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Alternative Way to
Develop Small-Signal
Models of Power
Converters
by Liyu Cao
T
he small-signal model of a switch-mode power converter is a useful
tool in understanding its dynamic behavior and designing a proper
compensator. In the power electronics community, people derive
small-signal models almost always by perturbation and then
extracting the small ac components manually from the whole
Digital Object Identifier 10.1109/MPEL.2024.3352244
Date of publication: 9 February 2024
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system equations. This process is not systematic and
potentially error prone. This article presents an alternative process to develop small-signal models based on a
well-established systematic method that has been used in
control system analysis and design for a long period. With
this math-centric approach, one just applies standard
math operation to linearize a nonlinear system described
by its state-space equation, and then uses a well-known
formula to derive the transfer functions of the linearized
system. This is a precisely defined and systematic process
with the advantage of deriving all the transfer functions
with a minimum effort.
Introduction
The small-signal model of a switch-mode power converter is a useful tool to engineers in understanding its
dynamic behavior and designing a proper compensator
to control its output voltage or current. The small-signal
modeling techniques have been known to the power
electronics community for decades. A small-signal mode
is usually the end target or final result of a more general
modeling technique for switched dynamic systems–the
averaging method. There are basically two kinds of averaging methods for switch-mode converters: 1) statespace averaging method; 2) PWM-switch circuit
averaging method. In the first method, a mathematical
averaging process is applied to the state-space equations
of the power converter that periodically switches
between two states. In the second method, a mathematical average process is applied only to the switching circuit components and then these components may be
replaced with non-switching components. With either of
the averaging methods, a non-switching but usually nonlinear model can be produced, which, with the first averaging method will stay in the form of state-space
equation; and it will be in the form of circuit model if
obtained by the second averaging method.
The non-switching, nonlinear model is also named
as large-signal model, suggesting it can be used for
analyzing the converter’s behavior under a large amplitude change with internal/external variables. Since this
model is usually nonlinear, it’s not easy to use for the
purpose of stability analysis or control loop design.
This is the main reason why the so-called small-signal
model is needed, which is a linearized version of the
large-signal model and can be represented in the form of
s-domain transfer function and directly used for control
loop analysis and design.
The central topic of this article is about how to derive
a small-signal model from the large-signal model, or
in other words, how to linearize a nonlinear system
described by its state-space equation. In the power electronics community, the linearization process described in
the power electronics textbook and papers [1], [2], [3], [4],
[5] and used in practice can be described simply as perturbation and then extracting the small ac components
manually from the whole system equations, which is not
systematic, and potentially an error-prone, and tedious
process. This article presents an alternative process to
develop small-signal models based on a well-established
systematic method that has been popular and widely
used in control system analysis and design since 1960s.
This alternative approach uses two mathematical tools to
facilitate the linearization process: the Jacobian matrix
to derive the linearized state-space equation and, the
inverse matrix method to derive the s-domain transfer
functions. With this matrix-centric approach, one just follows standard partial derivative operation to linearize a
nonlinear system described by its state-space equation,
and then uses a well-known formula to derive the transfer functions of the linearized system. This is a precisely
defined and systematic process with the advantage of
deriving all the transfer functions with a minimum effort.
As far as this author knows, [9] is the only publication
that has used this approach to do small-signal modeling
on power converters, and it is virtually unknown to the
power electronics community. The purpose of this article
is to introduce in detail the alternative modeling method
to a large audience, and through the Buck converter
example to demonstrate how this modeling method may
be used in linearizing the state-space equation and deriving the frequency domain transfer functions for a switching-mode power converter.
In recent years, more rigorous small-signal modeling
methods have been proposed and developed for switchmode power converters, such as based on the Description
Function and Harmonic State-space methods [11], [12].
The main goal of these advanced modeling methods is to
improve the accuracy of the traditional averaging method
in frequencies close to half of the switching frequency,
which is beyond the scope of this article. Readers interested in these advanced topics may refer to [11] and [12] and
the references there.
Review of Traditional Way to Derive Small-Signal
Model of Buck Converter
In this section, using a Buck converter as example, the traditional way of perturbation-based linearization is reviewed.
Here the PWM switch averaging method is used to derive
the large-signal model.
The equations describing the averaged voltage and current of the PWM switch in continuous-conduction mode
(CCM) are given by
vcp = dvap (1)
ia = dic (2)
where d is the PWM duty-cycle. One can imagine that the
above equations describe the circuit elements of controlled
voltage and current sources, or an ideal transformer [1] with
the turn ratio of d. By replacing the PWM switch in Figure 1
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63
FIG 1 Circuit diagram of Buck converter.
FIG 2 Circuit model of the Buck converter after averaging.
with an ideal transformer, the Buck converter in Figure 1
becomes the circuit shown in Figure 2, which doesn’t have
switching elements and is easier to analyze.
Although the circuit in Figure 2 doesn’t have a switching element, or is not time-varying, but it is a nonlinear system (this will be clear later when its state-space equation is
presented) and needs to be linearized so it can be analyzed
using the Laplace transformation method. The traditional
way of linearization starts by perturbing the average equation as follows.
Vcp + vˆ cp = ( D + dˆ ) ( Vap + vˆ ap ) (3)
I a + iˆa = ( D + dˆ ) ( I c + iˆc ) (4)
where the variables in capital letters are their dc components or equilibrium, and the variables superscripted with
“^” are the small ac variations. By equalizing the ac components on both sides of (3) and (4), and also ignoring the 2ndorder ac components or the products of ac quantities, we
can obtain the following equations.
ˆ ap (5)
vˆ cp = Dvˆ ap + dV
ˆ c (6)
iˆa = Diˆc + dI
These equations describe the small-signal behavior of the PWM switch with three terminals a, c and p.
Once again, in terms of circuit elements these equations
represent controlled voltage and current sources. Based
on these equations, the small-signal circuit model of a
Buck converter can be constructed as shown in Figure
ˆ ap .
3, and with the added controlled source of d̂I c and dV
As the name suggests, the small-signal mode in Figure 3
describes small-signal ac behaviors of the Buck converter,
64
FIG 3 Small-signal circuit model of the Buck converter.
where the variables with the circumflex mark ^ represent
their ac variations.
The circuit in Figure 3 seems to be more complicated
than that in Figure 2, but it is a linear circuit since Ic and
Vap are constants. To get the transfer functions, as shown
in [4] one can convert the circuit into the frequency domain
(the s-domain), and then work out the transfer functions by
using circuit analytic techniques.
With the state-space averaging method described in
[1], [2], [4], and [5], the averaging operation is applied to 2
sets (or 3 sets in case of discontinuous conduction mode)
of state-space equations that describe the sub-switching
cycles behaviors of a power converter, and leads to 1 set
of averaged state-space equations. In terms of linearization, the same perturbing approach is used by this method,
which leads to even more demanding mathematical operations since more equations need to be perturbed.
Alternative Approach to Derive Small-Signal Model
With the proposed new way to derive the small-signal
model, only the large-signal circuit model shown in Figure 2
will be needed, and the manually perturbing operations will
not be required. In fact, modern circuit simulation tools
such as PSpice can do the linearization work for a nonlinear
circuit shown in Figure 2. By setting the so-called AC Analysis with PSpice, one can get the small-signal model’s frequency responses directly without touching the small-signal
circuit model shown in Figure 3.
Although a simulation tool like PSpice can do some
kind of small-signal analysis numerically, it does not provide desirable analytical formular, such as the s-domain
transfer functions. This kind of transfer function is very
useful, since it shows the model’s pole/zero positions in
terms of the circuit parameters and operation condition.
Therefore, there is a need to have a systematic way to
derive the transfer function model based on the largesignal circuit model, besides the heavily manual operation involved perturbation-based linearization process
described above.
First of all, a brief introduction is provided on a standard
process to linearize a nonlinear system described by its
state-space equation, and on the relationship between the
state-space description and transfer function description.
In general, a nonlinear system is described by
X = f ( X ,U ) (7)
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where X is the n×1 state-space variable vector, U is the
input vector of l×1, and f represents a n×1 vector of nonlinear functions. To linearize the nonlinear system, or derive its
small-signal model, first of all one needs its steady-state
solution or the equilibrium, which is the solution of the following equation obtained by setting the derivative vector
X = 0 given U = U0,
Appendix A1
The Jacobian Matrix of A 2 × 1 Vector.
As has been stated previously, for modeling widely used power
converters we may treat them as having 2 state variables and 2
input variables. Here we just consider a nonlinear system with 2
state variables x1 and x2, as well as 2 input variables u1 and u2. In
general, this nonlinear system can be described by
(8)
x1 = f1 ( x1 , x 2 , u1 , u2 ) A1
As can be found in textbooks on Nonlinear System Analysis, for example in [6], the linearized system at the equilibrium X = X0, which is usually called small-signal model in
the context of power converters, can be obtained by using
the concept of Jacobian Matrix, and the linearized statespace equation is represented by
x 2 = f2 ( x1, x 2 , u1, u2 ) A2
f ( X 0 ,U ) = 0 x = Ax + Bu (9)
where A is the n×n Jacobian matrix and obtained by taking
partial derivatives on the vector of nonlinear function f with
respect to X and evaluated at the equilibrium X = X0 under
U = U0
A=
∂f
(10)
∂X X = X0 , U =U0
and B is another Jacobian matrix by taking partial derivatives of f with respect to U and evaluated at the equilibrium
X = X0 under U = U0
∂f
B=
∂U X = X0 , U =U0
(11)
The system’s output vector y of m × 1, is usually written
in terms of the state variable x.
y = Cx (12)
where C is called the output matrix of m × n.
Equations 9 and 12 are the so-called state-space representation (or model) of a linear system at the operating
point X = X0. The Jacobian matrix defined in Equation 10
(also in Equation 11) is a key element in the proposed smallsignal modeling method, and this matrix is a well-defined
mathematic concept and requires partial derivative operation on the vector of function f which defines the original
nonlinear system.
For the most popular and widely used power converters such as Buck, Boost, and Buck-Boost, they have two
state variables as the capacitor voltage and the inductor current, therefore the dimension of f is 2 × 1. The
Jacobian matrix for a 2 × 1 vector is shown in detail in
Appendix A1.
The above linearization approach based on Jacobian matrices can be applied to any nonlinear system
Then for the 2×1 vector f defined as
 f (x , x , u , u )
f ( x1, x 2 ) =  1 1 2 1 2 
 f2 ( x1, x 2 , u1, u2 ) 
Its Jacobian matrix with respect to x1 and x2 is given by the
following 2 × 2 matrix
 ∂f1
∂f  ∂x1
A=
=
∂x  ∂f2
 ∂x1
∂f1 
∂x 2 

∂f2 
∂x 2 
A3
And its Jacobian matrix with respect to u1 and u2 is also a 2 × 2
matrix
 ∂f1
∂f  ∂u1
B=
=
∂u  ∂f2
 ∂u1
∂f1 
∂u2 

∂f2 
∂u2 
A4
equations and certainly not limited to those derived from
the circuit averaging method, which is demonstrated in
this article. As we can see Equation 7 is a very general
representation of nonlinear state-space equations and
includes those derived by using the state-space averaging method.
With the linearized state-space equations determined,
all the transfer functions from the system input u to its
output y can be derived as shown below. First by applying
Laplace transforms on x to (9) and then taking matrix operation, one can get the following equation
x = ( sI − A )
−1
Bu (13)
In the above equation, I is an identity matrix with the
same dimension as A, and (sI-A)-1 represents the inverse
matrix of (sI-A). Now by substituting (13) into (12) one can
get the following matrix equation relating the output vector
y to the input vector u,
y = C ( sI − A )
−1
Bu
The above equation defines a matrix of transfer functions from the vectors u to y, as given by
G ( s ) = C ( sI − A )
−1
B
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(14)
65
Note that G(s) is a m × l matrix, and its dimension
depends on the numbers of inputs and outputs. To avoid
using the seemingly complex concept of matrix of transfer functions, one can always make (14) be just a scalar
function by using the concept of single-input and singleoutput system, that is, by considering just one input and
one output at a time. To make this happen, B needs to
be a n × 1 column vector, and C needs to be a 1 × n row
vector.
For a power converter, it is usually considered to have
two inputs: the input voltage and the PWM duty cycle. The
output voltage is most widely considered to be the output
variable, which, in the most popular converter topologies like Buck and Boost, is directly associated with the
state variable of the capacitor voltage1; and sometimes
the inductor current also needs to be considered as an
output, for example in modeling and designing a currentmode converter. If we consider a power converter has two
inputs and two outputs, then the transfer function matrix
G(s) defined by Equation (14) will be 2 × 2, or will have
four transfer functions. One of the advantages of the proposed modeling method based on the state-space equation framework is that it can treat and derive all of the
transfer functions of a converter altogether in a systematic way.
To derive any of the transfer functions using (14), the
first thing needed is the inverse matrix of (sI-A), which can
be calculated by using the following inverse matrix formula.
( sI − A )−1 =
1
adj ( sI − A ) det( sI − A)
(15)
where det (sI-A) is the determinant of sI-A or the characteristic polynomial of A, and adj(sI-A) represents its adjoint
(also called adjugate) matrix. Some practicing engineers
may not be familiar with the concept of matrix determinant
or adjoint matrix. Fortunately, for the most popular and
widely used power converters such as Buck, Boost and
Buck-Boost, the corresponding matrix A is 2 × 2, for which
the inverse matrix formula is quite simple and can be calculated without much effort as shown in Appendix A2.
Equation (15) tells us all the transfer functions have
the same poles, which are the roots of the characteristic
polynomial of A. Furthermore, we can see that the poles
are only dependent on the matrix A, and not dependent on
how the inputs and outputs are selected. These insights are
made available with the proposed new approach to derive
small-signal models, and are not provided with the traditional approach of small-signal modeling.
In the following, the small-signal model or the transfer
functions of the Buck converter shown in Figure 2 will be
derived based on Equation (14), by which the main steps
and operations involved with the proposed modeling
1
If the ESR of the capacitor is zero, then the output voltage is equal to the
capacitor voltage.
66
Appendix A2
The Determinant and Inverse of A 2 × 2 Matrix.
For a 2×2 matrix defined by
a b
A = 
 c d 
A5
det A = ad − bc A6
Its determinant is equal to
If det A is not equal to zero, then the inverse of A exists and can
be calculated as
A−1 =
1 d
det A  −c
−b 
a 
A7
As Equation A7 shows, computation of the inverse of a 2 × 2
matrix is quite straightforward.
method will be illustrated. Based on Figure 2, one can write
the following equations
iL = C
dvc vo
dv
v
R C dvc
+
=C c + c + c
(16)
dt
R
dt
R
R dt
vcp = dvin = L
diL
dv
+ RL iL + vc + RcC c dt
dt
(17)
To analyze a circuit based the state-space method, the
inductor current iL and capacitor voltage vc are considered
to be the state-space variables, and by their nature the input
voltage vin and duty cycle d are defined as the input variables, thus we have
v 
x =  c ,
 iL 
v
u =  in 
d 
It is worth to note that here the input u is treated as a
vector which includes the duty cycle in addition to the input
voltage vin, while in the traditional approach of using statespace equation to model power converters [2], [3], [5], only
vin is taken as the input variable. The advantage of treating
the input u as a vector is that, all the transfer functions normally used for power converters can be derived and analyzed by the unified formula of (14), and this will become
evidenced as we move forward from here. To get the standardized form of state-space equation shown in Equation
(7), Equations (16) and (17) are rewritten as
dvc
1
1
=−
vc +
iL R
dt
C ( R + Rc )

C  1 + c 
R 

diL
R
R + RL
d
=−
vc − 2
iL + vin dt
L( Rc + R )
L
L
where R2 is the parallel combination of R and Rc,
R2 =
RRc
.
R + Rc
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(18)
(19)
As can be seen from Equations (18) and (19), the only
nonlinear item d × vin is about the input variables vin and
d, not about the state-space variables vc and iL. Therefore,
linearization is actually only needed to the inputs vin and d.
By applying Equations A3 and A4 provided in Appendix A1
to Equations (18) and (19), the Jacobian matrices A and B
can be easily obtained as
1

− C ( R + R )
c
A=
R

 − L( R + R )

c
B = [ B1
R

C ( Rc + R ) 

R2 + RL 
−

L

0
B2 ] =  D

L
0 
Vin  
L 
(20)
(21)
where B1 and B2 are column vectors and associated with vin
and d respectively. The linearized (small-signal) state-space
equation in the matrix form is given by
 dvˆ c 
 dt 
 vˆ c 
 vˆ in 
 ˆ  = A  ˆ  + B  ˆ . i
di
d 


 L
L
 dt 
(22)
R

C ( Rc + R ) 
 . (23)
1

s+
C ( R + Rc ) 
where p(s) is the determinant of (sI-A) or the characteristic
polynomial of A defined by
p ( s ) = det ( sI − A )
R + RL (24)
1
 R + RL

= s2 +  2
+
 s + ( R + R ) LC
L
R
R
C
+
(
)


c
c
To use (14) for deriving the transfer function from d̂ to v̂c,
we need the column vector B2 associated with d̂, and a row
vector for C. The following equation defines the required
row vector C.
 vˆ 
y = vˆ c = Cx = [1 0 ]  c  .  iˆL 
vˆ c
−1
= C ( sI − A ) B2
dˆ
R + RL

s+ 2
L
1 

= [1 0 ]
−R
p ( s) 
 L( R + R )

c
R

0 
C ( Rc + R )  
  Vin 
1


s+
 L 
C ( R + Rc ) 
After completing matrix operations in the above equation, one can get
1
vˆ c
RVin
=
p ( s ) LC ( R + Rc )
dˆ
(25)
(26)
The transfer function from duty cycle d̂ to the output
voltage v̂o instead of v̂c, which is more used in design practice, can be obtained based on the following relationship
vˆ o
= 1 + RcCs vˆ c
Gdo =
vˆ o vˆ o vˆ c
1 RVin (1 + RcCs )
=
=
=
p ( s ) LC ( R + Rc )
dˆ vˆ c dˆ
Using Equation (14) one can derive any transfer function
from the input variables (vin and d) to the selected outputs,
such as the converter’s output voltage and inductor current.
In the following, the transfer functions from the duty cycle
d to output voltage and inductor current are to be derived to
demonstrate the proposed small-signal modeling method.
To calculate the transfer functions using Equation (14), the
first step is to compute the inverse matrix (sI-A)-1. For the
2 × 2 matrix A given in Equation (20), one can get the following inverse matrix by using Equations A6 and A7 provided
in Appendix A2
R + RL

s+ 2

L
1

( sI − A )−1 =
−R
p ( s) 
 L( R + R )

c
Then we can get the transfer function from the duty
cycle d̂ to the capacitor voltage as,
(27)
Vin RcCs + 1
R
1 + L p1 ( s )
R
(28)
where p1(s) is a different expression of the characteristic
polynomial of A given by
( R + RC ) LC
p ( s)
R
(29)
R + RC
L 

LCs 2 + C ( Rc + R3 ) +
s
=
+
1
R + RL
R + RL 

p1 ( s ) =
where R3 is the parallel combination of R and RL,
R3 =
RRL
R + RL
It can be seen that at dc (s = 0), p1(s) = 1, so the dc gain of
Gdo is equal to the input voltage Vin divided by 1+RL/R. At dc
Equation 28 can be written as
DVin
Vo =
(30)
R
1 + RL
Equation (30) shows how the inductor parasitic resistance RL changes the ideal conversion factor of a Buck converter, as discussed in [2].
By comparing Equation (28) with the corresponding
transfer function derived in references such as Equation
(2.59) in [2], one can see it is the same as what has been
derived based on the traditional approach.
Next let us consider another transfer function from d̂ to
the inductor current îL . When taking îL as the output, the
output equation is given by
 vˆ 
y = iˆL = Cx = [0 1]  c   iˆL 
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(31)
67
Note the difference between Equations (25) and (31).
And keep in mind that no matter which transfer function is
in consideration, the inverse matrix given in Equation (23)
is always the same. By applying Equation (14) again, one
can derive the transfer function from d̂ to îL as follows.
iˆL
−1
= C ( sI − A ) B2
dˆ
R + RL

s+ 2

L
1

= [ 0 1]
−R
p ( s) 
 L( R + R )

c
R

0 
C ( Rc + R )  
  Vin 
1


s+
 L 
C ( R + Rc ) 
By completing matrix operations in the above equation,
one can get
1
Gdi =
iˆL s + C ( R + Rc ) Vin [( R + Rc ) C ] s + 1 Vin
=
=
(32)
R + RL
p ( s)
L
p1 ( s )
dˆ
Equation (32) is named as a wide-frequency transfer
function from the duty cycle to the inductor current in [7],
where it is derived based on the small-signal circuit model.
At dc (s = 0), since p1(s) = 1, so the dc gain of Gdi is equal to
the input voltage Vin divided by the resistance sum of R+RL.
When the parasitic resistance RL and RC are zero, it can be
seen that Equation (32) is the same as what is provided by
Table 10.2 in [3]. The transfer function of (32) is particularly
useful in the average current-mode control, as has been discussed in detail in [8].
As can be seen, once the inverse matrix of (sI-A) has
been worked out, derivation of a transfer function is quite
straightforward. From Equation 14, we see that any transfer
function of the power converter includes the inverse matrix
of (sI-A)-1 given by (23), which determines the poles of the
transfer function. Therefore, we see that the difference
among different transfer functions may be in their dc gains
and zeros, but they all share common poles determined by
the characteristic polynomial of A. This is the insight provided by the new way of small-signal modeling and is not
available with the traditional modeling approach.
the same load condition is taken in the following calculation and simulation. Equation (28) is used to calculate the
frequency response for the Buck converter parameterized
in Table 1, and the large-signal model shown in Figure 2 is
simulated in the ac sweep mode with PSpice to generate
the frequency response for the same Buck converter. The
s-domain transfer function Gdo can be easily calculated by
using MathWorks’s MATLAB, or by using Microsoft’s Excel
with a little bit more of effort.
The larger-signal mode shown in Figure 2 represents
the averaged PWM switch with an ideal transformer, which
needs to be replaced with available circuit elements from
the simulation tool PSpice. Figure 4 is the averaged PWM
switch model implemented and tested in PSpice by this
author, whose port voltages and currents satisfy Equations
1 and 2. Compared with Figure 2, the terminal d is added to
represent the duty cycle command, which is another input
to the PWM switch and may be modeled with a voltage
source in PSpice simulation.
Figure 5 provides the calculated Bode plots of the
s-domain transfer function Gdo and also simulated ones
based on the large-signal PSpice model of Figure 4. With
no surprise, the plots calculated based on the signal-signal
s-domain transfer function align completely with those
simulated based on the large-signal model, which is another
evidence that the small-signal model derivation method
Table––1. Parameters of the Buck Converter Example.
Parameter
Value
Vin
15V
Vo
5V
L
33μH
RL
90mΩ
C
230μF
Rc
40mΩ
An Illustrative Example
So far only analytical results have been derived and presented in terms of state-space matrix equations and
s-domain transfer functions, and it could be beneficial to
apply those analytical results to a real example. In a case
study demonstrated in [9], the small-signal frequency
response of a Buck converter is provided based on theoretical calculation and real measurement. Here that Buck converter will be used as a numerical example, by which the
analytical s-domain function derived previously can be converted into graphical Bode plots and compared with simulation result provided through PSpice large-signal simulation.
The parameters of the Buck converter of [9] are provided
in Table 1.
In [9] the calculated and measured Bode plots of Gdo
are provided for the load resistance value of 1MΩ. Here
68
FIG 4 Averaged PWM switch model in PSpice.
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ogies like Buck and Boost converters, the state-space
equations are second-order or have two state variables.
3) Linearize the large-signal state-space equation by calculating partial derivatives with respect to the state variables and the input variables, and result in the Jacobian
matrices A and B. See details in Appendix A1.
4) Compute the inverse matrix (sI-A)-1 using Equation
(15). For popular converter topologies, this involves
with computing the inverse of a 2 × 2 matrix. See
details in Appendix A2.
5) Determine what are the input and output variables for
which the transfer function needs to be derived, and correspondingly derive the output row vector C. Then use
Equation (14) to derive the needed transfer function by
doing matrix operations.
The above procedure is systematic in nature by following the well-established procedure that has been used
in control system analysis and design for a long time. It is
the hope of this author that the power converter designers
may become familiar with this alternative linearization and
transfer function deriving method and start to use it in their
design practice.
About the Author
FIG 5 Bode plots of Gdo from s-domain transfer function and
PSpice simulation.
Liyu Cao (liyucao@yahoo.com) received the Ph.D. degree
in electrical engineering from Tsinghua University, Beijing,
China. He is a Senior Engineering Manager with XP Power,
San Jose, CA, USA, where he is engaged with power supply
product development and testing.
References
showcased in this article is in line with the well-established
linearization method adopted in commercial simulation
tools like PSpice. This expected correspondence between a
small-signal model’s frequency response and its associated
large-signal model-based simulation suggests that PSpice
(or other similar tool) simulation based on the large-signal
model could be used as a way to assess if a derived smallsignal model is reasonably good or not.
By comparing the plots of Figure 5 with those corresponding plots for the example Buck converter—Figure 5
of [9], one can see that they match very well.
Summary
In this article, a new approach to small-signal modeling and
transfer function derivation has been introduced. This
approach is based on the state-space equation of large-signal model either through circuit averaging or state-space
averaging process, and as has been demonstrated above,
can be described as the following steps:
1) Derive the large-signal circuit model based on the PWMswitch averaging technique. For widely used converter
topologies, this kind of model is already available in the
literature.
2) Identify the state variables of the circuit model and then
derive its state-space equations. For most popular topol-
[1] R. Erickson and D. Maksimovic, Fundamentals of Power Electronics,
2nd ed. Norwell, MA, USA: Kluwer Academic, 2001.
[2] R. P. Severns and G. Bloom, Modern DC-to-DC Switchmode Power Converter Circuits. San Rafael, CA, USA: e/j Bloom Associates, 1985.
[3] B. Choi, Pulsewidth Modulated DC-DC Power Conversion-Circuits,
Dynamics and Control Designs. Hoboken, NJ, USA: Wiley, 2013.
[4] R. D. Middlebrook, “Small-signal modeling of pulse-width modulated
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[8] D. K. Saini, “True-average current-mode control of DC-DC power converters: Analysis, design, and characterization,” Ph.D. thesis, Dept. Elect. Eng.,
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[9] L. Cao, “Small signal modeling for phase-shifted PWM converters with a
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[10] P. Smeets, “Digital compensators—Designing A FPGA based digital
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[11] Y. Qiu et al., “Multifrequency small-signal model for buck and multiphase buck converters,” IEEE Trans. Power Electron., vol. 21, no. 5, pp.
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[12] X. Yue, X. Wang, and F. Blaabjerg, “Review of small-signal modeling
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