Generalized Small-Signal Dynamical Modeling of multi-port dc dc converter.pdf

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Generalized Small-Signal Dynamical Modeling

of Multi-Port Dc-Dc Converters

David C. 13amill

School of Electronic Engineering, Information Technology and Mathematics

University of Surrey,GuildfordGU25XH, United Kingdom

[email protected]

Abstract — A general method is presented for modeling multi-

port de-de converters. It copes with multiple inputs, multiple

outputs and bidirectional ports, and is based on an averaging

formulation. A matrix description is adopted, so the technique

can be extended to converters with any number of ports. As an

example, a three-port forward-flyback converter is analyzed

using symbolic computation software (Maple V).

I. INTRODUCTION

compared with the attention paid to two-port conver-

ters, little consideration has been given to modeling

multi-output de-de converters. Important characteris-

tics such as static and dynamic cross regulation have been

little explored in the literature, and then usually only on an

ad-hoc basis — e.g. [1], [2]. Furthermore, very little work

has been done on generalized multi-port de-de converters.

For example, a personal computer power supply could com-

prise a single converter with multiple dc outputs as usual,but two dc inputs: one fed from rectitied ac mains, the other

from a secondary battery. The battery port could be bidirec-

tional, to allow recharging. A generalized de-de converter

can have multiple input ports, multiple output ports and

bidirectional ports.

This paper describes a general method for small-signal

modeling of any multi-port de-de converter. It is based on

matrices, for several reasons: they provide a compact nota-

tion; the results can be applied to converters with any num-

ber of ports; numerical matrix computations are easily

programmed using standard linear algebra packages, work-

sheets such as MathCad, and even spreadsheets; and matrixalgebra can be automated with symbolic computation pack-

ages such as Maple, Mathematical and Macsyma.

Most analyses of two-port de-de converters start by assum-

ing a stiff voltage source at the input and a resistive load at

the output. For good reasons the proposed approach does not.

At its input, a converter is often fed from a filter, or at least

via some line impedance decoupled by a capacitor. At the

output, the load’s dc characteristic many vary from constant

voltage to constant current, or be nonlinear, so a linear re-

sistive load is a special case. With other loads, the small-

signal damping will difler from that predicted using an

equivalent load resistance. Worse, the load could be induc-

tive or capacitive, and this will greatly affect the overall

system dynamics. For these cases, models developed using astiff voltage source and a resistive load will give misleading

results. Instead, the aim should be to model the converter in

isolation from its surrounding circuit. Provided its terminal

characteristics are properly represented, such a model can

subsequently be embedded within a complete power system,

allowing the interactions to be assessed with ease. This is the

basis of two-port circuit theory, adapted here to multiple

ports.

First, a generalized model of an open loop de-de converter

is developed, assuming an averaged description. Next, the

characteristics are linearized around a quiescent operating

point. Finally, the open loop model is embedded within a

feedback control loop. The method is subject to the usual

limitation of linear models: it may be inaccurate for large

signals. Nevertheless, l inearized average models have proved

popular with engineers because they allow the application of

standard linear systems control theory.

II. THE OPEN LOOP CONVERTER

A de-de converter has two or more power ports. If power

always flows into the converter it is an input por~ if power

always flows out, it is an output por~ if power can flow in

either direction, it is bidirectional. However, for generality,

the standard circuit-theory sign convention is adopted here:

the reference direction of current is into the positive terminal

of each port, as shown in Fig. 1. No distinction is made

between input and output ports; if the power at a port is

positive, it is acting as an inpuq if negative, as an output.

A. The State Vector

The model presented is based on four essential vectors, the

first of which is the state vector x(t). A general description of

a dynamical system is:

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where + distinguishes a particular system. The dynamics of

an nrth-order converter can be characterized by m internal

state variables, usually the inductor currents and capacitor

voltages. These state variables can be formed into an instan-

taneous state vector, x,~l(t) q ll?~. This gives an exact de-

scription of the time varying, nonlinear circuit including, for

example, ripple at the switching frequency.

In most converters, some components of the state vector

will be “fast” (comparable to the switching frequency) and

others will be “slow”. In many cases the model can be sim-

plified by overlooking the fast variables. A suitable process

converts the time varying mth order model into a time in-

variant rzth order one, n < m. The instantaneous state vector

x,.$,(t) e RY is changed into an equivalent “low frequency”

state vector x(t) e R“.

Two approaches for eliminating the fast variables aresampling and averaging. In the first, x,.,,(t) is sampled at the

switching frequency J, and the fast variables are neglected.

(Though not attempted here, the method presented could be

adapted to a sampled data description.) Alternatively, any of

the averaging methods developed for two-port converters can

in principle be used to obtain x(t). These include the original

circuit averaging process [3], state space averaging [4],

injected-absorbed currents [5], the PWM switch model [6],

Bogoliubov averaging [7], and switching-frequency depen-

dent averaging [8]. Of particular interest are symbolic com-

putational methods [9], Discussion of the pros and cons of

particular averaging processes is outside the scope of this

papeq it is assumed that some satisfactory process exists for

mapping X,ml(f) to x(f), and that the dynamics of the conver-

ter are adequately described by the result.

B. The Port Vectors

Suppose the de-de converter has N power ports. The port

currents i, and voltages v,, r = 1 ... N, comprise a set of 2N

port variables. For each port we choose either i, or v, and

form the chosen quantities into a vector of independent vari-

ables, w(t) G RN. The remaining quantities are then formed

into a vector of dependent variables, y(t) e E%N.

There are many ways in which the port variables can be

assigned to the two vectors, some of which are more helpful

than others. For each pofi it must be decided which variable,

i or v, is to be regarded as the independent one. For instance,

if the converter is designed to deliver a constant voltage to a

load, the output port’s current should be taken as the inde-

pendent variable, because the load, not the converter, deter-

mines the current drawn. The current is independent of the

converter so it should be the independent variable. Converse-

ly, if the converter is meant to deliver a constant current, e.g.

as a battery charger. the load voltage should be taken as the

‘1 ‘N+~ ~+

VI ‘Iv—

de-de‘2 ‘N–1

+~ converter ~+

“2 ‘N–1

—

0me

other ports

Fig. 1: Generalized multi-port de-de converter, showing

reference directions of current and voltage.

independent variable — the battery’s voltage is independent

of the converter and changes according to the state of charge.

Moving to the input port, the supply is usually approximated

by a variable voltage source, so voltage should be chosen as

the independent variable, with the converter’s input current

as the dependent variable. In control system terminology, the

components of w are disturbances to the system, whale the

components of y represent its response.

A very common situation, termed here the Ordinary Case,

is when the converter has a single input port and N – 1 out-

put ports. For this case the independent vector w best con-

sists of the input voltage and the output currents, while the

dependent vector y comprises the input current and the out-

put voltages.

C. The Control VectorLet the converter have M control variables. For example,

these might include signals that determine the duty ratio or

frequency of a switching device, or control magnetic amplifi-

ers. The control signals form a vector u(f) e lR”.

It might be thought that if M’ = N all the dependent vari-

ables of y could be individually controlled by u. This is not

necessarily so. In an ideal lossless converter, with the sign

convention adopted the total power entering the converter

must be zero. This removes one degree of freedom, so only

N – 1 components of y can be controlled by u.

Often, M < N – 1; then it is impossible to control more

than M dependent variables, and the rest must rely upon

cross regulation. If the converter is allowed losses, the total

power entering the converter is no longer constrained to

zero: it is positive and equal to the losses. Now all N depen-

dent variables can be individually controlled, for instance by

including linear regulators in the converter. (This adversely

affects efficiency.) In the Ordinary Case it is only necessary

to control the N – 1 output ports, while the input current

goes where it must.

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D. Large-Signal Model loop stability, the converter will settle to a steady state where

The open loop converter can be characterized by two non-

linear vector equations linking the essential vectors u, w, x

and y:

$ x(t)= (#)[x(t),u(t), w(t)](2)

y(t) = ~[x(f), u(t), w(t)] (3)

The state equation, (2), governs the converter’s large-signal

dynamics as it reacts to the control signals and independent

port variables. Equation (3), the response equation Uoutput

equation” in control terms), describes how the dependent

port variables respond. Functions + and v are attributes of a

particular converter.

E. Linearizat ion

We next find the steady state. Let u(t)= const = U (i.e. the

control signals are held steady) and w(t) = const = W (i.e.the independent variables are dc quantities). Assuming open

TASLE I NOMENCZ.ATTJRIFORTHEGENERALIZSDMODEL

SCALARS:

?1 dimension of the averageddynamical system

M number of control signals

Iv number of ports

VECTORS

ii(s) (M x 1) small-signal control vector*

*(S) (N x I) small-signal independentport vector *

i(s) (n x 1) small-signal statevector *

;(s) (N x 1) small-signal dependentport vector *

4 (n x 1) RHS of the stateequation

v (N x 1) RHS of the responseequation

* x isa large-signalquantity,i isasmall-sigualquautity

F (Nx N) @l&v ~

G(s) (N x N) closed loop port-to-port transfer function matrix

H(s) (N x M) open loop control-t~port tmnsfer function matrix

J(s) (N x N) open loop port-to-port transfm timction matrix

K(s) (M XN) controller matrix

t emluatedattheoperat ingpoint

x(t) = const = X and y(t) = const = Y. Substituting for u, w,

x and y in (2) and (3), and setting dX/dt = O (since X =

const), X maybe found in terms of U, W and Y. This gives

the steady state operating point, Q = {U, W, X, Y}.

Now consider small perturbations around Q. Adapting the

usual notation for de-de converters to the vector case, let

x(t) = X-t ~(t) , etc., where i(t)s a small perturbation from

the steady state equilibrium. Provided they are smooth, the

nonlinear fimctions 1$ and v may each be expanded by a

multivariable Taylor series; truncating after the

(2) and (3) become

: i(t)= A i(t)+ B ii(t)E ti(f)

j(t)= C i(t)+ D i(t) + Fti(t)

linear terms,

(4)

(5)

where A, B, C, D, E and F are real, constant matrices:

Here the Jth entry of the sensitivity matrix Z3$/i3x is @,/dx,,

etc. Equations (4) and (5), valid for small perturbations only,

are an augmented version of the standard state space descrip-

tion of a multivariable linear system. Fig. 2 shows the equa-

tions as a block schematic. The notation follows that of [10]:

matrices A, B, C and D have their usual linear-systems

meaning. Of particular importance is the system matrix A,

whose eigenvalues govern the dynamics. Matrices E and F

represent direct feed-through of disturbances.

1? Open Loop Small-Signal Model

Taking Laplace transforms, (4) and (5) become

s~(s) = A i(s) + B ii(s) + E +(s) (7)

~(s) = C ~(.s) + Dii(s) + Fti(.s) (8)

where, with some abuse of notation, @ is to be understood

as the Laplace transform of ~(f), etc. Finding i(s) from (7)

and substituting into (8) yields the open loop small-signalmodel:

j(s) = H(s) ii(s) + J(s) +(s) (a)

where H(s) = C(SI – A)”-l B + D (b)

and J(s) = C(SI – A)-l E + F (c)

(9)

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(i(s)J

B D :(s) E H(s)

++-

+- 2(s)

> f(s)> ;(s)

$ (s] E J(s) +E + F

6’(s) ? ?“

Fig. 2: Block schematic of the open loop small-signal Fig. 3: The model of Fig. 2 may be reduced to two

model of a de-de converter. frequency dependent blocks and an adder.

(I is the N x N identity matr)ix. In (9a) the internal state

vector ;(s) has been eliminated and the constant matrices A

to F have been replaced by frequency dependent matrices.

Equation (9a) isshown in block schematic form in Fig. 3,a

complete “black box” small-signal model of a generalizedopen loop N-port de-de converter. The entries of the N x M

matrix H(s) aretransfer functions from thek.f control signals

to the Ndependent port variables. and the entries of the NX

Nmatrix J(s)are transfer fanctions from theNindependent

portvariablesto the Ndependentport variables.

III. THECLOSEDLOOPCONVERTER

Anobjective ofa de-de converter isto regulate against

disturbances: tomaintain certain components of y constant

despite variations in w. Ideally, ~(s)=O. This goal is ap-

proachedby applying closed loop control. In the feedback

system of Fig. 4, ~(s) iscompared tozeroto produce an errorvector, which is processed by controller K(s) to produce the

control vector i(s), thus closing the loop:

i i(s) = –K(s) j(S) (lo)

In general K(s) allows cross coupling, so that any component

of y can influence any or all components of u. From (9) and

(10) the closed loop small-signal model is obtained:

j(s) = G(s) ti(S) (a)

(11)

where G(s) = [1 + H(s) K(s)]-’ J(s) (b)I I

A condition for validity is that I + H(s)K(s) must be nonsin-

gular, which is usually true. A block schematic is shown in

Fig. 5. Note that if K(s) = O, (1 la) reduces to G(s) = J(s).

A. The Ordinary Case

The Ordinary Case is particularly important in practice.

When the N independent variables are chosen from a set of

2N port variables, there are (2N) !/(2N! ) possible combina-

tions, excluding row swaps. This number determines the

different forms of matrix G. (E.g. there are six varieties of

two-port parameters, but 60 varieties of three-port parame-

ters.) In the analysis of large multi-port networks, the usual

choice is to group all the currents into one vector and all the

voltages into the other. Depending on which vector is chosen

as independent, the resulting matrix contains either im-

pedances (z-parameters) or admittances (y-parameters).

However, these are not very useful for the present purposes,

because the y and z-parameters for an ideal converter are all

infinite.

The two-port g-parameters have been suggested by Mand-

hana [11] as a small-signal model for de-de converters:

where g,, is the converter’s input admittance, g,~ its reverse

current gain, gzl its forward voltage gain, and gzj its output

impedance. This description can be extended to the Ordinary

Case of a multi-port de-de converter if the port variables are

segregated into dependent and independent vectors as sug-

gested above, with port 1 taken as the input. Then (1 la) may

be written as

lZI=[AV(S‘out(s)1:1Here ~(s) and *(s) have been partitioned into voltage and

current parts, and the matrix G(s) partitioned accordingly.

Scalar Y,.(s) = g,, =; 1/~ 1 is the converter’s input admittance.

Row ve$tqr A,(;) : k,, ... glNl comprises reverse current

gains, ilhz . . . z1/iN , and describes how output current

changes affect the input. Column vector A,(s) = ~21 . .. gm]~

comprises forward voltage gains ~Jo 1 ~.. ~IV/~ I It describes

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Reference +

o[

Open loop converter

Fig. 4: Block schematic of the closed loop small-signal

model ofadc-dc converter.

how input voltage changes affect the output voltages, i.e. the

dynamic line regulation of the converter

(audiosusceptibility). Ideally all its entries would be zero.

Matrix Zout(s)= ~g], i, j = 2.. .N, comprises self and mutual

~ . The leading-diagonal elements (i =j) arempedances ~,111

thesource impedances ofeach output port, and describe the

dynamic load regulation of the converter. The off-diagonal

terms (i #J) are mutual impedances relating the port-i cur-

rent to the port~” voltage, and describe the dynamic cross

regulation. Ideally all the entries of ZOU1(S)would be zero.

For cases other than the Ordinary Case, different ways of

choosing the independent and dependent variables might be

more appropriate, and they should be considered on their

merits.

Thus the N x N transfer function matrix G(s) gives a com-

plete small-signal description of the dynamics of any N-portde-de converter. For an open loop converter (with its control

signals held constant), G(s) is identical to J(s). For a closed

loop converter under the control scheme shown, G(s) is

given by (1 la). Other forms of control (e.g. inner/outer

loops) will result in a different form for (1 lb).

IV. EWLE

As an example, an Ordinary Case converter is analyzed

using the symbolic computation package Maple V [12]. The

method can easily be applied to more complicated conver-

ters, the computer handling the increased complexity of the

algebra.

A. Open Loop Model

The three-port converter shown in Fig. 6 [13] can be sepa-

rated into two “semi-converters”: it is basically a forward

converter but, instead of the normal energy-recovery reset

winding, the transformer has a flyback winding. The forward

semi-converter operates in continuous mode, so its output

voltage depends on the duty ratio but is independent of the

switching frequency. The flyback semi-converter operates in

‘(S)*’(S)

Fig. 5: The model of Fig, 4 can be reduced to a single

frequency dependent block, forming a small-signal “black

box” model of a multi-port de-de converter.

discontinuous mode, so both the duty ratio and the switching

frequency tiect its output voltage. For this example n = 5

(state variables), N = 3 (ports) and h’ = 2 (control signals).

Let the four essential vectors be:

State vector x = [i~l i~z vcr vcz va] ~

Independent port vector w = [vf iol im]T

Dependent port vector y = [i, Vol va]T

Control vector u = [a jy

where i~l is the current in L1, etc., 8 is the duty ratio andj is

the switching frequency. Other quantities are defined in Fig.

6. The circuit has the following parameter values: L, =50~.

L,= 600p.H, L, = 60pH, Cl = 47pF, C, = 470@, C, = 470@,

F.= 49kHz, A = 0.3, NJNP = 20/14, ~ = 28V, 101 = –1A, Io,

= –1A. The output voltages are intended to be VO1= 12V, V02

= 12V.

The “low frequency” state equation may be found by a

naive averaging process as:

—

(V1- vcl)/L1

(VC18N,11NP - v@Lz

(iLl - 6iLzN8~/NP - 82v1/2f,LP)lCl

(i~z + zol )IC2

( )~ti212LPfSvc3 i- io2, IC3

J

(14)

which corresponds to the state equation (2). Likewise,

I1!Jl iL]

y = Vol = VC2, (15)

V02 VC3

corresponds to the response equation (3).

The operating point is found by letting ill = 1~1, i~2 = I~j,

vCl=Vcl, vc2=Va, vG=Vm, il=ll, vol=Vol, vm=Vm, v1

= VI, iol =Iol, im=Im, 6= A,~=F~, setting dXldt=O and

solving for X:

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*I A A

Forward (CCM)

Fig. 6: Two-output forward-flyback converter used in the

example,

x=

IL1

IL2

v~, —

.

(16)

Substituting numerical values for the parameters, the

operating point Q is found as

U = [A FJ’ = [0.3 49 OOOHZ]’ (given)

W = [~ 101 ]@]’= [28V -1A -lA]’(given)x = [~~1L2 Vcl J’C2 VC31T

= [0.857A 1A 28V 12.OV 12.0VIT

Y = [i, VO,vJ’ = [0.857A 12.OV 12.OVl’

Matrices A through F are easily computed from their

definitions in (6), by calling Maple’s jacobian fimction six

times. For example, A = (@/ax)Q is found by

j acobian (pbivec, xvec) :

A := subs(Qset, “) ;

which yields

A=

01E

00 0

00 00–2F.LPI;2

V;A2C3

(17)

For space reasons only the 5 x 5 A matrix is shown here. B

is5x2, Cis3x5, Disthe3x 2zeromatrix, Eis 5x3,

and F is the 3 x 3 zero matrix.

The H(s) and J(s) matrices are found by computing (10)

and (11) in Maple:

Hmat:-.. (Clnat&*inverse (s * &*() -Amat) &*Rnat+bt) ;

Jlllat:- —evalm(Cinat&*iwerse (s * &*() -Amat) &*13mt+Fmat) ;

H(s) is 3 x 2; J(s) is 5 x 5.

The full expressions for H(s) and J(s) are too complex to

reproduce here but, substituting numerical parameter values

and settings = O, t heir dc values are found as

~ 4.286 -0.875x 10-51

[

H(0) = 40 0

1

(18)

80 –2.45 X10-4

[

0.0153 –0.4286 O

J(0) = 0.4286 0 0

1

(19)

0.8571 0 12.00

At other frequencies the entries will be complex numbers.

Columns 1 and 2 of H(0) indicate the effect of duty ratio

and switching frequency variations, while rows 1, 2 and 3

refer to the input current and the two output voltages. Thus,

for example, if the duty ratio is increased from 0.30 to 0.31,

the input current will increase by 4.286 x 0.01 = 42.86mA,

the forward output voltage will increase by 40 x 0.01 =

400mV, and the flyback output voltage will increase by 80 x

0.01 = 800mV. If the switching frequency is increased from

49 to 50kHz, the input current will decrease by 0.875x10-5

x 1000 = 8. 75mA, the forward output voltage will be unaf-

fected, and the flyback output voltage will decrease by 2.45 x

10-4 x 1000 = 245mV (all dc values).

Thejll = 0.0153 entry of J(0) means the incremental input

conductance is 15 .3mS. The jlz = –0.4286 entry is the cur-

rent gain from the forward output current to the input cur-

rent (negative because all currents are defined as flowing

into the converter). Increasing the flyback output current

does not affect the input current, as shown byjl~ = O, due to

the open loop discontinuous mode operation. The voltage

gains from the input voltage to the two output voltages arejzl

= 0.4286 and j,] = 0.8571 (poor line regulation). The for-

ward output resistance is jzz = 0f2 (perfect load regulation),

while the flyback output resistance is j~t = 12f2 (poor load

regulation). The two mutual resistances jj~ = J“qz= Of2 mean

that this idealized converter has perfect cross regulation.

B. Closed Loop Model

Suppose the loop is closed by adding a simple proportional

controller with no cross coupling:

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hrmwd

0.

-20-

dEI-40-

-60-

40-

0 2000 4000 * 6000 8!300 10000

Flyback

-10-

-20-

-30-

dB-40-

+o-

-so-

-70-

0 2000 4000f

6000 8000 10000

Fig. 7: Dynamic line regulation (audiosusceptibili ty) of the

example converter, m open loop and closed loop,

‘(S)=[:W1 (20)

The 3 x 3 G(s) matrix can now be calculated from (14):

-t := evalm(mverse( &*() + Hmat &* Kraat) &* Jinat) ;

The resulting (extremely large) expression may be simplified

by substituting numerical values for the parameters. Inverse

Laplace transformation can be used to find the response to

steps and other fhnctions, or the matrix may be evaluated in

the frequeney domain by setting s = jm. Fig. 7 shows the

dynamic line regulation of the two semi-converters, in open

loop and closed loop with K] = 5, Kz = –50. (No evaluation

of stability was made, but in practice this must be done.)

To see the effect of intlnite dc loop gain (e.g. by including

ideal integrators in the controller), s was set to zero and the

limit taken as K, -+ m, K2 -+ -m. The new value of G is

[

–0.0306 –0,4286 -0.4286

G(0) = O 0 0

1

(21)

o 0 0

which may be compared with the equivalent open loop ma-

trix, J(0). The inlinite loop gain has made the input conduc-

tance negative, g,, = –30.6mS. The reverse current gains are

now both -0.4286. The line, load and cross regulation are

perfect, as shown by the zeroes in rows 2 and 3.

As in [13], the duty ratio was employed to regulate the

forward semi-converter and the switching Iiequeney was

used for the flyback semi-converter. The two control loops

interact, see Fig. 7. For better results the loops should be

decoupled; this will be discussed in a planned future paper.

The full Maple listing of this example is available from

the author on request.

v. CONCLUSION

A technique has been presented for modeling a general-

ized N-port converter, in isolation from its sources and loads.

The starting point is an averaged large-signal state equation,

obtained by any applicable method. The outcome is a fidl

open loop model. This can be embedded within a control

loop to give a complete small-signal dynamical model, G(s),

for any multi-port de-de converter. The matrix formulation is

particularly suited to automatic computation, either numer-

ical or symbolic.

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[2]

[3]

[4]

[5]

[6]

[7]

[8]

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[10]

[11]

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Y.T. Chen, D.Y Che~ Y.P. Wu and F.Y. St@ “Small-signal modeling

of multiple-output forward converterswith current-mode control”, L%%?

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G.W. Wester and R.D. Middlebrook, “Low-fi-equency characterization

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Documents

Generalized Small-Signal Dynamical Modeling of multi-port dc dc converter.pdf