Cascade controller for DC/DC buck convertor

K.M. Tsang and W.L. Chan

Abstract: In order to deal with the increasing demand for good load and supply-voltage regulationof DC/DC convertors, a new cascade controller is proposed. The control of DC/DC buckconvertor is first decomposed into a primary voltage-control loop and a secondary current-controlloop. The cascade controller is then implemented based on the buck-converter settings with thedynamics of the secondary loop much faster than these of the primary loop. A robustness analysisof the cascade controller against load changes and supply changes is presented. Experimentalresults are included to demonstrate the effectiveness of the proposed control scheme over aconventional single-loop PI controller. The proposed controller can be implemented as a low-costaddon to a conventional single-loop controller. Detail design equations are presented.

1 Introduction

The rapid increase in the computer and telecommunicationmarkets has led to the introduction of many powerful andsophisticated devices. However, it also placed a highdemand on switching-mode power supplies [1]. Powerconvertors are now subjected to larger load and supply-voltage variations. The dynamic response of a conventionalpower-supply system is generally too slow to follow thesudden load changes introduced by modern microprocessorsystems. Fast load changes presented by high-speedmicroprocessors have drastically increased the importanceof the dynamic response of power supplies. Moreover,distributed power systems have come into widespread use inthe communications and related industries over the lastdecade. In these systems, the electrical power-distributionnetwork has been changed from one central power supplywith cables or buses to a number of smaller power-processing units that are placed throughout the host system.The usual intent is to bring the power processing closer tothe subsystems where the power is used. The most commonway is to distribute a 48V bus across a back plane [2].Several choices exist for selecting the intermediate busvoltage from the output of the 48V converter, depending onpower level and the quantity of rails. The most common on-board distribution schemes today are 3.3, 5 and 12V.

Buck convertors and synchronous-buck convertor arestep-down switching-mode power convertors. They arepopular because of their high efficiency and compact size.They are used in place of a linear voltage regulator at arelatively high output power. Buck convertors are the mostwidely used type of power convertor in battery-poweredapplications. There is an increasing need for goodcontrollers to obtain tight output-voltage regulation underdifferent supply and load conditions. Conventional DC/DCconvertors use simple single-loop voltage-mode control, in

which the PID family of controllers uses only outputvoltage as the feedback. However, their performances arenot satisfactory under parameter variation, nonlinearity,large supply and load disturbances [3]. To improve thesituation, state-feedback control [4] was introduced butaccurate mathematical models are required. The introduc-tion of robust control [5], sliding-mode control [3] andfuzzy-logic controllers [3] solved the problem of parametervariation and these are able to perform well under impreciseinputs or high nonlinearity. However, the dynamic responsewill be sacrificed, especially in the presence of suddenincreases in supply voltage. Moreover, the knowledge ofparameter-variation range is required in order to ensurestability. A simple way to improve transient response is tointroduce saturation logic on the duty ratio, reduce theinductor value and increase the capacitance of the outputcapacitor [6]. However, the capacitor could be too big andthe capacitance value may be impractical. This approachalso causes large inductor-current ripple and results in highswitching losses. To improve the transient response, anmultiphase interleaved parallel convertor [7] was introducedbut the cost and the complexity of control are the majorpitfalls of such topology.

Recently, dual-loop control techniques have been intro-duced to solve the problem of single-loop control. Thetraditional current-mode control scheme in which there aretwo feedback loops, the inner-inductor-current loop and theoutput-voltage loop has the advantage of very good supply-voltage regulation. However, it also has the problem ofsubharmonic instability [8] when the duty ratio is over 50%.Recently, dual-loop one-cycle control [9] was proposed toavoid the current-sensing requirement. Instead of usinginductor current, the integration value of the diode voltagewas used. The load transient response can be better if V 2

control [10] is used. The voltage across the equivalent seriesresistance (ESR) of the output capacitor was used. Thedifference between V 2 control and current-mode control isthat V 2 method uses load ripple voltage while current-modecontrol uses inductor current as feedback variable. How-ever, the ESR is the critical factor that affects the convertorperformance.

In this paper, cascade control of a DC/DC buckconvertor is proposed. The highly underdamped DC/DCbuck convertor can be resolved to two first-order systemsdescribing the voltage and current dynamics. Simple

The authors are with the Department of Electrical Engineering, The HongKong Polytechnic University, Hung Hom, Kowloon, Hong Kong

E-mail: steve.tsang@polyu.edu.hk

r IEE, 2005

IEE Proceedings online no. 20045198

doi:10.1049/ip-epa:20045198

Paper first received 26th October 2004 and in revised form 6th January 2005.Originally published online: 20th April 2005

IEE Proc.-Electr. Power Appl., Vol. 152, No. 4, July 2005 827

proportional-plus-integral controllers can easily be designedfor the two first-order systems. The overall design will becomposed of two control loops with the voltage loopoutside the inner-current loop. As long as the dynamics ofthe inner-current loop are much faster than those of theouter voltage loop, cascade control can be implemented. Inthe case of cascade control, the output of the primarycontroller is used to manipulate the setpoint of thesecondary controller as if it were the final control element.Better disturbance rejection can be obtained using cascadecontrol because the overall system can be tuned to work at afaster speed and much larger gain can be used because thesystem under control has been simplified to two first-ordersystems. Experimental examples are included to demon-strate the effectiveness of the proposed algorithm. Theproposed controller can be implemented as a low-costaddon to a conventional single-loop controller.

2 Model of DC/DC buck convertor

A schematic diagram for the DC/DC buck convertor isshown in Fig. 1 and the state-space averaging modeldescribing the voltage and current dynamics is given by

L_iLðtÞ þ voðtÞ ¼ dðtÞviðtÞ

C _voðtÞ ¼ iLðtÞ voðtÞ

R

9>=>; ð1Þ

where L is the inductance, C is the capacitance, R is theloading resistance, iL(t) is the inductor current, vo(t) is theoutput voltage, vi(t) is the supply voltage and d(t) is the dutyratio, respectively. The model is reasonably accurate forlarge-signal analysis under the continuous-conductionmode. Although under the discontinuous-conduction modethe modelling error will be increased, the analysis result isstill useful and able to give the correct trend. Note that DC/DC buck convertors are highly underdamped systems. Toavoid excessive oscillations in closed-loop control underload or supply-voltage changes, either proportional con-trollers or proportional-plus-integral controllers with theproportional gain set to rather low levels are used.Derivative actions are seldom used to avoid the differentia-tion of the switching actions. When proportional controllersare used, a fast speed of response can be obtained insacrificing the steady-state errors. When proportional-plus-integral controllers are used, comparatively slower systemswill result but steady-state errors can be removed.

3 Cascade-controller design

From (1), the control of a buck convertor can bedecomposed to an outer-voltage-control loop and aninner-current-control-loop with the voltage dynamics gov-erned by

C _vo tð Þ ¼ iL tð Þ vo tð ÞR

and the current-loop dynamics governed by

L_iL tð Þ þ vo tð Þ ¼ d tð Þvi tð Þ

3.1 Voltage-control loopIf the inductor current iL(t) is taken as the control input tothe convertor, the transfer function between the outputregulated voltage and the inductor current becomes

Gv sð Þ ¼ Vo sð ÞIL sð Þ ¼

RRCsþ 1

ð2Þ

where VoðsÞ is the Laplace transform of voðtÞ, IL(s) is theLaplace transform of iL(t) and s is the Laplace variable.Consider a proportional-plus-integral (PI) controller of theform

GPI1 sð Þ ¼ K1 R1Csþ 1ð ÞR1s

ð3Þ

where K1 and R1 are coefficients. The closed-loop transferfunction becomes

HB sð Þ ¼ Vo sð ÞVr sð Þ ¼

GPl1 sð ÞGv sð Þ1þ GPIl sð ÞGv sð Þ

¼ K1R1RCsþ K1RR1RCs2 þ R1 þ K1R1RCð Þsþ K1R

ð4Þ

where VrðsÞ is the Laplace transform of the referencevoltage vrðtÞ. The system characteristic equation is given by

D sð Þ ¼ R1RCs2 þ R1 þ K1R1RCð Þsþ KlR ð5Þwith undamped natural frequency

on ¼ffiffiffiffiffiffiffiffiffiffiffiffiK1R

R1RC

r¼

ffiffiffiffiffiffiffiffiffiK

R1C

rð6Þ

and the damping ratio z governed by

2zon ¼R1 þ K1R1RC

R1RC¼ 1

RCþ K1 ð7Þ

Clearly, the undamped natural frequency on is independentof the loading resistor R. Carrying out the design based onthe nominal load with R¼R1 and the closed-loop system tobe critically damped with z¼ 1 gives

2on ¼1

R1Cþ K1 ð8Þ

and

on ¼K1

R1C

rð9Þ

Solving (8) and (9) for K1 yields

K1 ¼1

R1Cð10Þ

and the undamped natural frequency

on ¼1

R1Cð11Þ

3.1.1 Worst-case analysis: For the voltage-controlloop, the uncertainty has arisen from different loadingconditions. To assess the robustness of the control designwith (3) and (10), two extreme cases are tested. With theloading resistor approaches zero, from (7)

2zon ¼1 ð12Þimplies that the system is well overdamped with z¼N. Ifthe loading resistor approaches infinity,

2zon ¼ K1 ) z ¼ 0:5 ð13Þ

L iL

VoViRC

Fig. 1 DC/DC Buck convertor

828 IEE Proc.-Electr. Power Appl., Vol. 152, No. 4, July 2005

The two tests indicate that the system is well under controlunder different loading conditions because, if the loadcurrent is above the nominal value, the system will beoverdamped resulting in a slower response system. Even ifthe load current is well below the nominal value, thedamping ratio will not fall below 0.5.

3.2 Current-loop controlFrom the voltage-control loop, if the output command ofthe PI controller is taken as the reference inductor current, acurrent-control loop can be designed based on the currentdynamics. If the supply voltage is fixed at Vi, the currentdynamics become

L_iLðtÞ þ voðtÞ ¼ VidðtÞand a block-diagram representation of the current-loopcontrol is shown in Fig. 2 where ir(t) is the referenceinductor current generated by the primary-loop PI con-troller. Since the dynamics of the current loop are verymuch faster than these of the primary voltage loop, theoutput regulated voltage vo(t) can be regarded as a constantdisturbance. To eliminate the constant load disturbance, aPI controller is included within the control loop. If the PIcontroller takes the form

GPI2ðsÞ ¼K2ðTsþ 1Þ

sð14Þ

where K2 and T are coefficients, the output inductor currentcan be determined by superposition theorem and becomes

ILðsÞ ¼ s

Ls2 þ K2TVisþ K2ViVoðsÞ

þ K2ðTsþ 1ÞVi

Ls2 þ K2TVisþ K2ViIrðsÞ ð15Þ

where IL(s) and Ir(s) are the Laplace transform of theinductor current and the reference current, respectively. Eq.(15) indicates that, if VoðsÞ is well regulated at a fixed level,the PI controller can eliminate the disturbance contributedby the output voltage. One important requirement forcascade control is that the secondary-loop process dynamicsmust be much faster than the primary-loop processdynamics. As a rule of thumb, secondary-loop processdynamics must be at least four times as fast as primary-loopprocess dynamics. If the undamped natural frequency of thecurrent loop is set to N times faster than the voltage loopsuch that

oI ¼ Non ¼K2Vi

L

rN44 ð16Þ

The required K2 is thus given by

K2 ¼N2o2

nLVi

ð17Þ

Again, if the current loop is set to be critically damped withthe damping ratio equal to 1,

2Non ¼K2TVi

L) T ¼ 2

Nonð18Þ

Hence for the current-loop PI controller of (14) with thesettings of (17) and (18), the undamped natural frequency ofthe current loop will be N times that of the voltage loop andthe current loop is critically damped under nominal supplyvoltage Vi.

3.2.1 Robustness analysis: For the current-con-trol loop, the uncertainly has arisen from the supplyvoltage. If the supply voltage is doubled, from (16) the

undamped natural frequency becomes ð 2pÞNon and the

damping ratio of the current loop becomes 2p

. When thesupply voltage is halved, from (16) the undamped natural

frequency becomes Non= 2p

and the damping ratio of the

current loop is ð 2pÞ=2. Even if there is a 50% reduction or

100% increase in the supply voltage, the current loop is stillwell under control.

4 Experimental setup and results

An experimental DC/DC Buck converter has been builtwith L¼ 1mH, C¼ 120mF, nominal supply voltageVi¼ 50V. The nominal load current was 1A. The outputregulated voltage was set to 10V and the nominal loadingresistor R1 was 10O. From (10),

K1 ¼1

R1C¼ 833:3

and the voltage-loop PI controller of (3) becomes

GPI1ðsÞ ¼ 0:1þ 83:33

sFor the current control-loop, the current-loop dynamicswere set to 20 times faster than the voltage dynamics withN¼ 20. From (17) and (18),

K2 ¼N2o2

nLVi

¼ 5555 T ¼ 2

Non¼ 1:2 104

and the current-loop PI controller of (14) becomes

GPI2ðsÞ ¼ 0:6666þ 5555

sFigure 3 shows the block diagram of the cascade controller,and the corresponding hardware implementation of thecascade control of DC/DC Buck convertor is shown inFig. 4. The switching frequency of the convertor was set to45kHz. The controller was implemented as an add-on to asingle-loop controller using a SG3525 PWM controller. Theadd-on PI controller was implemented using a low-cost dualoperational amplifier TL072. The closed-loop character-istics of the single-loop PI controller as a DC/DC powerconvertor have been presented in [11, 12].

4.1 Experimental resultsTo compare the performance of the proposed cascadecontroller with that of a conventional single-loop PIcontroller, arrangements with changing loads and changingsupply voltage were tested. The transfer function for thesingle-loop PI controller was given by

GPIðsÞ ¼ 0:0001þ 1

swhich is implemented with the SG3525 PWM controller.The single-loop PI controller was obtained in such a waythat the closed-loop step response under nominal working

Vi1Ls

−++

−

PI controller

vo(t)

d(t)GPI2(s)

ir (t) iL(t)

Fig. 2 Current-control loop

IEE Proc.-Electr. Power Appl., Vol. 152, No. 4, July 2005 829

conditions did not have any overshoots and the closed-loopsystem was well damped even under light load conditions.The closed-loop characteristic using a single-loop PIcontroller possessed similar damping characteristic to thoseof the closed-loop characteristic using a cascade controller.This ended up with a slower response system because thecontroller gain could not be too large.

4.1.1 Load-regulation performance: The refer-ence output voltage was set to 10V and the load demandwas switched between 1A and 0.01A by changing theloading resistor with a MOSFET which was driven by asquare-wave signal generator. Fig. 5 shows the performanceof the conventional single-loop PI controller with channel 1indicating the AC-coupled DC output-regulated voltage

and channel 2 indicating the current demand of the load.The disturbance on the output-regulated voltage went ashigh as 5V and it took more than 100ms to settle down.Fig. 6 shows the performance of the cascade controller withthe same load demand. The disturbance on the output-regulated voltage decreased to 1V and it took around 10msto settle down. Clearly, the performance of the cascadecontroller was far superior to conventional single loop PIcontroller with better disturbance rejection and faster speedof response.

4.1.2 Switching supplyThe reference output voltage was set to 10V with constant-current demand of 1A. However the supply voltage was

s s

referencevoltager (t)

referencecurrent

ir (t)

dutyratiod(t)

+−

+−

0.1s + 83.33 0.6666s + 5555DC/DC

buckconverter

inductor current iL(t)

output voltage o(t)

Fig. 3 Cascade control of DC/DC Buck convertor

+−

+−

+−

10 v 100 kΩ

100 kΩ

100 kΩ

100 kΩ

20 kΩ

10 kΩ

0.5 kΩ

37.5 kΩ

3.75 kΩ

R (10 Ω)

10 kΩ

50 V

1 mHIRF 840

BY399

0.12 µF

120 µF

0.012 µF

TLO72aTLO72b

+

1 3IN OUT

SG3525 and MOSFET driver

Fig. 4 Circuit diagram of cascade controller for DC/DC Buck convertor

100ms CH1 +DC Stp 1.50 0.00V?

T1

2

1:5.0 V 2 : 50 mV f = 1.2915 Hz PWR

Fig. 5 Performance of single-loop PI controller with changingloads

10ms CH1 +DC Run 220m 0.00V?

T1

2

2 : 50 mV f = 57.237 PWR1: 500mV Hz

Fig. 6 Performance of cascade controller with changing load

830 IEE Proc.-Electr. Power Appl., Vol. 152, No. 4, July 2005

switched between 27V and 50V by changing the outputvoltage of a LM338 linear voltage regulator with a square-wave signal generator. Fig. 7 shows the performance of theconventional single-loop PI controller with channel 1indicating the AC-coupled DC output-regulated voltageand channel 2 indicating the supply voltage. The dis-turbance on the output-regulated voltage went as high as10V when the supply voltage was changed from 27V to50V and it took around 50ms for the output to settle down.Fig. 8 shows the performance of the cascade controller withthe same changing supply. The disturbance on the output-regulated voltage went down to 1V when the supply voltagewas changed from 27V to 50V and it took around 10ms tosettle down. Again, the cascade controller outperformed theconventional single-loop PI controller with better distur-bance rejection and a faster speed of response.

5 Conclusions

A new cascade controller has been implemented successfullyfor the control of a DC/DC Buck convertor. The controllersettings can easily be obtained from the Buck-convertorsettings. The proposed cascade controller outperforms aconventional single-loop PI controller with better distur-bance rejection and a faster speed of response. Thecontroller has been shown to be robust against loadchanges and supply changes. Robustness analyses of thecascade controller against load changes and supply changeshave been presented. The proposed controller can beimplemented as a low-cost add-on to a conventional single-loop controller. Detail-design equations were presented forpractising engineers.

6 Acknowledgment

The authors gratefully acknowledge the support of HongKong Polytechnic University.

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100ms CH1 +DC Stp 1.50 0.00V?

T1

2

1:5.0 V 2 : 20 V f = 1.3935 Hz PWR

Fig. 7 Performance of single-loop PI controller with changingsupply

10 ms CH1 +DC Run 1.48 0.00V?

T1

2

1:1.0 V 2 : 20 V f = 508.93 Hz PWR

Fig. 8 Performance of cascade controller with changing supply

IEE Proc.-Electr. Power Appl., Vol. 152, No. 4, July 2005 831