1

Modelling Approaches for DC-DC Converters

With Switched Capacitors
J.C. Mayo-Maldonado, Student Member, IEEE, J.C. Rosas-Caro, Member, IEEE and P. Rapisarda

Abstract—In this paper we review relevant problems output impedances. The approach in [14] provides state-
in the modelling of DC-DC converters with switched space models by solving numerically the loss equations
capacitors. We study several approaches that overcome the depending on the position of the switches. The results
exposed modelling difficulties, addressing ideal and non- are used to calculate the steady state gain and a steady
ideal cases and using dynamic equations that are valid in
state equivalent resistance. Approaches considering ideal
a large signal domain.
switches and consequently, discontinuities on the volt-
Index Terms—Averaging, DC-DC converters, modelling, ages across capacitors have been discussed in [15], [16].
switched capacitors; switched systems. Parasitic resistances have played also an important role
in the study the dynamic performance of SC convert-
I. I NTRODUCTION ers [17], since energy losses during charge/discharge
Switched capacitor (SC) converters offer several ad- processes permit the elaboration of accurate dynamic
vantages such as light weight, small size, high power models (see [11],[18],[19],[20]).
density and large voltage conversion ratios [1], which In this paper we gather and discuss theory and princi-
results important for a large number of applications (see ples regarding the operation of SC converters. Moreover,
e.g. recent developments in [2], [3], [4], [5], [6]). For we present a systematic exposition of three modelling
these reasons, increased interest has been given to their approaches for DC-DC converters with SCs. The pro-
design, modelling and control. Many control techniques cedures are illustrated using a Fibonacci SC converter
for power converters are based on nonlinear models, obtained from [21] and the three switch high-voltage
see for instance the extensive compendium of control converter proposed in [22]. The methods are similar to
techniques presented in [7]. Such models and conse- the classical averaging techniques that consider equiva-
quently their associated nonlinear controllers are able to lent circuits depending on the position of the switches.
perform well in wide ranges of operation compared with However, instead of using the real ESR lumped in the
linearised models. However, the conventional approach circuit (see for instance [8],[23]), we consider: 1) the
for the regulation of SC based converters is a linear case with ideal switches where discontinuous signals
feedback control that is based on approximate small- are allowed, 2) an average loss modelling based on the
signal linearised models of the circuit topologies. In results provided in [18], and 3) a reduced order model
some applications, this approach does not make the con- based on a voltage balancing property.
verters able to respond well to requirements of regulation
II. P RELIMINARIES
in the presence of a wide range of input voltages and load
variations [8]. Power electronics devices with two linear dynamic
Different approaches have been proposed for the mod- modes and ideal switches can be modelled using the
elling of SC converters, such as incremental graph ap- following switched linear system structure (see [24],
proaches [9], useful for determining steady state voltage [25])
d
gains. In [10], [11], [12], approaches for modelling SC x = Au x + Bu ; u = 0, 1 ; (1)
are given by considering the inherent losses produced dt
when capacitors are connected in parallel. A steady where x(t) ∈ Rn×1 is called the state function; Au ∈
state modelling approach is provided [13] in which Rn×n , Bu ∈ Rn×1 are the matrices that define the
SC converters are analysed by considering equivalent physical laws of the dynamic modes, and u = 0, 1, a
binary index term that denotes which of the two modes
J.C. Mayo-Maldonado and P. Rapisarda are with the CSPC is active according to a specified switching signal. If
group, School of Electronics and Computer Science, University of
we assume that the switching signal is periodic and
Southampton, Great Britain, e-mail: jcmm1g11,pr3@ecs.soton.ac.uk,
Tel: +(44)2380593367, Fax: +(44)2380594498. J.C. Rosas-Caro is that the trajectories of the system variables are every-
with Universidad Panamericana Campus Guadalajara, Mexico. where continuous, we can approximate the dynamics of
 2

the switched linear system into averaged quantities by

considering a duty cycle denoted by D. This action is
equivalent to approximate a switched linear system into a
nonlinear system where the so-called current and voltage
ripples of the converter are neglected. The averaging
technique allow us to obtain the following structure
d
xav = [DA0 + (1 − D)A1 ] xav + DB0 + (1 − D)B1 ;
dt
(2)
where xav (t) ∈ Rn×1 is the averaged state function.
When we consider the duty cycle D as an input, the
averaged system can be conveniently written in state
d
affine nonlinear form dt xav = f (xav ) + g(xav )D ; with
f (xav ) := A1 xav + B1 and g(xav ) := (A0 − A1 )xav +
B0 − B1 . The latter structure is usually the starting
Fig. 1. Fibonacci switched-capacitor converter: (a) Full schematic;
point for the dynamic analysis and control of DC-DC Equivalent circuits when (b) the switches “1” are closed and when
converters, since it can be derived almost directly from (c) the switches “2” are closed.
the switched linear system structure (1) and it allows
to apply a wide number of nonlinear control techniques
(see for instance the compendium of controllers in [7]). A. Dynamic modelling with ideal switches
Unfortunately, as we will expose in the following section,
In order to describe the dynamics of the converter
the traditional averaging technique cannot be applied to
considering ideal switches, we proceed as usual; we
systems with discontinuous trajectories, as it is the case
model the dynamic modes for each equivalent circuit
of switched-capacitor converters. In other words, the par-
using current and voltage laws. Let us consider the case
allel connection between capacitors, which is the main
in Fig. 1(b). We obtain the following set of equations
feature of such converters, produces discontinuities on 
the voltages across their terminals at switching instants. 
 v1 − E = 0 ,
In mathematical terms, such situation is derived from

v1 + v2 − v3 = 0 ,



the introduction of algebraic constraints to the dynamic Σ1 : (3)
v3 − v4 = 0 ,
modes of the converter. Consequently, an averaged state 

 C2 d v2 + (C3 + C4 ) d v4 + v4 = 0 .

affine nonlinear model cannot be directly obtained. In


order to overcome this issue, we study several modelling dt dt R
alternatives for this type of converters. Moreover, the set of equations corresponding to the
circuit in Fig. 1(c), are the following

III. M ODELLING APPROACHES 
 v1 − v2 + E = 0 ,

 C1 v 1 + C2 d v 2 = 0 ,
d


In order to discuss the modelling procedures, we

 dt

dt
consider the Two-phase Fibonacci SC converter depicted Σ2 : d (4)
in Fig. 1, corresponding to a simplified version1 of the  C 3 v 3 = 0 ,
dt


SC converter in Fig. 1(b) of [21]. We aim at showing



 d v4
a detailed exposition that provides sufficient insight to

 C4 v 4 + =0.
dt R
extend the discussed approaches to other topologies with
SCs. Note that since the switching produces parallel con-
nections among the capacitors and the source, there
The converter in Fig. 1(a) has two possible modes
exist algebraic equations expressing the corresponding
depending on the position of the group of switches “1”
equalities for their voltages. As studied in the previous
and “2” illustrated by blocks and whose operation is
section, when we model standard power converters, the
complementary. Fig. 1(b) and Fig. 1(c) show the two
following step usually consists in obtaining one single
possible equivalent circuits of the converter.
set of equations by considering the duty cycle and the
1 average value of the state variables as in equation (2).
For ease of exposition, the third “Fibonacci cell” of the SC
converter in Fig. 1(b) of [21] has been omitted, and a parallel RC However, if we try to follow such a method, we find that
load is considered. the structure (2), which is based in a set of first order
 3

differential equations, cannot be satisfied since zero-th which together with the algebraic constraint v2 (t+ s) −
order equations (i.e. algebraic constraints) are involved in v1 (t+ ) = E(t + ) = E(t− ), we can determine the reset
s s s
the dynamics of the converter. Moreover, we eventually rule
find two additional issues:
 E(t+ ) 
s
   E(t−s ) 
+
Ce 2
0 0 0 0 −

1. The voltages across capacitors are discontinuous  vv1 (ts )

1 −C2 C1 C2 0 0  v1 (ts ) 
 2 (t+s+ )  = C  C1 C1 C2 0 0   v2 (t−s− )  , (6)
v3 (ts ) e2 0 0 0 10 v3 (ts )
at switching instants, i.e. there exist instantaneous 0 0 0 01
v4 (t+
s ) v4 (t−
s )
“jumps” in the trajectories of the system variables.
2. The value of the voltages at the switching instants is where Ce2 = C1 + C2 .
not uniquely determined, i.e. there exist more than The voltage ripples of the converter using equations2
one possible consistent choice of initial conditions (4)-(6) and a periodic switching signal are shown in Fig.
that satisfy the equations at switching instants. 2. Note that the discontinuous voltages are concatenated
via the reset rules after the switch.
Since the value of the capacitors need to be uniquely
specified at switching instants, we consider a model
addressing instantaneous values, instead of using the
traditional averaging technique. We proceed to complete
the dynamic model of the converter by introducing a
reset rule for the system variables acting at the switch-
ing instants. In order to do so, we use the notation
f (t− ) := limτ %t f (τ ) and f (t+ ) := limτ &t f (τ ), to
define the limit of a time function taken from the left
and from the right respectively.
In order to provide a realistic set of initial conditions at
the switching instants, the reset must respect the “prin-
ciple of conservation of charge” (cf. the redistribution
of charge in [15], [19]). We proceed by considering
the capacitors whose voltage is subject to algebraic
constraints, for instance when we switch from Σ2 to
Σ1 , the total charge in the capacitors that exhibits a
redistribution due to parallel connections must be the
same before and after every switching instant ts , i.e.
Fig. 2. Voltage ripples of the Fibonacci SC converter.
C2 v2 (t−
s) + C3 v3 (t−
s) + C4 v4 (t−
s)
= C2 v2 (t+
s) + C3 v3 (t+ +
s ) + C4 v4 (ts ) . Equations (4)-(6) represent an alternative dynamic
model based on instantaneous values for the discussed
Additionally, the algebraic constraints of Σ1 dictate that SC converter. Note also that the model describes dynam-
v1 (t+ + + + + +
s ) = E(ts ), v2 (ts ) = v3 (ts ) − v1 (ts ), v3 (ts ) = ics in a large signal domain.
+
v4 (ts ).
After straightforward algebraic manipulations and as- Remark 1. The presented approach allows to study the
suming without loss of generality that the voltage E is dynamics of the SC converters in the sense of switched
constant (and consequently E(t− + linear systems (see e.g. [24], [25], [26]). Consequently,
s ) = E(ts )), we obtain
the following reset rule their overall dynamic properties such as stability, stabil-
isability and control can be studied in this setting.
 E(t+ ) 
s
 Ce1 0   E(t− ) 
0 0 0 s
+
 vv1 (ts ) C 1 0 0 0 0 v1 (t−
s )
Remark 2. Note that the need to specify reset rules cor-
1  −C3e−C
 2 (t+s+ )  = −
, (5)

4 0 C2 C3 C4   v2 (ts )  responds to a more general dynamic modelling approach
v3 (ts ) Ce1 C2 0 C2 C3 C4 v3 (t−
s )
v4 (t+
s )
C2 0 C2 C3 C4 v4 (t−
s )
than that of traditional converters whose trajectories are
continuous at switching instants. For instance, in the
where Ce1 = C2 + C3 + C4 . Similarly, when we switch case of the switched linear system in equation (1), it
from Σ1 to Σ2 , the physical redistribution of charge is assumed that x(t− +
s ) = x(ts ) and consequently, the
establishes that for every switching instant ts we have matrices associated to the reset rules are trivially equal
that to the identity.
C1 v1 (t− − + +
s ) + C2 v2 (ts ) = C1 v1 (ts ) + C2 v2 (ts ) .
2
For this simulation, we used the parameters specified in Sec. III-3.
 4

B. Average losses-based modelling Ce = (C1 C2 )/(C1 + C2 ) is the equivalent capacitance.

The study of the large signal dynamics of SC con- From the above equation, considering average values, an
verters considering averaged quantities can be performed the equivalent resistance Re of the circuit can be defined
in the nonlinear systems setting by considering non- as a function of the duty cycle, i.e. Re : [0, 1] → R,
ideal switches. In this case, the issue regarding voltage where
1 + e−β
 
1
discontinuities vanishes. Re (D) := , (8)
2fs Ce 1 − e−β
We now discuss the concept of average dynamic
modelling with non-ideal switches. The approach has with β = (DTs )/(Rs Ce ).
been studied in detail in [18] for the modelling of Remark 3. In [27], it has been determined that the value
purely capacitor-based converters and in [20] for a hybrid of the dissipated power (7) in steady state is bounded by
multiplier converter. Such approach applies the method the Slow/Fast Switching Limits, corresponding respec-
of average current between capacitors in parallel, rather tively to the dissipated power considering ideal switches
than the use of instantaneous values. The method is first and the absence of parasitic resistances; and to the
discussed in relation to Fig. 3(a) in which a capacitor case when non ideal elements appear and the switching
C1 is connected via a switch S to a second capacitor frequency is fast enough to consider constant voltages
C2 , while the initial voltages across capacitors C1 and across capacitors during the switching period.
C2 are different, i.e. V2 (0) 6= V1 (0).
We can conclude that the average behaviour of the
circuit can be described by the average model of Fig.
3(c) in which voltages and currents are assumed to
be constant within the switching period Ts , which is
consistent with the concept of average models. In this
basic circuit, the average current Iav can be expressed
as,
V1av − V2av
Fig. 3. Electric diagram of a simple SC circuit and its equivalent Iav = .
circuits. Re (D)
Following the losses-based averaging technique dis-
When the switch is closed, assuming V1 (0) > V2 (0), cussed above, we obtain the following set of equations
an electrical current flow from C1 to C2 . The shape for the Fibonacci SC converter in Fig. 1 by applying
of the charging/discharging current will depend on the current laws with respect to each capacitor

total loop resistance Rs , see Fig. 3(b), which consists of d E − V1av E + V1av − V2av
C1 V1av = −


the sum of the switch resistance and capacitors’ ESR. dt Re1 (D) Re3 (D)




Hence, the actual circuit can be described by Fig. 3(b). V + V − V3av


 1av 2av
The instantaneous circuit of Fig. 3(b) can be converted

 − ,
Re2 (D)



into an average equivalent form by calculating the power

d E + V1av − V2av


C2 V2av =


loss of the circuit in Fig. 3(b). The intuition behind this 
 dt Re3 (D)


method is to obtain an off-line expression derived from 
 V1av + V2av − V3av
the computation of the average amount of dissipated Σloss : − ,
 Re2 (D)
energy according to the duty cycle of the converter, then 
d V1av + V2av − V3av



such expression is associated to the value of a variant 
 C 3 V 3av =

 dt Re2 (D)
equivalent series resistor. The time domain solution 

V3av − V4av


of the energy loss P dissipated by the loop resistor 

 − ,
Rs in Fig. 3, can be expressed in terms of decaying


 Re4 (D)

exponentials as (see [18], p. 3343)

 d V3av − V4av V4av
 C4 V4av = − .


dt Re4 (D) R
2 1 + e−β
 
Iav (9)
P = , (7)
2fs Ce 1 − e−β with
1 + e−βi
 
where β := (DTs )/(Rs Ce ), fs is the switching fre- 1 DTs
Ri (D) := −β
; βi = ;
quency, Iav is the average current of the circuit (over 2fs Cei 1 − e i Rsi Cei
the switching period Ts = 1/fs ), D the duty cycle that where Rsi , Cei , i = 1, ..., 4, are the total resistance and
sets the charge/discharge time period DTs (when switch capacitance respectively of the loops where the average
S in Fig. 3(a) is closed), for the circuit in Fig. 3(b), and currents are analysed.
 5

Remark 4. The dynamic model (9) presents several

advantages: 1) The model describes the dynamics of
the converter in a large-signal domain. 2) The averaged
equations permit the computation of the converter gain
in a standard way, i.e. we can consider the derivative
of the state variables to be equal to zero, and after
straightforward computations it follows that V3av ≈ 3E
where the approximation accounts the power losses of
the circuit. 3) The model is able to capture additional
situations including: complete charging, partial charg-
ing and no effective charging; depending on the ratio
(DTs )/(Rs Ce ) (cf. [18]).
Fig. 4. Comparison of the output voltage “v4 ” simulation and the
Remark 5. The main disadvantage of the presented proposed dynamic models.
model is its very high nonlinear structure. Due to this is-
sue, standard nonlinear control techniques that have been
set up for state affine nonlinear systems as in equation
(2) cannot be applied in a straightforward way using this IV. DYNAMIC MODELLING OF HYBRID CONVERTERS
type of models and a more sophisticated mathematical
treatment is required. Moreover, the accurate estimation SCs can be also combined with inductor/capacitor
of the equivalent loop resistances Rs may not be an stages that are not necessarily of discontinuous nature,
easy task, however a detailed exposition of the modelling we call these type of topologies hybrid SC converters.
of non-ideal elements in SC loops including non-ideal An example of a hybrid topology is the three switch
switches has been provided in [18]. high-voltage converter depicted in Fig. 5, that was firstly
proposed in [22] and recently used for current-ripple can-
cellation topologies [28]. Although the topology presents
C. Simulation results a basic principle of operation and a reduced number
of components, only its small-signal dynamic model is
In Fig. 4, we show the comparison between the available in the literature (see [22]). According to the
circuit-based simulation of the output voltage v4 the
SC converter in Fig. 1 using the software Synopsys
Saber, and that of the simulations of the model with
ideal switches in (4)-(6) and the average power loss-
based model in (9) using Matlab. We consider the
parameters E = 10V , fs = 50kHz , C1 = C2 = C3 =
C4 = 10µF and R = 100Ω. The total loop resistances
are Rs1 = 0.21Ω, Rs2 = 0.043Ω, Rs3 = 0.032Ω,
Rs4 = 0.022Ω, which are for this case the sum of the
switch on resistances Rsw = 0.01Ω of the transistors and
the capacitors ESR Rc = 0.001Ω, i.e. Rs1 = 2Rsw + Rc ,
Rs2 = 4Rsw + 3Rc , Rs3 = 3Rsw + 2Rc , and Rs4 =
2Rsw + 2Rc .
Fig. 5. Three Switch High Voltage Converter: (a) Full schematic;
Remark 6. The simulation shows that both, the dis- Equivalent circuits when (b) the switch is closed and (c) the switch
continuous and the average power loss-based models is open.
provide reliable information regarding the dynamics of
the circuit. Since both the switched linear- and the
nonlinear- systems frameworks offer powerful tools for material discussed in the previous sections, we show
dynamic analysis and control, the selection of the more two large-signal dynamic models for the three switch
appropriate approach relies on the application, i.e. where high-voltage converter. The model with ideal switches
either instantaneous or averaged values can be of special derived from the material in Sec. III-A encompasses the
interest. following modes according to the equivalent circuits in
 6

Fig. 5(b) and Fig. 5(c) respectively. of the SCs are much faster than those of the overall

d converter, consequently a zero-th order approximation

 L i = E − v1 , on the voltages across the SCs is used. Consider the SC
 dt


sub-circuit in Fig. 3(c) which shows an equivalent circuit

d
Σ1 := C1 v1 = i , that considers the average current between capacitors.

 dt
In such circuit, the magnitude of the equivalent resistor
 C2 d v 2 = − v 2 .



dt R Re may vary arbitrarily by modifying the duty cycle

d or the switching period. Moreover, from equation (8)
L i=E, we can conclude that at higher frequencies, the average


dt


losses inherent in the SCs decrease and the voltage across

Σ2 : d v1
(C1 + C2 ) v1 = − , capacitors tends to be constant during the switching
dt R


period Ts (see [30]). Consequently, the voltage across


 v2 = v1 .
C1 and C2 tend to be the same with the average
The reset rule when we switch from Σ2 to Σ1 at ts is difference being the voltage across the resistor Re , i.e.
given by V1av − V2av = Re Iav for the circuit in Fig. 3(c). In the
E(t+ E(t− case of the converter in Fig. 5, if we assume that the
    
s) 1 0 0 0 s)
 i(t+  +  average voltage across capacitors C1 and C2 is the same,
 s+  = 0
) 1 0 0
  i(ts−)  ,
 
v1 (ts ) 0 0 1 0 v1 (ts ) i.e. V1av = V2av , we automatically neglect the nonlinear
v2 (t+ v2 (t− terms associated to the power losses by considering the
s) 0 0 0 1 s)
sum of the dynamic equations for C1 and C2 in (10).
Similarly, when we switch from Σ1 to Σ2 at ts we have Thus we obtain the following reduced order average
E(t+ E(t−
    
s) 1 0 0 0 s)
dynamic model
 i(t+  + 
 s+)  = 0 1 0 0    i(ts−)  .
 
d

v1 (ts ) 0 C1 C2  
0 C1 +C2 C1 +C2 v1 (ts ) 
 L I = E − (1 − D)Vo ,
v2 (t+ C1 C2 v2 (t− ΣRO : dt (11)
s) 0 0 C1 +C2 C1 +C2 s)
 (C1 + C2 ) d Vo = (1 − D)I − Vo .

Applying the the power-loss based modelling in Sec. dt R
III-B we obtain the following set of average nonlinear
where I is the input current and Vo the output voltage
dynamic equations for the converter in Fig. 5.
(the voltage across C2 ). The model provides an approxi-
d


 L ILav = Vin − (1 − D)V1av , mation with a reduced number of equations considering


 dt an ideal case (without losses), which can easily adopt

 d the standard state affine nonlinear form (2). The “open
 C1 dt V1av = (1 − D)Iav




 loop” comparison of the output voltage considering the
1 − e−β
  
circuit simulation and equations (11) is depicted in

ΣAv : − 2fs Ce (V1av − V2av ) ,

 1 + e−β Fig. 6. The parameters used for the simulations are
d V2av C1 = C2 = 50µF , R = 50Ω and L = 300µH .


C2 V2av = −





 dt R
1 − e−β

 


 + 2fs Ce (V1av − V2av ) .
1 + e−β
(10)
with β = (DTs )/(Rs Ce ), where Rs is the total loop
resistance between the SCs and Ce = (C1 C2 )/(C1 +C2 ).
Moreover, a particular property in hybrid SC con-
verters allow us to propose a third modelling method
based on a reduced order approximation. This method
offers additional structural advantages with respect to the
previously discussed approaches, such as basic mathe-
matical representations, i.e. allowing standard state affine
nonlinear forms, and a reduced number of equations.
In order to obtain such model, we recall the voltage Fig. 6. Comparison of the output voltage “v2 ” circuit simulation and
the reduced order model.
balancing property (cf. [23],[29]). The intuition behind
this method is to exploit the fact that the dynamics
 7

Remark 7. Note that the reduced order model is methods were illustrated using a Fibonacci SC converter
based on the voltage-balancing assumption. Conse- and a three switch high voltage converter. The method
quently, cases such as soft-switching (see e.g. [31]) discussed in Sec. III-A allows the study of the converter
can be considered as long as the assumption holds. in a large signal domain allowing discontinuous signals,
However the introduction of new dynamic elements such taking into account instantaneous values; the latter re-
as small series inductors will increase the differences in sults convenient for the analysis of current and voltage
the dynamics on the voltages across switched capacitors, ripples. The approach in Sec. III-B provides an average
reducing the level of accuracy of the model. nonlinear model that captures non-ideal features such as
power losses. Finally, the method in Sec. IV allows the
Remark 8. Fig. 4 and Fig. 6 correspond to open-loop
use of basic nonlinear models with a reduced number of
simulations of the circuit topologies and the proposed
equations that results convenient for control purposes.
models. The simulations corroborate the “energy trans-
Future research directions include the development of
fer” principles that have been studied in this paper
control techniques using the approaches discussed in this
and illustrates the discussed features of the proposed
paper. Associated with the approach in Sec. III-A, new
approaches. Closed-loop implementations can be used
theoretical developments are under study, where issues
to test the discussed models under more challenging
such as modularity, i.e. the incremental development and
scenarios e.g. in the presence of arbitrary load and input
combination of mode dynamics, are of special interest,
voltage variations. However, since the theory and the
see e.g. [26] and [32].
technical issues that arise from those implementations
are part of an important research area in control systems
that needs to be studied in detail, we have reserved such R EFERENCES
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