Interleaving of a soft-switching Boost Converteroperated in Boundary Conduction Mode

D. Gerber and J. BielaLaboratory for High Power Electronic Systems, ETH Zurich

Email: gerberdo@ethz.ch

Abstract—This paper presents the interleaved operation of asoft-switching boost converter operated in boundary conductionmode. First, the operating principle of the converter as wellas the basic concept of the interleaving is presented. Then,the dynamic behaviour is modeled by using the z-transform toobtain a converter model which is independent of the switchingfrequency. With the model, the stability of the closed loop systemwith a PI-controller is analyzed. It is shown that an adaptive PI-controller can easily be implemented to a minimal settling timeover a wide operating range.

Finally, the controller is validated with two converters with40 kW nominal output power and an output voltage of 3 kV. Thetests at different output voltages under different load conditionsshow a stable interleaved operation.

I. INTRODUCTION

A compact and cost effective X-ray free electron lasersystem (SwissFEL) is currently built at the Paul ScherrerInstitute (PSI) in Switzerland [1]. This laser system requiresmodulators with a pulse power of 127 MW for 3µs. The highpulse power is provided by a capacitor bank which is rechargedto 3 kV by two 40 kW boost converters with 1.3 kV inputvoltage between two consecutive pulses.

In the considered system, two interleaved converters (Fig. 1)[2], are used to charge the capacitor bank. The interleavedoperation results in a reduced input current ripple. In PWMcontrolled systems with fixed switching frequency, the inter-leaving can be easily controlled by shifting the PWM signalsrelative to each other. However, this method does not work forconverters operating in boundary conduction mode (BCM) thatresults in a variable switching frequency. Different open-loopand closed-loop control methods for the operation in BCMhave been investigated (e.g. [3], [4]). However, those methodsdo not guarantee a soft-switched operation when the systemis perturbed.

In this paper, a control strategy for interleaved boost con-verters operated in BCM is presented. The proposed con-troller guarantees zero voltage switching during the interleavedoperation including startup and the synchronization of theconverters.

The basic operation principle of the converter and the inter-leaving is presented in section II. In section III, the convertermodel and the controller model are introduced. Two differentcontrollers are investigated and the implemented controller isshown. Finally, measurements with the proposed controller andtwo 40 kW, 3 kV interleaved boost converters are presented.

Inductor

InputCapacitor

SnubberCapacitor

Heat Sinkwith Power

Semiconductors

24V SupplyOutput

Capacitor

Fig. 1. Prototype of the 1.3 kV to 3 kV, 40 kW boost converter.

L1

+-

VinCin Cout Vout

D1a

S1n CSn,S1n RSn,S1n

S1a CSn,S1a RSn,S1a

CSn,D1a

RSn,D1a

D1n

CSn,D1n

RSn,D1n

S1

D1

iL

Ts

T1 T2 T3 T4

i

t

iLp

-iLp,min

iLp,min

i2Lp-i2Lp,min

T5

t

vds

Ts

T1 T2 T3 T4 T5

vout

Fig. 2. Boost converter circuit and the corresponding BCM waveforms.

II. CONVERTER OPERATION

The basic circuit of the converter including the snubbercircuits is shown in Fig. 2. Details of the converter and thefeedback controller are presented and analyzed in [2] and [5].In the following, the converter operation in BCM and theinterleaved operation are explained.

A. Converter Operation

The series connected switches S1a to S1n are turned on atthe beginning of interval T1 (cf. Fig. 2). The input voltage Vinis applied across inductor L1 and inductor current iL starts torise. When inductor current iL reaches the level iLp, switchesS1a to S1n are turned off at the beginning of interval T2 (peakcurrent control). Diodes D1a to D1n are not conducting at thattime since capacitors CSn,D1a to CSn,D1n are still charged.Inductor L1 and snubber capacitors CSn form a resonantcircuit. Since inductor current iL is positive, the capacitorsin parallel to the diodes D1a to D1n are discharged to zeroand the capacitors across the switches are charged to Vout. Assoon as the voltage across the diodes reaches zero, they startto conduct (interval T3). At that point of time, the difference

iL,0

Ts,0

i

t

iLp,0iLp,1

∆Tn

iL,1

∆Tn+1

Ts,1

MasterSlave

Fig. 3. Interleaved inductor current waveforms.

between Vout and Vin is applied across L1 and current iLdecreases. At the beginning of interval T4, iL reaches zeroand the diodes stop conducting. The snubber capacitors andthe inductor form again a resonant circuit. The voltage acrossthe inductor is the difference between Vout and Vin. Hence, theinductor current becomes negative and the capacitors acrossthe switches are charged to zero if the difference betweenVout and Vin is large enough. At then end of T4, the bodydiodes of the switches start to conduct. At that point of time,the switches have to be turned on before the inductor currentbecomes positive and the next switching cycle starts.

The operation in BCM results in a switching frequencywhich depends on the input and output voltage, the outputpower as well as the inductance value.

B. Interleaved Operation

The inductor current waveforms for two interleaved convert-ers are shown in Fig. 3. Since the switching frequency dependson iLp, the phase shift ϕ = ∆Tn

Ts,0can be controlled by reducing

or increasing iLp,1 relative to iLp,0. The phase shift can bemeasured by detecting the inductor current zero crossings. Oneconverter is used as a reference (master) whereas the other isused to adjust the phase shift (slave).

In an ideal system with identical converters, the switchingfrequency for both converters are identical for the same currentiLp. This is not the case in a real system because of componenttolerances, temperature drifts, etc. In BCM, non-equal inductorvalues result in unbalanced inductor currents. Fig. 4 showsthe relative deviation of the average inductor current betweenthe master and the slave depending on the inductor mismatchat the nominal operating point of the investigated system(Table I). The deviation is smaller if the slave’s inductor islarger than the one of the master. Therefore, the converter withthe smallest inductor value should be selected as the master.

In order to model the interleaved operation, a general modelwith non-identical converters has to be developed.

III. FEEDBACK CONTROLLER

In this section, the interleaving model and the controllermodel are derived. Also, the optimal controller parameters arecalculated. Afterwards, the implemented phase shift measure-ment and the controller are presented.

A. System Model

1) Plant Transfer-Function: The inductor current wave-forms are shown in Fig. 3. It is assumed that the input- andoutput-voltage as well as iLp,0 remain constant. The dynamic

−10 −5 0 5 10−15

−10

−5

0

5

10

15

∆L (%)

∆iL,avg (%

)

Fig. 4. Current balancing of the average inductor currents depending oninductance mismatch ∆L =

Lslave−LmasterLmaster

.

behaviour of the time offset between the two inductor currentsis modeled by calculating the time offset for the next switchingcycle ∆Tn+1 depending on the offset ∆Tn and the switchingperiod lengths of the converters Ts,0(iLp,0) and Ts,1(iLp,1):

∆Tn+1 = ∆Tn − Ts,0(iLp,0) + Ts,1(iLp,1) (1)

The switching periods Ts,k(iLp,k) can be calculated ana-lytically by solving the differential equations describing thedynamic behaviour of the converter. These expressions arenon-linear because of the resonant transitions during the timeintervals T2 and T4. In order to simplify the model, Ts,1(iLp,1)is linearized at iLp = iLp,0:

Ts,1(iLp,1) ≈ Ts,1(iLp,0) +dTs,1(iLp)

diLp

∣∣∣∣iLp,0

∆iLp (2)

with ∆iLp = iLp,1 − iLp,0.Since the continuous offset ∆Tn is discrete in time, the

z-transform is applied:

Z∆Tn+1 = zZ∆Tn = z∆T (z) (3)

It is assumed that iLp,0 is constant. Hence, Ts,0(iLp,0),Ts,1(iLp,0) and dTs,1(iLp)

diLp

∣∣∣iLp,0

are constant. By using (1) -

(3) one obtains:

∆T (z) = Gd(z) +Gp(z)∆ILp(z) (4)

with the disturbance transfer-function

Gd(z) =1

z − 1[Ts,1(iLp,0)− Ts,0(iLp,0)] =

1

z − 1∆Ts

and the plant transfer-function

Gp(z) =1

z − 1

dTs,1(iLp)

diLp

∣∣∣∣iLp,0

=1

z − 1T ′s,1(iLp,0)

Gc Gp

Gd

ΔILpe ++

-

+ΔTset ΔT

Plant

Fig. 5. Closed loop control system of the interleaving model.

2) Closed Loop Transfer-Function: In this section, theclosed-loop system shown in Fig. 5 is analyzed. The systemmodel including the controller is:

∆T (z) =Gc(z)Gp(z)

1 +Gc(z)Gp(z)∆Tset(z)+

Gd(z)

1 +Gc(z)Gp(z)

=Gcl(z)∆Tset(z) +Gdis (5)

where Gcl(z) represents the plant closed loop transfer-functionand Gdis the disturbance closed loop transfer-function.

B. Proportional-Integral Controller

It is known from the control theory that a system is stable ifand only if its poles are inside the unit circle of the complexplain. Since a proportional (P) controller is not able to drivethe system to the desired phase-shift without a static deviation,a proportional-integral (PI) controller is investigated.

1) Stability: The closed loop transfer-function for a PI-controller Gc(z) = KP +KI

zz−1 is

Gcl,PI(z) =(KI,n +KP,n)z −KP,n

z2 + ((KI,n +KP,n)− 2)z −KP,n + 1(6)

KP,n =KPT′s,1(iLp,0) (7)

KI,n =KIT′s,1(iLp,0) (8)

Hence, the dynamic behaviour of the system depends on KP,n

and KI,n, providing an operating point independent dynamicbehaviour. Gcl,PI(z) has two poles at

p1,PI = −KP,n+KI,n−

√(KP,n+KI,n)2−4KI,n

2+1 (9)

and

p2,PI = −KP,n+KI,n+

√(KP,n+KI,n)2−4KI,n

2+1. (10)

The system is stable for

0 < KI,n < 4− 2KP,n (11)

and

0 < KP,n <4−KI,n

2. (12)

2) Settling Time: The settling time of the system with PI-controller is shown in Fig. 6. The plot shows a minimumsettling time of 2 switching cycles around KP,n = 1 andKI,n = 1.

5

4

3

46

6 4

4

4 5 6

7

8

910

5 6 7

8

2

Kp,n

Ki,n

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

Fig. 6. Settling time in cycles of the closed loop system with PI-controller.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.5

0

0.5

1

1.5

2

2.5Step Response

Am

plitu

deTime (cycles)

Fig. 7. Step response for KP,n = 1 and KI,n = 1.

C. Feedback Controller Implementation

The closed loop transfer-functions Gcl,p and Gcl,pi showthat the system dynamics depend on the product of T ′s,1(iLp,0)and the controller coefficients. Hence, the optimal controllerparameters depend on the operating point. They can be calcu-lated with

Kp =Kp,n

T ′s,1(iLp,0)(13)

and

Ki =Ki,n

T ′s,1(iLp,0). (14)

The term T ′s,1(iLp,0) can be calculated analytically. Thismeans, that it is possible to calculate the controller coefficientsonline. Since the sampling frequency of the system modelis equal to the switching frequency, the integral part of thecontroller is sampled with the switching frequency of theconverters. A block diagram of the controller is shown inFig. 8.

Since the derived model depends on the linearized systemmodel, it is necessary to limit the controller output to assurethat the linearized model is a good approximation of thesystem dynamics. This also makes sure, that the switchingfrequency of both converters are close enough to each otherin order to prevent one converter to skip switching cyclesor switch multiple times during one switching cycle of theother converter. The slave set-point iLp,1 is limited after the

Linearization

vin vout iLp,0

KP,n KI,n

KP KI

Controller

Phase ShiftMeasurement

∆Tsetφset

Ts

∆T

+

-

+ +iLp,1

∆iLp

Ts’(iLp,0)

Fig. 8. Block diagram of the implemented controller.

TABLE ISPECIFICATIONS FOR THE PROTOTYPE SYSTEM

Input Voltage 1.3 kVOutput Voltage 3 kVOutput Power 40 kW (2 Interleaved)

Switching Frequency 70 kHz - 200 kHzEfficiency approx. 97 %

Power density approx. 2 kW/dm3

controller output to ensure ZVS and to avoid excessive thermalstress.

IV. MEASUREMENTS

A. Test System

The interleaved operation is tested with two 40 kW, 3 kVconverters. The converter parameters are shown in Table I.The two converters are connected in parallel to an ohmic loadas shown in Fig. 9.

B. Results

The interleaved operation was successfully tested with twodifferent ohmic loads of 1.4 kΩ and 215 Ω. The output voltageswas set between 750 V and 3 kV. The operation at low outputpower is more critical than the operation with high outputpower because of the higher switching frequency and becausethe relative error of the linearized term is bigger at inputcurrents close to the free-wheeling current. As an example,the measured current waveforms of the inductor current andthe resulting phase shift at 2.5 kV with a 1.4 kΩ load are shownin Fig. 10.

The deviation of the set-point during the operation can beexplained by the high switching frequency and the jitter of thezero-crossing detection and the accuracy of the internal current

L1

S1 CS1Cin Cout

D1

CD1

iL400V MainsLoadConverterIsolated

PFC

Fig. 9. Setup for interleaving measurements.

0 10 20 30 40 50 60 70 80 90 100−20

−10

0

10

20

Time (µs)

Cur

rent

(A)

Inductor Current

0 10 20 30 40 50 60 70 80 90 1000

100

200

300

Time (µs)

Phas

e Sh

ift (d

egre

e)

Phase Shift

Master Slave

Fig. 10. Measured inductor current waveforms and phase shift at an inputvoltage of 1020 V, an output voltage of 2.5 kV and an output power of 4.5 kW.

measurement of the converter. A phase-shift of 10 corre-sponds to a shift of approximately 200 ns at 140 kHz switchingfrequency. The used current sensor has a measurement rangeof 100 A. An accuracy of 0.5 % already corresponds to adeviation of 50 ns of the on-time of the switch at the measuredoperating point.

V. CONCLUSION

In this paper, the interleaved operation of a soft-switchingboost converter operated in boundary conduction mode isinvestigated. The soft-switching operation as well as the in-terleaving concept is presented.

A general model for interleaved operation is derived byusing the z-transform in order to obtain a switching frequencyindependent model. Afterwards, the stability of the closed loopsystem with a PI-controller is investigated. Also, it is shownthat the dynamic behaviour of the small signal model of thesystem is independent of the operating point for normalizedcontroller coefficients. An adaptive interleaving controller ispresented.

Finally, the controller has been implemented and success-fully tested with two converters at different loads at an outputvoltage ranging from 750 V to 3000 V.

ACKNOWLEDGEMENTS

The authors would like to acknowledge Ampegon AG andAmpegon PPT GmbH for the construction of the convertersand the tests with the 215 Ω load.

REFERENCES

[1] (2014, March) SwissFEL. [Online]. Available: http://www.psi.ch/swissfel/[2] D. Gerber and J. Biela, “Charging precision analysis of a 40-kW 3-kV

soft-switching boost converter for ultraprecise capacitor charging,” IEEETrans. Plasma Sci., vol. 42, no. 5, pp. 1274–1284, 2014.

[3] L. Huber, B. Irving, and M. Jovanovic, “Open-loop control methods forinterleaved DCM/CCM boundary boost PFC converters,” IEEE Trans.Power Electron., vol. 23, no. 4, pp. 1649–1657, 2008.

[4] H. Choi and L. Balogh, “A cross-coupled master–slave interleavingmethod for boundary conduction mode (BCM) PFC converters,” IEEETrans. Power Electron., vol. 27, no. 10, pp. 4202–4211, 2012.

[5] D. Gerber and J. Biela, “40kW, 3kV soft-switching boost converter forultra precise capacitor charging,” in Proc. of the 4th Euro-Asian PulsedPower Conference, 2012.