUT C L M I COU R COUT RC C T RC KM RI KM RI L ω = ω ω ω C ×R C T RC C TT×RCC C T RC =KKKKMMMM××RRRRIIII 1 and ω and ω ω ω .

To include the sampling-gain term in the control to out-put transfer function, the term

1+ s L ω is replaced with 1 2 2 + × + s Q s N N ω ω in the low-frequency equations. This represents the closed-current-loop sampling-gain term. Inclusion of this term in the line-to-output equations will not produce the same accuracy of results. For peak or valley current mode with a ﬁ xed slope-compensating ramp,ωωωωωωωωωωωωωωωωωωωωωωωωωLLLLLLLLLLLL= ×= ×= ×= ×= ×= ×QQQQQQQQQ ωωωωωωωωωωωωωωωωωωωωωωωωωNNNNNNNNNNNN.

### 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
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Fig. 2. *For a buck regulator, sampling gain H _{P}(s) is placed in the forward *

*path (upper circuit), and sampling gain H(s) is placed in the*

*closed-current feedback loop path (lower circuit).*

### CURRENT-MODE CONTROL

### CURRENT-MODE CONTROL

### CURRENT-MODE CONTROL

### CURRENT-MODE CONTROL

function, the effective sampled-gain inductor pole is given by: f Q T Q Q L f QL f Q( ) f Q( ) f Q = (( (( ) ) × ×T Q × ×T Q×××(((( +++(((( ×××Q )Q )−−− )) 1 4 ××(((((( 111++((((((444 )) 1)) 2 × 2 − × − .

This is the frequency at which a 45-degree phase shift
occurs because of the sampling gain. For Q = 0.637, f_{L}
occurs because of the sampling gain. For Q = 0.637, f_{L}
occurs because of the sampling gain. For Q = 0.637, f (Q)
occurs at 24% of the switching frequency. For Q = 1, f_{L}
occurs at 24% of the switching frequency. For Q = 1, f_{L}
occurs at 24% of the switching frequency. For Q = 1, f (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

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
ω_{R}. For the peak-current-mode buck with a ﬁ xed
slope-com-pensating ramp, the effective sampled-gain inductor pole is
only ﬁ xed 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 compensa-tion to increase the voltage-loop bandwidth, an

**emulated-Mode** **Slope _{compensation}**

**SE, SN**

**mC, Q**

**KM, K**

**KE**

Peak current mode Fixed slope VSL S T VSL S V =SE× V =S × V =SEE× S V T E SL VSL V = S V D R L N AP VAP D V D I =VV ×DD′ × m S S C E N = +1 = +1 = + Q C = × × ′ 1 0 5 π××××××((((((((((mmmmmmmmmmmmmmmmmmCCCCCCCC××××××DDDDDDDDDDDDDDDDDD′′ −−0 50 5)... K D R T L V V M I SL VSL V AP VAP V = D R −D ×R −D ×R −D ×R × +× + 1 0 5 ( .0 5 ( .0 5−−DDDDDDDD)))))××RRRRRRRR K R T L D D I K= ×R K 0 5 R × ×× ×D DD D× ′ K 0 5 R K R K= ×R K 0 5 R K= ×R K 0 50 5. R K 0 5 R K . R K 0 5 R K R K= ×R K 0 5 R K= ×R K . R K= ×R K 0 5 R K= ×R K R KE=0 Peak current mode Proportional slope K R T L SL KSL R K =RI× K =R × K =RII× For Q = 0.637 (single-cycle damping) S V D K T E AP VAP D V D SL =VV × ×× ×DD S V D R L N AP VAP D V D I =VV ×DD′ × m S S C E N = +1 = +1 = + Q C = × × ′ 1 0 5 π××××××((((((((((mmmmmmmmmmmmmmmmmmCCCCCCCC××××××DDDDDDDDDDDDDDDDDD′′ −−0 50 5)... K D R T L K D M I S L I S L K D I KKKSLL DDD = −D ×R −D ×RIIIIII× +× +× + ×××××KKKKKKKKKKSSSSSSLL×××××DDDDDDDDDD 1 0 5 II 22 SS ( .0 5 ( .0 5−−DDDDDD)))××RRRRRR K R T L D D K D I S L I S L D D I D D SL K= ×R K RIIIIIIII× ×× ×× ×D DD DD DD DD DD D×× ′ +++++KKKKKKKKKKSSSSSSSSLL×××××DDDDDDDDDD K 0 5 R K 0 5 R K 0 5 R K R K= ×R K 0 5 R K= ×R K R 2 0 5. 0 5 K 0 5 R K . R K 0 5 R K R K= ×R K 0 5 R K= ×R K . R K= ×R K 0 5 R K= ×R K R K K D L R E S KE KS K KL I K = −K K K × ×× ×DD Valley current mode Fixed slope VSL S T VSL S V =SE× V =S × V =SEE× S V T E SL VSL V = S V D R L N AP VAP D V D I =VV × ×× ×DD m S S C E N = +1 = +1 = + Q C = × × 1 0 5 π××××××(((((((((mmmmmmmmmmmmmmmmmmCCCCCCCC××××××DDDDDDDDDDDDDDDDDD−−0 50 5)... K D R T L V V M I SL VSL V AP VAP V = D− ×R D R × +× + 1 D 0 5 R D 0 5 R D 0 5 R D R D− ×R D 0 5 R D− ×R D R (D . R (DDD− . ×RRR (DDDD−0 5. ×RRRR (DD 0 50 5. RR (DD 0 50 5. RR (DDDDDD−−0 50 5. ××RRRRRR (DDDDDDDDDDDD−−−−0 5. )))))××××RRRRRRRRRRRR K R T L D D I K= − R K ×R × K ×R × K 0 5×RII× × ×× ×D DD D′ K 0 5 R K 0 50 5. R K 0 5 R K . R K 0 5 R K R KE=0 Valley current mode Proportional slope K R T L SL KSL R K =RI× K =R × K =RII× For Q = 0.637 (single-cycle damping) S V D K T E AP VAP D V D SL =VV ×DD′ × S V D R L N AP VAP D V D I =VV × ×× ×DD m S S C E N = +1 = +1 = + Q C = × × 1 0 5 π××××××(((((((((mmmmmmmmmmmmmmmmmmCCCCCCCC××××××DDDDDDDDDDDDDDDDDD−−0 50 5)... K D R T L K D M I S L I S L K D I KKKSLL DDD = D− ×R D RIIIIII× +× +× + ×××××KKKKKKKKKKSSSSSSLL×××××DDDDDDDDDD′ 1 0 5 D 0 5 R D R D− ×R D 0 5 R D− ×R D RII 22 SS (D . R (DDD− . ×RRR (DD−0 5. ×RR (DD 0 50 5. RR (DDDDDD−−0 50 5. ××RRRRRR (DDDDDDDDDD−−−−0 5. )))××××RRRRRRRRRR K R T L K D I S L I S L L K= − R K 0 5 R ′ ′ K 0 5 R K R 2 . ( . ( K . R ( K .... RII TT D DD D KKSS ((((DD L I S L . ( L I S L D D I D D S . II D DD D KKSS (DD . I KKSL (DD . KKLL (DD . × × KKL (DD . × × ( K ×R × K . R ( K ×R × K 0 50 5... ××RIIIIIIIIIIIIIIIIIIII×× × ×× ×× ×× ×× ×× ×× ×× ×D DD DD DD DD DD DD DD DD DD DD DD D′ −′−−−−−−−−−KKKKKKKKKKKKKKKKKKKKSSSSSSSSSSSSSSSSSSSSLLLL××××××××××((((((((((((DDDDDDDDDDDDDDDDDDDD K 0 5 R K . R ( K 0 5 R K R ) K K D L R E S KE KS K KL I K = −K K K × ′ × Emulated-peak current mode Fixed slope VSL S T VSL S V =SE× V =S × V =SEE× S V T E SL VSL V = S V R L N AP VAP R V RI =VV ×RR m S S C E N = Q C = × − 1 0 5 π××××××(((((((mmmmmmCCCCCC−−−−−−0 50 5)... K D R T L V V M I SL VSL V AP VAP V = D− ×R D R × +× + 1 D 0 5 R D 0 5 R D 0 5 R D R D− ×R D 0 5 R D− ×R D R (D . R (DDD− . ×RRR (DDDD−0 5. ×RRRR (DD 0 50 5. RR (DD 0 50 5. RR (DDDDDD−−0 50 5. ××RRRRRR (DDDDDDDDDDDD−−−−0 5. )))))××××RRRRRRRRRRRR K R T L D D I K= − R K ×R × K ×R × K 0 5×RII× × ×× ×D DD D′ K 0 5 R K 0 50 5. R K 0 5 R K . R K 0 5 R K R KE D T KE D K D K = −D K D× Emulated-peak current mode Proportional slope K R T L SL KSL R K =RI× K =R × K =RII× For Q = 0.637 (single-cycle damping) S V K T E AP VAP K V KSL =VV ×KK S V R L N AP VAP R V RI =VV ×RR m S S C E N = Q C = × − 1 0 5 π××××××(((((((mmmmmmCCCCCC−−−−−−0 50 5)... K D R T L K M I S L I S L K I KSL = D− ×R D RII× +× +× + SS 1 D 0 5 R D 0 5 R D 0 5 R D R D− ×R D 0 5 R D− ×R D R (D . R (DDD− . ×RRR (DDDD−0 5. ×RRRR (DD 0 50 5. RR (DD 0 50 5. RR (DDDDDD−−0 50 5. ××RRRRRR (DDDDDDDDDD−−−−0 5. )))××××RRRRRRRRRR K R T L D D K D I S L I S L D D I D D SL K= − R K ×R × K ×R × K 0 5×RIIIIIIIIII× × ×× ×× ×× ×D DD DD DD DD DD D′ +++++KKKKKKKKKKSSSSSSSSSSLL×××××DDDDDDDDDD K 0 5 R K 0 50 5. R K 0 5 R K . R K 0 5 R K R KE D T KE D K D K = −D K D×

peak-current-mode buck with proportional
slope-com-pensation switching circuit was implemented in SIMPLIS.
A standard type-II 10 MHz error ampliﬁ er was used for
frequency compensation. With T/L = (5 µs/5 µH) and
R_{I} = (0.1 V/A), the best performance was achieved with
Q = 0.637 for a crossover frequency of 40 kHz and 45-degree
phase margin. By setting Q = 1, a crossover frequency of
50 kHz was achieved, again with 45-degree phase margin but
reduced gain margin. This appears to be the practical limit

### CURRENT-MODE CONTROL

### CURRENT-MODE CONTROL

### CURRENT-MODE CONTROL

### CURRENT-MODE CONTROL

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 func-tions are presented.

### 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 veriﬁ ed 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
inde-pendent of topology, and written in terms of the terminal
voltage (V_{AP}

voltage (V_{AP}

voltage (V ) and duty cycle (D).

The coefﬁ cients for the linear model of the buck regulator
shown in **Fig. 3** are:

V V D V V V V M D I V M R AP VAP V V VIN V VIN V V VVOUOUT IN VIN V OU VOU V T IN VIN V C AP VAP M V M OUT = = V =V = V =V DD= ′ − − = = M D= = M D I= I = VV ×MM , , D =D =′′ (1 D)=V , IN VIN V , and F K V M FM F M AP VAP V = .

The control-to-output simpliﬁ ed transfer function is:
v
v
R
R K
s
s s
OUT
C
OUT
D
Z
P L
=
R ×K
R K ×
+
s s
s s
P L
P L
× +
s s
s s
P L
P L
1
R1 K
R K
1
1 s 1 s
1 s 1 s
1 1
1 s 1 s
1+ s 1 s
1+ 1
1 ss 1 ss
1 ss 1 ss
1 1
1 _{}_{} 1
1 _{} 1
1 1
× +
1 × +1
1 ss 1 ss
1 ss × +× +1 ss
1 × +× +× +1
× +
1 × +1
× +
× +_{}
× +
1 × +× +_{}_{}1
1 × +× +× +× +_{}_{}1
1 × +× +× +× +× +_{}1
× +_{}
× +
× +
1 × +1
× +_{}
× +
× +
ω
ωP ωL
ωP ωL
ωP ωL
ωP ωL
ωP ωL
ωP ωL
ωPP ωLL
ωPP ωLL
ω ω
ωPP ωLL
ωPP ωLL
ω ω
,

and the line-to-output simpliﬁ ed transfer function is:

, v v R D K R K s

and the line-to-output simpliﬁ ed transfer function is:_{s}
and the line-to-output simpliﬁ ed transfer function is:

s s
OUT
IN
O N
RO D N
RO D KKN
I D
RI KD
R K
Z
P L
=RRRRRROOOO× ×× ×× ×× ×DDDDDD NNNN
R ×K
R K
RI KD
R ×K
RI KD
R K ×
+
s s
s s
P L
P L
× +
s s
s s
P L
P L
1
1 s 1 s
1 s 1 s
1 s 1 s
1 s 1 s
1 1
1 s 1 s
1+ s 1 s
1+ 1
1 ss 1 ss
1 ss 1 ss
1 1
1 _{}_{} 1
1 _{} 1
1 1
× +
1 × +1
1 ss 1 ss
1 ss × +× +1 ss
1 × +× +× +1
× +
1 × +1
× +
× +_{}
× +
1 × +× +_{}_{}1
1 × +× +× +× +_{}_{}1
1 × +× +× +× +× +_{}1
× +_{}
× +
× +
1 × +1
× +_{}
× +
× +
ω
ωP ωL
ωP ωL
ωP ωL
ωP ωL
ωP ωL
ωP ωL
ωPP ωLL
ωPP ωLL
ω ω
ωPP ωLL
ωPP ωLL
ω ω
where
K R
K R K
K
D
D
OUT
M I
KM RI
K R M
= +
K ×R
K R
KM RI
K ×R
KM RI
K R
1
= +1
= + , KKKKKKNN==== 1 −−−− and
ωP
OUT OUT M I
COU R
COUT RO
C T RO KKMM RRII
= ×
= × +
K ×R
KM RI
KMM×RII
KM RI
T O
T O
C R
C T RO
C TT ROO
C T RO
C R
C R
C R
C R
1 1
1 1 1
.

The coefﬁ cients for the linear model of the current-mode
boost regulator shown in **Fig. 4** are:

V V V V V V V M D I V M AP VAP V V VOU V VOU V V T OU VOU V V T VI V T VI V VN OU VOU V T IN VIN V OU VOU V T C AP VAP M V M = = V =V = V V VV −VV ′ = − = = ′ = V ×M V M , DD ,, V D , V OU D OUT I , D T IN, D N, = D= , = = D , ( D) ( D) = −( ) = −((1 )) , = −( ) = −1 = −( ) = − 1 R R R F K V OUT M FM F M AP VAP V and = .

The control-to-output simpliﬁ ed transfer function is: v v R D R K s s s OUT C OU ROU D R T D R T D R D I D RI KD R K R Z P =RR ×DD′ R ×K R K RI KD R ×K RI KD R K ×

The control-to-output simpliﬁ ed transfer function is:

The control-to-output simpliﬁ ed transfer function is:

The control-to-output simpliﬁ ed transfer function is:

The control-to-output simpliﬁ ed transfer function is:

s s s s R Z R Z × +

The control-to-output simpliﬁ ed transfer function is:

The control-to-output simpliﬁ ed transfer function is:

s s s s R Z R Z

The control-to-output simpliﬁ ed transfer function is:

The control-to-output simpliﬁ ed transfer function is:

1 1
1 s 1 s
1− s 1 s
1− 1
1 ss 1 ss
1 ss 1 ss
1 1
1 _{}_{} 1
1 _{} 1
1 1
× +
1 × +1
1 ss 1 ss
1 ss × +× +1 ss
1 × +× +× +1
× +
1 × +1
× +
× +_{}
× +
1 × +× +_{}_{}1
1 × +× +× +× +_{}_{}1
1 × +× +× +× +× +_{}1
× +_{}
× +
× +
1 × +1
× +_{}
× +
× +
1 1
1 s 1
1+ s 1
1+ 1
1 1
1 1
1 _{}_{} 1
1 _{} 1
1 1
×
1 × 1
ωR ωZ
ωR ωZ
ωRR ωZZ
ωRR ωZZ
ω ω
ωRR ωZZ
ωRR ωZZ
ω ω
ω +++
1 1
1 1
1 1
1 1
1 _{}1
1 1
1 1
1 _{}1
1 1
1 1
1 1
1 _{}1
1 1
1 1
s
L
ω
,
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Fig. 3.* The low-frequency linear model for this buck regulator was made *
*using SIMetrix. *

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and the line-to-output simpliﬁ ed transfer function is: v v R D K R K s

and the line-to-output simpliﬁ ed transfer function is:_{s}
and the line-to-output simpliﬁ ed transfer function is:

s s
OUT
IN
OU
ROU D
R T D N
R T D N
R T D KKN
I D
RI KD
R K
Z
P L
=RRRRRR TTTT××DDDDDD′ ×× NNNN
R ×K
R K
RI KD
R ×K
RI KD
R K ×
+
s s
s s
P L
P L
× +
s s
s s
P L
P L
1
1 s 1 s
1 s 1 s
1 s 1 s
1 s 1 s
1 1
1 s 1 s
1+ s 1 s
1+ 1
1 ss 1 ss
1 ss 1 ss
1 1
1 _{}_{} 1
1 _{} 1
1 1
× +
1 × +1
1 ss 1 ss
1 ss × +× +1 ss
1 × +× +× +1
× +
1 × +1
× +
× +_{}
× +
1 × +× +_{}_{}1
1 × +× +× +× +_{}_{}1
1 × +× +× +× +× +_{}1
× +_{}
× +
× +
1 × +1
× +_{}
× +
× +
ω
ωP ωL
ωP ωL
ωP ωL
ωP ωL
ωP ωL
ωP ωL
ωPP ωLL
ωPP ωLL
ω ω
ωPP ωLL
ωPP ωLL
ω ω
,
where
K R D
R K
K
D K K
R
R D
R
D
OU
ROU D
R T D
R T D
R D
I M
RI KM
R K N
M RO D
M RRROUTUT DDD
R
OUT
= +RR ×DD′ ×× ++
′
× +
× +
R K
RII KMM
RI KM
R K
RI KM
R ××××K ++++
× +
×_{} +
× +
× +
R K
R K
R K
R ××××K ++++
== ++RR ×DD′
= × ′
2
= +2
= + 2 111 KK 111 1
2
, ,
ω DDD
L
C R
D
R K
K
D
P
ou
Cou R
C t RO
C t RO
C R UT RRII KKMM
2
2
1 2 1
and
ω =P
ω =P ××× +++ ′ ××× +++
′
× +
× +
R K
RII KMM
RI KM
R K
RI KM
R ××××K ++++
× +
×_{} +
× +
× +
R K
R K
R K
R ××××K ++++
1 2
1 2
× +
× +
C _{}R
C tt ROO
C t RO
C _{}R
C t RO
C ××××_{}R ++++
× +
×_{} +
× +
× +
C _{}R
C R
C _{}R
C ××××_{}_{}R ++++
.

The coefﬁ cients for the linear model of the current-mode
buck-boost regulator shown in **Fig. 5** are:

V V V D V
V V
V
V V
M D
AP
V_{AP} V
V V_{IN}
V V_{IN}
V V VVVV_{OU}_{OU}_{T}_{T} DDDD VVOUOUT
IN
V_{IN} V
V V_{OU}
V V_{OU}
V V _{T}
IN
V_{IN}
V
IN
V_{IN} V
V V_{OU}
V V_{OU}
V V _{T}
= +
V =V +
V =V_{IN}_{IN}+
V V_{IN}
V =V +
V V_{IN}
V V =
V +V
V V ′ = − =VV +VV
=
′
,
V , D
V D , D
D ,
( D)
( D)
= −( )
= −((1 )) ,
= −( )
= −1
= −( )
= −
III V M
R F
K
V
C AP
V_{AP} M
V M
OUT
M
FM
F M
AP
V_{AP}
V
=VV ×MM and = .

The control-to-output simpliﬁ ed transfer function is:

v v R D R K s s s OUT C OU ROU D R T D R T D R D I D RI KD R K R Z P =RR ×DD′ R ×K R K RI KD R ×K RI KD R K ×

The control-to-output simpliﬁ ed transfer function is:

The control-to-output simpliﬁ ed transfer function is:

The control-to-output simpliﬁ ed transfer function is:

The control-to-output simpliﬁ ed transfer function is:

s s s s R Z R Z × +

The control-to-output simpliﬁ ed transfer function is:

The control-to-output simpliﬁ ed transfer function is:

s s s s R Z R Z

The control-to-output simpliﬁ ed transfer function is:

The control-to-output simpliﬁ ed transfer function is:
1 1
1 s 1 s
1− s 1 s
1− 1
1 ss 1 ss
1 ss 1 ss
1 1
1 _{}_{} 1
1 _{} 1
1 1
× +
1 × +1
1 ss 1 ss
1 ss × +× +1 ss
1 × +× +× +1
× +
1 × +1
× +
× +_{}
× +
1 × +× +_{}_{}1
1 × +× +× +× +_{}_{}1
1 × +× +× +× +× +_{}1
× +_{}
× +
× +
1 × +1
× +_{}
× +
× +
1 1
1 s 1
1+ s 1
1+ 1
1 1
1 1
1 _{}_{} 1
1 _{} 1
1 1
×
1 × 1
ωR ωZ
ωR ωZ
ωRR ωZZ
ωRR ωZZ
ω ω
ωRR ωZZ
ωRR ωZZ
ω ω
ω +++
1 1
1 1
1 1
1 1
1 _{}1
1 1
1 1
1 _{}1
1 1
1 1
1 1
1 _{}1
1 1
1 1
s
L
ω
,

and the line-to-output simpliﬁ ed transfer function is:

v
v
R D D K
R K
s s
s
OUT
IN
OU
ROU D
R T D N
R T D N
R T D DD KKN
I D
RI KD
R K
K Z
P
=RRRRRR TTTTTT× ×× ×× ×× ×DDDDDD DDDDDDDD′′××KKKKKKKKNNNNNN
R ×K
R K
RI KD
R ×K
RI KD
R K ×
s s
s s
K Z
K Z
× +
s s
s s
K Z
K Z
+
1 s 1 s
1 s 1 s
1 s 1 s
1 s 1 s
1 1
1 s 1 s
1+ s 1 s
1+ 1
1 ss 1 ss
1 ss 1 ss
1 1
1 _{}_{} 1
1 _{} 1
1 1
× +
1 × +1
1 ss 1 ss
1 ss × +× +1 ss
1 × +× +× +1
× +
1 × +1
× +
× +_{}
× +
1 × +× +_{}_{}1
1 × +× +× +× +_{}_{}1
1 × +× +× +× +× +_{}1
× +_{}
× +
× +
1 × +1
× +_{}
× +
× +
1
ωK ωZ
ωK ωZ
ωKK ωZZ
ωKK ωZZ
ω ω
ωKK ωZZ
ωKK ωZZ
ω ω
ω
× +
× +
× +
× +
× +
× +_{}
× +_{}
× +
× +_{}
× +
× +
× +
× +_{}
× +
× +
1
× +1
× + s
L
ω
,
where
K D R D
R K
K
D
K
K
K
D
R D
R D
D
KD D
K D RRRROUOUTT DDDD
I M
RI KM
R K
N
M
I
RI D
R D
OU
ROU D
R T D
R T D
R D
K = +D
K D+RR ×DD′ ××× +++
′
× +
× +
R K
RII KMM
RI KM
R K
RI KM
R ××××K ++++
× +
×_{} +
× +
× +
R K
R K
R K
R ××××K ++++
= −
= − + RR ×DD
R ×D
R D′
K 1 D
K D
K = +D
K 1 D
K = +D
K D 1
1
2
2
,
, ωRRR
OUT
ROU D
ROUT D
R T D
L D
=RR ×DD′
L D×
L D
2
,
ω
ω
K
OUT N
P
OUT OUT I M
ROU D
ROUT D N
R TTT D KKNNN
L K
C
D
R
T RO
T O
D
RI KM
RI KM
K
D
=RRRR TTTTTT××DDDD′ ×× NNNNNN
L K×
L K
= ×
= × + + ′ ××× +++
′
× +
× +
I M
I M
R K
RI KM
RII KMM
RI KM
× +
× +
×_{} +
×_{} +
× +
×_{} +
× +
× +
R K
R K
R K
R ××××K ++++
2
2
1 1 1
and
1 1
1 1
1 1
1 1
T O
T O
.

### 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
sampling-gain term H_{P}(s). To use the forward-path sampling-gain
term, slope compensation was implemented with a ﬁ xed
ramp. The results will be slightly different if a proportional
ramp is used, as this modiﬁ es the modulator gain term K_{M}
and feed-forward term K. For an actual boost-converter
implementation with a ﬁ xed ramp, it is only possible to get

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Fig. 5.* The low-frequency linear model for this buck-boost regulator *
*was made using SIMetrix.*

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 f_{SW }
the two models at half the switching frequency, where f_{SW }
the two models at half the switching frequency, where f
= 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 de-gree 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 switching-model simulation results. The low-frequency switching-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 denomi-nator 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.

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Fig. 4.* The low-frequency linear model for this boost regulator was *
*made using SIMetrix.*

### CURRENT-MODE CONTROL

### CURRENT-MODE CONTROL

### Uniﬁ ed Modulator Modeling

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 identiﬁ ed. The importance of
the concept of K_{M} as the modulator voltage gain cannot be
overstated. Most linear models for current-mode control
have allowed the math to deﬁ ne the model. In reference 1,
an intuitive understanding of the modulator was used to
drive the math. By algebraic manipulation, both the averaged

model and continuous-time model were redeﬁ ned to ﬁ t the form of the uniﬁ ed model. Combining the uniﬁ ed-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
deﬁ ned that accommodates any ﬁ xed-frequency peak- or
valley-derived operating mode. Limitation of the
forward-sampling-gain term has been identiﬁ ed, 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 Uniﬁ ed Model for
*Current-Programmed Converters,” IEEE Transactions on *
Power Electronics, Vol. 10, Issue 4, pp. 397-408, 1995.
Fig. 6.* This comparison of control-to-output transfer functions for a peak-current-mode boost converter using ﬁ xed-slope compensation *
*reveals the switching and linear models behave similarly, with a slight discrepancy at 100 kHz, which is half the switching frequency.*

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### CURRENT-MODE CONTROL

### CURRENT-MODE CONTROL

### CURRENT-MODE CONTROL

### CURRENT-MODE CONTROL

**Topology**

**iL**

**LE**

**E(s)**Buck iiiiiLL=iiiii LLLLLEE=LLLLL

_{E s}V D OF VOF V F ( ) E s( ) E s = Boost iiiiiLL=iiiiiG LE= L ′ ( )D ( )D′ ( )′2 E s V s L D V I OF VOF V F OFFF IIOON ( ) E s( ) E s ( / D (V / D (VOFOF / D VOF D (V / D VOF D (V FF/ OO =V × =VOF × =VVOFOF × =VVOFF× = F× − s L× s L ′ D′ V D V D × V D V 1 Buck-boost iiiiiiLLLLL= += += += += += += += += +iiiiii iiiGGGGG

_{L}L E= ′ ( )D ( )D′ ( )′2 E s V D s L D V I OF VOF V F OFFF IIOON ( ) E s( ) E s ( / D (V / D (VOFOF / D VOF D (V / D VOF D (V FF/ OO = × = OF × = OFF× = F×

_{−}s Ls L× ′ D′ V D V D × V D V 1 Notes: VVVVVVVOFOFFF======V ; VVVVVVAAPPPPPPPP ; ; IIION======IIICCCCCCCC