maximum power without calculus

19

Upload: djpsychoscientz

Post on 09-Sep-2015

7 views

Category:

Documents


1 download

DESCRIPTION

Maximum power derivation tutorial. Uses in electrical engineering.

TRANSCRIPT

  • !

    "#$%&'()*(*+-./01'20#$#324.5'607*7

    8#9./:/'$+3./:4.#/.7

    ;?&'/29/0@42A84?-?

    422BCDDE1)'/29/0@42?@##@(.B'@.+?)#7D

    F)4##(#3F)0.$).+'$G:.)4$#(#@^BC@AFOHMFC@MPB_A@SHMJPML

    JPZIED@B@BCKBOB@FL@H@ABCRDMFKZMFCBC@BNFGPCFX[AFCFGH?LJPZ\]F@AHL>>^SHKKHJCP

    RMHGFLDMF@AP@GP?EF_F?FMPKB`FL@HPKKHJ@AFLFMBNP@BH?HS@AF@AFHMFOSHMaUGBMGDB@CI

    PCLF@PBKFLB?bFG@BH?>>>cP?LbFG@BH?>>>UXaKCHI]F@AHL>>DCFCPdFZMFCDK@@AP@BC

    LFNFKHRFLB?]F@AHL>X

    ]F@AHL>

    eF@

    V

    W fWg hCHISMHO\i^I

    j k

    i

    i

    X

    V

    fW

    l m

    fW W

    g

    n

    \!^

    oHJFNFMIEZSPG@HMB?_HD@WB?@AFLF?HOB?P@HMI\!^GP?EFMFJMB@@F?PC

    j k

    i

    i

    X

    p

    V

    f m

    l

    W

    f

    g

    n

    \q^

    UKFPMKZ@AF?I@HSB?L@AFOPQBODONPKDFHS\q^IJFODC@SB?L@AFOPQBODONPKDFHS

    j k

    i

    p

    p

    X

    i

    p

    f

    f f

    f

    r

    g g

    n n

    n

    \s^

    tFFRB?OB?L@AP@fBCRHCB@BNFXUKFPMKZ@AF?I@HOPQBOB`F\s^IJFODC@OB?BOB`F

    p

    f f

    r

    n X

    uHM@D?P@FKZIB@BCC@MPB_A@SHMJPML@HSB?L@AFOB?BODONPKDFHS

    p

    f f

    r

    n JB@AHD@GPKGDKDCX

    >?SPG@I

    p

    if f

    r

    n v X[HRMHNF@ABCJFDCF@AFB?FwDPKB@Z

    j k

    i

    p xfr v XoF?GFI

    i

    i p xf fr nv HM

    p

    i xf f

    r

    r n v XyFPMMP?_B?_@AFKP@@FMB?FwDPKB@Z_BNFC@AFLFCBMFLMFCDK@I

    BXFX

    p

    if f

    r

    n v X

    [AFMFSHMFI pfgOB?BOB`FC

    p

    f f

    r

    n JABGA@AF?OPQBOB`FC\q^XoF?GFI I

    V

    W Wg PCMFwDBMFLX

    aLLB@BH?PKKZICDEC@B@D@B?_ pfgB?@H\s^_BNFC@AFOPQBODORHJFM@MP?CSFMMFLPC

    j k

    i i

    OPQ

    i

    p

    X

    q

    pp

    V

    m m

    l

    W W

    g g

    n

    \z^

  • !

    "#$%%&'()*(+(,$'-.+&'+/01(#2%#((,+*3'#)3+42#.+)*'*5&67($#6#5(,+%#*4

    )+'$'(*67+8",$'2%#($'9$:+6$6;$980-.,+)+.+,*:+*''&? @*64(,+*22%$+4:#%(*9+$'AB=C8D%+*)%E-(,+

  • !

    "#

    $%&'())*+,%-.*&',-/0*1*2(%13*456%-(%2&'*-*%,.(%6&,7,89:;&,2*&/

    :

    : :

    :

    :

    :

    /

    ?5

    "

    #

    %

    (

    (

    * +

    ,

    - .

    / 0

    9@C>DC@>44E56DC>5F5B@>:C

    G$() H5AD59I;7L:6AD5;I:C5

    L;J7D5LC;J4BE5?57;9>$5$@5@6A:6A>B564F;4:6M5L;J7D5$

    N5::>AM G() M>F5LJL:C544JL:76:5:C>L9@5JL51##2:;K4;::C56EL;7E5BK;@576L6IJAD:>;A;I:C5>A:57A64

    75L>L:6AD5$OC>LK4;:>LM>F5A>AP>M$Q9@C575@5C6F56LLJL:6AD5>L

    #() R6AB:C56KK4>5BF;4:6M5>LST#U$34567489:C5A6M755:C1#"26AB;DDJ7L@C5A G

    %

    () R96L>:LC;J4B$

    W WXY Z ZXY [ [XY \ \XY ] ]XY Y

    W

    WXZ

    WX[

    WX\

    WX]

    WXY

    WX^

    WX_

    WX`

    WXa

    Z

    bcdefcghfejkjdgclemn

    R

    o

    pqref

    gsjqfset

    su

    dve

    hqgt

    nwo

    P>M$Q$x;@576EL;7E5B6L6IJAD:>;A;I:C5>A:57A6475L>L:6AD596LLJAM:C6::C54;6B

    75L>L:6AD5>L#R6AB:C56KK4>5BF;4:6M5>LST#U$

    yz{{|}~}||~}}z}|}|

    z}z}}{|{z{}

    }|}|z}}{}|z|{|{z{}

    }|}}|~

  • !

    """#%&'()*+(,-,.+/&0*1(2324,5&'6'*-7.&'6/&,'&2.,'899(':3(+7

    ;?@@AB=?CAJKL?JMJNCEAH=HKACHAJOCHPQB?HBM?=@D

    E?=>CM==>AM@ACOBKFBMFM@OCH=>HAABK@A@R

    S?TE>ACC@A=>ABCJNFA=AFCKG?JNAGK=>AFCKGHA@?@=CH

    KAFCKGHAKB=KAKIA

  • !

    "#%&'()*)+,-./0/&123./04.,/.)-(546,7.25/(85(,1,154.9,&:;)?@ABCDE?FGEAHIAIG>??JK>JII@G?E@JLMN?@AOGMLAJB

    P

    QRACBIAIG>??@AMJGN

    CSTANG>FA?JGREJDRSGUCSLSTJIADBDJS?@AEJLDFAV=>?@CEFGEAHIAGDABDAA?JF@JJEA

    RJ?@?@AMJGNDAECE?G>FAG>NMJGNDAGF?G>FAV

    WMAGDMXH?JSGUCSCYAZ[\]H

    ^ _

    `

    P

    a ab E@JLMNRASC>CSCYANVc@CEIGEGMEJDAFJd>CYANC>

    efHTdVg[hiG>NejHTdVkh`iVlA>FAHIASGKA?@AMJGNDAGF?G>FA?@A>AdG?COAJB?@A

    C>?AD>GMZc@AOA>C>]DAGF?G>FAHCVAV V

    P

    a amn oC?@?@CEF@JCFAHZ[\]RAFJSAEZ`]HI@CF@

    IGESGUCSCYANIC?@ V

    P

    p pm c@ADABJDAHC?CEFMAGD?@G??@AMJGNCSTANG>FAE@JLMNRA

    AqLGM?J?@AFJ>rLdG?AJB?@Ac@AOA>C>CSTANG>FAJB?@AEJLDFAHGECEIAMMsK>JI>BDJS

    ?@AFGMFLMLENADCOG?CJ>V

    tLD?@ADSJDAH?@ASGUCSLSTJIAD?DG>EBADDANCEGMEJdCOA>RXZj]HI@ADAuCE>JI?@A

    DJJ?sSAG>sEqLGDAZvw]OGMLAJB?@AEJLDFAV

    cJCMMLE?DG?A?@CEHIALEAZ[\]?JTMJ??@AGREJDRANTJIADGEGBL>F?CJ>JB?@AMJGN

    DAECE?G>FAG>N?@AMJGNDAGF?G>FAVc@CETMJ?CEdCOA>GEGFJ>?JLDdDGT@C>tCdVfHI@ADA

    IA@GOAGEELSAN?@G??@AC>?AD>GMCSTANG>FAJB?@AEJLDFACE [ [Q xmb yG>N?@A

    GTTMCANOJM?GdACEz [{m| }VWMAGDMXH?@ASGUCSLSTJIADGREJDRANCE{V`foC>

    GdDAASA>?IC?@Zj]G>NJFFLDEI@A> [ [

    P

    Q xmn yHGEC?E@JLMNV

    ~*))&/04.,&:()

  • !"

    #

    $

    #

    %

    #

    $

    #

    %

    #

    $

    #

    %

    #

    $

    #

    %

    #

    $

    #

    %

    #

    $

    #

    %

    #

    $

    &

    #

    $

    &

    #

    $

    &

    #

    $

    &

    #

    $

    &

    #

    $

    &

    #

    $

    &

    #

    $

    &

    %

    #

    $

    &

    %

    #

    $

    &

    %

    #

    $

    &

    %

    #$'

    #$'

    ()*+-./0/1*23.-

    (

    4

    5

    6

    ()*+

    -.*31*23.

    7

    (

    4

    5

    6

    8#$'%

    # & ' 9 : % ; < =

    >=

    >;

    >:

    >'

    #

    '

    :

    ;

    =

    ?@ABCBDEFGEHIAIJKLEMGLNKEONIJPQEIPNRPSGLNTEJRJQGLNTEJRINJUGJFUNJFRTEJR

    INQ@QGJFUNJINVJI@NRBWLN@FGNIFJT@XKNRJFUNEMGLNQEHIUN@Q ! !Y Z[\ 5JFR

    ] !"[^ _BWLNFHXPNIQEFGLNUEFGEHIT@FNQJINKEONITNVNTQBDTNJITS`GLNXJa@XHX

    JPQEIPNRKEONI@Q"BbCcJFREUUHIQOLNF ! !

    d

    Y Z[e 5B

    fgijklmnmopqrstsjuvwrstxrpsrmqxruypjz{rj|}ju|r|juup}~rjslrz~n}

    }xrypjz{rvlv}ju|r|ju~rxpvru

    FGL@QQNUEFRUJQN`ONJINMINNGEULEEQNGLNEKG@XHXVJTHNEM `

    d

    PHGGLNVJTHNEM

    d

    UJFFEGPNULJFANRBWEM@FRGLNEKG@XHXVJTHNEM `

    d

    ONOI@GN!JQ

    b

    b b

    b

    b

    b b

    B

    b

    d

    d

    d d

    d

    d d d

    [

    \ \ \

    [

    \ \ \ \

    !C

    @V@R@FAGLNFHXNIJGEIJFRRNFEX@FJGEIEM!CPS

    b

    b

    d

    \ \ A@VNQ

  • !!

    " #

    " #

    $

    $

    $

    $

    $

    $

    %

    $

    !

    &

    &

    &

    & &

    &

    '

    ' ( (

    ) *

    ' ''

    ' ( (

    + +

    ,

    +

    +

    + +

    -!./

    01234356237895325:;3-!./8@30

    @A

    6B3

    -./0

    9C;

    D2

    -

    E$

    :

    F

  • !"

    # $

    %

    %

    % %

    &'( )

    % &'( (*+

    ,

    , ,

    , , , ,

    -

    . /

    - - 0 1 0 1

    2

    2 2

    3

    4 4 4 4

    5%!6

    7*8*9*+:;?=@A;'@A+99=+'?*+A;'@'B5%!6CD

    % %

    0 14 :*8=(

    # $

    % %

    %

    %

    % % % %

    &'( )

    % &'( (*+

    !

    ,

    , ,

    , , ,,

    -

    0 1

    . /

    - 0 1-

    0 1 0 1

    2

    2 2

    4

    3

    4

    4 4

    4 4

    5%%6

    E'F=8=@GF=&A+@=F@*;=5%%6A(

    # $

    %

    % %

    %

    % %

    % %

    % % % %

    &'(

    )

    % &'( (*+

    !

    ,

    ,

    ,

    , ,, ,

    -

    /

    0 1

    .

    0 1- -

    0 1

    0 1

    0 1 0 1

    2

    2 2

    4

    3

    4

    4

    4 4

    4

    4 4

    5%H6

    7=B*+*+:

    % %

    ,

    -

    I

    0 1

    3

    4

    G5%H6C=&'?=(

    # $

    # $

    %

    % %

    %

    % %

    %

    % %

    !

    % %

    &'(

    % &'( (*+

    !

    &'(!

    )

    % &'( (*+

    ,

    ,

    , ,

    ,

    , ,

    /I

    .

    0 1

    0 1

    I I

    0 1

    /

    0 1

    0 1

    I I

    0 1

    2

    2 2

    2

    2 2

    J

    3

    4

    4

    4 4

    4

    3

    4

    4

    4 4

    4

    5%"6

    KL=A@LDG*+'@9=@B'@5%"6;'C=A?AM*?>?G

    !

    I I

    J

    4 ?>(;C=A?*+*?>?)E'F=8=@GF=

    AL;';9='B;

  • !"

    # $

    %

    &'(

    % %

    % %

    %

    % %

    )*+!

    % )*+ +,-

    %

    .

    )*+ +,-

    /

    %)*+

    0

    0

    0 0

    0 0

    0

    1

    2

    3 4

    3 4

    3 4

    1

    3 4 3 4

    5

    5 5

    5 5

    5

    6

    7

    7

    7

    7

    6

    8 9

    7 7 7

    : ;

    : ;

    < =

    >%"?

    @*ABAC'A,DACBE*'F,+GHIBEJIB+,+A,KBL,.B. M

    0

    56 L'-F,DACB,-ABI-'EIB')A'-)B,+NBI*L

    >%"?IBFH)B+A*>O?'+,A+C*HEF.

    PHIACBI&*IBL,DACBE*'F'-QEB,+BRH'EA*ACB-BQ'A,KB*DACB'-QEB*DACB,-ABI-'E

    ,&GBF'-)BL,.B.

    !

    LSCBIB A'- L

    0

    4

    3

    5 5 5

    T

    6T 6 ACB-ACBE*'FS,EEUB*GA,&H&L+,-)BACB

    E*'FS*HEFUBACB)*-VHQ'AB*DACB,-ABI-'E,&GBF'-)B.WB-)BL>%"?+C*HEFIBFH)BA*>O?

    D*IAC,+)'+B'+SBEE.

    X*+BBAC'AAC,+C'GGB-+LIB)'EEAC'A

    % %

    )*+3 3 4 56 7 '-F

    % %

    +,-4 3 4 56 7 .

    W*SBKBIL

    0

    5 56T UJ'++H&GA,*-.WB-)BL+HU+A,AHA,-QAC,+,-A*ACBAS*GIBK,*H+

    BRH'A,*-+Q,KB+

    % %

    )*+

    0

    3

    3 4

    56

    7

    '-F

    % %

    +,-

    0

    4

    3 4

    5

    T

    6

    7

    .P,-'EEJLSBGHAACB+B,-A*

    >%"?A*QBA

    %

    &'(

    % %

    % % % %

    % %

    .

    /

    %

    0

    1

    2

    3 4

    3 4 3 4

    3 4 3 4

    3

    3 4

    6

    T

    8 9

    7 7 7

    : ;

    7 7

    : ;

    : ;

    : ;

    7

    < =

    >%O?

    YHEA,GEJ,-QACB-H&BI'A*I'-FFB-*&,-'A*I*DACBE*SBIDI')A,*-,->%O?UJ

    % %

    3 47

    Q,KB+ACBFB+,IBFIB+HEAL,.B.

    %

    &'(

    % % % %

    %

    /

    %

    .

    /

    0

    1

    2

    3 4 3 4

    3

    1

    3

    6

    8 9

    7 7 T

    : ;

    < =

    6 >%Z?

  • !"

    $%&''()*+,*-*.&)/0-()-1234*%5'%**.-,6)%+6-75%0-+,),8(9:*&%9%8*.-;,

  • !"

    ###$&'()*+,-'(

    ./012345467801695:29;9467016

  • !"

    # $

    !

    %

    %

    %

    %

    ! &

    %

    '

    '

    ' '

    '

    ' '

    '

    ' '

    ' '

    '

    '

    (

    )

    * * *

    * )

    * * *

    * * * *

    )

    * * *

    * * * * )

    * * * * *

    * * )

    * * *

    + +

    ,

    +

    ,

    - .

    , / /

    +

    0 1

    ,

    2 3

    - .

    , /

    + /

    0 1

    , ,

    2 3

    - .

    /

    + /

    0 1

    ,

    2 3

    45%6

    789:8;?@8ABCD?AC8ECF

  • !"

    #$%&'()*+'(,-./.,01234562378042,9

    ):

    ':*+*;),??,

    @AA'(B>::?'C*D'(,EF,9GGH.

    #I%C;JJ*)K,L.&.>):-*??'(,M.N.,3252OP2PQ782R,H

    (:

    ':*+*;),

    ST;UAK;)V'?U>(W'>()*)X,9GG$.

    #Y%F;T)K;),V./.,&*?JZ(),F.W.,F;T)K;),F.C.>):B=;++,::?'C*D'(,EF,!""I.