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    Embedded Electronic Systems

    draNolte,

    GesineMarwedel,2003

    Graphics:

    Alexan

    Power Issues

    Battery Management

    Processing units

    Need for efficiency (power + energy):

    Wh worr about

    Power is considered as the most important constraint in

    embedded systems[in: L. Eggermont (ed): Embedded Systems Roadmap 2002, STW]

    energy and power?

    Energy consumption by IT is the key concern of

    green computing initiatives

    (embedded computing leading the way)http://www.esa.int/images/earth,4.jpg

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    Low Power vs. Low Energy Consumption

    Minimizing the power consumption is important for

    the design of the power supply

    the design of voltage regulators

    the dimensioning of interconnect

    short term cooling

    Minimizing the energy consumption is important due to

    restricted availability of energy (mobile systems) limited battery capacities (only slowly improving)

    ver hi h costs of ener solar anels, in s ace

    cooling high costs

    limited space

    dependability

    long lifetimes, low temperatures

    Power Consumption in CMOS Digital Logic

    Dynamic power consumption

    charging and discharging capacitors

    Short circuit currents

    short circuit path between supply rails duringswitching

    Leakage

    leaking diodes and transistors

    problem even when in standby!

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    Power Consumption in CMOS Digital logic

    Dynamic power Power due to

    short-circuit Power due to

    P = ACV2 f + AIsw Vf + Ileak V

    where

    current duringtransition

    leakage current

    A = activity factor (probability of 0 1 transition)C = total chip capacitanceV = total voltage swing, usually near the power supply voltage

    f = clock frequency

    Isw = short circuit current when logic level changes

    Ileak = leakage current in diodes and transistors

    Dynamic Power Consumption

    V Supply voltage

    Trend: has been dropping

    with each successive fab

    C Total capacitance

    seen by the gates output s

    Function of wir e lengths,

    fACV2

    A - Act iv it y o f gates

    How often on average do

    wires switch?

    f clock frequency

    Trend: increasing ...

    , ...

    Reducing Dynamic Power

    1) Reducing V has quadratic effect; Limits?

    2) Lower C - shrink structures, shorten wires

    3) Reduce switching activity - Turn off unused parts or

    use design techniques to minimize number of transitions

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    Short-circuit Power Consumption

    fAVI

    Finite slope of the input signal

    causes a direct current path

    between VDD and GND for a

    short period of time during

    switching when both the

    NMOS and PMOS transistors

    Vin Vout

    CL

    Ishort

    are conducting

    Reducing Short-circuit

    1) Lower the supply voltage V

    2) Slope engineering match the rise/fall time of the input and output signals

    Leakage Power

    leakVI

    Sub-threshold current grows exponentially with

    increases in temperature and decreases in Vt

    Sub-threshold

    current

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    How can we reduce

    power consumption?Dynamic power consumption Reduce the rate of charge/discharge of highly loaded nodes Reduce spurious switching (glitches) Reduce switching in idle states (clock gating) Decrease frequency Decrease voltage (and frequency)

    Static power Consumption Smaller area (!) Reduce device leakage through power gating Use higher-threshold transistors when possible

    Power performance tradeoffs!

    Why not simply lower V?

    Total P can be minimized by lower V

    lower V are a natural result of smaller feature sizes

    But transistor speeds decrease dramatically as V is

    reduced to close to threshold voltage

    performance goals may not be met

    td = CV / k(V-Vt) where is between 1-2

    Why not lower this threshold voltage?

    makes noise margin and Ileak worse!

    Need to do smarter voltage scaling!

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    Approaches to Energy Efficiency

    P = C V2 f

    Continuous Event-Driven

    Latency is ImportantOnly Throughput is

    Important

    Reduce V

    (Burst throughput)

    Increase h/w andMake f low or 0

    Shutdown when

    e.g., Speech CodingVideo Compression

    e.g., X Display Servera gor m c concurrency

    Disk I/OReduce CEnergy efficient s /w CommunicationSystem partition ing

    nac ve

    Efficient Circuits & Layouts

    Speed vs. Voltage

    7.0

    3.0

    5.0Normalized

    Delay

    1.0 1.5 2.0 2.5 3.0

    Supply Voltage, V

    1.0

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    Reducing the Supply Voltage: an Architectural

    Approach

    Operate at reduced voltage at lower speed

    Use architecture optimization to compensate for slower

    e.g. concurrency, pipelining via compiler techniques

    Architecture bottlenecks limit voltage reduction

    degradation of speed-up

    interconnect overheads

    Trade-off AREA for lower POWER

    Fundamentals of dynamic voltage scaling (DVS)

    Power consumption of CMOS

    circuits (ignoring leakage):

    2

    Delay for CMOS circuits:

    fre uencclock:

    voltagesupply:

    ecapacitancload:

    activityswitching:

    f

    V

    C

    dd

    L

    ddL

    )than(

    voltagethreshhold:

    with2

    ddt

    t

    tdd

    ddL

    VV

    V

    VVCk

    Decreasing Vdd reduces P quadratically,

    while the run-time of algorithms is only linearly increased

    E=P x t decreases linearly

    (ignoring the effects of the memory system and Vt)

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    Voltage scaling: Example

    Vdd[Courtesy, Yasuura, 2000]

    Tframe Tframe

    Fixed Supply Variable Supply

    Varying the Supply Voltage

    Active Idle

    Efixed = 1/2 CVdd2

    Active

    Evar = 1/2 C(Vdd/2)2 = 1/4Efixed

    0.6

    0.8

    1.0

    Normalized Fixed Supply

    0 0.2 0.4 0.6 0.8 1.00

    0.2

    0.4

    Normalized Workload

    Variable Supply

    from [Gutnik96](VLSI Symposium)

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    Low-power Software

    Wireless industry Constantly evolving standards

    Systems have to be flexible and adaptable

    Significant portion of system functionality is implemented as

    software running on a programmable processor

    Software drives the underlying hardware Hence, it can significantly impact system power consumption

    software design.

    Low-power Software Strategies

    Code running on CPU

    Code o timizations for low ower

    Code accessing memory objects

    SW optimizations for memory

    Data flowing on the buses

    I/O coding for low power

    CPU

    Cache

    Compiler controlled power management

    Memory

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    Dynamic Power Optimization

    Two main steps:

    Workload allocation to processing elements: task mapping and

    schedulin

    Af ter workload allocation, resource of p rocessing elements

    should be adapted to the required performance to minimize

    energy consumption

    shut-down voltage scaling

    Shut-Down

    When the system is idle the processor can be placed into a

    -

    reactivity power level-core clock gating (waked-up by timer interrupt)

    - core power gating (waked-up by on-chip peripherals)

    no need for context restore

    -chip power gating (waked-up by external, on board

    interrupts)

    nee or con ex res ore

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    Shutdown for Energy Saving

    Blocked

    Active

    Subsystems may have small duty factors

    CPU, disk, wireless interface are often idle

    Tblock Tactive ideal improvement = 1 + Tblock/Tactive

    uge erence e ween on o power Some Low-Power CPUs: StrongARM 400mW (active)/ 50 mW (idle) / 0.16 mW (sleep)

    2.5 Hard Disk: 1.35W (idle spinning) / 0.4W (standby) / 0.2W (sleep) / 4.7W (start-up)

    Example: SA-1100 CPU

    400 mW

    IDLE

    CPU stopped when not inuse

    Monitoring for interrupts

    RUN

    10 s 10 s

    90 s

    160 ms

    SLEEP

    Shutdown on-chip activity IDLE SLEEP

    50 mW 0.16 mW

    90 s

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    When is DPM useful?

    Ptr Ttr

    If Ttr=0, Ptr=0 then DPM policy is trivial Stop a component when it is not needed

    Off On

    Ptr Ttr

    If, as is usual, Ttr!=0, Ptr!=0 shutdown only when idleness is going to be long enough to

    make it worthwhile

    Complex decision if the time spent in state is not deterministic

    Breakeven Point

    Breakeven point: minimum idle time that would make it

    worthwhile to shutdown

    DPM worthwhile when TBE < Average Tidle

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    Problems in Shutdown

    Cost of restarting: latency vs. power trade-off

    increase in latency (response time) e. . time to save restore CPU state, s in u disk

    increase in power consumption e.g. higher start-up current in disks

    When to Shutdown

    Optimal vs. Idle Time Threshold vs. Predictive

    When to Wakeup

    Optimal vs. On-demand vs. Predictive

    Cross-over point for shutdown to be effective

    Frequency/Voltage Scaling

    DFVS:

    Frequency must be scaled with voltage to keep circuit

    functionality

    Dynamic power goes with the square of Vdd andlinearly with clock speed

    Scaling V and F by a factor of s -> power scales as s3

    fVCP ddeff 2

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    Shutdown vs. Variable Voltage

    Example: task with 100ms deadline, requires 50ms CPU

    time at full s eed

    When Voltage Reduction is Better

    normal system gives 50ms computation, 50ms idle/stopped

    time

    half speed/voltage system gives 100ms computation, 0ms

    idle

    same number of CPU cycles but 1/4 energy reduction

    T1 T2 T1 T2

    Speed

    Time

    Idle

    Same work,

    lower energyTask

    Task

    Problem with Voltage Reduction

    Voltage gets dictated by the tightest (critical) timing constraint

    not a problem if latency not important ,

    parallelism etc.

    but, real systems have bursty throughput and latencycritical tasks

    Solution: dynamically vary the voltage!

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    Barriers to Future Voltage Scaling

    Voltage scaling requires threshold voltage Vt to be scaled as well (15%

    per generation)

    this increases sub-threshold leakage current

    impact on power consumption and circuit robustness

    scaling V and Vt poses serious challenges to special circuits such as

    domino logic, sense amplifiers

    Leakage power

    total leakage current goes up 7.5x per generation

    leakage power power by 5x

    active power remains constant for constant die size

    leakage power, and therefore total power, can be substantially

    reduced by cooling Essential to control die temperature

    power density (W/cm2): 0.6 micron chips surpassed a hot plate!

    Energy Use scaling while it

    Scaling vs. Shutdown

    Region of

    scaling

    Region of

    shutdown

    Emin

    If more time is

    allowed, scale

    down to the

    minimum energy

    point and

    subsequently use

    shutdown

    Power

    time

    allowed time

    Execution time = t*

    Power

    time

    allowed time

    Execution time = t*

    Power

    time

    allowed timeExecution time

    t

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    Energy in Radio

    x: ece ver

    ChannelIncoming

    informationOutgoing

    information

    Tx

    elecE Rx

    elecERFE

    Transmit ReceivePower

    Wireless communication subsystem consists of threecomponents with substantially different characteristics

    Their relative importance depends on the transmissionpowerof the radio

    Tx Electronics Rx Electronics

    txElecstarttx PEP rxElecstartrx PEP

    Path Loss

    Prcvd P1m dn

    Communication Energy Model

    P P 2Digital Processing

    ProtocolMAC

    Link

    Radio

    Tx

    ProtocolMAC

    Link

    Radio

    Rx

    Power Amp Efficiency

    amp amp amp out bit bit DD

    VDDIleak(VDD,VTH) Tbit(VDD,VTH)

    CbitIleakTbitVDDVTH

    Estartn

    P1mPoutPrxElecPtxElec

    Switched capacitance per bit

    Leakage current

    Processing time per bit

    Supply Voltage

    Transistor Threshold Voltage

    Startup (turn-on) energy

    Path loss exponent

    Attenuation over one meter

    Output power from power amp

    Receiver static power

    Transmitter static power

    Min et. al., Mobicom 2002 (Poster)

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    Energy Consumption of Transmit ting a Packet

    overheadbitselectronictransmitpacket EHLTPPE

    Power consumed by the power amplifier, depends on the required

    performance and the wireless channel (distance, fading, etc.)P

    transmit

    Power consumed by the electronic circuitry for filtering,

    upconverting, modulation, frequency synthesis, etc.P

    electronics

    Energy consumption that is independent of the packet size andE

    modulation scheme (startup cost, fixed encoded header, etc.)over ea

    Time to transmit one bit (depends on modulation and symbol rate Rs)Tbit

    Size of packet payloadL

    Size of packet headerH

    Energy Consumption of Transmitting a Bit

    Minimize header

    size

    Minimize

    overhead

    : Optimize

    modulation

    Independent of

    bitE

    LL

    TPPL

    overheadbitselectronictransmit

    packet 1

    modulationModulation

    Scaling

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    Energy per Bit

    JEbit

    Region of modulation scalingPtransmit

    4-QAM 9 mW

    electronics

    Rs 1 Mbaud

    Beyond Energy Efficiency: Battery-aware Design

    Theoretical capacity of battery is decided by the amount of the activematerial in the cell

    batteries often modeled as buckets of constant energye.g. halving the power by halving the clock frequency is assumedto double the computation time while maintaining constant

    computation per battery life

    In reality, delivered or nominal capacity depends on how the battery isdischarged

    sc arge ra e oa curren

    discharge profile and duty cycle

    operating voltage and power level drained

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    Problem Formulation

    Due to battery properties, the waysat which the batteries are

    time

    Total Service, S

    U discharged have impact on theactual lifetime

    Static Approach: Fix U(t), find theoptimal U that maximize S

    T

    t

    APP dttUS0

    )(

    Dynamic Approach: Heuristicapproach to find optimal U(t)

    Power and energy are related to each other

    dtPE

    EE'

    t

    In many cases, faster execution also means less energy, but

    the opposite may be true if power has to be increased to allow

    faster execution.

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    Battery Characteristics

    Important characteristics: energy density (Wh/liter) and specific energy (Wh/kg)

    ower densit W/liter and s ecific ower W/k

    open-circuit voltage, operating voltage

    cut-off voltage (at which considered discharged)

    shelf life (leakage)

    cycle life

    The above are decided by system chemistry

    significant changes in older systems carbon-zinc, alkaline manganese, NiCd, lead-acid

    new systems primary and secondary (rechargeable) Li

    secondary zinc-air, Ni-metal hydride

    Modeling the Battery Behavior

    Theoretical capacity of battery is decided by the amount of the

    active material in the cell

    e.g. halving the power by halving the clock frequency is

    assumed to double the computation time while maintaining

    constant computation per battery life

    In reality, delivered or nominal capacity depends on how the

    battery is discharged

    sc arge ra e oa curren

    discharge profile and duty cycle

    operating voltage and power level drained

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    Battery Model: Physical Picture

    Battery Capacity

    100

    Current in C rating:

    load current normalized to batterys capacity

    e.g. a discharge current of 1C for a capacity of 500 mAh is 500 mA

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    Battery Capacity vs. Discharge Current

    Amount of energy delivered is decreased as the current (rate

    at which power is drawn) is increased

    at a specific rate to a specific cut-off voltage primary cells rated at a current which is 1/100th of the

    capacity in ampere hours (C/100)

    secondary cells are rated at C/20 or C/10

    At high currents, the diffusion process that moves new active

    concentration of active material at cathode drops to

    zero, and cell voltage goes down below cut-off

    even though active material in cell is not exhausted!

    Battery Capacity vs. Discharge Current

    Peukerts Formula

    Peukerts Law expresses the capacity of a battery

    in terms of it discharge.

    CP = Ikt

    CP is the capacity according to Peukert (A.h).I discharge current.

    k Peukert constant .

    t time of discharge.

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    Ragone Plots (log-log plot )

    Specific Power

    W/kg

    Specific Energy

    Wh/kg

    Al ternate Equivalent View of the Battery

    Manufacturers often give battery efficiency (%) vs. discharge rate

    (or discharge current ratio)

    dischar e rate = I /I

    Battery cannot respond to instantaneous changes in current so, a

    time constant used to calculate Iave

    Given actual energy drawn by the circuit, one can use the battery

    efficiency to calculate the actual depletion in the stored energy in

    Example:

    battery efficiency is 60% and its rated capacity is 100 mAh @ 1V

    computed average DC-DC current of 300 mA would drain the

    battery in 12 min, while at 100% efficiency it would last 1 hr

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    Modeling Battery Efficiency

    rated

    aveI

    I

    IR

    cycle

    batT

    N

    batN

    c cleII1

    cyclebatN 0

    cyclebatavebatbat TVIE )1(

    from [Simunic01]

    Capacity & Variable Discharge Current

    Constant vs. Pulsed

    Capacity can be extended by draining power in short discharge

    also works with constant background current

    Battery relaxes and partially recovers the active material lost

    during the current impulse

    longer the rest period, the better is the recovery

    longer rest period needed as the discharge depth becomes greater

    battery voltage also goes back up

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    Benefits of Pulsed Discharge

    Higher specific power for a given specific energy

    impulses of several times the limiting current value

    rest periods

    Higher specific energy for a given specific power

    ideally, want specific energy = theoretical capacity

    depends on pulse and rest periods

    Exploiting Pulse Discharge

    Gain in battery life if system shutdown is done taking into

    account the pulse discharge

    Examples:

    protocols in case of radios where power during

    transmission is a lot higher than during receive andidle periods

    shutdown of CPUs and variable speed CPUs

    shutdown of disks

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    Rate Capacity Effect

    Capacity depends on the discharge rate

    Summary of Battery

    Non-Idealities

    Relaxation Effect

    When discharge current is either cutoff or reduced, the batterys

    capacity is recovered

    2.66

    2.68

    2.7

    2.72

    V)

    0 100 200 300 400 500 600

    time (sec)

    2.56

    2.58

    2.6

    2.62

    2.64

    Voltag

    e( .

    1.9 mA

    Battery Modeling

    Predict battery lifetime given a load profile

    Model electrochemical processes in the battery Solve system of PDEs, e.g. Berkeleys DUALFOIL Accurate but long simulation times and large number of

    parameters (e.g. > 50 with DUALFOIL) Not easy to use in an optimization tool

    Abstract representation of batteries E.g. Markov chain

    Not easy to use in an optimization tool

    Analytical models Capture key factors of battery performance E.g model by Rakhmatov & Vrudhula @ U. Arizona

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    Lithium Battery Charging

    Example of Rechargeable Battery management

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    Example of Rechargeable Battery management

    Example of Non-Rechargeable Battery management

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    Battery Charging Method

    Trickle Charging -dT/dt cutoff

    -V cutoff Time controlled charging

    Al ternatives to Bat teries?

    Small batteries are the only choice for consumer productsup to 20W

    But heavy expensive

    expire without warning require replacement (disposal problem) or recharging

    (time problem)

    Are there alternatives?

    YES, we canharvest the energy!!!!

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    How to Select and Use Power

    Supplies and dc/dc Converters for

    Your Applications

    Introduction to Power Supplies and dc/dc

    Converters

    Available/Raw Power Sources

    Un-regulated (changes with load, prime source, etc.)

    Voltage (different level, polarity, isolation)

    Non-protected (against over load, fault, temp., etc.)

    Load Demand

    Regulated (against load, prime source, etc.)

    Voltage (different level, polarity, isolation)

    Protected (against over load, fault, temp., etc.)

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    Desired power out

    Introduction to Power Supplies and dc/dc

    Converters

    Power & Electronic

    Circuits

    Raw power in(V, I, P, F)

    To loads:

    Electronic ckts

    Motor

    Computer

    Battery

    Fuel Cell

    AC Out let

    Solar

    Power Supply qu pmen

    Control

    Power Suppl ies and dc/dc Converters

    Types & Technologies

    AC-DC Power Supply (or AC Adapter)

    Change ac power into regulated dc power, e.g., a typical AC Adapter takes 120

    V ac input and converter it to regulated 5 Vdc.

    DC-DC Converters

    Change dc at one voltage potential to a dc at a different voltage potential

    DC-AC Power Supply (for example, UPS, 12Vdc-

    120Vac ada ter

    AC-AC Power Supply/Regulator (for example, line

    regulator)

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    Linear Regulators

    + Simple, inexpensive

    - Vout < Vin

    - Poor Efficiencye.g., if Vin = 6V, Vout = 3V, Efficiency = %?

    - Can be physically large

    Switching Regulators

    Also known as DC-DC Converters or

    Switchmode Regulators

    + Wide range of input voltages

    + Multiple output voltages possible+ High Efficiency (sometimes >90%!)

    + Compact

    - ,

    - Electrically noisy (not pure DC)

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    Switching Regulators

    A few types

    Buck (step-down)

    oost step-up

    Flyback (supports multiple outputs; transformer needed)

    Buck-boost (step-up or down)

    SEPIC, Zeta, Cuk (specialized buck-boost)

    Charge Pump (no inductor)

    Switching Regulators

    Buck Converter

    Continuos mode Discontinuos Mode

    D = Duty Cycle

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    Switching Regulators

    Boost Converter

    on nuos mo e scon nuos o e

    D = Duty Cycle

    Switching Regulators

    Buck Boost Converter

    D = Duty Cycle

    Continuos mode Discontinuos Mode

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    A.K.A. charge pump or flying capacitor

    Switching Regulators

    Switched Capacitor

    n uc or ess ow cos

    Work well for loads up to ~100mA

    Produce output voltage equal to Vin or 2 x Vin

    Output noisy and poorly regulated,

    but OK for certain applications

    AC-DC Power Suppl ies

    Circuit Selection and Design

    Using Linear Regulators

    Step-downXfmer

    Regulator120 VAC

    For low power (some watts or below) applications.

    Low effic iency, large size and weight

    (bulky step-down line transformer)

    Low cost

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    AC-DC Power Suppl ies

    Circuit Selection and Design

    Using Switching-Mode

    High efficiency

    Small size and light weight

    For high power (density) applications

    Selecting the Right dc/dc Converter

    The Need for dc/dc Converters

    E.g., a single AA alkaline battery produces 1.5 V when fully charged and its voltage

    drops to as low as 0.9 V when becoming depleted.

    Dc/dc Converter Types

    Buck

    Boost

    Buck-Boost

    Dc/dc Converter Technologies

    Linear Regulators

    Switching Regulators

    Charge Pumps

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    Selecting the Right dc/dc Converter

    Dc/dc converter technology comparison

    Linear Switchingregulator regulator

    Efficiency Low High Medium

    EMI Noise Low High Medium

    Output current Low to medium Low to High Low

    Boost (step-up) No Yes Yes

    Buck (step-down)

    Yes Yes Yes

    Solution size small Large Medium

    Selecting the Right dc/dc Converter(example mobile phone)

    VBAT = 3.7 V nom, Vcircuit= 1.2 V

    Load Current = 600 mA

    Power delivered to load = 600 mA * 1.2 V = 720 mW

    Power converted to heat =

    Linear regulators:

    Inexpensive

    m . . - = , mTotal power consumed =

    720 mW + 1,500 mW = 2,200 mW

    32% goes to work , 68% goes to heating user hand and earwhen using a Linear Regulator for a mobi le device

    VBAT = 3.7 V nom; Vci rcui t = 1.2 V

    Load Current = 600 mA

    low part count

    low noise

    high ripple rejection

    Switching regulators:Converter efficiency = 90%

    Power delivered to load = 600 mA * 1.2 V = 720 mW

    Power conver ted to heat =720 mW * ((1/0.9)-1)=80 mW

    Total power consumed = 720 mW + 80 mW = 800 mW

    90% goes to work , 10% goes to heating user hand and ear

    When using a Switch-mode regulator for a mobile device.

    a bigger footprint

    higher part count,

    more cost

    prone to conducted and

    radiated EMI.

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    Specs, Performance and Protection

    Voltage ripple (+-50 mV, or 5%)

    Isolation e. . 1 500 V ac for 1 min.

    Load regulation (e.g., 3%)

    Dynamic response (transients, wake-up time, etc.)

    Short circuit protection

    OC protection (Overcharge)

    OV protection (Overvoltage)

    OT protection (Overtemperature)

    Power Losses and Thermal Design

    For example, a 7815 linear regulator with input voltage of 20 V and

    output current of 1 A.

    The power loss is (20-15)Vx(1 A)=5 W.

    From the chip to the ambient, Ti can be calculated according to thethermal circuit using Ohms law (R=V/I), where R is the thermal

    resistance, V is the temperature and I is the power dissipation.

    case ambientThcase ambient

    T TR

    Where:

    Tcase is case Temp.Tambient is ambient Temp.

    issipation

    outdissipation in out out

    op

    PP P P P

    Pdissipation is power loss

    Pin is input power

    Pout is output powerop is efficiency undergiven operating conditi ons

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    Power Losses and Thermal Design

    A more detailed thermal circuit

    W : Device power loss

    Tj : Junction temperature of device

    Tc : Device case temperature

    Tf : Temperature of heatsink

    Ta : Ambient temperature

    Rth(j-c) : Thermal resistance between junction and case,

    specified in datasheet

    Rth(c-f) : Contact thermal resistance between case and

    ea s n , spec e n a as ee

    Rth(f-a) : Thermal resistance between heatsink and

    ambient air, specified by the heatsink manufacturer

    Power Losses and Thermal Design

    = - +

    Tc=W{Rth(c-f) + Rth(f-a)}+Ta

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    Examples

    Device : 7815 (Linear regulator)

    Vin=20V, Vo=15V, Io=1A

    - Rth(j-c) : 5 C/W

    Rth(c-f) : 0.5 C/W, Greased surface

    Rth(f-a) :20 C/W

    Ta=25 CAn assortment of 78XX ser ies

    Tc=5(0.5 + 20)+25=127.5 C

    Tj=51+127.5=132.5 C

    Tj=82.5-25=107.5CAn assortment of heatsinks