Analysis of a Flying Capacitor Converter:A Switched Systems Approach

Alex Ruderman, Boris Reznikov, and Michael Margaliot

Abstract

Flying capacitor converters (FCCs) attract considerable interest because of their inherent naturalvoltage balancing property. Several researchers analyzed the voltage balance dynamics in FCCs usingfrequency domain methods.

Recently, considerable research attention has been devoted to switched systems, i.e. systems com-posed of several subsystems, and a switching law that determines which subsystem is active at everytime instant.

In this paper, we propose a new approach to the analysis of an FCC. The analysis is performed inthe time–domain, treating the FCC as a switched system. The subsystems are the various configurationsobtained for each state of the circuit switches, and the switching law is determined by the modulation.

We demonstrate this new approach by using it to analyze a single–phase single–leg three–levelFCC. The switched systems approach provides simple closed–form expressions for the system behavior.We show that the natural balancing property is equivalent to the asymptotic stability of a certain matrix.We also show that it is possible to rigorously analyze properties such as the capacitor time constantand relate them to the parameter values of the load, carrier frequency, and duty ratio.

Index Terms

Multilevel converters; flying capacitor; affine switched systems; periodic switching law.

I. INTRODUCTION

Multilevel converters (MCs) are recently attracting considerable interest. When compared toconventional converters, MCs allow higher power ratings, higher efficiency, and lower harmonicdistortion. The topology, modulation strategy, and performance of MCs have been extensivelystudied over the last two decades (see, e.g. [1], [2], [3]). Although initially suggested forhigh–voltage power applications, MCs may also be interesting for low– and medium–powerconverters [4], [5].

Three basic MC architectures are: multiple point clamped (MPC) (or diode clamped), flyingcapacitor, and cascaded H–bridge with separate DC sources. A shortcoming of the MPC archi-tecture is that it operates in a limited (modulation index/load power factor) envelope [6]. Themaximal possible modulation index is achieved only for a pure reactive load. As the load powerfactor approaches one, the modulation index is reduced due to limitations imposed by the needto actively balance the capacitor voltages (see also [7]).

An abridged version of this paper was presented at the 13th International Power Electronics and Motion Control Conference(EPE–PEMC’08), Poland, 2008.

AR (aruderman@elmomc.com) is with Elmo Motion Control Ltd, Israel; BR (reznikovb@spb.gs.ru) is with theGeneral Satellite Corporation, Russia; MM (michaelm@eng.tau.ac.il) is with the School of Electrical Engineering–Systems, Tel Aviv University, Israel.

2

Several strategies for addressing the balancing problem in diode clamped DC/AC MCs withpassive front ends have been proposed (see [8] and the references therein). More generally,voltage balancing is regarded as the most important problem in the field of high–voltage MCs [9].

The flying capacitor converter (FCC) is a multilevel pulse–width modulated (PWM) converterwhose internal architecture automatically guarantees the voltage balancing property for passiveloads. It thus provides an attractive alternative to the MPC. Furthermore, a single–leg FCCmay be used for both DC/DC and DC/AC conversion, whereas an MPC converter cannotserve as a DC/DC converter. This is so because a DC current at a clamping point permanentlycharges/discharges the DC bus capacitors, so no voltage balance is possible in principle. Evenfor the case of DC/AC conversion, significant low frequency clamping point voltage oscillationsmay appear for some operating conditions [6].

Several authors studied the dynamic behavior of FCCs (see, e.g., [9], [10], [11], [12], [13]).The analysis is usually based on representing the switching strategy using a piecewise constantfunction and then applying frequency domain methods. This is probably due to the importantrole of the load current high–order harmonics in the voltage balancing process. Indeed, consideran ideally smooth, ripple–free load current, and an appropriate phase–shifted voltage modulationstrategy with relatively high switching frequency. Then the capacitor will be charged and dis-charged by the same amount of load current on time intervals of equal durations. The capacitorvoltage will then oscillate around some average value, determined by the capacitor initial voltage,and no voltage balancing will take place in this case.

However, the application of frequency domain methods in this context is non trivial. Ittypically requires a double Fourier transform, lengthy computations, and many simplificationsand approximations. Analysis in the frequency domain also makes it more difficult to gain anintuitive understanding of the physical mechanisms underlying the self balancing property.

Recently, considerable attention is devoted to the analysis of switched systems [14]. Consider mcontinuous–time systems described by:

x(t) = f i(x(t)), i = 1, . . . , m. (1)

Here x(·) ∈ Rn is the state vector, and f i(·) : R

n → Rn describes the dynamics of system i. A

switched system is a mathematical model in the form:

x(t) = fσ(t)(x(t)), (2)

where σ(·) ∈ 1, . . . , m is the switching–law. This models a system that can switch betweenthe m subsystems (1). The switching–law determines which subsystem is active at which timeinstant. For example, if σ(t) = 1 for t ∈ [0, 10), and σ(t) = 2 for t ∈ (10, 15), then the solutionof (2), starting at time 0, follows the dynamics x = f 1(x) for the first 10 seconds, and thenfollows x = f 2(x) for the next 5 seconds.

Switched systems are useful for modeling the combination of continuous–time dynamics withdiscrete switchings. The underlying philosophy is to explicitly take into account the coexistenceof both the continuous– and discrete–time dynamics. Numerous examples and applications canbe found in [14], [15], [16]. In particular, switched systems provide suitable models for electriccircuits that contain on/off switches. Here typical state-variables are the capacitor voltages andinductor currents. Each possible configuration of the set of switches induces a continuous–timedynamics of the state variables. The dynamics of the system thus changes every time a switchopens or closes. In other words, the modulation of the switches determines the switching–law.

February 5, 2009 DRAFT

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It is important to note that even when all the subsystems (1) are linear, i.e. f i(x) = Aix,with Ai ∈ R

n×n, the switched system may demonstrate a highly nonlinear behavior [15]. Thissuggests that frequency–domain approaches may not be appropriate in the analysis of switchedsystems.

In this paper, we demonstrate the switched systems perspective by applying it to analyze asingle–capacitor single–leg three–level FCC. Here each subsystem, corresponding to a possibleconfiguration of the switches, is a second-order affine linear system. Our analysis is based on“stitching” together the analytical solutions of the subsystem trajectories, corresponding to eachmodulation phase of the FCC.

This approach provides considerable insight into the FCC dynamics. In particular, we showthat the natural balancing property is equivalent to the asymptotic stability of a certain matrix.Furthermore, we obtain explicit formulas for the capacitor time constant.

In an FCC, the switching is performed at a frequency that is much higher than the naturalfrequency of the equivalent RLC–circuit. This is needed in order to guarantee that the currentand voltage ripples are relatively low (which is always true for a practical converter). Using thisfact, it is possible to use a small parameter approximation that further simplifies the analyticexpressions. This reveals how the capacitor charge rate depends on the inductive load parameters,the carrier frequency, and the duty ratio. To the best of our knowledge, this is the first time thatsuch formulas are derived. Numerical simulations of the FCC demonstrate excellent agreementwith the analytic results.

The remainder of this paper is organized as follows. Section II reviews the FCC topology andmodulation strategy. Section III derives the mathematical model describing the FCC behavior inthe case of DC–modulation. The asymptotic and transient behavior of this model is rigorouslyanalyzed in Sections IV and V, respectively. Section VI demonstrates how the analytic resultsobtained for the DC–modulated case can be used to analyze the case of AC–modulation. Thefinal section concludes and describes some possible directions for future research.

II. TOPOLOGY AND MODULATION

We consider a single–phase single–leg three–level FCC with an RL load (see Fig. 1). Theconverter is composed of two equal voltage sources, four (ideal) switches, and a capacitor C.The load is modeled by an inductor connected in series with a resistor. Wilkinson et al. [11]refer to this topology as a two–cell inverter.

The four switches operate in two complementary pairs: when switch Si is on (off), switch Si

is off (on). The FCC can thus be in one of four possible configurations.In the DC–modulated case, the modulation strategy is based on comparing a constant command

signal VCOM(t) ≡ VCOM to a triangular wave s(t). The state of the switches is determinedaccording to:

S1 =

on, if s(t) < VCOM(t),

off, otherwise,

S2 =

on, if s(t) > −VCOM(t),

off, otherwise.(3)

The resulting modulation is periodic. In each period the FCC switches between four possiblephases. In phases P1 and P3 the switch configuration is S1S2 (that is, both S1 and S2 are

February 5, 2009 DRAFT

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1S

1S

2S

2S

+-

+-

2/DCV

2/DCV

C L R+-

CvLi

Fig. 1. Single–phase three–level FCC with an RL load.

S1S2S1S2S1S2 S1S2 S1S2

P4P2 P3P1P4

M

0−VCOM

VCOM

s(t)

4t1 4t2 4t3 4t4

TPWM

t

Fig. 2. DC modulation and corresponding switch states.

on); in phase P2 the configuration is S1S2; and in phase P4 it is S1S2. The transition orderis P1 → P2 → P3 → P4 (see Fig. 2).

Let TPWM denote the time of one period, and let 4ti denote the time spent in phase Pi duringone period, so that

4∑

i=1

4ti = TPWM .

February 5, 2009 DRAFT

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R

iL

+−

L

VDC/2

(a)

R

iL

L

−

−

+

+

VDC/2

vc

(b)

R

iL

L

+−VDC/2

vc

+ −

(c)

Fig. 3. FCC topology: (a) in phases P1 and P3; (b) in phase P2; (c) in phase P4.

A simple calculation shows that for the modulation strategy defined above:

4t1 = 4t3 = DTPWM/2,

4t2 = 4t4 = (1 − D)TPWM/2, (4)

where D, the PWM duty ratio, is given by

D := M/(VCOM + M)

with M := maxt s(t) − VCOM .The converter topology in each phase is depicted in Fig. 3. Phases P1 and P3 correspond to

the same topology. Here the capacitor C is disconnected, so its voltage, vc(t), remains constantduring these phases. The configurations during phases P2 and P4 are similar, except for the factthat voltages across the source and capacitor have opposite polarities.

It is desirable to regulate the converter such that the capacitor is charged to a value VDC/2.With this principle in mind, it is possible to calculate the voltage on the load in each possibleconfiguration (see Fig. 3), and to use this to determine the duty ratio yielding the intendedfunctionality of the converter.

The main advantage of the FCC is that the average value of the voltage across the capacitorconverges to VDC/2 for a large range of initial conditions and duty ratios. The next exampledemonstrates this.

Example 1 Consider the FCC depicted in Fig. 1 with: VDC = 100V , R = 1Ω, L = 0.25∗10−3H ,C = 100 ∗ 10−6F and TPWM = 300 ∗ 10−6 sec. The initial conditions (capacitor voltage andinductor current) are zero. Fig. 4 depicts the behavior of the inductor current (iL) and capacitorvoltage (vc) as a function of time, for various values of the duty ratio D.1

Let iL (vc) denote the average value of iL (vc). It may be seen that in each of the four cases:

iL → VDCD/(2R), vc → VDC/2.

Here are a few more observations concerning the behavior of the FCC that can be deduced

1All the simulations in this paper were performed using MATLAB’s ODE45 numerical integration procedure.

February 5, 2009 DRAFT

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0 0.005 0.01 0.015 0.02 0.025−20

−10

0

10

20

30

40

50

60

Time, s

vc

iL

(a)

0 0.005 0.01 0.015 0.02 0.025−20

−10

0

10

20

30

40

50

60

Time, s

vc

iL

(b)

0 0.005 0.01 0.015 0.02 0.025−20

−10

0

10

20

30

40

50

60

Time, s

vc

iL

(c)

0 0.005 0.01 0.015 0.02 0.025−20

−10

0

10

20

30

40

50

60

Time, s

vc

iL

(d)

Fig. 4. Load current and capacitor voltage for different duty ratios: (a) D = 0; (b) D = 0.2; (c) D = 0.4; and (d) D = 0.8.

from Fig. 4. A good approximation of the average current through the inductor is given by:

iL(t) ≈ (1 − exp(−t/TL))VDCD/(2R), (5)

where the time constant isTL := L/R.

To demonstrate this, Fig. 5 depicts iL(t) for the case D = 0.8 and the exponential curve givenby (5).

Note that this implies that we can regulate the current through the load by changing the dutyratio D. Note also that (5) is independent of the capacitance C.

A careful inspection of vc(t) in Fig. 4(a) suggests that this signal is composed of two

February 5, 2009 DRAFT

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0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.0160

5

10

15

20

25

30

35

40

45

Time, s

Fig. 5. iL(t) vs. the exponential curve given by Eq. (5) (solid line).

exponentials. A fast exponent that appears as spikes in the initial part of the response, anda slower exponent that determines the overall rise of vc(t) toward its asymptotic value. The timeconstant T2 of this slower exponent depends on D. As D is increased, the rise time increases,i.e. T2 increases. For D ∈ [0, 1/2], T2 changes slowly with D. The change is more dramatic forvalues D > 0.6.

The increase in the capacitor time constant has a clear physical interpretation. Indeed, as Dincreases to one, the time intervals 4t2 and 4t4 go to zero (see (4)). This means that thecapacitor is never connected, so its time constant must go to infinity.

In the AC–modulated case, the state of the switches is again determined by (3), but thecommand signal is sinusoidal: VCOM(t) = M sin(ηt). Note that in this case, a new phase,corresponding to the state S1S2, appears. The corresponding topology is similar to that ofphase P1, but the polarity of the voltage source is reversed.

The frequency of the command signal is usually much lower than that of the triangular wave, sothat in every cycle [kTPWM , (k +1)TPWM) of the triangular wave, VCOM(t) ≈ VCOM(kTPWM).

Example 2 Consider the FCC with the parameter values as in Example 1. Fig. 6 depictsthe behavior of the inductor current (iL) and capacitor voltage (vc) as a function of time,for VCOM(t) = 0.5 sin(1000t). It may be seen that vc again converges to VDC/2 = 50. Theinductor current iL is sinusoidal with the same frequency as the command signal. The amplitudeof iL can be controlled by changing the amplitude M of VCOM(t).

Summarizing, the average value of vc(t) converges exponentially to the desired value VDC/2,and then the FCC operates as desired. This is true for various initial conditions and for bothDC– and AC–modulation. The time constant of this exponential behavior is of great importance.The reason for this is that it provides a measure on how quickly the FCC returns to normaloperation after some perturbation of its state-variables.

The remainder of this paper is devoted to developing a suitable time–domain approach foranalyzing the FCC. We first analyze the DC-modulated case and then use the results to analyze

February 5, 2009 DRAFT

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0 0.005 0.01 0.015 0.02 0.025−30

−20

−10

0

10

20

30

40

50

60

Time, s

vC

iL

Fig. 6. Load current and capacitor voltage for AC modulation.

also the AC–modulated case. As we will see below, this approach provides an analytic explanationfor many of the features demonstrated in the simulation results.

III. MATHEMATICAL MODEL

Denote the state variables of the converter by x(t) = (x1(t), x2(t))′, where x1(t) = iL(t) (the

inductor current) and x2(t) = vc(t) (the capacitor voltage).Applying Kirchhoff’s laws yields a differential equation for each of the four modulation phases

in the form:x(t) = Aix(t) + biVDC/2, i = 1, . . . , 4, (6)

with A1 = A3 =

(

−1/TL 00 0

)

, A2 =

(

−1/TL 1/L−1/C 0

)

, A4 =

(

−1/TL −1/L1/C 0

)

, b1 =

b3 = b4 =

(

1/L0

)

, and b2 =

(

−1/L0

)

.Note that (6) and the PWM implies that the FCC dynamics is described by a second–order

affine switched system with a periodic switching law.We derive an expression for the solution of (6) using the standard variation of constants

method. Let y(t) := exp(−Ait)x(t). Then (6) yields

y(t) = −Ai exp(−Ait)x(t)

+ exp(−Ait)(Aix(t) + biVDC/2)

= exp(−Ait)biVDC/2,

February 5, 2009 DRAFT

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so for any two times t and τ , with t ≥ τ :

y(t) = y(τ) + (VDC/2)

∫ t

τ

exp(−Ais)bids

= exp(−Aiτ)x(τ) + (VDC/2)

∫ t

τ

exp(−Ais)bids.

Thus,

x(t) = exp(Ait)y(t)

= exp(Ai(t − τ))x(τ) + ci(t − τ), (7)

where ci(t − τ) := (VDC/2)∫ t

τexp(Ai(t − s))bids.

Denote α := 1/(2TL), w0 := 1/√

LC and w :=√

w20 − α2. We consider from here on the

case where the capacitance is sufficiently small so that:

R < 2√

L/C. (8)

Note that this implies that w is real. This assumption is made for concreteness only. All theresults below hold for the case R < 2

√

L/C once w is replaced by jw, with j =√−1.

A calculation yields

exp(A1t) = exp(A3t) =

(

exp(−t/TL) 00 1

)

,

exp(A2t) = exp(−αt)

(

s−(wt) 1wL

sin(wt)− 1

wCsin(wt) s+(wt)

)

,

exp(A4t) = exp(−αt)

(

s−(wt) − 1wL

sin(wt)1

wCsin(wt) s+(wt)

)

,

where s−(l) := cos(l) − αw

sin(l), s+(l) := cos(l) + αw

sin(l), and

c1(t) = c3(t) =VDC

2R

(

1 − exp(−t/TL)0

)

,

c2(t) =VDC

2

(

− 1wL

sin(wt) exp(−αt)1 − s+(wt) exp(−αt)

)

,

c4(t) =VDC

2

(

1wL

sin(wt) exp(−αt)1 − s+(wt) exp(−αt)

)

.

Let tk denote the switching times, i.e., tk :=∑k

i=1 4ti (so in particular t4 = TPWM ). Supposethat our initial state is x(0) and that we repeatedly apply the phase sequence P1P2P3P4. Then (7)

February 5, 2009 DRAFT

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yields

x(t1) = exp(A14t1)x(0) + c1(4t1),

x(t2) = exp(A24t2)x(t1) + c2(4t2),

x(t3) = exp(A34t3)x(t2) + c3(4t3), (9)x(t4) = exp(A44t4)x(t3) + c4(4t4),

x(t5) = exp(A14t1)x(t4) + c1(4t1),

...

Combining the first four equations yields

x(t4) = Ax(0) + b, (10)

where

A := exp(A44t4) exp(A34t3)

× exp(A24t2) exp(A14t1), (11)b := exp(A44t4) exp(A34t3) exp(A24t2)c1(4t1)

+ exp(A44t4) exp(A34t3)c2(4t2)

+ exp(A44t4)c3(4t3) + c4(4t4).

A calculation yields

A = q×(

r2s2−(z) + r

w2LCsin2(z) sin(z)

wL(rs−(z) − s+(z))

r sin(z)wC

(rs−(z) − s+(z)) 1w2LC

sin2(z)r + s2+(z)

)

,

where

r := exp(−αDTPWM),

z := w(1 − D)TPWM/2,

q := exp(−α(1 − D)TPWM). (12)

Since the modulation is periodic, (10) yields

x(t4(k+1)) = Ax(t4k) + b, k = 0, 1, . . . . (13)

Let pA(λ) := det(λI − A) denote the characteristic polynomial of the matrix A. A calculationyields

pA(λ) := λ2 − ((s+(z) − rs−(z))2 + 2r)qλ + r2q2. (14)

It is clear that the dynamic behavior of the FCC is determined by (13). In particular, for smallvalues of k, (13) can be used to yield the time constants of the exponential behavior of vc(t)and iL(t). For k → ∞, (13) describes the asymptotic behavior of the FCC as t → ∞. The nexttwo sections analyze these two cases.

February 5, 2009 DRAFT

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IV. ASYMPTOTIC BEHAVIOR

In this section, we consider the periodic behavior of x(t) as t → ∞. The switching frequencyis usually much higher than the natural frequency of the equivalent RLC–circuit, i.e., TPWM 1/w. Hence, we assume that

z = w(1 − D)TPWM/2 1. (15)

Lemma 1 There exists a constant p > 0 such that for all z ∈ [0, p) the eigenvalues of thematrix A satisfy |λi| < 1, i = 1, 2.

In other words, the matrix A is asymptotically stable for all sufficiently small z.Proof. See the Appendix.

Theorem 1 Suppose that the matrix A is asymptotically stable. Define

x0 := (I − A)−1b. (16)

Then, as t → ∞ the state–vector x(t) converges to a periodic solution. Each period is describedby the solution of:

x(t) =

A1x(t) + b1VDC/2, t ∈ [0, t1),

A2x(t) + b2VDC/2, t ∈ [t1, t2),

A3x(t) + b3VDC/2, t ∈ [t2, t3),

A4x(t) + b4VDC/2, t ∈ [t3, t4),

(17)

with x(0) = x0.

Proof. Eq. (13) is a linear difference equation for the state at times t4k, whose solution is:

x(t4k) = Akx(0) +

k−1∑

j=0

Ajb. (18)

Taking the limit as k → ∞, and using the assumption that A is an asymptotically stable matrixyields limk→∞ x(t4k) = (I − A)−1b. Using (6) completes the proof.

Note that Eqs. (16) and (11) imply that x0 depends on the modulation strategy and theparameter values of the converter and the load, but is independent of the initial condition x(0).This implies that perturbations in the values of the state–variables will have no effect on theasymptotic behavior of the FCC. In other words, the natural balancing property is completelyequivalent to the asymptotic stability of the matrix A.

The fact that x0 is independent of the initial condition x(0) has an additional implication.Recall that we considered the result of repeatedly applying the phase sequence P1P2P3P4.Now suppose that starting from x(0), we repeatedly apply, say, the phase sequence P2P3P4P1.Let x := exp(−A14t1)(x(0)−c1(4t1)). Then this is equivalent to repeatedly applying the phasesequence P1P2P3P4 with x(0) = x. Since our results are independent of the initial condition,this implies that we will obtain exactly the same asymptotic behavior.

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0 0.5 1 1.5x 10−3

0

10

20

30

40

50

60

Time, s

VC

IL

Fig. 7. Asymptotic behavior for D = 0.2. Here x(0) = x0.

Example 3 Consider the FCC with the parameters given in Example 1 and D = 0.2. In thiscase, a calculation yields:

A =

(

0.4029 −0.1359−0.3014 0.8493

)

, b =

(

11.731410.5030

)

.

The eigenvalues of A are λ1 = 0.3248, λ2 = 0.9274, so A is asymptotically stable. Eq. (16)yields

x0 =

(

6.946255.7891

)

.

Fig. 7 depicts the periodic solution of (17) for this case. It may be seen that this is in agreementwith the behavior depicted in Fig. 4(b) for large values of t.

For the particular case of zero duty ratio, x0 admits a particularly simple form.

Proposition 1 If D = 0, then x0 =

(

0VDC/2

)

.

Note that this agrees, of course, with the simulation results depicted in Fig. 4(a).Proof. For D = 0, 4t1 = 4t3 = 0 and 4t2 = 4t4 = TPWM/2. Hence, (11) simplifies to

A = exp(A4TPWM/2) exp(A2TPWM/2),

b = exp(A4TPWM/2)c2(TPWM/2) + c4(TPWM/2).

A calculation shows that I − A has the form:(

∗ exp(−αTPWM) 1wL

(s+(z) − s−(z)) sin(z)∗ 1 − exp(−αTPWM)(s2

+(z) + 1w2LC

sin2(z))

)

,

February 5, 2009 DRAFT

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where ‘∗’ denotes values that are not important for the proof. Also,

b = (VDC/2)

×(

1wL

exp(−αTPWM)(s+(z) − s−(z)) sin(z)1 − exp(−αTPWM)(s2

+(z) + 1w2LC

sin2(z))

)

,

where z = wTPWM/2. Using these expressions, it is easy to verify that (I−A)

(

0VDC/2

)

= b.Using (16) completes the proof.

Theorem 1 provides considerable information on the behavior of the FCC as t → ∞. Wenow consider the analysis of the transient behavior of x(t) and, in particular, the associated timeconstants.

V. TRANSIENT BEHAVIOR

Our approach is based on relating the eigenvalues of the matrix A to the time constantsof the original, continuous–time, system. To do so, suppose for a moment that A is diagonal,i.e. A =

(

λ1 00 λ2

)

and b = 0. Then, the solution of (10) satisfies

xi(TPWM) = λixi(0), i = 1, 2. (19)

The response of a continuous–time system with a diagonal matrix and time constants Ti, i = 1, 2,is xi(t) = exp(−t/Ti)xi(0), so

xi(TPWM) = exp(−TPWM/Ti)xi(0), i = 1, 2.

Comparing this with (19) yields λi = exp(−TPWM/Ti). We thus define the time constants as:

Ti := −TPWM/ ln(λi), (20)

where λi are the eigenvalues of the matrix A.For the sake of simplicity, we first consider the case D = 0, and only then turn to the case

of an arbitrary duty cycle D ∈ [0, 1).

A. Zero duty ratioConsider the case where D = 0. Then

4t1 = 4t3 = 0, 4t2 = 4t4 = TPWM/2.

In other words, the PWM period consists of only two time intervals corresponding to phases P2

and P4. In this case, r = 1 and the characteristic polynomial of A given in (14) simplifies to

pA(λ) = λ2 − 2(1 + 2δ2)qλ + q2, (21)

withδ :=

α

wsin(z), z = wTPWM/2, q = exp(−αTPWM). (22)

Since we consider the case where 0 < z 1 (see (15)), δ > 0 and we may treat δ as a smallparameter.

February 5, 2009 DRAFT

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The roots of (21) are:λi = qsi, i = 1, 2, (23)

withs1 = 1 + 2δ2 − 2δ

√1 + δ2, s2 = 1 + 2δ2 + 2δ

√1 + δ2. (24)

Note that 0 < s1 < 1 < s2. Since, by definition, q ∈ (0, 1), this implies that λ1 ∈ (0, 1).Also, λ2 ∈ (0, 1) whenever qs2 < 1.

Using (24) yields:

ln(s1) = −2δ + δ3/3 + O(δ5),

ln(s2) = 2δ − δ3/3 + O(δ5).

Using (20), (22), (23) and the Taylor series sin(x) = x − x3

3!+ O(x5) yields the time constants:

T1 ≈L

R, T2 ≈ 48

L

R

LC

T 2PWM

. (25)

The first time constant T1 = TL is, as expected, the one associated with the load. This agrees,of course, with the transient behavior of iL(t) (see (5) and Fig. 5).

The second time constant can be expressed as T2 = 48LCT1/T2PWM . Our simulations indicate

that an exponential with time constant T2 provides a very good approximation of the transientbehavior of the capacitor’s voltage vc(t). In other words,

vc(t) ≈ (1 − exp(−t/T2))(VDC/2). (26)

The next example demonstrates this.

Example 4 Consider the FCC with the parameter values given in Example 1 and D = 0. Inthis case T2 = 1/300. Fig. 8 depicts the behavior of vc(t) vs. the function given by (26). Itmay be seen that after a short initial transient, vc(t) and the exponential curve are practicallyindistinguishable.

Summarizing, the analysis yields simple and explicit expressions for the two time constants T1 =TL and T2 that determine the behavior of iL and vc, respectively. Note that the fact that thetime constant of iL is just TL, and is independent of the capacitance C, has a simple physicalinterpretation. Indeed, the opposite symmetry of the two phases P2 and P4 (see Fig. 3) suggeststhat on average the capacitor has no effect on the current iL.

B. General caseWe now consider the more general case where D ∈ [0, 1). The analysis runs along the same

lines as in Section V-A, but with somewhat lengthier calculations.We begin by deriving approximations of the two eigenvalues of (14). Denote b := −q((s+(z)−

rs−(z))2+2r). Note that r = exp(−αDTPWM) = exp(−αµz/w), with µ := 2D/(1−D). Using

February 5, 2009 DRAFT

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0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.0160

5

10

15

20

25

30

35

40

45

50

Time, s

Fig. 8. vc(t) vs. the exponential curve given by Eq. (26) (solid line).

the geometric series for the functions sin, cos, and exp yields:

b/q = −2 + 2µα

wz − 2(2 + µ(2 + µ))

α2

w2z2

+4

3µ(3 + µ(3 + µ))

α3

w3z3 (27)

+α2(2 + µ)(2 + 3µ)w2 − α4µ2(9 + 2µ(4 + µ))

3w4z4

+ O(z5).

The discriminant of (14) is

4 : = b2 − 4r2q2

= q2(s+(z) − rs−(z))2((s+(z) − rs−(z))2 + 4r),

and expending yields:

4 = 4q2(2 + µ)2 α2

w2z2(1 + p(z)) + O(z5),

where

p(z) := −2µα

wz

+α2(6 + µ(9 + µ(18 + 7µ))) − (2 + 3µ)w2

3(2 + µ)w2z2.

Using the series√

1 + x = 1 − x/2 + x2/8 − . . . yields√

4 ≈ 2q(2 + µ)α

wz(1 − p/2 + p2/8). (28)

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.81

2

3

4

5

6

7

8

9

10

D

Fig. 9. The function f(D) as a function of D.

The roots of (14) are λ1,2 = (−b ±√4)/2 and using (27) and (28) yields

λi = qsi, (29)

with:

s1 = 1 − 2(1 + µ)αz/w + O(z2),

s2 = 1 + 2αz/w + 2α2z2/w2

− (α/6)(3α2(µ − 2) + (2 + 3µ)w2)z3/w3 + O(z4).

Using the series ln(1 + x) = x − x2/2 + x3/3 + O(x4) yields

ln(s1) = −2(1 + µ)αz/w + O(z2),

ln(s2) = 2αz/w − (α/6)(2 + 3µ)(a2 + w2)z3/w3 + O(z4).

Using (20), (12), and (29) yields the time constants:

T1(D) ≈ L/R,

T2(D) ≈ 48

(1 − D)2(1 + 2D)

L

R

LC

T 2PWM

. (30)

Note that T1, i.e. the time constant associated with iL is just TL. The second time constantsatisfies T2(D) = f(D)T2(0), with f(D) := 1

(1−D)2(1+2D)(see (25)). Fig. 9 depicts the func-

tion f(D). It may be seen that f(D) increases with D. The increase is relatively moderate forthe range D ∈ [0, 1/2], and becomes more dramatic for values D > 0.6. This agrees well withthe changes in the behavior of vc as a function of D (see Fig. 4). As D → 1, T2(D) → ∞. Thisis reasonable since for D = 1 the capacitor is always disconnected, so its time constant mustgo to infinity.

Exponentials with time constants Ti(D), i = 1, 2, provide an excellent approximation of theaverage behavior of the state–variables of the converter. The next example demonstrates this.

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0 0.005 0.01 0.015 0.02 0.025−10

0

10

20

30

40

50

60

Time, s

vc

iL

Fig. 10. Behavior of the FCC for D = 0.5 and exponentials with time constants Ti (solid lines).

Example 5 Consider the FCC with the parameters given in Example 1 and D = 0.5. Fig. 10depicts the FCC state–variables with x1(0) = x2(0) = 0. Also depicted are the exponentials (5)and (26) with T1(D) and T2(D) given in (30). It may be seen that the exponentials provide anexcellent approximation of the average behavior of the state–variables.

Note that for R → 0, T1, T2 → ∞, suggesting that for a purely reactive load the naturalbalancing property disappears. This agrees with the observations in [11].

VI. AC MODULATION

Consider now the case of AC modulation, i.e. VCOM(t) = M sin(ηt), with M ∈ [0, 1]. Asnoted above, the frequency of the command signal is usually much lower than that of thetriangular wave, so that in every cycle [kTPWM , (k+1)TPWM) of the triangular wave, VCOM(t) ≈VCOM(kTPWM). This suggests that we may be able to approximate the time constants in thiscase by averaging the time constants of the DC case over one period of VCOM(t).

It turns out that it is actually easier to average the reciprocal of the time constant, namely, g(D) :=1/T2(D) = (1 − D)2(1 + 2D)/T2(0). Note that in our analysis of the DC-modulated case, weassumed that D ∈ [0, 1] (see (4)). Thus, we average g(D(t)) = g(M sin(ηt)) on [0, π/η] (ratherthan [0, 2π/η]) as this is the interval for which D(t) ∈ [0, 1]. A calculation yields

g :=1

π/η

∫ π/η

0

g(M sin(ηt))dt =16M3 − 9M2π + 6π

6πT2(0).

Thus, the averaged time constant is

T2(M) :=1

g=

T2(0)

1 − 1.5M2 + 8M3/(3π),

so our approximation in the AC–modulation case is

vc(t) ≈ (1 − exp(−t/T2(M)))(VDC/2). (31)

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0 0.005 0.01 0.015 0.02 0.025 0.030

10

20

30

40

50

60

Time, s

Fig. 11. Capacitor voltage vs. the exponent (31) (solid line) as a function of time.

Extensive simulations indicate that (31) provides an excellent approximation of the averagebehavior of vc(t) and that this holds for a large range of values of M and η. The next exampledemonstrates this.

Example 6 Consider the FCC with the parameter values as in Example 1. Fig. 11 depicts thecapacitor voltage vC(t), as a function of time, for VCOM(t) = M sin(1000t) with M = 0.5. Inthis case, T2(M) = 1/219.331. Fig 11 also depicts the approximation (31) for this value of g. Itmay be seen that vc(t) indeed provides an excellent approximation of the average value of thecapacitor voltage.

VII. DISCUSSION

Multilevel converters combine continuous–time elements and on/off switches. This makes theiranalysis and design highly non trivial. In this paper, we suggested an analysis approach that isbased on treating such converters as a switched system.

We demonstrated this approach for the case of a simple FCC. We analyzed this FCC inthe time–domain by combining the effects of the subsystems that correspond to the variousswitching configurations. The analysis provides considerable information on the circuit behaviorand in particular on its natural balancing property. The latter property is completely equivalentto the asymptotic stability of a matrix A. By analyzing the eigenvalues of A, we obtained simpleand explicit expressions for the time constants that determine the FCC behavior for both DC–and AC–modulation.

Topics for further research include extending the analysis to the case of more complex loads,and the analysis of other, more complicated, MCs.

Finally, the PWM modulation considered here may be viewed as an open–loop control strategyfor the FCC. The design of closed–loop controllers is recently attracting considerable interest(see [17], [18], [19], [20] and the references therein). The time–domain model developed heremay also be useful in this context.

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ACKNOWLEDGMENTS

We thank the anonymous reviewers for their helpful comments. The first and second authorsare grateful to the management of Elmo Motion Control and General Satellite Corporation,respectively, for their encouragement and support.

APPENDIX: PROOF OF LEMMA 1For z = 0, the characteristic polynomial of the matrix A (see (14)) becomes

p0A(λ) := λ2 + a1λ + a0, (32)

with a1 := −((1 − r)2 + 2r)q and a0 = r2q2.It is well known [21, Fact 11.18.2] that a necessary and sufficient condition for asymptotic

stability of the polynomial p0A is that

|a0| < 1 and |a1| < 1 + a0.

The condition |a0| < 1 clearly holds since, by definition, r, q < 1. The second condition isequivalent to

q(1 − exp(−αDTPWM))2 < (1 − exp(−αTPWM))2.

Since α > 0, q < 1 and D ∈ [0, 1), this inequality indeed holds.We conclude that the polynomial p0

A is asymptotically stable, and a continuity argumentcompletes the proof.

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