Modelling of switched mode power converters usingbond graph

A.C. Umarikar and L. Umanand

Abstract: Modelling of the switched mode power converter (SMPC) involves obtaining the largesignal, steady-state and small signal representations. Transfer functions and state equations areused for mathematical representation of the SMPC. Obtaining these dynamic models using amathematical or graphical approach sometimes lead to loss of an intuitive understanding of thesystem dynamics. A unified graphical method of modelling SMPC using bond graphs is presented.Only the PWM DC–DC converters are considered. The concept of switched power junctions isused for modelling of switching phenomenon in the SMPC. This approach uses a logical process ofgraphical reduction, which ultimately results in the state equations of the large signal and smallsignal AC models of the SMPC including the steady-state equations of the SMPC. This method isapplicable to SMPC, in both continuous conduction mode and discontinuous conduction modeThe proposed modelling approach is illustrated by a few examples.

1 Introduction

Switched mode power converters (SMPCs) are in a class ofvariable structure systems. The topology of the dynamicalelements of the system depends on the instantaneouspositions of the power switch. Several analytical andgraphical methods have been proposed in the literature onmathematical modelling of the SMPCs. Among analyticalmethods, large signal discrete time models for computersimulation and continuous time models for physical under-standing has been proposed [1]. Method of state spaceaveraging has been discussed along with obtaining the smallsignal model of the switching converter [2]. Graphicalmethods, like the circuit-averaging method [3] and PWMswitch model method [4, 5] have also been employed toobtain the averaged large signal model of a converter. Theswitching flow graph nonlinear modelling technique hasbeen used for modelling of switching converters [6]. Thisapproach is a modification of the signal flow graph methodfor modelling of switching systems. It unifies the method ofobtaining large signal, steady-state and small signal modelsof the switching converter.

This paper explores the bond graph technique to modelswitched mode power converters. Modelling of PWM DC–DC converters is considered. The concept of switchedpower junctions [7] is used to model the switchingphenomenon in the converters. There is one-to-onecorrespondence between the physical system and bondgraph elements. This model visually indicates the cause andeffect relationships between energy variables, which enhancethe visual understanding of the system. The bond graphunifies the method of obtaining the large signal, steady-stateand small-signal model for switched mode power converter.The large signal, steady-state and small signal models are all

derived graphically. Further, the bond graph can handlemultidomain systems without sacrificing the ease of derivingthe state space representation of physical systems byinspection.

2 Bond graph modelling and concept of switchedpower junction

2.1 Bond graph overviewA bond graph is a labelled and directed graphicalrepresentation of a physical system. The basis of bondgraph modelling is power/energy flow in a system. Asenergy or power flow is the underlying principle for bondgraph modelling, there is seamless integration acrossmultiple domains. As a consequence, different domains(such as electrical, mechanical, thermal, fluid, magnetic etc.)can be represented in a unified way. The power or theenergy flow is represented by a half arrow, which is calledthe power bond or the energy bond [8, 9]. The causality foreach bond is a significant issue that is inherently addressedin bond graph modelling. As every bond involves twopower variables, the decision on setting the cause variableand the effect variable is by natural laws. This has asignificant bearing on the resulting state equations of thesystem. Proper assignment of power direction resolves thesign-placing problem when connecting sub-model struc-tures. The causality will dictate whether a specific powervariable is a cause or the effect. Using causal bars on eitherend of the power bond, graphically indicate the causality forevery bond. Once the causality is assigned, the bond graphdisplays the structure of the state space equations explicitly.Examples of engineering systems using bond graphmodelling elements are discussed in [8, 9].

2.2 Switching systems in bond graphs:an overviewModelling and simulation of switching systems using abond graph is one of the topics of research. Various modelshave been proposed. One model uses the ideal switchelement [10]. In this model, discontinuous change is handledby enforcing either zero flow (OFF) or zero effort (ON) on

a junction. However, this approach The shows the bondgraph of only one structural mode at any given instant.Also it is unnatural to consider switches as bond graphelements as, unlike other bond graph elements, they do nothave energy-related functions. Another approach [11]proposes a switched system description using several bondgraph models that depend on switch positions, where thepower switch is not considered as a model element.However, for converters containing more switches, thismethod is cumbersome since the number of combinations is2n where n is the number of switches. Petri net, or finite statemachine, is one of the methods to determine the transitionsacross the modes. This approach is used in [12] for averagedmodelling of switching converters. However all relevantdiscrete systemmodes and the transition between them haveto be explicitly determined before modelling. For n switchesin the system, there are 2n potential system modes in whichall combinations are not physically realisable. In addition,as the switching between the systems is not represented atthe energy level, this approach lacks the causality insightthat bond graph inherently provides. Use of a modulatedtransformer MTF to model switching in switching con-verters is proposed in [13]. It first finds different bond graphmodels depending on the number of switch positions andthen connects these models using MTF elements atappropriate places. However, as the number of switchesincreases, the model becomes complex. A controlledjunction concept has been proposed [14, 15]. In the ONstate, the controlled junction behaves like a normal 0- or 1-junction. In the OFF state the 0-controlled junction forcesthe effort value at all connected bonds to become 0.Similarly The 1-controlled junction forces the flow value atall connected bonds to become 0. This implies that there isno energy transfer across this junction. The control logic fordetermining the ON and the OFF state for a controlledjunction is implemented as a finite state automata andrepresented as a state transition graph or table. Here theactual switch mechanism is not modelled as a bond graphelement. However, changes in configuration across thevarious operating modes of the system results in reassign-ment of causality. In [16–18], the switch is modelled ascurrent and voltage source according to the switch state.The OFF condition is modelled as a small current source(leakage current) and the ON condition is modelled as asmall voltage source (forward voltage drop). There is acausality change associated with this change of state of theswitch. The ‘causality resistors’ are introduced to ‘‘take thecausality change’. Appropriate rules are defined to fix therange of the value of causality resistors. A causalityswitching algorithm is proposed for swapping of bondvariables in the junction structure to represent change incausality and change in the state of the switch. The conceptof a junction structure matrix is used to implement thisalgorithm. The suitability of the algorithm for forced andnaturally commutated converters is also presented withexamples and computer simulations. The user can modelthe switching systems without knowing the functionalaspects of the switching circuit.

This paper proposes the modelling of switching con-verters using the concept of switched power junctions [7]where the causal structure remains unchanged even onswitching. This type of structure is especially useful forderiving state equations.

2.3 Concept of switched power junctionSwitched power junction is a generalisation of the junctionelements in bond graph, [7]. There are two junctionelements in the bond graph. the 0-junction and the 1-

junction. The 0-junction represents Kirchoff’s current law(KCL) and the 1-junction represents Kirchoff’s voltage law(KVL) in the electrical domain. Equations and causalassignments for the 0 and 1-junction are discussed in theAppendix (Section 8).

At the junctions, there is no dissipation or storage of thepower. Incoming power is equal to outgoing power at thejunction. If there is an instantaneous discontinuous changein power through the junction instantaneously, then itrepresents power switching at the junction. This can beachieved by switching flow at the 1-junction and effort atthe 0-junction. At the 0-junction only one bond decides theeffort. If there are two effort-deciding bonds at the 0-junction and the effort of these bonds is active at the 0-junction only at mutually exclusive instants of time then itrepresents the effort switching at the 0-junction. Such a 0-junction, that inherently incorporates temporally mutuallyexclusive multiple effort deciding bonds, is denoted as a 0s-junction. Similarly, a 1-junction where temporally mutuallyexclusive multiple flow deciding bonds are handled isdenoted as a 1s-junction. A 0s-junction is shown in Fig. 1ain which there are two effort-deciding bonds and a 1s-junction is shown in Fig. 1b with two flow-deciding bonds.However, 0s/1s-junctions can be generalised to any numberof effort/flow decider bonds.

The equations of effort and flow at 0s- and 1s-junctionsare:

For 0s-junction

e3 ¼junction effortjunction effort ¼U1 � e1 þ U2 � e2

f1 ¼U1 � f3

f2 ¼U2 � f3

ð1Þ

For 1s-junction

f1 ¼junction flowjunction flow ¼U1 � f2 þ U2 � f3

e2 ¼U1 � e1e3 ¼U2 � e1

ð2Þ

Effort is denoted as e and flow is denoted as f.U1 and U2 are the control signals used to switch the

junction effort or flow. Boolean algebra governs themultiplication and addition of U1 and U2. The controlsignal to switch the junction variable is derived from thecontrol logic, which is specific to that system. U1 and U2 areplaced at the corresponding effort or flow decider bonds,whose effort or flow is active when either U1 is 1 or U2 is 1.The example considered two effort/flow deciding bonds.Figure 1 is given for two effort/flow decider bonds,however, 0s/1s junctions can be generalised to any numberof effort/flow decider bonds.

e e e e

ee

U

f

U

UU

1

3 f3f1f1

f2

2

2f2

2

2

3 31

1

10s

a b

1s

Fig. 1 Switched Power Junctionsa 0s-Junctionb 1s-Junction

2.4 Switched power junctions and powerelectronic systemsPower semiconductor switches such as diode, MOSFET,power transistor etc. are an integral part of switchingconverters in power electronic systems. All powersemiconductor switches are basically a single pole singlethrow (SPST) type of switch. To analyse the functionalperformance of converters, the semiconductor switches canbe considered as ideal SPST switches. The state ofthe switch can be controlled by generating signals thatdepend on the converter topology and the properties of thesemiconductor device. By using the SPST switch as thebasic unit, one can realise all switching circuits in powerelectronic systems.

The SPST switch can be modelled as a 1s-junction withtwo flow decider bonds as shown in Fig. 2. U and U arecontrol signals that control the junction flows. When theswitch is closed, i.e. when U¼ 1 junction flow is determinedby the system. When the switch is open, i.e. when U ¼ 1,flow at as the junction is zero. Therefore one flow deciderbond is modelled as the zero current source and the otherflow decider bond is connected to the system. Thus thefunction of the switch is emulated.

Combinations of SPST switches can be modelled using0s-and 1s-junctions. For example, the H-bridge as shown inFig. 3a is shown as combination of SPST switches. Thecircuit in Fig. 3a is a single-phase inverter using a controlledswitch such as aMOSFET. However, the H-bridge can be adiode bridge rectifier with source and load positioninterchanged which is not considered in this example.All these switches can be modelled as SPST switches.The bond graph of the H-bridge is shown in Fig. 3b.Individual SPST switch function is modelled as 1s-junctionand the points O1 and O2 are modelled as 0s-junctions sincethere is effort switching. The purpose of this example is toshow the generality of application of switched powerjunctions to power electronic systems. It is important tonote that the user should know the functional aspects of thecircuit for using switched power junctions for modelling.This concept is now applied to model PWM DC–DCpower converters.

3 Modelling methodology

The switched model power converter circuit containsswitches and supposedly linear circuit elements. Switchesare responsible for the nonlinear behaviour of theconverters. In most converters, there is a combination ofswitches in the form of a single pole double throw (SPDT)switch. The SPDT switch is a combination of two SPSTswitches. In most converter circuits combinations of oneactive switch (transistor or MOSFET) and one passiveswitch (diode) realise the SPDT switch. Figure 4 shows anSPDT switch, its SPST realisation and two combinations ofMOSFET and diode, which realise the SPDT switch. Thebond graph model for this basic switching block is theobject of focus here.

A single switch is modelled as a SPST switch. Usingswitched power junctions, switch network is modelled as abond graph as shown in Fig. 5. The SPST switch ismodelled as a 1s-junction and the node O is modelled as a0s-junction. Here continuous conduction mode (CCM) isconsidered. Discontinuous conduction mode (DCM) op-eration is discussed in Section 5. Switches S1 and S2 areclosing mutually exclusive in time with respect to eachother. The control signals U and U (instead of U1 and U2)activate the switches S1 and S2, respectively, i.e. when U¼ 1,S1 is ON and when U ¼ 1, S2 is ON. U and U are theBoolean signals and are complement of each other such thatU þ U ¼ 1. These signals are derived from a control logic,which is derived according to the switch property. Forexample, if S1 is a MOSFET, U is given from the duty cyclecontroller and if S2 is a diode, U is derived from the dutycycle controller as well as the current in S2. In the bondgraph, one can incorporate these signals in switched power

system system system systemf

U

S = 0

U U

U

f = 0

f

fe

a b

e1s

Fig. 2 SPST Switcha SPST switchb bond graph model of SPST switch

VDC

VDC

VGDC

VGDC

U

U

U

U

UU

UU

U

UUUU

U U U

0

0

0s 0s

1s 1s

1s1sSf = 0 Sf = 0

Sf = 0Sf = 0

O2O1

a b

Fig. 3 H-bridge inventor.a H-Bridge single phase inverterb bond graph model of H-bridge inverter

C C

CC

B

B

A

A

A

A

OBU

U UU

U

U

Bi i

ii

�A

�B�B

�A

�B

�A�B

�A

S1

S2

a b

c

Fig. 4 SPDT switcha SPDT switchb SPST switch realization of SPDT switchc examples of SPDT switch realization using circuit elements

e1 = U�A + U�B

i

�A

�B1

2

3

4

7

6

50s 1s

1sSf = 0

Sf = 0

UU

UU

f5 = Ui

f2 = Ui

Fig. 5 Bond graph realisation of SPDT switch

IEE Proc.-Electr. Power Appl., Vol. 152, No. 1, January 2005 53

junctions as in Fig. 5. It should be noted that the bonds 2and 5 in Fig. 5 are not switching bonds. The presence of U

and U is merely a notational convenience. It means that thejunction effort and/or flow is taken from these mutuallyexclusive bonds at U and U times.

The main objective here is to establish the averaged largesignal, steady-state and small signal models by graphicalreduction. This would ultimately enable one to obtain thelarge and small signal state equations by inspection. Thebond graph model of the switch shown in Fig. 5 describesthe large signal behaviour of the switch network. To obtainthe averaged large signal model of the switch network, thefollowing graphical steps are applied.

Step 1 Inspect the bonds that have zero power. In Fig. 5, thezero power bonds are 3 and 6 since flow in them is zero.Remove these bonds from the graph. The resulting graph isshown in Fig. 6a.

Step 2 Referring to Fig. 6a, merge bonds 2 and 4 and alsomerge bonds 5 and 7 as the 1s-junctions have less than threebonds connected to them. The resulting graph is shown inFig. 6b.

Referring to Fig. 6b, the causality on bonds 2,4 and 5,7indicates that the bond is effort causal at the 0s-junction.The Boolean terms will appear as a multiplier in the ’effect’variables, which in this case is the flow. The flow at bond 1is the effect from the external circuit but the effort is decidedeither by bond 2,4 or bond 5,7 depending on the Booleansignals. This dependency is seen in the effort equation ofbond 1 as indicated in Fig. 6b.

Step 3 For CCM operation, U and U are each othercomplement and are rectangular pulse trains of duty cycle dand 1-d, respectively. To obtain the averaged model, replaceU and U in Fig. 6b by d and 1-d where d and 1-d are timeaveraged values of U and U . The resulting graph that isshown in Fig. 7a is the averaged large signal bond graphmodel of the switch network.

Step 4 To obtain the steady-state model, in Fig. 7a replace dby D, e by E and f by F, where D, E and F are the steady-state operating points of the duty cycle, the effort and the

flow, respectively. The switch network in steady state isshown in Fig. 7b.

Step 5 To obtain the small signal ACmodel, the principle ofapplying a small perturbation in the neighbourhood of thesteady-state operating point is used graphically.

Introduce small perturbations [2] around the operatingpoint in d, e and f of each bond as

d ¼Dþ dd

e ¼E þ ee

f ¼F þ ff

where dd; ee and ff are small perturbations in duty cycle d,effort e and flow f around operating point D, E and F,respectively.

Step 6 Referring to Fig. 7a, the product of two quantitieswith perturbation like d*e and d*f can be written as

d � e ¼ ðDþ ddÞ � ðE þ eeÞ ¼ DE þ Deeþ Edd þ eedd ð3Þ

d � f ¼ ðDþ ddÞ � ðF þ ff Þ ¼ DF þ Dff þ F dd þ ff dd ð4Þ

By neglecting second-order perturbations eedd; ff dd and byremoving DC terms DE and DF from (3) and (4), the first-order small signal AC terms are obtained. Referring toFig. 7a, the bond 1 effort equation is given by

ee1 ¼ DvvA þ VAdd þ ð1� DÞvvB � VBdd ð5ÞIt is evident from (5) that new perturbation terms are addedto the effort equation at bond 1 due to the product of twoquantities that includes the perturbation effect. The newterms can be equivalently represented as input effort sourcesdue to perturbation in duty cycle as indicated in Fig. 8.

Referring to Fig 7a, it can be seen that the bond 2 flowequation is given by

ff1 ¼ Diiþ Idd ð6Þ

From (6) the new perturbation terms that are added to theflow equation at bond 2 can be equivalently represented asinput flow sources due to the perturbation in the duty cycleas indicated in Fig. 8.

Referring to Fig. 7a, the bond 3 flow equation is given by

ff2 ¼ Dii� Idd ð7Þ

From (7) the new perturbation terms that are added to theflow equation at bond 3 can be equivalently represented asinput sources due to the perturbation in the duty cycle asindicated in Fig. 8. Figure 8 now gives the small signal ACmodel of the switch network.

1

1i

i

i

i U

Sf = 0

Sf = 0

UU

UVA

VB

2,4

5,72

3

4

56

71s

1s

0s

0s

�A

�Be1 = UVA + UVB

a b

Fig. 6 SPDT switch bond grapha removal of zero power bonds from SPDT switch bond graphb bond graph of SPDT switch after merging two bonds

ii

1 0s 0s2

3 3

2

1

e1 = d�A + (1−d )�B E1 = DVA + (1−D)VB

VB

VA

(1−D)I

DIdi

I(1−d )

a b

�B

�A

Fig. 7 Switch networka averaged large signal modelb steady-state model

11

0

0s 0

2

3

4

5

6

7

8

9

10

VB d

e4 = D�A + (1−D)�B

(1−D)�B

�A

VA dId

Id

Di

^

^ ^

^

^^

^

^

^

^−

−

i

i

Fig. 8 Small signal ac model of the switch network

In some converters, combination of switches as shown inFig. 9 occurs (e.g. Cuk and SEPIC) frequently. The bondgraph for this combination is shown in Fig. 10, where theadditional 0s-junction is formed by bonds 12–14. When S1is ON, v is connected to the ground with respect to pointO1. When S2 is ON v is connected to ground with respect topoint O2. This is to provide the reference ground to thesystem modelled using the bond graph. Once the model isformed one can apply a graphical reduction algorithm to it.Applying steps 1–3 discussed above, one obtains theaveraged large signal bond graph model as shown inFig. 11. It should may be noted that one can remove bonds12–14 in the second step since its significance for obtainingthe averaged large signal, steady-state and small signalmodelling is null.

The small signal AC model is also obtained along similarlines, as discussed previously from steps 4–6. The smallsignal model in this case is shown in Fig. 12.

As mentioned earlier, these switch networks are the basicbuilding blocks in SMPCs. In general, one has to trace thepoints where effort and flow switches. By placing theappropriate switching power junction, the bond graphmodel of the converter can be obtained which is large signalmodel. If there is switching in the reference ground asshown in Fig. 9, then one has to put the additional junctionto provide the appropriate reference ground according toswitching state as in Fig. 10. Once the large signal bondgraph model is obtained, the averaged large signal, steady-state and small signal models of the SMPCs can be obtainedby using the graphical reduction process illustrated in thisSection. Using the large signal and the small signal bondgraph models, the state equations representing the SMPCsdynamics can be obtained directly by inspection.

4 Modelling of switching converters in CCM

This Section discusses the modelling of switching convertersin CCM. Two basic converters namely boost and Cukare taken as examples. Figure 13 shows a boost converter.Switches S1 and S2 are connected at point O, whichforms the SPDT switch as described in section 3. S1represents the MOSFET and S2 represents the diodeconnected as shown in Fig. 13. Figure 14 shows a boostconverter large signal bond graph model. There aretwo state variables in the converter, iL and vO. Referringto the bond graph of Fig. 14, it is seen that the flow of bond2 is the iL state variable and effort at bond 11 is the otherstate variable, vO. Therefore bond 2 is the flow decider forthe corresponding 1-junction and effort at bond 2 gives thefirst state equation, which is given by KVL at thecorresponding 1-junction as

e2 ¼ LdiLdt¼ vg � iLRL � �UUvO ð8Þ

Similarly bond 11 is the effort decider for the corresponding0-junction. Therefore flow at bond 11 is the second stateequation, which is given by KCL at the corresponding 0-junction as

f11 ¼ CdvO

dt¼ �UUiL �

vO

Rð9Þ

O2O1

S1 S2

U U U U

V2

VGND

i2i1

+ −�

Fig. 9 Commonly occurring switch network

e5 =U (v −VGND) e7 = U (v −VGND)

e14 = UVGND + UVGND

−

e1 =UVGND + U (v −VGND)−

e11 = U (v −VGND) + U VGND−

−

−

−

f2 = U (i 1 - i 2)

Sf = 0 Sf = 0

f2 = U (i 1 - i 2)

U

U

U

U

U

−U−U

−U

1314

5

2

1

3

4

6 8

7

12

11

10

9

0s

0s0s

1s

1s

1s

i 1 i 2

v

VGNDVGND

Fig. 10 Bond graph of the switch network in Fig. 9

e1 = dVGND + (1−d )(�−VGND)e1 =d(�−VGND) + (1−d )VGND

f4 = diL2 + (1−d )iL1

1

2

34

5

6

7d

VGNDVGND

1s 0s0s i2(1−d )

(1−d )d

Fig. 11 Averaged large signal bond graph model of the switchnetwork of Fig. 9

1

2

1 0s 0s1s3

4

5

6

7

8

0I2d I1d11

12 14 1 16

1513

Vd VdVGVD VGVD

e3 = (1− D)� e14 =D�

(1−D) Di2i1

�

Di2 + (1-D)i1

^ ^

^

^^

^ ^

^ ^

^

^

Fig. 12 Small signal AC bond graph model of the switch networkof Fig. 9

U

UUU

R

RCS

SL i

V

L L

1

2 0

g+ +

−−

GND= 0

�

�

Fig. 13 Boost converter

e4 = UV GND + UvO f10 = U ∗ 0 + UiL

Sf = 0

UU

UU

L

RC

1s0s

1s

10 0

76

1

2

14

3 5

8

9

11 12

VGND = 0

RL

iL

vg

v0

S f = 0

Fig. 14 Bond graph model of the boost converter

As discussed in Section 3, we apply the steps for graphicalreduction to obtain the averaged large signal, steady-stateand the small signal graphs. Applying step 1, the zero powerbonds 6 and 9 are removed from the bond graph.According to step 2 bonds 5,7 and 8,10 are merged. Theresulting graph is shown in Fig. 15a. After executing step 3,the averaged large signal model is given in Fig. 15b. Thestate equation for the large signal averaged model shown inFig. 15b is given by

L 00 C

� �ddt

iL

vO

� �¼ �RL �ð1� dÞð1� dÞ �1=R

� �iL

vO

� �

þ 10

� �vg ð10Þ

After executing step 4, one obtains the steady-state model ofthe converter. This model is useful to calculate theperformance indices such as efficiency and conversion ratio.In the steady state, the derivatives of states iL and vO arezero since these terms are DC. Therefore the LHS of (10) iszero. Equating the terms of the LHS to zero and replacingall the variables by their DC values, one obtain the steady-state equations as

� ILRL � ð1� DÞVO þ Vg ¼ 0

ð1� DÞIL �VO

R¼ 0

ð11Þ

For instance, e.g. one can obtain the voltage conversionratio from (11) as

VO

Vg¼ ð1� DÞ

RLR þ ð1� DÞ2

ð12Þ

Now, to obtain the small signal AC model, executestep 5 and step 6. The small signal ac bond graph modelis given in Fig. 17. The small signal AC state space equationis given as

L 00 C

� �ddt

iiLvvO

� �¼ �RL �ð1� DÞð1� DÞ � 1

R

� �iiLvvO

� �

þ 10

� �vvg þ

VO

�IL

� �dd ð13Þ

There is a cross on bond 6, which indicates that bond 6 isthe zero power bond in this case and hence is neglected forobtain the small signal model.

In this way the large signal, steady-state and small signalAC models of the boost converter are obtained graphically.It is clear from the procedure that in a few steps, startingfrom the large signal model bond graph, the steady stateand small signal AC models are obtained. When one isfamiliar with the graphical procedure one can obtain thesemodels in just three steps.

The example of the Cuk converter in CCM given belowconsolidates this approach. The Cuk converter is given inFig. 18, and its bond graph model is shown in Fig. 19. Thisswitch network is same as that described in Fig. 9.

After executing steps 1–3, one obtains the large signalaveraged model as shown in Fig. 20. The bonds 18–20 are

e4 = UVGND + UvO

e4 = (1− d ) ∗vO

f8 = (1− d ) ∗iL

f8 = U ∗ iL

L

2

L

2

3 U

U

C RVGND = 0

4 00s

11 128,10

5,7

1 1vg

iL

RL

v0

3 d

C RVGND = 0

4 00s

11 128,10

5,7

1 1vg

iL

RL

v0

(1−d)

a

b

Fig. 15 Boost convetera Reduced bond graphb Averaged large signal model

E4 = (1−D ) ∗ V0

F6 = (1−D ) ∗ IL( 1−D)

VgIL

RLV0

V GND = 0

1 1

2

4 067 83 5

0S

D

L

C R

Fig. 16 Steady state bond graph model of boost converter

e5

= (1− D ) ∗ v0

f7

= (1− D ) ∗ iL

∧

∧

∧

∧

∧

∧

∧

∧

∧

d

D

C RRL

v0

V0

iL

vg

3

1 1

4 6

0

987

10

5

2

L IL d

0s

VGND = 0

(1− D )

Fig. 17 Small signal ac bond graph model of boost converter

+−

S 1 S 2

C 2

L 2C 1L 1

v g

i L 1i L 2

v C 1

v 0

R

VGND = 0

UU U U

Fig. 18 Cuk converter

e3 = Uvc1

f 7 = UiL1f 9 = UiL2

e 13 = UvC1

L2C1

iL21415113 0

1610

209

19187

4

1211

17

L1

iL1

UiL2UiL1

C2

21 1

56

3vg

vC 1

v0

Sf = 0

VGND = 0 VGND = 0

0s 1s1s

1s 0s0s

UU

RUU

U U

UU

8

Sf = 0

Fig. 19 Bond graph model of Cuk converter

removed since they are zero power bonds as well as havingnull significance in obtaining the averaged model.

L1 0 0 00 L2 0 00 0 C1 00 0 0 C2

2664

3775 d

dt

iL1

iL2

vC1

v0

2664

3775

¼

0 0 � �UU 00 0 �U �1�UU U 0 00 1 0 � 1

R

2664

3775

iL1

iL2

vC1

v0

2664

3775þ

1000

26643775vg ð14Þ

Equation (14) is the state equation of the large signal modelof the Cuk converter.

L1 0 0 00 L2 0 00 0 C1 00 0 0 C2

2664

3775 d

dt

iL1

iL2

vC1

v0

2664

3775

¼

0 0 �ð1� dÞ 00 0 �d �1

ð1� dÞ d 0 00 1 0 � 1

R

2664

3775

iL1

iL2

vC1

v0

2664

3775

þ

1000

26643775vg ð15Þ

Step 4 gives the steady-state model. After executing step 5and 6 one obtains the small signal AC model of the Cukconverter as shown in Fig. 21. The state equations for thesmall signal AC model are obtained from the small signalgraph of Fig. 21 as

L1 0 0 00 L2 0 00 0 C1 00 0 0 C2

2664

3775 d

dt

iiL1

iiL2

vvC1

vv0

2664

3775

¼

0 0 �ð1� DÞ 00 0 �D �1

ð1� DÞ D 0 00 1 0 � 1

R

2664

3775

iiL1

iiL2

vvC1

vv0

2664

3775

þ

1000

26643775vvg þ

VC1

�VC1

�IL1þ IL2

0

2664

3775dd ð16Þ

The modelling procedure for SMPCs in CCM is illustratedthrough the boost and Cuk converter examples. A switchedpower converter of any complexity can be modelled usingthe above procedure.

5 Modelling of switching converters in DCM

The graphical modelling procedure of PWM DC-DCpower converters in DCM is discussed in this Section.The discontinuous inductor current mode (DICM) isdiscussed here. When the PWM converter operates inDICM, the inductor current reaches zero before oneswitching cycle is over. Conventionally this phenomenonis modelled using a reduced-order model of the converter.The problems with state-order reduction models arediscussed in [4, 19, 20]. To prevent the order reduction inthe converter model, a fictitious device called the discontin-uous mode switching device (DMSD), which is a four-terminal device has been proposed [19]. Its main functionlies in the topological support to the formulation of the full-order state space model. Along similar lines, this paperintroduces the virtual switch to prevent the order reductionin the state space model. As an example, the virtual switch

e3

= (1−d )vc1

f6

= di L 2

+ (1−d )iL 1

f8

= d (iL 1

− iL 2

)f4

= d (iL 1

− iL 2

)

e9 = dvc1

vg

v0

vC11

231

45

6 101 0119

78 12 13

0s0s

0s

1s

RC 2

C 1 L 2L 1iL1

iL2

VGND = 0 VGND = 0

Fig. 20 Averaged large signal bond graph model of Cuk converter

e3 = (1−D )VC 1

f 6 = DiL 2

+ (1−D )iL 1

∧

∧∧

∧

∧

∧ ∧

∧e12 = Dv

C 1

L1

vg

L2

C2

C1

v0

IL2d

∧

∧∧

IL1d

−VC1d

∧

∧

VC1d

iL1

vC1

iL2

RVGND = 0VGND = 0

7 98

2 6

5311

410 12

13

11 1415

16 170

0

11S3 0S40S1

Fig. 21 Small signal ac bond graph model of Cuk converter

U2 + U3

U1 + U2

U2 + U3

U1 + U3

U2 + U3

U1 + U3

U1 + U2

U1 + U3

U1 + U2

U1

U3

U1

U3

U2

U2

U1

iL

iL

iL

v2

v0

v2

v0

v2

v0

U3

U2

S3

S3

S1

S1

S3

S2

S2

S1 S2

virtual switch

virtual switch

virtual switch

R

R

R

L

L

C

C

C

L

+−

−

−

+

a

b

c

+

Fig. 22 Equivalent circuits of basic converters in DCM by usingvirtual switchinga buck converterb boost converterc buck–boost converter

IEE Proc.-Electr. Power Appl., Vol. 152, No. 1, January 2005 57

position in three basic converters is shown in Fig. 22. Abuck converter example is chosen for modelling (Fig. 22a).In DCM when the inductor current reaches zero, both thecontrolled switch S1 and diode S2 obtains OFF. The voltageat point O becomes vO. This virtual switch only closeswhen both switches become OFF. U1, U2 and U3 are theBoolean signals, which are mutually exclusive in timewith respect to each other. These Boolean signals controlthe states of the switches S1, S2 and S3 shown in Fig. 22a.The bond graph model of converter in Fig. 22a is as shownin Fig. 23. Fig. 23 is the large signal model of the converterin DCM. State equations for the large signal model aregiven as

L 00 C

� �ddt

iL

vO

� �¼ 0 �ðU1 þ U2ÞðU1 þ U2Þ � 1

R

� �iL

vO

� �

þ U1

0

� �vg

ð17ÞThe rest of the procedure to obtain the large signal averagedmodel, steady-state and small signal AC model is thesame as was described for the CCM in Section 4. Theonly change will be that, instead of U and �UU , there will bethree control signals U1, U2 and U3. The sum of U1, U2

and U3 i.e. U1+ U2+U3¼ 1. The time average of U1, U2

and U3 are d1, d2 and (1-d1-d2). The duty cycle controllergives d1. d2 is a function of the parameters of the circuit,switching frequency as well as d1. d3 is a function of d1and d2. After executing steps 1–3 one obtains the largesignal averaged model as shown in Fig. 24a. The stateequations are

L 00 C

� �ddt

iLvO

� �¼ 0 �ðd1 þ d2Þðd1 þ d2Þ � 1

R

� �iLvO

� �

þ d10

� �vg

ð18ÞTo obtain the get steady-state model, execute step 4.Figure 24b shows the steady-state graph. The equationsobtained from this graph give the steady-state performanceindices.

D1Vg � ðD1 þ D2ÞVO ¼ 0

ðD1 þ D2ÞIL �VO

R¼ 0

ð19Þ

The voltage conversion ratio obtained from (19) is

VO

Vg¼ D1

D1 þ D2ð20Þ

To obtain the small signal AC model, execute steps 5 and 6.

The small signal AC bond graph model is shownin Fig. 25. The small signal AC state space equationsare

L 00 C

� �ddt

iiLvvO

� �¼ 0 �ðD1 þ D2ÞðD1 þ D2Þ � 1

R

� �iiL

vvO

� �

þ D1

0

� �vvg þ

Vg � V0

�IL

� �dd1

þ �V0

�IL

� �dd2

ð21Þ

Note the appearance of the term dd2, it is considered as a

control input. and can be expressed in terms of dd1 and othervariables [20].

Therefore using the virtual switch concept,switched power converters in DCM can also be modelledusing the bond graphical approach without reducingthe state order of the converter model. For calculationof d2, however, one has to use analytical approach describedin [20].

Sf = 0

Sf = 0

Sf = 0

U1 + U

2

U1 + U

3

U2 + U

3

U1

U3

U2

iL

vg

VGND = 0

1S

1S

0S

v0

e7

= U1vg + U2vGND + U3v0

C R

9 0

1110

8

4

5

6

3 7 1

2

14

13

12

L1s

Fig. 23 Bond graph model of buck converter in DCM

f 8 = (1 − d1− d

2)iL

F 8 = (1 − D1− D

2)IL

e3 = d1v

2 + (1 − d

1− d

2)v0

E3 = D1V

2 + (1 − D

1− D

2)V0

f1 = d1iL

F1 = D1ILF2 = D2IL

f2 = d2iLVGND = 0

VGND = 0

v2

V2

d3

D3

v0

V0

i L

I L

76

05

4

3

2

1

8

1

1 0

8

4

31

56 7

0s

0s

C R

L

L

C R

a

b

Fig. 24 Bond graph modela averaged large signal bond graph modelb Steady state bond graph model

f15

= (1−D1−D

2)i

L

e3

= D1vg + (1 − D

1− D

2)v

0

∧

∧

∧

∧

∧

∧∧

∧ ∧ ∧

∧

f1 = D1iL

vg 12

30 00s

VGND = 0

ILd

1∧

ILd

1 ∧ILd

2

∧V

0d

2∧

V0d

1

∧V

gd

1

D1

D2

D3

v0

iL

15

65 1

L

84 9

10 11 12

1314

7

C R

Fig. 25 Small signal AC bond graph model of buck converter inDCM

6 Conclusions

The bond graph approach to model switched mode powerconverters has been presented in the paper. The Switchingphenomenon is modelled using the concept of switchedpower junctions. Bond graph modelling of the switchedpower converter is described to obtain the large signalaveraged, steady-state and small signal AC model of theswitched power converters.

The bond graph of the converter is the large signal modelof the converter. A graphical procedure is proposed thatgives the averaged large signal, steady-state and small signalAC models. The procedure is suitable for both converters inCCM and in DCM.

Modelling of converters in CCM is illustrated usingexamples of boost and Cuk converters. For modelling inDCM, the concept of the virtual switch is used to model theconverter using the bond graph. The concept of the virtualswitch topologically supports the formulation of the full-order model of the converter in DCM. The DCM case isillustrated using the buck converter.

Using the above procedures, converters of any complex-ity can be modelled incorporating all the advantagesof bond graph modelling. Before using this method,one needs to be familiar with the functionality of theconverter. The proposed method enhances thevisual understanding of the cause–effect relationshipbetween voltage–current variables in the circuit. The bondgraph of the converter displays all the physically realisableswitching states in a single graph. Further, the bond graphof the converter explicitly displays the structure of thedynamic equations of the converter.

7 References

1 Erickson, R.W., Cuk, S., and Middlebrook, R.D.: ‘Large signalmodelling and analysis of switching regulators’. Proceedings of IEEEPESC, Cambridge, MA, USA, 1982, pp. 240–250

2 Cuk, S., and Middlebrook, R.D.: ‘A general unified approach tomodelling switching-converter power stages’. Proceedings IEEEPESC, Cleveland, OH, USA, 1976, pp. 18–34

3 Wester, G.W., and Middlebrook, R.D.: ‘Low frequency characteriza-tion of switched DC-DC converters’, IEEE Trans. Aerosp. Electron.Sys., 1973, 9, pp. 376–385

4 Vorperian, V.: ‘Simplified analysis of PWM converters using themodel of PWM switch: Parts I and II’, IEEE Trans. AerospaceElectron. Syst., 1990, 26, pp. 490–505

5 Van Dijk, E., Spruijt, H.J.N., Sullivan, D.M.O., and Klaassens, J.B.:‘PWM switch modelling of DC-DC converters’, IEEE Trans. PowerElectron., 1995, 10, (6), pp. 659–665

6 Smedley, K., and Cuk, S.: ‘Switching flow-graph nonlinearmodelling technique’, IEEE Trans. Power Electron., 1994, 9, (4),pp. 405–413

7 Umanand, L.: ‘Switched junctions in bond graph for modelling powerelectronic systems’, J. Indian Inst. Sci., 2000, pp. 327–332

8 Karnopp, D.C., and Rosenberg, R.C.: ‘System dynamics a unifiedapproach’, (Wiley Interscience Publications, USA, 1975)

9 Mukherjee, A., and Karmakar, R.: ‘Modelling and simulation ofengineering systems through bond graphs’, (Narosa Publishing House,India, 2000)

10 Stromberg, J.E., Top, J., and Soderman, U.: ‘Variable causalityin bond graph caused by discrete effects’. Proceedings ofSCS Conference ICBGM’95, San Diego, CA, USA, 1993,pp. 115–119

11 Demir, Y., and Poyraz, M.: ‘ Derivation of state and output equationsfor systems containing switches and a novel definition of switchusing the bond graph model’, J. Franklin Inst., 1997, 3343, (2), pp.191–197

12 Allard, B., Helali, H., Lin, C., and Morel, H.: ‘Power converteraverage model computation using the bond graph and petrinet techniques’. Proceedings of IEEE PESC, Atlanta, GA, USA,1995, pp. 830–836

13 Delgado, M., and Sira-Ramirez, H.: ‘A bond graph approach to themodelling and simulation of regulated dc-dc power supplies’, Simul.Pract. Theory Vol. 6, 1998, 12, pp. 631–646

14 Mosterman, P.J., and Biswas, G.: ‘Behaviour generation using modelswitching– a hybrid bond graph modelling technique’. International

Conference on Bond graph modelling, Las Vegas, NV, USA, 1995,pp. 177–182

15 Mosterman, P.J., and Biswas, G.: ‘A theory of discontinuitiesin physical systems model’, J. Franklin Inst., 1998, 335B, (3), pp.401–439

16 Asher, G.M., and Eslamdoost, V.: ‘Power electronic circuit simulationand bond graph techniques’. EPE Aachen, Germany, Oct 1989, Vol.2, pp. 807–812

17 Asher, G.M., and Eslamdoost, V.: ‘A novel causality changingmethod for the bond modelling of variable topology electronicswitching circuits’. Proceedings of IMACS Symposium, Lille, Belgium,1991, pp. 371–377

18 Asher, G.M.: ‘The robust modelling technique of variable topologycircuits using bond graphs’. Proceedings of ICBGM 93, San Diego,CA, USA, pp. 126–131

19 Femia, N., and Tucci, V.: ‘On the modelling of PWM converters forlarge signal analysis in discontinuous conduction mode’, IEEE Trans.Power Electron., 1994, 9, (5), pp. 487–496

20 Sun, J., Mitchell, D.M., Greuel, M.F., Krein, P.T., and Bass, R.M.:‘Averaged modelling of PWM converters operating in discontinuousconduction mode’, IEEE Trans. Power Electron., 2001, 16, (4), pp.482–492

8 Appendix

Structure and equations of the 0-junction and1-junction8.1 0-Junction.Figure 26 shows the 0-junction. At the 0-junction, only onebond at the junction decides the effort at the junction andhence the effort at all bonds. The flow at all the bondsexcept the effort deciding bond decides flow at the effortdeciding bond. In Figure 26, bond 1 decides the effort at allbonds and bonds 2–6 decide the flow at bond 1. In equationform

e2 ¼e1e3 ¼e1e4 ¼e1e5 ¼e1e6 ¼e1f1 ¼f2 þ f3 þ f4 þ f5 þ f6

8.2 1-Junction.Figure 27 shows the 1-junction. At the 1-junction, onlyone bond at the junction decides the flow at thejunction and hence the flow at all bonds. The effort at allthe bonds except the flow deciding bond decides effort atflow deciding bond. In Fig. 27, bond 3 decides the flow atall bonds and bonds 1,2 and 4 decide effort at bond 3. In

f 6

f 5

f 4

f 3

f 2

f 1

e1

1

6 5

4

3

2

0

Fig. 26 0-junction

IEE Proc.-Electr. Power Appl., Vol. 152, No. 1, January 2005 59

equation form as

f1 ¼f3

f2 ¼f3

f4 ¼f3

e3 ¼e1 þ e2 þ e4

e 4

e 1

e 2

f 3

11

3

4

2

Fig. 27 1-Junction