##  1 Introduction

Small-signal modeling of DC-DC converters has been a subject of research for
several decades now, resulting in two main families of modeling techniques:
state-space averaging [22] and switch or circuit averaging [39, 10]. State-
space averaging has become a broadly used modeling approach for DC-DC
converters [14], and is considered the most effective method for building
small-signal models [43]. It continuously received extensions, for example,
for converters in discontinuous conduction mode (DCM) [11, 28] or variable-
frequency operation [26], and new theoretical results to this day [21].

The typical workflow using state-space averaging (SSA) includes applying
Laplace transform and deriving frequency-domain transfer functions
characterizing input/output impedance or control-oriented behavior [24], which
must then be maintained individually. Thus, whenever the converter is being
extended, for example, by control loops, this leads to the tedious and
potentially error-prone work of having to adapt all frequency-domain models
separately [38]. Even worse, the underlying state-space model might have to be
derived again if circuit elements (such as an input filter) are added to the
converter. Basso [7] calls this the “pain of the SSA technique.”

For easier modeling, the idea of modular approaches has emerged in the last
two decades, be it for single converters [40], or even working towards
automatic modeling solutions for microgrids [12, 13]. Building and analyzing
models for multi-converter configurations has of course been of increasing
interest since the advent of DC microgrids [14, 15], but has been the subject
of research long before, especially when studying converter interactions in a
cascaded source-load setup [16, 18, 23]. Matters of particular interest are
impedance analysis [2, 4, 38], implications for controller design [1, 16, 25]
and, of course, stability [3, 18, 23]. It has been pointed out [8, 36] that
studying detailed dynamic interactions between source and load converters is
important for stability analysis, going beyond constant power load
assumptions.

Many existing studies adopt an inverse-hybrid two-port description of
converters (also known as g-parameters) [20], which lends itself well to the
behavioral modeling of converters and converter systems [35, 42], especially
in the case of unterminated converter models [5, 9, 32].

Departing from the prevalent approach of deriving individual frequency-domain
transfer functions for the different aspects of converter (system) modeling,
this article proposes creating and maintaining a single state-space model in
the time domain. It encompasses both control-oriented and electrical
characteristics in a two-port description with inverse-hybrid parameters
(g-parameters), following existing good practice and enhanced by additional
control inputs as necessary. Whenever needed, of course, all relevant
frequency-domain transfer functions can easily be extracted at any modeling
stage from this model, superseding the necessity of updating individual
frequency-domain transfer functions.

In order to relieve the “SSA pain” when building larger models, standardized
(and therefore less error-prone) operations are being proposed for attaching
controllers, closing control loops and creating series-connections of
converter subsystems. These subsystems can be passive circuits such as input
filters, open-loop or controlled converters, or even series-connections of
converters themselves. In that manner, the actual modeling effort is reduced
to modeling smaller building blocks which can later on be reused and combined
to create larger converter system models.

The remainder of this article is organized as follows: The general modeling
approach is presented in Section 2. Modeling examples for building blocks such
as passive components, converters and controllers are given in the Appendixes
A–C, respectively. From these ingredients, a complete model of a converter
system can be built up incrementally. For this purpose, new model connection
operations are introduced in Section 3 (adding control loops to converters)
and Section 4 (series-connecting converter subsystems), which form the main
contribution of this article. Finally, examples for modeling a multiloop-
controlled converter and a two-stage converter system are presented in Section
6.

##  2 Modular Modeling Approach

###  2.1 Model for Passive Components

For passive components of a converter system, such as an input filter or an
electrical load, the two-port state-space model is given in general form in
(1). The number of state variables, and therefore the size of the matrices
$\bm{A}$, $\bm{B}$, $\bm{C}$ depends on the order of the respective system,
that is, the number of storage elements.

$\begin{split}\bm{\dot{x}}(t)&=\bm{A}\cdot\bm{x}(t)+\begin{pmatrix}\bm{B}_{1}&\bm{B}_{2}\end{pmatrix}\cdot\begin{pmatrix}v_{\mathrm{in}}(t)\\\
i_{\mathrm{out}}(t)\end{pmatrix}\\\ \begin{pmatrix}i_{\mathrm{in}}(t)\\\
v_{\mathrm{out}}(t)\end{pmatrix}&=\begin{pmatrix}\bm{C}_{1}\\\
\bm{C}_{2}\end{pmatrix}\cdot\bm{x}(t)+\begin{pmatrix}D_{11}&D_{12}\\\
D_{21}&D_{22}\end{pmatrix}\cdot\begin{pmatrix}v_{\mathrm{in}}(t)\\\
i_{\mathrm{out}}(t)\end{pmatrix}\end{split}$ (1)

Figure 1 illustrates the definition of input and output voltages and currents
for a generic two-port system. Examples for modeling passive subsystems of a
converter system are given in Appendix A.

Figure 1: Generic two-port network. Currents are defined as flowing into the
input and output ports.

###  2.2 Model for Controlled Converters and Converter Systems

Based on Section 2.1, the model for a converter (system) gains at least one
additional input: the control signal $\mathit{ctl}(t)$. For a single DC-DC
converter, this will be the duty cycle in most cases. Equation (1) is
therefore enhanced to (2). Examples are given or referenced in Appendix B.

$\begin{split}\bm{\dot{x}}(t)&=\bm{A}\cdot\bm{x}(t)+\begin{pmatrix}\bm{B}_{1}&\bm{B}_{2}&\bm{B}_{3}\end{pmatrix}\cdot\begin{pmatrix}v_{\mathrm{in}}(t)\\\
i_{\mathrm{out}}(t)\\\ \mathit{ctl}(t)\end{pmatrix}\\\
\begin{pmatrix}i_{\mathrm{in}}(t)\\\
v_{\mathrm{out}}(t)\end{pmatrix}&=\begin{pmatrix}\bm{C}_{1}\\\
\bm{C}_{2}\end{pmatrix}\cdot\bm{x}+\begin{pmatrix}D_{11}&D_{12}&D_{13}\\\
D_{21}&D_{22}&D_{23}\end{pmatrix}\cdot\begin{pmatrix}v_{\mathrm{in}}(t)\\\
i_{\mathrm{out}}(t)\\\ \mathit{ctl}(t)\end{pmatrix}\end{split}$ (2)

Equation (2) is a very general model. Depending on both the approach for
controlling the converter (such as voltage mode or current mode) as well as
the modeling stage during the elaboration of the model, there are different
possible meanings or functions the control signal $\mathit{ctl}(t)$ can
possess: duty ratio, peak or average current reference (in current mode
control, if the model only describes the current loop so far), voltage
reference (in voltage mode or if the outer voltage control loop around a
current-controlled converter has been closed), or even the respective open-
loop inputs if the control loop has not yet been closed in the model.

Additionally, (2) is very general in the regard that the model may contain a
(series) connection of several controlled converters. In this case,
$\mathit{ctl}(t)$ is a vector containing the individual control loop inputs
for the converters in this model.

When connecting input filters or loads to the model according to (2), its
structure remains the same, therefore (2) is being used both for modeling
“bare” converters as well as the final result, that is, a converter system
with one or more controlled converters as well as possible filter and load
subsystems.

###  2.3 Model for Controllers

While controllers—such as a controller for the output voltage or an underlying
average current mode controller—will be modeled in this article in a state-
space representation as well, they require a different treatment and do not
fit into the two-port model category. This is due to the fact that even though
the controllers may be realized in analog form by means of electric circuits,
they represent an information flow rather than a power flow. The controller
model employed here will be a continuous-time state-space model given in (3),
with a single input (the control error value $e$) and a single output (the
control signal $u$ of the respective controller). Examples of controllers
typically used for converters are given in Appendix C.

$\begin{split}\bm{\dot{x}}_{\mathrm{C}}(t)&=\bm{A}_{\mathrm{C}}\cdot\bm{x}_{\mathrm{C}}(t)+\bm{B}_{\mathrm{C}}\cdot
e(t)\\\
u(t)&=\bm{C}_{\mathrm{C}}\cdot\bm{x}_{\mathrm{C}}(t)+\bm{D}_{\mathrm{C}}\cdot
e(t)\end{split}$ (3)

###  2.4 Making Connections

The most simple form of a practical converter system consists of the actual
“bare” converter (such as a buck or boost converter), an (optional) input
filter and a load, cf. Fig. 2. A more complicated case arises when a second
converter is attached as the load system of a first converter.

Figure 2: Two-port network description consisting of three subsystems: input
filter, converter (with a control input $\mathit{ctl}$), and the actual load.

In this article, a solution is presented to create a complete system model for
arbitrary series connections of converter subsystems. The generic model
ingredients have already been presented in Section 2.1 (passive components),
Section 2.2 (converters), and Section 2.3 (controllers). Two different methods
of connecting these model ingredients to a complete system mode are necessary:
(1) connecting a converter with one or more controllers, which will be
presented in Section 3, and (2) connecting (controlled) converter subsystems
or passive subsystems in a series connection, which will be presented in
Section 4.

A typical workflow of using the models and methods presented in this article
is as follows: (1) create a converter model (examples in Appendix B), (2)
create a controller model (examples in Appendix C) and connect the controller
(Section 3) for every control loop of the converter, (3) make series
connections to further controlled converters (if present, Section 4), (4)
create models for input filter and load (examples in Appendix A) and connect
them (Section 4). The latter may, of course, also be connected before
connecting the controller in order to tune the controller according to
possible interactions of the converter with source and load impedances,
especially if the system consists only of a single converter.

##  3 Adding Control Loops to Converter Models

In the following, a general procedure is being described that combines a two-
port converter model featuring one (scalar) control input $\mathit{ctl}(t)$ as
given in (2) with a model of the controller according to (3). Firstly, an
open-loop model is being constructed, followed by the closed-loop model, both
of which maintaining the same interface of the general two-port state-space
model (2).

The control input $\mathit{ctl}(t)$ of the given converter could be the duty
cycle or the set-point of an inner control loop, such as an inner current
control loop. Through the stages of connecting the converter model with the
controller model, the meaning of the model’s control input changes from its
original meaning to the control error value $e(t)$ (open-loop case) and
finally the set-point (reference signal) $r(t)$ of the closed control loop.
The interfaces of the respective two-port models are shown in Fig. 3.

(a)

(b)

Figure 3: Interfaces of the two-port network models with control inputs when
adding a controller to a converter. (a) Open-loop case. (b) Closed-loop case.

In order to obtain the open-loop model, the state variable vector of the
converter model must be enhanced by the state variables of the controller
model. In general, a common model structure of the open-loop and closed-loop
models can be given as follows in (4), with subscripts OL and CL
distinguishing the open-loop and closed-loop case.

$\begin{split}\begin{pmatrix}\bm{\dot{x}}(t)\\\
\bm{\dot{x}}_{\mathrm{C}}(t)\end{pmatrix}&=\bm{A}_{\mathrm{OL/CL}}\cdot\begin{pmatrix}\bm{x}(t)\\\
\bm{x}_{\mathrm{C}}(t)\end{pmatrix}+\bm{B}_{\mathrm{OL/CL}}\cdot\begin{pmatrix}v_{\mathrm{in}}(t)\\\
i_{\mathrm{out}}(t)\\\ \mathit{ctl}_{\mathrm{OL/CL}}(t)\end{pmatrix}\\\
\begin{pmatrix}i_{\mathrm{in}}(t)\\\
v_{\mathrm{out}}(t)\end{pmatrix}&=\bm{C}_{\mathrm{OL/CL}}\cdot\begin{pmatrix}\bm{x}(t)\\\
\bm{x}_{\mathrm{C}}(t)\end{pmatrix}+\bm{D}_{\mathrm{OL/CL}}\cdot\begin{pmatrix}v_{\mathrm{in}}(t)\\\
i_{\mathrm{out}}(t)\\\ \mathit{ctl}_{\mathrm{OL/CL}}(t)\end{pmatrix}\\\
\text{with}\quad{ctl}_{\mathrm{OL/CL}}(t)&=\begin{cases}{ctl}_{\mathrm{OL}}(t)=e(t)&\quad\text{open-
loop case (control error)}\\\ {ctl}_{\mathrm{CL}}(t)=r(t)&\quad\text{closed-
loop case (reference signal)}\end{cases}\end{split}$ (4)

The main step required to obtain the $\bm{A}$, $\bm{B}$, $\bm{C}$, $\bm{D}$
matrices of the open-loop model is to account for the changed meaning of the
control input of the model. The (now internal) control input of the converter
must be connected to the output of the controller, cf.
(LABEL:eqn:ConverterController_OpenLoop).

$\displaystyle\bm{A}_{\mathrm{OL}}=\begin{pmatrix}\bm{A}&\bm{B}_{3}\cdot\bm{C}_{\mathrm{C}}\\\
\bm{0}^{m\times n}&\bm{A}_{\mathrm{C}}\end{pmatrix},$
$\displaystyle\bm{B}_{\mathrm{OL}}=\begin{pmatrix}\bm{B}_{1}&\bm{B}_{2}&\bm{B}_{3}\cdot
D_{\mathrm{C}}\\\ \bm{0}^{m\times 1}&\bm{0}^{m\times
1}&\bm{B}_{\mathrm{C}}\end{pmatrix},$ (5)
$\displaystyle\bm{C}_{\mathrm{OL}}=\begin{pmatrix}\bm{C}_{1}&\bm{0}^{1\times
m}\\\ \bm{C}_{2}&\bm{0}^{1\times m}\end{pmatrix},$
$\displaystyle\bm{D}_{\mathrm{OL}}=\begin{pmatrix}D_{11}&D_{12}&D_{13}\cdot
D_{\mathrm{C}}\\\ D_{21}&D_{22}&D_{23}\cdot D_{\mathrm{C}}\end{pmatrix}$

In (LABEL:eqn:ConverterController_OpenLoop), $n$ refers to the number of state
variables in the converter model, and $m$ to the number of state variables in
the controller model.

When deriving the closed-loop model in the following, it will be assumed that
there is no (or only a negligible) direct feedthrough from the original
control input to the desired control variable. If, for example, the output
voltage is the desired control variable, $D_{23}\approx 0$ should hold, which
is the case for all converter types mentioned in this article.

The $\bm{B}$, $\bm{C}$, $\bm{D}$ matrices of the open-loop and closed-loop
case are identical, that is, $\bm{B}_{\mathrm{CL}}=\bm{B}_{\mathrm{OL}}$,
$\bm{C}_{\mathrm{CL}}=\bm{C}_{\mathrm{OL}}$,
$\bm{D}_{\mathrm{CL}}=\bm{D}_{\mathrm{OL}}$. The $\bm{A}$ matrix of the
closed-loop model is computed via (6), which is an equation known from
classical state feedback control.

$\bm{A}_{\mathrm{CL}}=\bm{A}_{\mathrm{OL}}-\bm{B}_{\mathrm{OL}}\cdot\bm{K}$
(6)

A suitable feedback gain matrix $\bm{K}$ must be chosen depending on which
control loop is being closed, such as an inner average current loop or an
output voltage loop. It is important to note that $\bm{K}$ is not being used
to design the controller (as in state feedback control), but only to select
and feed back the desired control variable when constructing the closed-loop
state-space model. The controller parameters are contained in the controller
model (3). To demonstrate that the choice of $\bm{K}$ is almost trivial in
most applications, the (probably) most important two cases will be shown in
Sections 3.1 and 3.2.

As a final side note, a sensor or filter with non-negligible dynamics in the
feedback path can rather easily be accounted for by extending the converter
model with the respective dynamics and choosing the output of the sensor or
filter as the control variable.

###  3.1 Average Current Mode

For closing a current loop, $\bm{K}$ must be chosen in order to extract the
desired current variable from the vector of converter system states
$\bm{x}(t)$ and feed it back to the third input of the open-loop converter
model, that is, $\mathit{ctl}_{\mathrm{OL}}(t)$. If we, for example, assume
for the converter model that the average current is the first state variable
(as in the boost converter example in Appendix B.2), $\bm{K}$ simply has to be
chosen as (7).

$\bm{K}=\begin{pmatrix}\bm{0}^{2\times 1}&\bm{0}^{2\times(n-1+m)}\\\
1&\bm{0}^{1\times(n-1+m)}\end{pmatrix}$ (7)

###  3.2 Voltage Mode or Outer Voltage Loops

If output voltage $v_{\mathrm{out}}(t)$ is the desired control variable, which
is the case for voltage mode as well as for an outer voltage loop around a
current-controlled converter, $\bm{K}$ has to be chosen as (8).

$\bm{K}=\begin{pmatrix}\bm{0}^{2\times n}&\bm{0}^{2\times m}\\\
\bm{C}_{2}&\bm{0}^{1\times m}\end{pmatrix}$ (8)

This assumes that $D_{23}\approx 0$ holds for the uncontrolled converter
model, which means that $v_{\mathrm{out}}(t)$ can approximately be extracted
from the state variable vector by
$v_{\mathrm{out}}(t)\approx\bm{C}_{2}\cdot\bm{x}(t)$.

##  4 Connecting Converter Models

In order to create a combined state-space model for a series connection of two
subsystems (denoted “source” and “load” here), the respective models for
source (superscript S) and load (superscript L) must be given in the following
form:

$\begin{split}\bm{\dot{x}}^{\mathrm{S/L}}(t)&=\bm{A}^{\mathrm{S/L}}\cdot\bm{x}^{\mathrm{S/L}}(t)+\bm{B}^{\mathrm{S/L}}\cdot\begin{pmatrix}v_{\mathrm{in}}^{\mathrm{S/L}}(t)\\\\[4.30554pt]
i_{\mathrm{out}}^{\mathrm{S/L}}(t)\\\\[4.30554pt]
\bm{\mathit{ctl}}^{\mathrm{S/L}}(t)\end{pmatrix}\\\
\begin{pmatrix}i_{\mathrm{in}}^{\mathrm{S/L}}(t)\\\\[4.30554pt]
v_{\mathrm{out}}^{\mathrm{S/L}}(t)\end{pmatrix}&=\bm{C}^{\mathrm{S/L}}\cdot\bm{x}^{\mathrm{S/L}}(t)+\bm{D}^{\mathrm{S/L}}\cdot\begin{pmatrix}v_{\mathrm{in}}^{\mathrm{S/L}}(t)\\\\[4.30554pt]
i_{\mathrm{out}}^{\mathrm{S/L}}(t)\\\\[4.30554pt]
\bm{\mathit{ctl}}^{\mathrm{S/L}}(t)\end{pmatrix}\end{split}$ (9)

Note that the connection operation introduced in this section works regardless
of the number of control inputs $\bm{\mathit{ctl}}(t)$ in the source and load
models, that is, even without control inputs, as is the case for passive
subsystems such as filters or loads. In these cases, the respective columns of
the $\bm{B}$ and $\bm{D}$ matrices must simply be left out in all of the
following equations.

The input and output ports of the combined system are the input port of the
source subsystem and the output port of the load subsystem, that is, the
following equations hold:
$i_{\mathrm{in}}(t)=i_{\mathrm{in}}^{\mathrm{S}}(t)$,
$v_{\mathrm{in}}(t)=v_{\mathrm{in}}^{\mathrm{S}}(t)$,
$v_{\mathrm{out}}(t)=v_{\mathrm{out}}^{\mathrm{L}}(t)$, and
$i_{\mathrm{out}}(t)=i_{\mathrm{out}}^{\mathrm{L}}(t)$, cf. Fig. 4. The result
is the following combined model (10).

$\begin{split}\begin{pmatrix}\bm{\dot{x}}^{\mathrm{S}}(t)\\\
\bm{\dot{x}}^{\mathrm{L}}(t)\end{pmatrix}&=\bm{A}\cdot\begin{pmatrix}\bm{x}^{\mathrm{S}}(t)\\\
\bm{x}^{\mathrm{L}}(t)\end{pmatrix}+\bm{B}\cdot\begin{pmatrix}v_{\mathrm{in}}(t)\\\
i_{\mathrm{out}}(t)\\\ \bm{\mathit{ctl}}^{\mathrm{S}}(t)\\\
\bm{\mathit{ctl}}^{\mathrm{L}}(t)\end{pmatrix}\\\
\begin{pmatrix}i_{\mathrm{in}}(t)\\\
v_{\mathrm{out}}(t)\end{pmatrix}&=\bm{C}\cdot\begin{pmatrix}\bm{x}^{\mathrm{S}}(t)\\\
\bm{x}^{\mathrm{L}}(t)\end{pmatrix}+\bm{D}\cdot\begin{pmatrix}v_{\mathrm{in}}(t)\\\
i_{\mathrm{out}}(t)\\\ \bm{\mathit{ctl}}^{\mathrm{S}}(t)\\\
\bm{\mathit{ctl}}^{\mathrm{L}}(t)\end{pmatrix}\end{split}$ (10)

Figure 4: Series connection of two two-port networks (source and load). The
control inputs $\bm{\mathit{ctl}}^{\mathrm{S/L}}(t)$ are optional and only
present if source or load contain at least one converter, respectively.

The equations of the resulting model can be found by replacing the inner
terminal variables of the two-port series connection, which are not present in
(10) anymore, cf. Fig. 4. For the inner voltage
$v_{\mathrm{out}}^{\mathrm{S}}(t)=v_{\mathrm{in}}^{\mathrm{L}}(t)$ and the
inner current
$i_{\mathrm{out}}^{\mathrm{S}}(t)=-i_{\mathrm{in}}^{\mathrm{L}}(t)$ the
following equations hold:

$\displaystyle v_{\mathrm{in}}^{\mathrm{L}}=v_{\mathrm{out}}^{\mathrm{S}}$
$\displaystyle=\bm{C}_{2}^{\mathrm{S}}\bm{x}^{\mathrm{S}}+D_{21}^{\mathrm{S}}v_{\mathrm{in}}^{\mathrm{S}}+D_{22}^{\mathrm{S}}i_{\mathrm{out}}^{\mathrm{S}}+\bm{D}_{23}^{\mathrm{S}}\bm{\mathit{ctl}}^{\mathrm{S}}$
$\displaystyle=\bm{C}_{2}^{\mathrm{S}}\bm{x}^{\mathrm{S}}+D_{21}^{\mathrm{S}}v_{\mathrm{in}}^{\mathrm{S}}+D_{22}^{\mathrm{S}}\left(-\bm{C}_{1}^{\mathrm{L}}\bm{x}^{\mathrm{L}}-D_{11}^{\mathrm{L}}v_{\mathrm{in}}^{\mathrm{L}}-D_{12}^{\mathrm{L}}i_{\mathrm{out}}^{\mathrm{L}}-\bm{D}_{13}^{\mathrm{L}}\bm{\mathit{ctl}}^{\mathrm{L}}\right)+\bm{D}_{23}^{\mathrm{S}}\bm{\mathit{ctl}}^{\mathrm{S}}$
$\displaystyle=\frac{1}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}\cdot\left(\bm{C}_{2}^{\mathrm{S}}\bm{x}^{\mathrm{S}}-D_{22}^{\mathrm{S}}\bm{C}_{1}^{\mathrm{L}}\bm{x}^{\mathrm{L}}+D_{21}^{\mathrm{S}}v_{\mathrm{in}}-D_{22}^{\mathrm{S}}D_{12}^{\mathrm{L}}i_{\mathrm{out}}+\bm{D}_{23}^{\mathrm{S}}\bm{\mathit{ctl}}^{\mathrm{S}}-D_{22}^{\mathrm{S}}\bm{D}_{13}^{\mathrm{L}}\bm{\mathit{ctl}}^{\mathrm{L}}\right)$
(11)

$\displaystyle i_{\mathrm{out}}^{\mathrm{S}}=-i_{\mathrm{in}}^{\mathrm{L}}$
$\displaystyle=-\bm{C}_{1}^{\mathrm{L}}\bm{x}^{\mathrm{L}}-D_{11}^{\mathrm{L}}v_{\mathrm{in}}^{\mathrm{L}}-D_{12}^{\mathrm{L}}i_{\mathrm{out}}^{\mathrm{L}}-\bm{D}_{13}^{\mathrm{L}}\bm{\mathit{ctl}}^{\mathrm{L}}$
$\displaystyle=-\bm{C}_{1}^{\mathrm{L}}\bm{x}^{\mathrm{L}}-D_{11}^{\mathrm{L}}\left(\bm{C}_{2}^{\mathrm{S}}\bm{x}^{\mathrm{S}}+D_{21}^{\mathrm{S}}v_{\mathrm{in}}^{\mathrm{S}}+D_{22}^{\mathrm{S}}i_{\mathrm{out}}^{\mathrm{S}}+\bm{D}_{23}^{\mathrm{S}}\bm{\mathit{ctl}}^{\mathrm{S}}\right)-D_{12}^{\mathrm{L}}i_{\mathrm{out}}^{\mathrm{L}}-\bm{D}_{13}^{\mathrm{L}}\bm{\mathit{ctl}}^{\mathrm{L}}$
$\displaystyle=\frac{-1}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}\cdot\left(D_{11}^{\mathrm{L}}\bm{C}_{2}^{\mathrm{S}}\bm{x}^{\mathrm{S}}+\bm{C}_{1}^{\mathrm{L}}\bm{x}^{\mathrm{L}}+D_{11}^{\mathrm{L}}D_{21}^{\mathrm{S}}v_{\mathrm{in}}+D_{12}^{\mathrm{L}}i_{\mathrm{out}}+D_{11}^{\mathrm{L}}\bm{D}_{23}^{\mathrm{S}}\bm{\mathit{ctl}}^{\mathrm{S}}+\bm{D}_{13}^{\mathrm{L}}\bm{\mathit{ctl}}^{\mathrm{L}}\right)$
(12)

Putting (11) and (12) in the source/load state equations of (9) will now lead
to the combined state equations in (10). Similarly, the output equation of the
connected system in (10) is obtained by putting (11) and (12) in the output
equations from (9) for $i_{\mathrm{in}}(t)=i_{\mathrm{in}}^{\mathrm{S}}(t)$
and $v_{\mathrm{out}}(t)=v_{\mathrm{out}}^{\mathrm{L}}(t)$. The resulting
detailed formulation of the matrix contents for $\bm{A}$, $\bm{B}$, $\bm{C}$,
$\bm{D}$ in (10) is given in (13).

$\begin{split}\bm{A}&=\begin{pmatrix}\displaystyle\bm{A}^{\mathrm{S}}-\frac{\bm{B}_{2}^{\mathrm{S}}D_{11}^{\mathrm{L}}\bm{C}_{2}^{\mathrm{S}}}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}&\displaystyle-\frac{\bm{B}_{2}^{\mathrm{S}}\bm{C}_{1}^{\mathrm{L}}}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}\\\\[15.00002pt]
\displaystyle\frac{\bm{B}_{1}^{\mathrm{L}}\bm{C}_{2}^{\mathrm{S}}}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}&\displaystyle\bm{A}^{\mathrm{L}}-\frac{\bm{B}_{1}^{\mathrm{L}}D_{22}^{\mathrm{S}}\bm{C}_{1}^{\mathrm{L}}}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}\end{pmatrix},\quad\bm{C}=\begin{pmatrix}\displaystyle\bm{C}_{1}^{\mathrm{S}}-\frac{D_{12}^{\mathrm{S}}D_{11}^{\mathrm{L}}\bm{C}_{2}^{\mathrm{S}}}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}&\displaystyle-\frac{D_{12}^{\mathrm{S}}\bm{C}_{1}^{\mathrm{L}}}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}\\\\[15.00002pt]
\displaystyle\frac{D_{21}^{\mathrm{L}}\bm{C}_{2}^{\mathrm{S}}}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}&\displaystyle\bm{C}_{2}^{\mathrm{L}}-\frac{D_{21}^{\mathrm{L}}D_{22}^{\mathrm{S}}\bm{C}_{1}^{\mathrm{L}}}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}\end{pmatrix},\\\
\bm{B}&=\begin{pmatrix}\displaystyle\bm{B}_{1}^{\mathrm{S}}-\frac{\bm{B}_{2}^{\mathrm{S}}D_{11}^{\mathrm{L}}D_{21}^{\mathrm{S}}}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}&\displaystyle-\frac{\bm{B}_{2}^{\mathrm{S}}D_{12}^{\mathrm{L}}}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}&\displaystyle\bm{B}_{3}^{\mathrm{S}}-\frac{\bm{B}_{2}^{\mathrm{S}}D_{11}^{\mathrm{L}}\bm{D}_{23}^{\mathrm{S}}}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}&\displaystyle-\frac{\bm{B}_{2}^{\mathrm{S}}\bm{D}_{13}^{\mathrm{L}}}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}\\\\[15.00002pt]
\displaystyle\frac{\bm{B}_{1}^{\mathrm{L}}D_{21}^{\mathrm{S}}}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}&\displaystyle\bm{B}_{2}^{\mathrm{L}}-\frac{\bm{B}_{1}^{\mathrm{L}}D_{22}^{\mathrm{S}}D_{12}^{\mathrm{L}}}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}&\displaystyle\frac{\bm{B}_{1}^{\mathrm{L}}\bm{D}_{23}^{\mathrm{S}}}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}&\displaystyle\bm{B}_{3}^{\mathrm{L}}-\frac{\bm{B}_{1}^{\mathrm{L}}D_{22}^{\mathrm{S}}\bm{D}_{13}^{\mathrm{L}}}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}\end{pmatrix},\\\
\bm{D}&=\begin{pmatrix}\displaystyle
D_{11}^{\mathrm{S}}-\frac{D_{12}^{\mathrm{S}}D_{11}^{\mathrm{L}}D_{21}^{\mathrm{S}}}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}&\displaystyle-\frac{D_{12}^{\mathrm{S}}D_{12}^{\mathrm{L}}}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}&\displaystyle\bm{D}_{13}^{\mathrm{S}}-\frac{D_{12}^{\mathrm{S}}D_{11}^{\mathrm{L}}\bm{D}_{23}^{\mathrm{S}}}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}&\displaystyle-\frac{D_{12}^{\mathrm{S}}\bm{D}_{13}^{\mathrm{L}}}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}\\\\[15.00002pt]
\displaystyle\frac{D_{21}^{\mathrm{L}}D_{21}^{\mathrm{S}}}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}&\displaystyle
D_{22}^{\mathrm{L}}-\frac{D_{21}^{\mathrm{L}}D_{22}^{\mathrm{S}}D_{12}^{\mathrm{L}}}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}&\displaystyle\frac{D_{21}^{\mathrm{L}}\bm{D}_{23}^{\mathrm{S}}}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}&\displaystyle\bm{D}_{23}^{\mathrm{L}}-\frac{D_{21}^{\mathrm{L}}D_{22}^{\mathrm{S}}\bm{D}_{13}^{\mathrm{L}}}{1+D_{11}^{\mathrm{L}}D_{22}^{\mathrm{S}}}\end{pmatrix}\end{split}$
(13)

It has to be noted that (13) will often be reduced to a simpler form in
practical cases since not all matrix elements of the subsystem matrices are
non-zero. Furthermore, (10) and (13) are primarily meant to be implemented in
a computer algebra system or a numerical software package only once and then
be used to conveniently build up a complete model of a DC-DC converter system
from simple subsystems in a fully or semi-automated manner. Compact matrix
formulations for (13) can be found by inspection, the corresponding set of
equations is presented in (14).

$\begin{split}\bm{A}&=\operatorname{diag}\left(\bm{A}^{\mathrm{S}},\bm{A}^{\mathrm{L}}\right)-\bm{M}_{1}\cdot\bm{M}_{C}\\\
\bm{B}&=\left(\begin{array}[]{c|c}\operatorname{diag}\left(\bm{B}_{1}^{\mathrm{S}},\bm{B}_{2}^{\mathrm{L}}\right)&\operatorname{diag}\left(\bm{B}_{3}^{\mathrm{S}},\bm{B}_{3}^{\mathrm{L}}\right)\end{array}\right)-\bm{M}_{1}\cdot\bm{M}_{D}\\\
\bm{C}&=\operatorname{diag}\left(\bm{C}_{1}^{\mathrm{S}},\bm{C}_{2}^{\mathrm{L}}\right)-\bm{M}_{2}\cdot\bm{M}_{C}\\\
\bm{D}&=\left(\begin{array}[]{c|c}\operatorname{diag}\left(D_{11}^{\mathrm{S}},D_{22}^{\mathrm{L}}\right)&\operatorname{diag}\left(\bm{D}_{13}^{\mathrm{S}},\bm{D}_{23}^{\mathrm{L}}\right)\end{array}\right)-\bm{M}_{2}\cdot\bm{M}_{D}\end{split}$
(14)

In (14), the abbreviations defined in (15) were used for recurring terms.

$\displaystyle\bm{M}_{1}$
$\displaystyle=\operatorname{diag}\left(\bm{B}_{2}^{\mathrm{S}},\bm{B}_{1}^{\mathrm{L}}\right)\cdot\begin{pmatrix}D_{22}^{\mathrm{S}}&-1\\\
1&D_{11}^{1}\mathrm{L}\end{pmatrix}^{-1}$ (15) $\displaystyle\bm{M}_{2}$
$\displaystyle=\operatorname{diag}\left(D_{12}^{\mathrm{S}},D_{21}^{\mathrm{L}}\right)\cdot\begin{pmatrix}D_{22}^{\mathrm{S}}&-1\\\
1&D_{11}^{\mathrm{L}}\end{pmatrix}^{-1}$ $\displaystyle\bm{M}_{C}$
$\displaystyle=\operatorname{diag}\left(\bm{C}_{2}^{\mathrm{S}},\bm{C}_{1}^{\mathrm{L}}\right)$
$\displaystyle\bm{M}_{D}$
$\displaystyle=\left(\begin{array}[]{c|c}\operatorname{diag}\left(D_{21}^{\mathrm{S}},D_{12}^{\mathrm{L}}\right)&\operatorname{diag}\left(\bm{D}_{23}^{\mathrm{S}},\bm{D}_{13}^{\mathrm{L}}\right)\end{array}\right)$

To summarize the connection procedure, all that is required to obtain the
connected model from two given source and load models is to arrange the input,
output, and state variables of the resulting system as given in (10), and
compute the $\bm{A}$, $\bm{B}$, $\bm{C}$, $\bm{D}$ matrices using (13).

##  5 Summary of the Building-Block Modeling Approach

For convenience, all building block types and connection operations of the
modular modeling approach presented in this article are summarized in this
section. Furthermore, the relation of the developed state-space model to well-
known frequency-domain models of converter systems is given.

###  5.1 Building Block Types

To start with modeling a DC-DC converter or converter system, the first step
is to collect all building blocks, such as models for the bare converter,
controllers, filters, loads and so forth. In the modeling approach presented
in this article, there are three different categories of building blocks in
terms of their state-space model representation. For convenience, they are
listed again in table 1. Concrete examples for these three building block
types are given in the Appendixes A–C.

Table 1: Summary of building block types in terms of their state space equation structure. Type | Equation | Definition
---|---|---
Passive components | (1) |  $\begin{aligned} \bm{\dot{x}}(t)&=\bm{A}\cdot\bm{x}(t)+\begin{pmatrix}\bm{B}_{1}&\bm{B}_{2}\end{pmatrix}\cdot\begin{pmatrix}v_{\mathrm{in}}(t)\\\ i_{\mathrm{out}}(t)\end{pmatrix}\\\ \begin{pmatrix}i_{\mathrm{in}}(t)\\\ v_{\mathrm{out}}(t)\end{pmatrix}&=\begin{pmatrix}\bm{C}_{1}\\\ \bm{C}_{2}\end{pmatrix}\cdot\bm{x}(t)+\begin{pmatrix}D_{11}&D_{12}\\\ D_{21}&D_{22}\end{pmatrix}\cdot\begin{pmatrix}v_{\mathrm{in}}(t)\\\ i_{\mathrm{out}}(t)\end{pmatrix}\end{aligned}$
Controlled converters | (2) |  $\begin{aligned} \bm{\dot{x}}(t)&=\bm{A}\cdot\bm{x}(t)+\begin{pmatrix}\bm{B}_{1}&\bm{B}_{2}&\bm{B}_{3}\end{pmatrix}\cdot\begin{pmatrix}v_{\mathrm{in}}(t)\\\ i_{\mathrm{out}}(t)\\\ \mathit{ctl}(t)\end{pmatrix}\\\ \begin{pmatrix}i_{\mathrm{in}}(t)\\\ v_{\mathrm{out}}(t)\end{pmatrix}&=\begin{pmatrix}\bm{C}_{1}\\\ \bm{C}_{2}\end{pmatrix}\cdot\bm{x}+\begin{pmatrix}D_{11}&D_{12}&D_{13}\\\ D_{21}&D_{22}&D_{23}\end{pmatrix}\cdot\begin{pmatrix}v_{\mathrm{in}}(t)\\\ i_{\mathrm{out}}(t)\\\ \mathit{ctl}(t)\end{pmatrix}\end{aligned}$
Controllers | (3) |  $\begin{aligned} \bm{\dot{x}}_{\mathrm{C}}(t)&=\bm{A}_{\mathrm{C}}\cdot\bm{x}_{\mathrm{C}}(t)+\bm{B}_{\mathrm{C}}\cdot e(t)\\\ u(t)&=\bm{C}_{\mathrm{C}}\cdot\bm{x}_{\mathrm{C}}(t)+\bm{D}_{\mathrm{C}}\cdot e(t)\end{aligned}$

###  5.2 Connecting Building Blocks

There are two different kinds of connections that can be made in order to
create a converter system model from smaller building blocks:

  1. 1.

_Connecting converters with passive components or other converters._ Since
both building blocks for passive elements as given by (1) and building blocks
for converters as given by (2) are two-port models, the same connection
operation can be used for all these cases. The resulting model (10) is
obtained by applying (13).

  2. 2.

_Creating control loops._ Connecting a controller, that is, a building block
type as given by (3), to a converter is done by creating an open-loop model
(4) using (LABEL:eqn:ConverterController_OpenLoop). For closing the control
loop, the model’s $\bm{A}$ matrix must be adapted using (6), with a feedback
gain matrix $\bm{K}$ chosen depending on the controlled variable, for example,
one of (7) or (8) for current or voltage mode control.

All resulting models (connected subsystems, open- or closed-loop models) are
of the same structure as the generic model (2), which means that frequency-
domain characteristics as listed in the following section can be extracted at
any modeling stage.

Since the structure and linearity of the resulting state-space models are
preserved by the connection operations, there is no impact on the small-signal
modeling accuracy of the building blocks.

###  5.3 Frequency-Domain Analysis

As mentioned before, one major benefit of the modular state-space modeling
approach presented in this article is that, when needed, frequency-domain
transfer functions can easily be obtained at any modeling stage. Table 2 lists
all frequency-domain models that can be extracted from the generic converter
or converter system model (2), regardless of the current modeling state (e.g.,
with or without closed-loop control, with or without filters and loads etc.).

Table 2: Transfer functions than can be obtained from the converter (system)
model (2), cf. Section 2.2, at any modeling stage (e.g., with or without
controller).

Name | Definition
---|---
Control-to-output $\dagger$ | $G_{\mathrm{co}}(s)$ | $=\displaystyle\frac{v_{\mathrm{out}}(s)}{\mathit{ctl}(s)}$ | $=\displaystyle\bm{C}_{2}\cdot\left(s\bm{I}-\bm{A}\right)^{-1}\cdot\bm{B}_{3}+D_{23}$
Output impedance | $Z_{\mathrm{out}}(s)$ | $=\displaystyle\frac{v_{\mathrm{out}}(s)}{i_{\mathrm{out}}(s)}$ | $=\displaystyle\bm{C}_{2}\cdot\left(s\bm{I}-\bm{A}\right)^{-1}\cdot\bm{B}_{2}+D_{22}$
Input admittance | $Y_{\mathrm{in}}(s)$ | $=\displaystyle\frac{i_{\mathrm{in}}(s)}{v_{\mathrm{in}}(s)}$ | $=\displaystyle\bm{C}_{1}\cdot\left(s\bm{I}-\bm{A}\right)^{-1}\cdot\bm{B}_{1}+D_{11}$
Forward voltage gain | $G_{\mathrm{v}}(s)$ | $=\displaystyle\frac{v_{\mathrm{out}}(s)}{v_{\mathrm{in}}(s)}$ | $=\displaystyle\bm{C}_{2}\cdot\left(s\bm{I}-\bm{A}\right)^{-1}\cdot\bm{B}_{1}+D_{21}$
Reverse current gain | $G_{\mathrm{i}}(s)$ | $=\displaystyle\frac{i_{\mathrm{in}}(s)}{i_{\mathrm{out}}(s)}$ | $=\displaystyle\bm{C}_{1}\cdot\left(s\bm{I}-\bm{A}\right)^{-1}\cdot\bm{B}_{2}+D_{12}$

  * $\dagger$

If there is more than one control input (after connecting several converters
to one model), the control-to-output transfer function for the $k$-th control
input is obtained using the $\bm{B}_{2+k}$ column of the $\bm{B}$ matrix and
$D_{2,(2+k)}$ instead of $\bm{B}_{3}$ and $D_{23}$.

##  6 Examples

In the following, three modular modeling examples are presented with special
emphasis on demonstrating the systematic methods to extend two-port oriented
state-space models of converters with control loops (method from Section 3,
example in Section 6.1) and to create an overall model from series-connections
of subsystems (method from Section 4, examples in Sections 6.2 and 6.3).

It should be noted that the parameterization of converters and controllers in
this section was chosen rather arbitrarily and solely to serve as numerical
examples for the modeling methods developed here. For simplicity, all
inductors and capacitors in the following examples are being modeled with a
uniform equivalent series resistance (ESR) value of
$r_{\mathrm{L/C}}=$10\text{\,}\mathrm{m\SIUnitSymbolOhm}$$, unless otherwise
noted.

As will be demonstrated, all models developed in these examples can be
constructed from already available modules, thus no manual effort is required
regarding the derivation of state-space equations. In order to illustrate the
validity of the models in an intuitive manner, comparisons will be provided
with time-domain results from circuit simulations.

###  6.1 Buck Converter with Multiloop (I2) Control

The systematic approach to extend a converter model by attaching a control
loop (presented in Section 3) opens up the possibility of studying
characteristics of converters with multiple control loops in an easy manner.
Therefore, as a first example, a buck converter in peak current mode control
will be examined, which will be extended by a second control loop for the
average current (also known as I2 control [41]) and a third outer voltage
control loop. Only three steps are necessary to create the complete model from
already available building blocks:

  1. 1.

As the starting point to build up the complete model, the two-port state-space
model for the PCM buck converter from Reference [27] is being used. The
example is parameterized as given in the following. Operating point:
$V_{\mathrm{in}}=$24\text{\,}\mathrm{V}$$,
$V_{\mathrm{out}}=$12\text{\,}\mathrm{V}$$,
$I_{\mathrm{out}}=$2.4\text{\,}\mathrm{A}$$. Power stage:
$f_{\mathrm{Switch}}=$50\text{\,}\mathrm{kHz}$$,
$L=$100\text{\,}\mathrm{\SIUnitSymbolMicro H}$$,
$C=$100\text{\,}\mathrm{\SIUnitSymbolMicro F}$$. Peak current mode controller:
external ramp of $1.5\text{\,}\mathrm{A}$ per switching cycle.

  2. 2.

For the I2 current loop, an additional controller for the average inductor
current is being added. A Type 1 controller with $K_{\mathrm{i}}=20$,$000$ is
employed, cf. Appendix C, which is connected to the state-space model as
described in Section 3.1.

  3. 3.

For the outer voltage control loop, a Type 2 controller with zero at
$0.3\text{\,}\mathrm{kHz}$, pole at $25\text{\,}\mathrm{kHz}$ and
$K_{\mathrm{i}}=3000$ is being used, cf. Appendix C. It is connected to the
model as described in Section 3.2.

To validate the model in the time domain, it is being compared against a
circuit simulation of this converter/controller configuration with three test
cases. Firstly, a reference step of the I2 current control loop (i.e., using
the model after step 2 as given above) is simulated using both the detailed
circuit simulation and the transfer function for the control loop of the
average inductor current, which can easily be obtained from the state-space
model (transfer function from the control input to the state variable of the
inductor current). The results are given in Fig. 5 and confirm that the
dynamics of the average inductor current under a combined average and peak
current mode (I2) control are very well captured by the state-space model.

Figure 5: Impact of a reference step
$$2.4\text{\,}\mathrm{A}$\rightarrow$2.0\text{\,}\mathrm{A}$$ at
$t=$0\text{\,}\mathrm{s}$$ on the inductor current of the buck converter from
example Section 6.1 with closed I2-current loop.

As a second test, a reference voltage step is performed using the outer
voltage control loop (i.e., with the complete model after step 3). The results
are given in Fig. 6 and, again, the circuit simulation and state-space model
agree very well.

Figure 6: Reference step
$$13\text{\,}\mathrm{V}$\rightarrow$12\text{\,}\mathrm{V}$$ at
$t=$0\text{\,}\mathrm{s}$$ of the buck converter from example Section 6.1 with
closed voltage loop and underlying I2 current control.

Finally, +25% load step is being applied. The results in Fig. 7 once more
confirm that the state-space model captures the dynamics of the converter
under this multiloop control configuration very well.

Figure 7: Impact of a 25% load step
$$2.4\text{\,}\mathrm{A}$\rightarrow$3.0\text{\,}\mathrm{A}$$ at
$t=$0\text{\,}\mathrm{s}$$ on the output voltage of the buck converter from
example Section 6.1 with closed voltage loop.

Again, it should be noted that frequency-domain characteristics such as
impedance plots or control-to-output transfer functions are all readily
available at any stage of the state-space modeling procedure, cf. table 2 but
omitted here for brevity. There is no necessity anymore to derive custom
models or transfer functions for the I2 current loop configuration, in
contrast to existing work [41, 17, 19].

###  6.2 Boost Converter with Input Filter

With the model connection approach presented in Section 4, building up higher-
order state-space models has become an easy task. As an example, the impact of
additional input circuitry on a voltage mode controlled boost converter will
be studied here. Four steps are necessary in order to obtain the final model
from already available modules:

  1. 1.

A linearized state-space model for a boost converter in CCM as given in
Appendix B.2. The example is parameterized as given in the following.
Operating point: $V_{\mathrm{in}}=$10\text{\,}\mathrm{V}$$,
$V_{\mathrm{out}}=$24\text{\,}\mathrm{V}$$,
$I_{\mathrm{out}}=$1.2\text{\,}\mathrm{A}$$. Power stage:
$f_{\mathrm{Switch}}=$100\text{\,}\mathrm{kHz}$$,
$L=$20\text{\,}\mathrm{\SIUnitSymbolMicro H}$$,
$C=$220\text{\,}\mathrm{\SIUnitSymbolMicro F}$$.

  2. 2.

A load resistance (model from Appendix A.1) corresponding to the operating
point is connected at the converter output using Section 4.

  3. 3.

A Type 3 controller (cf. Appendix C) is being employed for the voltage control
loop, with zeros placed at $10\text{\,}\mathrm{kHz}$, poles at
$0.1\text{\,}\mathrm{kHz}$ and $50\text{\,}\mathrm{kHz}$, and
$K_{\mathrm{i}}=10$. It is connected to the model as described in Section 3.2.

  4. 4.

Finally, an LC filter model (cf. Appendix A.2) accounting for both an input
filter and wiring is connected at the input of the converter model using
Section 4, with $L=$5\text{\,}\mathrm{\SIUnitSymbolMicro H}$$,
$r_{\mathrm{L}}=$50\text{\,}\mathrm{m\SIUnitSymbolOhm}$$, and
$C=$1\text{\,}\mathrm{\SIUnitSymbolMicro F}$$.

As a first step, the influence of the input filter on the frequency-domain
characteristics will be examined. The input and output impedance transfer
functions can easily be obtained from our two-port oriented state-space model,
cf. table 2. Figure 8 shows the impedance magnitude and phase plots for the
voltage mode controlled boost converter both with and without input filter. In
the input impedance plots one can clearly notice the negative resistance
behavior of the controlled converter at low frequencies. Furthermore one can,
for instance, observe a dampening effect of the input filter around
$1\text{\,}\mathrm{kHz}$ in the output impedance, which has an impact on load
step dynamics.

Figure 8: Input and output impedance of the boost converter from example
Section 6.2, with closed voltage loop, both with input filter (solid lines)
and without input filter (dashed lines).

Therefore, a time-domain simulation of a +100% load step at the output will be
performed with and without input filter and compared against the results of
detailed time-domain circuit simulations. The results in Fig. 9 not only
confirm the good accuracy of the state-space model, but also clearly
demonstrate the dampening effect of this particular input circuitry on the
load step dynamics, as predicted in the frequency-domain analysis.

Figure 9: Impact of a 100% load step
$$1.2\text{\,}\mathrm{A}$\rightarrow$2.4\text{\,}\mathrm{A}$$ at
$t=$0\text{\,}\mathrm{s}$$ on the output voltage of the boost converter from
example Section 6.2 with closed voltage loop, both with input filter (solid
lines) and without input filter (dashed lines). For better comparability, the
circuit simulation results were filtered (moving average) to remove the
switching ripple voltage.

###  6.3 Series-Connected Boost and Buck Converter Stages

Since the model connection approach developed in Section 4 is not restricted
to attaching subsystems with passive components, one can just as easily
connect models of controlled converters in a source/load configuration.
Therefore, as a final example, we will create a series-connected configuration
of the boost and buck stages examined in Sections 6.1 and 6.2, respectively,
cf. Fig. 10. The following (almost trivial) steps are required to obtain the
complete model:

  1. 1.

Reuse the final model from Section 6.2 (voltage-controlled boost converter
including input filter) but without the load resistance, since the buck
converter will constitute the new load to the boost stage.

  2. 2.

Reuse the final model from Section 6.1 (voltage-controlled buck converter with
underlying I2 current control).

  3. 3.

Connect the buck converter model at the output of the boost converter, as
described in Section 4.

Figure 10: Two-port view of a series connection of boost and buck converter
stages. In both stages, the filter elements $L$ and $C$ are being modeled with
equivalent series resistances (ESR).

As a small example of the possibilities offered by the all-encompassing state-
space model, the interactions between the two power stages will be examined in
the following test case. Again, a +25% load step is applied at the output of
the buck stage and the impact on the intermediate output voltage of the boost
stage is observed. From the state-space model, a transfer function from the
output current (i.e., the load) to the state variable representing the output
capacitor voltage of the boost stage can easily be extracted. It is, once
more, being compared against a time-domain circuit simulation containing the
series-connected converter configuration.

Figure 11: Impact of a 25% output load step
$$2.4\text{\,}\mathrm{A}$\rightarrow$3.0\text{\,}\mathrm{A}$$ at
$t=$0\text{\,}\mathrm{s}$$ of the series-connected buck stage from example
Section 6.3 on the intermediate output voltage of the boost stage. For better
comparability the circuit simulation result was filtered (moving average) to
remove the switching ripple voltage.

From the results in Fig. 11 one can conclude that the overall state-space
model is very well able to predict the essential dynamics. Since the buck
stage has much faster dynamics than the boost stage, the load step dynamics in
Fig. 11 closely resemble Fig. 9 in this case but it has to be emphasized again
that the results in Fig. 11 fully incorporate small-signal interactions of the
two connected power stages with all their respective controllers.

##  7 Conclusions

The rederivation of state-space models of converter systems when adding input
filters or non-trivial loads such as other converters was, up to now,
considered a painful process [7]—and rightfully so. With the systematic
approach of extending such state-space models presented in this article,
adding control loops and building up system models from series-connected
modules such as filters, loads, or converters have become straightforward
tasks.

This modular approach leverages already existing models for common converter
or filter topologies by incorporating them as building blocks, such that users
do not have to manually derive larger sets of state-space equations anymore in
many application cases. If not yet available, the modeling efforts required
are only limited to the respective subsystem.

The small-signal modeling accuracy of the building blocks is not compromised
by the proposed connection operations, the resulting models in all modeling
stages preserve their structure and always remain linear.

The validity of the resulting small-signal models has been illustrated and
compared in the time-domain against circuit simulations.

At all stages of the modeling procedure proposed in this article, the benefits
of a two-port oriented state-space description become obvious, since
input/output impedance and control-to-output or control-to-state transfer
functions are easily obtainable from one single model. This also allows for
the design of controllers starting from the innermost loop in a systematic
fashion, while incorporating interactions with non-trivial source or load
elements.

##  Appendix A Building Blocks: Passive Component Models

The generic two-port modeling approach for passive subsystems of a converter
system has already been introduced in Section 2.1. In this section, two
typical examples often found in a converter system will be given: an (ideal)
resistive load, and an LC input filter. These are also used in the examples
presented in Section 6.

###  A.1 Resistive Load

With no storage elements, there are no dynamics in a resistive system as shown
in Figure 12a, which means that $\bm{A}=0$, $\bm{B}=0$, $\bm{C}=0$ hold. Only
the $\bm{D}$ matrix elements of a two-port state-space model according to (1)
are present, and one obtains the model (16).

$\begin{pmatrix}i_{\mathrm{in}}(t)\\\
v_{\mathrm{out}}(t)\end{pmatrix}=\begin{pmatrix}\frac{1}{R}&-1\\\
1&0\end{pmatrix}\cdot\begin{pmatrix}v_{\mathrm{in}}(t)\\\
i_{\mathrm{out}}(t)\end{pmatrix}$ (16)

###  A.2 LC Filter

In (17), the two-port state-space model of an LC filter as depicted in Figure
12b is being given. In addition to the circuit in Figure 12b, the effects of
equivalent series resistances (ESR) in both inductor (ESR $r_{\mathrm{L}}$)
and capacitor (ESR $r_{\mathrm{C}}$) have been accounted for in (17).

$\begin{split}\begin{pmatrix}\dot{i}_{\mathrm{L}}(t)\\\
\dot{v}_{\mathrm{C}}(t)\end{pmatrix}=&\begin{pmatrix}-\frac{r_{\mathrm{L}}+r_{\mathrm{C}}}{L}&-\frac{1}{L}\\\\[4.30554pt]
\frac{1}{C}&0\end{pmatrix}\cdot\begin{pmatrix}i_{\mathrm{L}}(t)\\\\[4.30554pt]
v_{\mathrm{C}}(t)\end{pmatrix}+\begin{pmatrix}\frac{1}{L}&-\frac{r_{\mathrm{C}}}{L}\\\\[4.30554pt]
0&\frac{1}{C}\end{pmatrix}\cdot\begin{pmatrix}v_{\mathrm{in}}(t)\\\\[4.30554pt]
i_{\mathrm{out}}(t)\end{pmatrix}\\\ \begin{pmatrix}i_{\mathrm{in}}(t)\\\
v_{\mathrm{out}}(t)\end{pmatrix}=&\begin{pmatrix}1&0\\\
r_{\mathrm{C}}&1\end{pmatrix}\cdot\begin{pmatrix}i_{\mathrm{L}}(t)\\\
v_{\mathrm{C}}(t)\end{pmatrix}+\begin{pmatrix}0&0\\\
0&r_{\mathrm{C}}\end{pmatrix}\cdot\begin{pmatrix}v_{\mathrm{in}}(t)\\\
i_{\mathrm{out}}(t)\end{pmatrix}\end{split}$ (17)

(a)

(b)

Figure 12: Two-port network subsystems for passive elements. (a) Resistive
load. (b) LC filter.

As a side note it can be stated that manually creating a model for a two-stage
LC filter is not necessary. A benefit of the connection method presented in
Section 4 will be that one can create a series connection of two LC filter
models from (17) if that is necessary.

More generally speaking: it is beneficial to create very simple building
blocks as demonstrated in this section, and bypass the error-prone process of
manually modeling a larger system by creating a complete system model from the
combination of these small subsystems instead. The necessary tools are
presented in Section 4.

##  Appendix B Building Blocks: Converter Models

###  B.1 Available Converter Models

The two-port state-space approach to converter modeling goes beyond most
state-space models found in the literature, which mostly do not account for
the output current $i_{\mathrm{out}}(t)$ as an additional input variable of
the model. This is probably due to the fact that state-space models have so
far mostly been used only as an intermediate step in order to derive transfer
functions in frequency domain.

Nevertheless, a variety of two-port oriented state-space models can already be
found in more recent work, and these can be used here almost in a plug-and-
play fashion. For example, models for buck, boost, and flyback converter in
peak current mode (PCM) control, all in continuous conduction mode (CCM), are
presented in [27]; and PCM-controlled buck and boost converters in DCM in [31,
33], respectively. Furthermore, ready-to-use state-space models for buck,
boost, and buck-boost converters can be found in [30, 34], which have to be
seen as the most comprehensive references devoted to systematic two-port
modeling of converters.

In this article, the recommended strategy is to model the load subsystem of
the converter separately (cf. Appendix A), and connect it to the converter
models afterwards (cf. Section 4). This implies that, ideally, the converter
modeled with (2) should be unterminated at first. However, most state-space
converter models found in the literature do feature a resistive load, but
these can be used here by letting $R\rightarrow\infty$, as done e.g., in [29].
The models presented in [30] do already describe unterminated converters.

###  B.2 Example: Boost Converter

For an unterminated boost converter operating in CCM (cf. Fig. 13), the
$\bm{A}$, $\bm{B}$, $\bm{C}$, $\bm{D}$ matrices for a linearized state-space
model (2) with small simplifications (e.g., ideal switches) are given in (18).
The power stage inductor $L$ and capacitor $C$ are being modeled with ESR
values $r_{\mathrm{C}}$ and $r_{\mathrm{L}}$. $D$, $I_{\mathrm{L}}$, and
$V_{\mathrm{out}}$ define the operating point for the duty cycle, the average
inductor current, and the output voltage, respectively. The two state
variables are the inductor current $i_{\mathrm{L}}(t)$ and the output
capacitor voltage $v_{\mathrm{C}}(t)$, i.e.,
$\bm{x}(t)=\begin{pmatrix}i_{\mathrm{L}}(t)&v_{\mathrm{C}}(t)\end{pmatrix}^{\mathrm{T}}$.

$\displaystyle\bm{A}=\begin{pmatrix}-\displaystyle\frac{((1-D)\cdot
r_{\mathrm{C}}+r_{\mathrm{L}})}{L}&-\displaystyle\frac{1-D}{L}\\\\[8.61108pt]
\displaystyle\frac{1-D}{C}&0\end{pmatrix},$
$\displaystyle\bm{B}=\begin{pmatrix}\displaystyle\frac{1}{L}&-\displaystyle\frac{(1-D)\cdot
r_{\mathrm{C}}}{L}&\displaystyle\frac{V_{\mathrm{out}}}{L}\\\\[8.61108pt]
0&\displaystyle\frac{1}{C}&-\displaystyle\frac{I_{\mathrm{L}}}{C}\end{pmatrix},$
(18) $\displaystyle\bm{C}=\begin{pmatrix}1&0\\\ (1-D)\cdot
r_{\mathrm{C}}&1\end{pmatrix},$ $\displaystyle\bm{D}=\begin{pmatrix}0&0&0\\\
0&r_{\mathrm{C}}&-r_{\mathrm{C}}\cdot I_{\mathrm{L}}\end{pmatrix}$

Figure 13: Two-port view of an unterminated boost converter stage.

This model is also used and compared against time-domain circuit simulations
in the examples presented in Section 6.2 (boost converter with input filter)
and Section 6.3 (series-connected boost and buck stages).

##  Appendix C Building Blocks: Controller Models

For the control of voltage and current in power converters, some of the most
prominent examples of controllers used in practice besides proportional-
integral controllers are the so-called Type 1, Type 2, and Type 3 controllers
described in [37, 6]. These are integral controllers with zero, one, or two
lead-lag elements, respectively, and are also being used in the examples in
Section 6.

In table 3, transfer functions and state-space models according to (3) are
given for these controller types. For all of them, the state-space models do
not possess direct feedthrough terms in (3), i.e., $\bm{D}_{\mathrm{C}}=0$.

Table 3: Transfer function $G_{\mathrm{C}}(s)$ and state-space model matrices $\bm{A}_{\mathrm{C}}$, $\bm{B}_{\mathrm{C}}$, $\bm{C}_{\mathrm{C}}$ according to (3) for controllers known as Type 1, 2, and 3. Type 1 | Type 2 | Type 3
---|---|---
$G_{\mathrm{C}}(s)=\displaystyle\frac{K_{\mathrm{i}}}{s}$ | $G_{\mathrm{C}}(s)=\displaystyle\frac{K_{\mathrm{i}}}{s}\cdot\frac{1+T_{\mathrm{z}}s}{1+T_{\mathrm{p}}s}$ | $G_{\mathrm{C}}(s)=\displaystyle\frac{K_{\mathrm{i}}}{s}\cdot\frac{(1+T_{\mathrm{z1}}s)(1+T_{\mathrm{z1}}s)}{(1+T_{\mathrm{p1}}s)(1+T_{\mathrm{p2}}s)}$
$\bm{A}_{\mathrm{C}}=0$ | $\bm{A}_{\mathrm{C}}=\begin{pmatrix}0&0\\\\[4.30554pt] 1&-\frac{1}{T_{\mathrm{p}}}\end{pmatrix}$ | $\bm{A}_{\mathrm{C}}=\begin{pmatrix}0&0&0\\\\[4.30554pt] 1&0&-\frac{1}{T_{\mathrm{p1}}T_{\mathrm{p2}}}\\\\[4.30554pt] 0&1&-\frac{T_{\mathrm{z1}}+T_{\mathrm{z2}}}{T_{\mathrm{p1}}T_{\mathrm{p2}}}\end{pmatrix}$
$\bm{B}_{\mathrm{C}}=K_{\mathrm{i}}$ | $\bm{B}_{\mathrm{C}}=\begin{pmatrix}\frac{K_{\mathrm{i}}}{T_{\mathrm{p}}}\\\\[4.30554pt] \frac{K_{\mathrm{i}}T_{\mathrm{z}}}{T_{\mathrm{p}}}\end{pmatrix}$ | $\bm{B}_{\mathrm{C}}=\begin{pmatrix}\frac{K_{\mathrm{i}}}{T_{\mathrm{p1}}T_{\mathrm{p2}}}\\\\[4.30554pt] \frac{K_{\mathrm{i}}\left(T_{\mathrm{z1}}+T_{\mathrm{z2}}\right)}{T_{\mathrm{p1}}T_{\mathrm{p2}}}\\\\[4.30554pt] \frac{K_{\mathrm{i}}T_{\mathrm{z1}}T_{\mathrm{z2}}}{T_{\mathrm{p1}}T_{\mathrm{p2}}}\end{pmatrix}$
$\bm{C}_{\mathrm{C}}=1$ | $\bm{C}_{\mathrm{C}}=\begin{pmatrix}0&1\end{pmatrix}$ | $\bm{C}_{\mathrm{C}}=\begin{pmatrix}0&0&1\end{pmatrix}$

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