Battery Management Systems

Battery Management Systems Design by Modelling

H.J. Bergveld

The work described in this thesis has been carried out at Philips Research Laboratories Eindhoven as part of the Philips Research programme.

Cover design: Hennie Alblas Figures: Hennie Alblas Printed by: University Press Facilities, Eindhoven

Explanation of the cover The cover shows a transparent battery as an illustration of the use of battery models for the design of Battery Management Systems.

© Royal Philips Electronics N.V. 2001 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written consent of the copyright owner.

CIP-Gegevens Koninklijke Bibliotheek, Den Haag Bergveld, Hendrik Johannes Battery Management Systems-Design by Modelling Proefschrift Universiteit Twente, Enschede, – Met lit. opg., - Met samenvatting in het Nederlands ISBN 90-74445-51-9 Trefw.: Batteries, Secondary Cells, Power Supplies to Apparatus, Modelling, Battery Management, NiCd, Li-ion, Power Amplifiers, Cellular Phones

BATTERY MANAGEMENT SYSTEMS DESIGN BY MODELLING PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit Twente, op gezag van de rector magnificus, prof.dr. F.A. van Vught, volgens besluit van het College voor Promoties in het openbaar te verdedigen op donderdag 28 juni 2001 te 16.45 uur.

door Hendrik Johannes Bergveld geboren op 17 maart 1970 te Enschede

Dit proefschrift is goedgekeurd door de promotoren: Prof.dr.ir. P.P.L. Regtien (Universiteit Twente) Prof.dr. P.H.L. Notten (Technische Universiteit Eindhoven, Philips Research Laboratories, Eindhoven) Samenstelling promotiecommissie: Prof.dr. H. Wallinga Voorzitter Prof.dr. J.M. Tarascon Université de Picardie Jules Verne, Amiens, Frankrijk Dr.ir. M.J.M. Pelgrom Philips Research Laboratories, Eindhoven Prof.dr.ir. P. Bergveld Universiteit Twente Prof.dr. J.F.J. Engbersen Universiteit Twente Prof.dr.ir. B. Nauta Universiteit Twente

Ter nagedachtenis aan mijn broer To the memory of my brother

BONIFACIO MARINUS BERGVELD 20 August 1973

30 January 1995

Simple example of Battery Management (by Franquin © MARSU, 2001, printed with permission, see www.gastonlagaffe.com)

Aan Peggy en mijn ouders

i

Table of contents List of abbreviations

vii

List of symbols

ix

1. Introduction

1

1.1 The energy chain 1.2 Definition of a Battery Management System 1.3 Motivation of the research described in this thesis 1.4 Scope of this thesis 1.5 References

2. Battery Management Systems 2.1 A general Battery Management System 2.2 Battery Management System parts 2.2.1 The Power Module (PM) 2.2.2 The battery 2.2.3 The DC/DC converter 2.2.4 The load 2.2.5 The communication channel 2.3 Examples of Battery Management Systems 2.3.1 Introduction 2.3.2 Comparison of BMS in a low-end and high-end shaver 2.3.3 Comparison of BMS in two types of cellular phones 2.4 References

3. Basic information on batteries 3.1 Historical overview 3.2 Battery systems 3.2.1 Definitions 3.2.2 Battery design 3.2.3 Battery characteristics 3.3 General operational mechanism of batteries 3.3.1 Introduction 3.3.2 Basic thermodynamics 3.3.3 Kinetic and diffusion overpotentials 3.3.4 Double-layer capacitance

1 3 4 5 6

9 9 10 10 14 18 19 19 22 22 22 25 29

31 31 33 33 35 36 43 43 44 45 50

ii

3.3.5 Battery voltage 3.4 References

4. Battery modelling 4.1 General approach to modelling batteries 4.1.1 Chemical and electrochemical potential 4.1.2 Modelling chemical and electrochemical reactions 4.1.3 Modelling mass transport 4.1.4 Modelling thermal behaviour 4.2 A simulation model of a rechargeable NiCd battery 4.2.1 Introduction 4.2.2 The nickel reaction 4.2.3 The cadmium reactions 4.2.4 The oxygen reactions 4.2.5 Temperature dependence of the reactions 4.2.6 The model 4.3 A simulation model of a rechargeable Li-ion battery 4.3.1 Introduction 4.3.2 The LiCoO2 electrode reaction 4.3.3 The LiC6 electrode reaction 4.3.4 The electrolyte solution 4.3.5 Temperature dependence of the reactions 4.3.6 The model 4.4 Parameterization of the NiCd battery model 4.4.1 Introduction 4.4.2 Mathematical parameter optimization 4.4.3 Results and discussion 4.4.4 Quality of the parameter set presented in section 4.4.3 under different charging conditions 4.4.5 Results obtained with a modified NiCd battery model and discussion 4.5 Simulation examples 4.5.1 Simulations using the NiCd model presented in section 4.2 4.5.2 Simulations using the Li-ion model presented in section 4.3 4.6 Conclusions 4.7 References

5. Battery charging algorithms 5.1 Charging algorithms for NiCd and NiMH batteries 5.1.1 Charging modes, end-of-charge triggers and charger features

52 52

55 55 58 59 67 82 86 86 89 92 97 102 103 107 107 108 113 117 118 118 124 124 126 131 138 144 149 149 155 162 165

169 169 169

iii

5.1.2

Differences between charging algorithms for NiCd and NiMH batteries 5.1.3 Simulation example: an alternative charging algorithm for NiCd batteries 5.2 Charging algorithm for Li-ion batteries 5.2.1 The basic principle 5.2.2 The influence of charge voltage on the charging process 5.2.3 The influence of charge current on the charging process 5.2.4 Simulation example: fast charging of a Li-ion battery 5.3 Conclusions 5.4 References

6. Battery State-of-Charge indication 6.1 Possible State-of-Charge indication methods 6.1.1 Definitions 6.1.2 Direct measurements 6.1.3 Book-keeping systems 6.1.4 Adaptive systems 6.1.5 Some remarks on accuracy and reliability 6.2 Experimental tests using the bq2050 6.2.1 Operation of the bq2050 6.2.2 Set-up of the experiments 6.2.3 Results and discussion 6.2.4 Conclusions of the experiments 6.3 Direct measurements for Li-ion batteries: the EMF method 6.3.1 Introduction 6.3.2 EMF measurement methods 6.3.3 Measured and simulated EMF curves for the CGR17500 Li-ion battery 6.3.4 Conclusions 6.4 A simple mathematical model for overpotential description 6.5 Proposed set-up for State-of-Charge system 6.5.1 The algorithm 6.5.2 Comparison with the bq2050 system 6.5.3 Comparison with systems found in the literature 6.6 Experimental tests with the system proposed in section 6.5 6.6.1 Introduction 6.6.2 Set-up of the experiments 6.6.3 Experimental results 6.6.4 Discussion of the results 6.6.5 Conclusions of the experiments

175 177 184 184 186 187 188 191 192

193 193 193 195 199 202 203 204 204 206 208 211 212 212 212 214 219 219 225 225 229 230 231 231 231 232 235 237

iv

6.7 Conclusions 6.8 References

238 239

7. Optimum supply strategies for Power Amplifiers in cellular phones 241 7.1 Trends in cellular systems 7.2 The efficiency control concept 7.2.1 Basic information on Power Amplifiers 7.2.2 Optimum supply voltage for optimum efficiency 7.3 DC/DC conversion principles 7.3.1 Linear voltage regulators 7.3.2 Capacitive voltage converters 7.3.3 Inductive voltage converters 7.3.4 EMI problems involved in capacitive and inductive voltage converters 7.3.5 Inductive voltage conversion for efficiency control 7.4 Simulation model derivation 7.4.1 DC/DC down-converter 7.4.2 Power Amplifier 7.5 Theoretical benefits of efficiency control 7.5.1 Simulation set-up 7.5.2 Results and discussion 7.5.3 Conclusions 7.6 Experimental results obtained with a CDMA PA 7.6.1 Measurement set-up 7.6.2 Measurement results and discussion of part 1: no DC/DC converter 7.6.3 Measurement results and discussion of part 2: with DC/DC converter 7.6.4 Estimation of talk time increase in a complete CDMA cellular phone 7.7 Application of efficiency control in a GSM cellular phone 7.7.1 GSM power control protocol 7.7.2 Modifications in the Spark GSM phone 7.7.3 Measurement results and discussion 7.7.4 Conclusions of the experiments 7.8 Conclusions 7.9 References

8. General conclusions and recommendations 8.1 General conclusions 8.2 Recommendations 8.3 References

241 245 246 250 251 252 253 255 258 258 258 258 260 261 262 263 265 266 266 267 269 271 274 274 276 279 281 281 282

285 285 287 289

v

List of publications and patents

291

Summary

293

Samenvatting

297

Dankwoord

303

Curriculum Vitae (English)

305

Curriculum Vitae (Nederlands)

306

vi

vii

List of abbreviations ACPI ACPR ADC AM BER BIOS BMS CAC CC CDMA Cd(OH)2 CHC CV DAC DCR DCS DMC DQPSK DSP DTC e EC ECC EDGE EM EMC EMF EMI ESR FDD FDMA FM FSK GMSK GSM H+ H2O H2SO4 HVIC ID IEC IIC KOH LED LCD LR Li-ion LiCoO2

Advanced Configuration and Power Interface Adjacent Channel Power Ratio Analogue-to-Digital Converter Amplitude Modulation Bit Error Rate Basic Input Output System Battery Management System Compensated Available Charge Constant Current Code-Division Multiple Access Cadmium hydroxide Charging Control Constant Voltage Digital-to-Analogue Converter Discharge Count Register Digital Cellular System Dimethyl carbonate Differential Quadrature Phase Shift Keying Digital Signal Processor DeskTop Charger Electron Ethylene carbonate Energy Conversion Control Enhanced Data rates for GSM Evolution Electro-Magnetic Ethyl methyl carbonate Electro-Motive Force Electro-Magnetic Interference Equivalent Series Resistance Frequency Division Duplex Frequency-Division Multiple Access Frequency Modulation Frequency Shift Keying Gaussian Minimum Shift Keying Global System for Mobile communication Proton Water Sulphuric acid High-Voltage IC Identification International Electrotechnical Commission Interface IC Potassium hydroxide Light-Emitting Diode Liquid-Crystal Display Linear Regulator Lithium-ion Lithium cobalt oxide

viii LiMn2O4 LiNiO2 LiPF6 LiC6 LMD LSE MSK NAC NADC NiCd NiMH Ni(OH)2/NiOOH NTC O2 OQPSK Ox PA Pb PbO2 PCB PEO PFC PM PTC PWM QAM QPSK Red RF Rx SBC SBD SBS SEI SHE SLA SMBus SMPS SoC SoH TCH TCM TDD TDMA Tx UMTS UPS VRLA Zn-MnO2 3G

Lithium manganese oxide Lithium nickel oxide Lithium hexafluorophosphate Lithium graphite Last Measured Discharge Least-Square Error Minimum Shift Keying Nominal Available Charge North American Digital Cellular Nickel-cadmium Nickel-metalhydride Nickel hydroxide/nickel oxyhydroxide Negative-Temperature Coefficient resistor Oxygen Offset Quadrature Phase Shift Keying Oxidized species Power Amplifier Lead Lead dioxide Printed-Circuit Board Polyethylene oxide Programmed Full Count Power Module Positive-Temperature Coefficient resistor Pulse-Width Modulation Quadrature Amplitude Modulation Quadrature Phase Shift Keying Reduced species Radio-Frequency Receive Smart Battery Charger Smart Battery Data Smart Battery System Solid Electrolyte Interface Standard Hydrogen reference Electrode Sealed lead-acid System Management Bus Switched-Mode Power Supply State-of-Charge State-of-Health Traffic Channel Timer-Control Module Time Division Duplex Time-Division Multiple Access Transmit Universal Mobile Telecommunication System Uninterruptible Power Supply Valve-regulated lead-acid Zinc-manganese dioxide Third Generation

ix

List of symbols Symbol A A Abat ACdmax ai airef aib ais ci cib cis Cch Cel Cth Cdl Codl CH CG-C Cpara Caprem Capmax CFi Di di Dj dV/dtlim E Ei Eieq Eeqbat Eeq* Eio Eobat Eapar Ech Eel Eth Emax

Meaning Electrode surface area Area of spatial element Battery surface area Surface area of cadmium electrode Activity of species i Activity of species i in the reference state Bulk activity of species i Surface activity of species i Concentration of species i Bulk concentration of species i Surface concentration of species i Chemical capacitance Electrical capacitance Thermal capacitance Double-layer capacitance Double-layer capacitance per unit area Helmholtz capacitance Gouy-Chapman capacitance Parasitic capacitance in DC/DC converter Remaining battery capacity Maximum possible capacity that can be obtained from a battery Cost function for output variable i Diffusion coefficient of species i Diffusion layer thickness of species i Anti-parallel diodes that model ButlerVolmer relation for reaction j Change of battery voltage in time, used as parameter in proposed SoC indication system Error in ‘battery empty’ prediction of an SoC indication system Potential of electrode i Equilibrium potential of electrode i Equilibrium potential of battery, or EMF Apparent equilibrium potential Standard redox potential of electrode i Standard redox potential of battery Activation energy of parameter par Chemical energy Electrical energy “Thermal energy” Maximum energy stored in capacitor or coil

Value

Unit m2 m2 m2 m2 mol/m3 mol/m3 mol/m3 mol/m3 mol/m3 mol/m3 mol/m3 mol2/J F J/K F F/m2 F F F Ah Ah m2/s m V/s

% V V V V V V J/mol J J JK J

x Symbol Eq F fc fRF Fswitch fTd f bkV,T,I

Ioj Ia Ic Idl Ilim Isup Ji Jch Jth ka,j

Meaning Energy term, normalized to current, used in simple overpotential description of (Eq. 6.4) Faraday’s constant Channel frequency RF frequency Switching frequency in DC/DC converter Relation between measured battery parameter and SoC in direct-measurement SoC system Function that translates coulomb counter contents into SoC based on the basis of V, T and I measurements in a book-keeping system Exchange current for reaction j Anodic current Cathodic current Double-layer current Current level that determines state in proposed SoC indiction system Supply current Diffusion flux of species i Chemical flow Heat flow Reaction rate constant for oxidation (anodic) reaction j

kc,j

Reaction rate constant for reduction (cathodic) reaction j

kb

Backward reaction rate constant

kf

Forward reaction rate constant

KO2

Oxygen solubility constant

lNi lpos lneg

Thickness of nickel electrode, grain size Thickness of positive electrode, grain size Thickness of negative electrode, grain size Thickness of electrolyte in Li-ion model Number of electrons in reaction

lelyt m

Value

Unit J/A

96485

C/mol Hz Hz Hz %

%

A A A A A A mol/(m2.s) mol/s W Unit depends on reaction Unit depends on reaction Unit depends on reaction Unit depends on reaction mol/ (m3.Pa) m m m m -

xi Symbol m

p P Pout Psup Pch Pel Pth Par(T)

Meaning Number of spatial elements in the electrolyte in Li-ion model Number of parameter sets in optimization process Molar amount of species i Molar amount in the reference state Molar amount of cadmium nuclei at t=to Molecular weight of cadmium Number of electrons in reaction Number of capacitors in capacitive voltage converter Number of nuclei Number of points taken into account in optimization process Number of spatial elements in a system Pressure Output power of PA in cellular phone Supply power for PA Chemical power Electrical power “Thermal power” Temperature-dependent parameter

paro

Pre-exponential factor for parameter par

Q Qth QCd,Max QNi,Max QCd(OH)2,Ni

Charge Heat Maximum capacity of cadmium electrode Maximum capacity of nickel electrode Overdischarge reserve of Cd(OH)2 at the nickel electrode Overdischarge reserve of cadmium at the cadmium electrode Maximum capacity of LiCoO2 electrode Maximum capacity of LiC6 electrode Radius of hemispherical particles Gas constant Chemical resistance Electrolyte resistance Resistance that models self-discharge in Li-ion model On-resistance of bypass switch in DC/DC converter ESR of coil in DC/DC converter

M mi mref moCd MCd n n N N

QCd,Cd QmaxLiCoO2 QmaxLiC6 r(t) R Rch Re Rleak Rbypass Rcoil

Value

Unit mol mol mol kg/mol Pa dBm W W W WK Unit depends on parameter Unit depends on parameter C J C C C C

8.314

C C m J/(mol.K) Js/mol2 Ω Ω Ω Ω

xii Symbol Rloss

Meaning Ohmic-loss converter

Rswitch

Uq(I)

On-resistance of switch in DC/DC converter Electrical resistance Transformed antenna impedance seen at collector/drain of final PA stage Thermal resistance Series resistance of switch S Ohmic and kinetic resistance Diffusion resistance Time constant diffusion overpotential Time constant kinetic overpotential Switch i in DC/DC converter First SoC in equilibrium state, just after re-entry from transition state First SoC when discharge state is entered Last SoC in transition state just before equilibrium state is entered Time Remaining time of use Experimental remaining time of use from the moment a discharge current is applied until the battery is empty Switching period in DC/DC converter Temperature Ambient temperature Operating time Period time of burst drawn by PA model Interaction energy coefficient for LiCoO2 electrode in phase i Interaction energy coefficient for LiC6 electrode in phase i Inverse step function

V Vbat Vcon VEoD Verror

Voltage Battery voltage Control voltage End-of-Discharge voltage Error voltage in behavourial PA model

Vnom Vsup Vsup,opt

Nominal supply voltage for PA Supply voltage Optimum supply voltage, as applied in efficiency control Minimum PA supply voltage at which linearity specification can still be met

Rel Rload Rth RS RΩ k Rd RdCd RkCk Si SoCE SoCS SoCt t trem t1,actual T T Tamb Toper Tperiod Upos,i Uneg,i

Vsup,min

Value resistance

in

DC/DC

Unit Ω Ω Ω Ω K/W Ω Ω Ω s s % % % s s s

s K K s s 1 for I≤0 and 0 for I>0

Verror=0 or Verror=1

-

V V V V V V V V V

xiii Symbol Vswitch V(t) Vo Vg

Meaning Switch drive voltage in DC/DC converter Volume of deposited material at time t Initial volume of hemispherical particles at time to Free gas volume inside battery

Vari

Output variable in optimization process

Wi,j

Normalizing and weighing factor for output variable Vari Distance Reaction order of species in reaction j Mol fraction of species i Mol fraction of Li+ ions at which phase transition occurs in positive electrode Mol fraction of Li+ ions at which phase transition occurs in negative electrode Valence of ionic species i Transfer coefficient Heat transfer coefficient Activity coefficient Fugacity coefficient

x xj, yj, zj xi xposphasetransition xnegphasetransition zi α αth γ γg

δ δ δH δPA ∆ Go ∆Go’ ∆ GO ∆ GR ∆GO

∆G R

∆H ∆S ∆x εo εr φ φe φs

Diffusion layer thickness Duty cycle in DC/DC converter Helmholtz layer thickness Duty cycle of PA = ON/OFF ratio Gibbs free energy change under standard conditions Gibbs free energy change under standard conditions, taking constant activity terms for e.g. OH- or H2O into account Change in (Gibbs) free energy in an oxidation reaction Change in (Gibbs) free energy in a reduction reaction Change in free electrochemical energy in an oxidation reaction Change in free electrochemical energy in a reduction reaction Change in a reaction’s enthalpy Change in a reaction’s entropy Thickness of a spatial element Permittivity in free space Dielectric constant Electrostatic or Galvani potential Electrostatic electrode potential Electrostatic electrode potential

Value

Unit V m3 m3 m3

1, see note 1

Depends on Vari Depends on Vari m -

00 and Eeq,- 0

eq, –

E =E

+

+

0 +

–

+

2

1

1

2

∆Hc =

1

0

– T∆Sc +

1

Oxidation – ∆GO = nFE < 0 –

∆Gd = ∆GR + ∆GO = – nF(E – E ) = – nFEbat < 0

2

∆Gc

∆Hd < 0 ∆Sd > 0

Reduction + ∆GR = – nFE < 0

Reduction – ∆GR = – nFE > 0

∆Gc = ∆GO + ∆GR = nF(E – E ) = nFEbat > 0

0

–

>0

–

Thermal –T∆S reservoir

(a)

eq, +

E =E

(b)

∆Gd

∆Hd

– T∆Sd

–2

= –1 +

–1

Figure 4.14: (a) Charging a battery (subscripts c mean charging) with an infinitesimally small current, so that overpotentials are zero, so no losses occur and the equilibrium potential does not change. The energy ∆Gc supplied during charging can be fully retrieved (∆Gd) (subscripts d mean discharging) during discharging. Changes in free energy (∆G), enthalpy (∆H) and entropy (∆S) of the battery are shown. The charge reaction has arbitrarily been chosen to be exothermic (∆Sc0) by definition

The change in entropy in the battery during charging will be negative (∆Sc0) and a change in heat from 0 to 1 (-T∆S =1 >0), which is delivered to the thermal reservoir. The battery will be discharged when the current source is replaced by a resistor. Again, an infinitesimally small current is assumed. Now, a reduction reaction will occur at the positive electrode and an oxidation reaction at the negative electrode. Both ∆GO and ∆GR will now be negative, as indicated below the battery in Figure 4.14b in which subscripts d mean discharging. The change in the battery’s free energy during discharging ∆Gd, which equals the work performed on the environment, is again inferred from the summation of ∆GO and ∆GR, leading to

84

Chapter 4

∆Gd=-nFEbat. This means that the minus sign convention (∆G=-nFE) that is often encountered in the literature actually holds for the spontaneous reaction direction. Note that ∆Gd=-∆Gc, which means that the energy supplied during charging will be retrieved completely. This is in agreement with the assumption of zero losses. As (Eq. 4.31) is of course still valid, the discharge reaction has to be endothermic, or ∆Sd>0. This means that the battery’s entropy will increase. This can be seen at the bottom of Figure 4.14b, in which the battery free energy changes from 2 to 0 (∆Gd =-2 Vout only) (b) Time-discrete with energy-storage element

252

Chapter 7

The dissipative element in Figure 7.5a remains connected between the battery and the load. The efficiency of the voltage conversion will always be lower than 100%, because of its dissipative nature. The energy-storage element in Figure 7.5b is first connected to the battery to store energy, after which it is connected to the load to supply this energy. The efficiency of the voltage conversion process is 100% in the theoretical case in which no energy is lost in the energy-storage element and switches. The type of time-discrete voltage converter employed and the characteristics of the employed components will determine the efficiency in practice. An energy buffer Cbuf is necessary, because of the time-discrete nature of converters of this type. 7.3.1 Linear voltage regulators The configuration of Figure 7.5a is commonly known as a linear voltage regulator [8]. A more detailed schematic representation of a linear voltage regulator is shown in Figure 7.6. It consists of a transistor, which may be of any type, controlled by a regulator, which compares a fraction of Vout with a reference voltage Vref. Linear regulators vary in the used type of transistor and its drive circuit. The transistor is operated in the saturation region in the case of FETs and the linear region in the case of bipolar transistors. This means that the output current Iout, and hence Vout, will hardly change when Vin changes. However, a certain minimum voltage difference Vin-Vout, which is the dropout voltage, has to be present for proper operation. A simple resistor represents the load in Figure 7.6. Figure 7.6 illustrates that the maximum efficiency η is the ratio of Vout and Vin. A decrease from this maximum value is caused by the current consumption of the linear voltage regulator itself, including the opamp, voltage reference and current through R1 and R2. Hence, Iout is smaller than Iin. The efficiency of the linear voltage regulator will be higher in the case of smaller differences between Vin and Vout, as can be understood from the ratio. Therefore, the term Low DropOut regulator (LDO) is often encountered in practice when a low voltage drop can be accommodated, the term ‘low’ being a relative term. Iin

Vin

Linear voltage regulator Vin > Vout

Vout R1 – Vref

Load Iout

η = Pout/Pin ≤ Vout/Vin (Vin – Vout)min = dropout voltage

R2

Figure 7.6: Basic schematic representation of a linear voltage regulator

Linear voltage regulators are encountered in portable devices when the difference between the battery and load supply voltages is not too high, because the efficiency will then still be acceptable. An advantage of linear regulators is that they are cheap and small because they do not need an energy-storage element, which usually takes

Optimum supply strategies for Power Amplifiers in cellular phones

253

up quite some volume. Linear regulators are often used as filters, because variations in Vin are suppressed in Vout. Although not included in Figure 7.6, small capacitors with values specified in the data sheet are added to the input and output of a linear voltage regulator in practice. The output capacitor improves the response to load changes and usually implements regulator loop frequency compensation. 7.3.2 Capacitive voltage converters A converter as shown in Figure 7.5b has to be used when there is a large difference between the battery voltage and the voltage needed by the load, or when the load voltage should be higher than the battery voltage. The energy-storage elements can either be capacitive or inductive. The basic principle of a capacitive voltage converter is shown in Figure 7.7. Such a configuration is commonly referred to as a charge pump. The switches are operated from a non-overlapping clock signal with periods Φ1 and Φ2. The capacitors are connected in parallel to the battery for the upconverter for period Φ1 and each capacitor is charged to the battery voltage. The capacitors are connected in series for period Φ2, added to the battery voltage, and connected to the output buffer capacitor Cout. At no load, this leads to an output voltage Vout=(n+1).Vin, where n is the number of capacitors.

Vin < Vout

Vin

Φ1

Φ1

Φ2

C

Φ1

Φ2

C

Vout Φ2

Φ2

C Cout

Φ1 (a)

Φ1

1

2

n

Φ1 Vin

Load

Φ1

Vin > Vout

Φ2

Φ2

Vout Φ2

C

Φ1

C

Φ1

C Cout

Φ2

Φ2

Load

Φ2

(b)

Figure 7.7: Basic schematic representation of a capacitive voltage converter: (a) Voltage upconverter (b) Voltage down-converter

The capacitors are connected in series for the down-converter for period Φ1 and the total series connection is charged to the battery voltage. The capacitors are connected in parallel for period Φ2 and connected to the output buffer capacitor Cout. This leads to an output voltage Vout=Vin/n at zero load, with n being the number of

254

Chapter 7

capacitors. By combining parallel and series connections for periods Φ1 and Φ2, non-integer conversion factors can be realized. Efficiency considerations Consider the charging of an ideal capacitor by a voltage source V through a switch S with an on-resistance RS. This is illustrated in Figure 7.8, from which the following equation can be derived for the energy EC(t) stored in C as a function of time t: −t  −τ2t  1 2  EC (t ) = ∫ VC (t ) I (t )dt = CV ⋅  e − 2e τ + 1 2 0   t

(Eq. 7.8)

And for the energy ER(t) dissipated in RS as a function of time t: t

E R (t ) = ∫ VR (t ) I (t )dt = 0

S

−2 t   1 CV 2 ⋅ 1 − e τ  2  

+ VR (t) – RS

I(t) =

(Eq. 7.9)

V –tτ e RS –t

+

I (t) C

V –

+ VC (t) –

VR(t) = V • e τ

–t

VC(t) = V • (1 – e τ ) τ = RSC

Figure 7.8: Charging a capacitor C by a voltage source V through a switch S with an onresistance RS

EC(t) and ER(t) both become Emax= ½CV2 when t approaches infinity. This means that, irrespective of the value of RS, equal amounts of energy are stored and dissipated. This also holds when RS is zero or is not constant. The latter case occurs, when the capacitor C is charged through a transistor. Table 7.3 shows the course in time of EC(t) and ER(t). For clarity, these energies have been normalized to the maximum energy Emax. Table 7.3: Normalized energies EC(t) and ER(t) and normalized voltage VC(t)/V at normalized times t/τ t/τ 0.25 0.38 0.69 1 1.23 2.97 3.68

VC(t)/V [%] 22.4 31.6 50 63.2 70.7 94.9 97.5

EC(t)/Emax [%] 5 10 25 40 50 90 95

ER(t)/Emax [%] 39.7 53.3 75 86.5 91.4 99.7 99.9

Table 7.3 shows that storing the first 5% of Emax causes an energy loss of 39.7% of Emax in RS. However, when the same amount of energy is stored starting at 90% (90% → 95% of Emax), the energy loss is only 0.2 % of Emax. Hence, although the

Optimum supply strategies for Power Amplifiers in cellular phones

255

energy storage occurs fast at the beginning of charging, because it only takes 0.25τ to store the first 5% of Emax, it is rather inefficient. The efficiency is 5/(5+39.7) = 11.2%. The efficiency increases to 5/(5+0.2)=96.2% when VC approaches its end value of V, but it takes 0.71τ to store 5% of Emax from 90% to 95%. This means that the capacitors should be charged (almost) to the source voltage V in order to have a high efficiency and the capacitance should be high, so that the voltage drop is minimal in the discharge period. Hence, only a fraction of the stored energy is active in the transfer process. The frequency at which the energy is transferred or the amount of the energy that is transferred each cycle need to increase when the required output power of the charge pump is high. The combination of high efficiency and high power leads to very large capacitance values, because storing energy takes more time at the end of the charging process. These large capacitors cannot be integrated with the switches and control circuitry on a single IC in practice. External capacitors should be used instead. When efficiency is not a big issue and/or the load current is small, smaller capacitors can be used, which creates the possibility of integrating them. 7.3.3 Inductive voltage converters The basic principle of using an inductive voltage converter is depicted in Figure 7.9. This type of converter is generally referred to as a DC/DC converter. As with capacitive converters, up-conversion and down-conversion of the battery voltage are possible. The time-discrete nature of the two converters can be clearly recognized by the switches.

Vin < Vout

Vin > Vout

S2 Vin

L

Vout

Vin

L

S1

Vout

D S1

Cout

(a)

Load

S2

Cout

D

Load

(b)

Figure 7.9: Basic schematic representation of an inductive voltage converter: (a) Voltage upconverter (b) Voltage down-converter

Switch S1 is closed first and energy is stored in the inductor for both the up- and the down-converters. The inductor current ramps up linearly in the ideal case of zero series resistance. Switch S1 is opened at a certain moment, depending on the desired ratio of Vin and Vout and the value of the load. The current will flow through the diode D, because the inductor current cannot change instantly, and the polarity of the voltage across the inductor changes, which leads to a decrease in the energy stored in the inductor and the current will ramp down accordingly. The energy that was stored in the inductor for the period when S1 conducted will now be transferred to the output buffer capacitor Cout. Switch S2 is not strictly necessary in either case.

256

Chapter 7

When it conducts in the second phase, the efficiency will be higher than without the switch, since the voltage across diode D will be lower than without the bypass switch. Bypassing the diode by switch S2 is commonly referred to as synchronous rectification. The waveform of the current through the inductor is shown in Figure 7.10 for both the up- and the down-converter. For simplicity, it is assumed that the inductor current is never zero. This is called continuous conduction mode, as opposed to discontinuous conduction mode, in which the inductor current decreases to zero each switching cycle. The depicted waveform is valid for a high-efficiency converter, in which the voltage drop across the switches during current conduction can be neglected.

T

IL

∆I δT

∆I =

VL∆t L

(I–δ)T t

Figure 7.10: Inductor current IL for the converters of Figure 7.9 in continuous conduction mode with ideal switches

The conduction period of S1 is δ.T. The duty cycle δ is defined as the ratio of the time for which S1 conducts and the total switching time T and has a value 0