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Waveform Vector representation

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B APhase shift = 90 degrees

B is ahead of A(B "leads" A)

A

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A

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B

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90 degrees

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180 degrees

Waveforms Phase relations Vector representations

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phase shift

angle

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VXGIFKGIKLNDkNiL_^pGgT_GbD{GIT_GIFaSbGw!ULOZM^cSIiGR^`T_i|uJPGbaFKGJjXNUWFLUWFLOZKGSbUWT_SIhKUxLrJPGSIiW^TOGRJuLON[VXGt`pTONhKFHJf YUL

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length = 6angle = 0 degrees

length = 8angle = 0 degrees

total length = 6 + 8 = 14

angle = 0 degreesUWsYUi\^`T_i|fUDkNixL$^`pGQONhHT_SIGQ}dULOZcL_ZKGQ_^`sYGjKZa^QOG^`FKpiG^`T_GSbNFHFKGSbLOGJcLONpGbL_ZKGITUWFQOGIT_UGRQIfLOZKGUT

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14 V 14 V

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length = 6 angle = 0 degrees

length = 8

total length = 6 - 8 = -2 at 0 degrees

angle = 180 degrees

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0 deg- + - +

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- +

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6 V8 V

2 V 2 V

0 deg

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0 deg

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8 V-+

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Test lead colors provide a frame of referencefor interpreting the sign (+ or -) of the metersindication.

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V A

AOFF

Source 1 Source 2

Total voltage?

The meter tells us +24 volts

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Source 1 Source 2

Total voltage?

24 V

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Source 1 Source 2

Total voltage?

The meter tells us -17 volts

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24 V 17 V

Total voltage = 7 V- +

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` nH0n0 n

180 deg

- +0 deg

+ -

redblack

blackred

"phasemeter"

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QjKZH^Q]GiWUvGLOZKU\Q FKN`L_UWSIGLOZKGtQOdUL_S$ZqUFqjXNi\^`T_UxL|

sc^`T_vUFKpQ

+ -

8 V180 deg

?V 3P *!A(j$A%\*

EDakGSbLONT$QdUxL_ZhHFHSbNsBsYNFB^FKpiWGQ^`T_G!^JKJPGRJflLOZHGIUWT}sY^pFKULOhaJPGQiGFKp`L_ZHQ

^JKJthHj

hKULOGJPUXGTOGFLOiW|gLOZH^F

LOZa^lLN`DQOS^`i\^`Tsc^`pFKUxL_hHJPGRQI

length = 6angle = 0 degrees

length = 8angle = 90 degrees

length = 10angle = 53.13

degrees6 at 0 degrees8 at 90 degrees+

10 at 53.13 degrees

Vector addition

EDLd!NkNiL_^pGQ

gNhPLN`DjKZa^QOG0^`T_Gt^JKJKGJL_NpGbLOZHGITrV|VXGIUWFKpqSbNFKFKGSbLOGRJUFeQ]GTOUWGQfXLOZKGUT

kNixL$^`pGBsc^pFKULOhHJKGQtJKNFKN`LJKUT_GSbLOiW|e^JKJmNTtQOhKVPL_T_^SLt^QgdUxL_ZQ_SI^`i\^`TgkNiL_^pGQUWFgwFHQ]LOGR^JfLOZHGQOG

KaY2aI

kNixL$^`pG

hH^FL_UxL_UGRQ^`T_GSbNsBjHiGI

hH^`FL_UxL_UGRQIf^FHJ~hHQ]LiUWvG!L_ZKG^`VXNlkG!kGRSL_NT$QIf`dZHUWS$Zc^JHJhHjBUWFc^rLOT_UpNFKN

sYGbL_TOU\SD^QOZKUWNFfa^

kNixLQONhKT$SbGg^`L^JKJPGJL_N^`F

kNixLrQ]NhKT_SIGg^lL

TOGRQ]hHixL$QUFRYkNixL$Q^lL^cjHZH^QOG

^`FHpiWGrND

`

wW

0 deg- + - +

90 deg

53.13 deg- +

6 V 8 V

10 V

nNsYjH^TOGRJuLONuSIUT$SbhHUxLg^`FH^i|PQOUWQfaLOZKU\QU\QrkGTO|qQL_T_^FKpGtUFHJKGIGJwN`L_GoLOZa^lLUxL

QrjXNQ_Q]UWVKiWGoLONNVPL$^`UWF

kNixL_sYGbLOGTUWFHJPU\SI^`LOUWNFHQN`D

^FHJ

kNiL_QfT_GQOjaGRSLOUWkGi|f^SbT_NQ_QLOZKGLdnNkNiL_^pGQONhKT$SbGRQIf|GbL!NFKiW|BTOGR^J

BkNiL_QnD{NT^L_N`L_^ikNixL$^`pG

ZKGTOGUWQ!FKNYQ]hHUxL$^`VKiWGg^FH^`iWNp|BD{NT!dZH^lLd!G

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jKZH^Q]GwgkNiL_^pGRQS^`FYNFHi|tJKUT_GSbLOiW|^`U\JBNTJPUWT_GSL_i|tNjKjXNQOGf`dULOZcFHN`LOZHUFKpgUWFYVaGILdnGGIFwULOZ[flLd!N

kNixL$^`pGQS^`FoVXG^`U\JPUFHpNTNjKjXNQOUWFKpNFKGn^FKN`L_ZKGIT

E`H#E`O$

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FKN`L$^lLOUWNF[LONJKGQ_SbT_UVXG

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hH^FL_UxL_UGRQUWFQO|stVXNiWUWST$^lL_ZKGITLOZH^F[pT$^`jHZP

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sYGbL_ZKNPJKQNDQO|stVXNiWUWQOsshHQLVXGohHQOGJuUD^`F|QOGIT_UNhHQSI^iWSIhKiW^`LOUWNFHQ^TOGL_NcVaGtjaGT]D{NTOsYGJNFuL_ZKGQOG

hH^`FK

LOULOUWGQw

b&%

kNiL_^pGRQSI^FNFKiW|gGIULOZKGTJKUT_GSbLOiW|t^`U\JoNTJKUT_GSbLOiW|gNjKjXNQOGnGR^S$ZN`LOZHGITdZKGIFYSbNFKFKGRSLOGRJoUWFBQOGIT_UWGQw

kNixL$^`pGQ!sc^z|^`U\JNTNjKjXNQOG

ElHE-__

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?V (P# %!A#BC\oArA%\(

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Q]L_^`FaJK^`T$JYsc^lLOZHGIsc^lL_UWS^`iHFHN`L_^`LOUWNFw}ZKGTOG^`T_GLdnNtVH^QOUWSD{NT_scQN`DSINsYjKiWGbYFhKstVXGIT!FKN`L$^lL_UNF

H

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w

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b-{#

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HI

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f

:IbE Z- H

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cb 0

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l-W

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nH0n0 n

8.49 45o

8.06 -29.74o(8.06 330.26o)

5.39 158.2o 7.81 230.19o(7.81 -129.81o)

Note: the proper notation for designating a vectors angleis this symbol:

L$^`FHJK^T_JNT_UGFL_^lL_UNFcD{NTkGSbLONT^`FHpiWGQ!UFqSIUT$SbhHUxLSI^iWSIhKi\^lLOUWNFaQ!JKGbHFKGRQ^QVXGIUWFKpBLONLOZKGgT_UWpZL

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fsc^vUWFKp

cQ]LOT$^`UWpZLohKjf

BLON0LOZKGiGIDLf^FHJ

[

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kGRSL_NT$Q^FKpiWGJ0JPNldFHSI^`FMZH^zkGB^`FKpiGRQT_GIjHTOGRQ]GFL_GJUWFejaNiW^TD{NTOs^QjXNQOULOUWkGBFhHstVXGIT$QUWFMGIKSbGQ_QN`D

KfNTrFKGp^lL_UkGtFhKsVaGT_QriWGQ_QL_ZH^`F

HwgKNTGIP^sYjKiGf^[kGSL_NT^`FKpiGRJ

[

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SI^F

^`i\QONgVaGQ_^`U\JLONoZa^zkG^`F[^FKpiWGN`D

w}ZKG^`VXNlkGkGRSLONTNFBL_ZKGT_UpZL

w

r`

SI^FY^iWQONgVXGJPGIFHN`LOGRJ

^Q

w

`

zw

0 o

90 o

180o

270o(-90o)

The vector "compass"

GRSL_^FKphHiW^TD{NTOsqfNFL_ZKGcN`L_ZKGITgZa^`FHJfUWQdZHGIT_GY^SINsYjKiWGbFhKsVaGTgUWQgJKGIFKNLOGJeV|0UL_QgT_GQOjXGSL_UkG

ZKNTOUINFL$^`iH^FHJckGITOLOU\SI^`iaSbNsYjXNFKGFL_QwFGQ_Q]GFHSbGfL_ZKG^FKpiWGJYkGSL_NTU\QL_^`vGIFYL_NtVXGLOZHGZ|jaNLOGIFhHQOGND^

T_UpZLLOT_UW^FKpiWGfXJPGQ_SbT_UVXGJqV|LOZKGiGFKp`L_ZHQNDLOZKGB^Jz~]^SbGIFL^FHJqNjHjaNQ]ULOGtQOU\JPGQw^lL_ZKGITLOZH^FJPGRQOSITOUWVKUWFKp

^[kGSbLONT

QiWGIFKpLOZe^`FHJ0JKUT_GSbLOUWNFV|JPGFKN`L_UFKpusc^`pFKUxL_hHJPGY^`FHJ^`FKpiGfXULUWQJPGRQOSITOUWVaGRJ0UWF0L_GIT_sYQN`D!`ZKNld

D^`TiWGbDL zT_UpZL_Y^`FHJu`ZHNldD^`ThHj!lJPNldFw

ZKGRQ]GLd!NtJPUWsYGIFHQOUWNFH^iHHphKT_GQZKNT_UINFL$^`ia^FHJYkGT]L_UWS^`i

^`T_GQO|stVXNiWUoGJcV|BLdnNtFhHsBGTOU\SI^iKHphKT_GQw

FBNT$JPGITLONJPU\QL_UFKphKU\Q]ZtLOZKGZKNTOUINFL$^`i^`FHJtkGT]L_UWS^`iJPUsYGFHQ]UWNFaQD{T_Ns2GR^S$ZN`LOZHGITRfzLOZKGkGT]L_UWS^`iUWQjKT_GbHGRJ

KaY2aI

dULOZ0^YiWNldnGT]SI^Q]G`U\{UWFjKhKT_GgsY^`LOZKGsc^lLOU\SIQ

NT~]{UWFGIiWGSbLOT_NFKU\SIQ

w!ZKGQOGgiWNldnGT]SI^Q]GriGIL]L_GIT$QJPNcFKNL

T_GIjKT_GQOGIFL^cjKZ|PQOUWS^`ikl^`T_U\^`VKiWGpQOhHS$Z0^QUFHQ]L_^FL_^`FHGINhaQSbhKT_T_GIFLfX^`i\Q]N[QO|stVXNiWUoGJuV|u^ciWNld!GITOES^QOGiWGbLOLOGIT

`U\

fVKhPLgT$^lLOZHGIT^`T_Gsc^lL_ZKGIsc^`LOU\SI^`i}NjXGIT$^lLONT_Q_YhHQOGJLONqJPU\QL_UFKphKU\Q]ZLOZKGYkGRSL_NT

QkGT]L_UWS^`iSbNsYjXNFKGFL

D{T_NsUxL$QZKNTOUINFL_^`iSbNsYjXNFKGFLwQ^SbNsYjKiWGbL_GSbNsYjKiGIgFhHstVXGITRfL_ZKGZHNT_UoNFL_^i^`FHJkGITOLOU\SI^i

hH^`FLOULOUWGQ

^`T_GdTOUL]L_GIFq^Q^cQOhKsq

4 + j4"4 right and 4 up"

In "rectangular" form, a vectors length and directionare denoted in terms of its horizontal and vertical span,"the first number representing the horixontal ("real") andthe second number (with the "j" prefix) representing thevertical ("imaginary") dimensions.

4 + j0"4 right and 0 up/down"

4 - j4"4 right and 4 down"

-4 + j0"4 left and 0 up/down"

-4 + j4"4 left and 4 up"

-4 -j4"4 left and 4 down"

+j

-j

+ "imaginary"

- "imaginary"

+ "real"- "real"

ZKGZHNT_UoNFL_^iSbNsYjaNFKGIFLU\QT_GbD{GTOT_GJrL_N^QLOZHG

O

SbNsBjXNFHGIFLflQOUFHSIGL_ZH^lLJPUsYGFHQ]UWNFtUWQSbNsYjH^`LOUWVKiWG

dULOZFHNT_sY^ifrQOS^`i\^`TU`T_G^iW

FhKstVXGIT$QwZHGMkGITOLOU\SI^`iSbNsYjaNFKGIFLqUWQTOGID{GIT_TOGRJLON^QLOZHG

MHIi-

SbNsYjaNFKGIFLfQ]UWFHSIGYLOZH^`LJPUWsBGFHQOUNFeiWUWGQgUWF^JPUxGIT_GIFLtJPUWT_GSL_UNFfLON`L$^`iWi|M^`iWUGFML_NqLOZKGQOS^`iWGBNDnL_ZKG[T_G^i

` nH0n0 n

FhKstVXGIT$Qw

ZKGTOGR^`i\^`U\QN`DL_ZKGpT$^`jKZYSbNTOT_GQOjXNFHJKQL_NgLOZKGD^sYUiWUW^TFhKstVXGITiUWFKGd!GQO^zdG^`T_iWUGTL_ZKGNFKGdUxL_Z

VXN`LOZjXNQOULOUWkGg^`FHJuFKGp^`LOUWkGgkl^`iWhKGQNFqUxLRwnZKGtUsc^pUWFH^`T_|Pt^lPU\QN`DLOZHGopT$^`jKZqSbNTOT_GQOjXNFHJKQnL_N[^`FKNLOZKGT

FhKstVXGITiWUWFKGoQ]ULOha^lLOGRJq^lL

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G^S$ZNLOZKGT

0$

1 2 3% 4 5&. . .. . .

-1-2-3-4-5

1

2'3%4(5&

-1

-2

-3

-4

-5

"real" number line

"imaginary"number line

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nNFkGIT$Q]UWNFVXGbLd!GIGFL_ZKGuLdnNmFKNL_^lL_UNFH^`i!D{NTOscQBUFkNiWkGQBQOUsYjKiWGuLOT_UpNFKNsBGILOT_|wNSbNFkGITOLtD{T_Ns

jXNi\^`TLONmTOGRSL$^`FKphKiW^TfHFaJL_ZKGqTOGR^`iSbNsBjXNFHGIFLBV|shKixL_UjHi|UWFKpL_ZKGqjaNiW^TYsY^pFKULOhaJPGV|mL_ZKGSbNQOUWFKG

N`D}L_ZKGY^`FKpiGfX^FHJqLOZHGUWsc^`pUWFH^TO|uSbNsYjXNFKGFLV|qsthKiLOUWjKiW|UWFKpLOZKGjaNiW^Tsc^pFKULOhHJKGtV|uLOZKGBQ]UWFKGtNDLOZHG

^`FHpiWGwZKU\Qsc^z|VXGthKFaJPGIT$QL_NNJqsYNTOGgT_G^JPUiW|V|qJPT$^zdUFKpYL_ZKG

hH^FL_UxL_UGRQ^QQOUWJPGRQND}^[TOUWpZLLOT_U\^`FKpiGf

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ZKNTOUINFL$^`iHSINFHQ]LOULOhKLOUWFKpgLOZHGjXNi\^`TD{NTOs

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KaY2aI

+j3

+4

length = 5

angle =36.87o

(polar form)

(real component)(imaginary component)

4 + j3 (rectangular form)

(5)(cos 36.87o) = 4(5)(sin 36.87o) = 3

5 ) 36.87o

NSbNFkGITOLaD{T_NsTOGRSL_^FKphHiW^TLONjXNi\^`TRfbHFHJrL_ZKGjXNi\^`Tsc^pFKULOhHJKGLOZKT_NhKpZL_ZKGhHQOG}N`DL_ZKG|L_ZH^`pNT_G^`F

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f^`FaJqLOZKGB^`FKpiGtV|uL$^`vUFHpcLOZKGY^`T$SL$^`FKpGIFL

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4 + j3* (rectangular form)

c =+ a, 2 + b2 (pythagorean theorem)

polar magnitude = 4* 2 + 32

polar magnitude = 5)

polar angle = arctan, 34*

polar angle =

(polar form)

36.87o

5 ) 36.87o

b&%

-

o-

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w

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\lb

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sYUWiGRQNhKLOZw

nH0n0 n

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kGSbLONT

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Z|jXN`L_GIFhHQOGiGFKp`L_Z10sY^pFKULOhaJPGP^`FHpiWGdUxL_Z[TOGRQ]jXGSbLLONZKNT_UINFL$^`iHQOUWJKG20^`FKpiG

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FhKstVXGITRwNsYGQOSIUGFLOUaSSI^`i\SbhHiW^`LONT$Q^TOGjKT_NpT$^`sYsYGJcLONYJPUT_GSbLOiW|cjaGT]D{NTOsL_ZKGQOGrNjXGIT$^lLOUWNFaQNF[Ld!NNT

sYNT_GcSbNsYjKiWGbMFhHstVXGIT$QIfVKhPLgLOZHGQOGcNjXGIT$^lLOUWNFaQgSI^`Fm^`i\QONuVXG[JPNFKGc`V|0ZH^FHJw ZKUWQtQ]GRSLOUWNFedUWiiQOZKNld

|Nh[ZKNldL_ZKGVH^QOU\SNjXGIT$^lL_UNFHQ!^`T_GjXGITOD{NT_sBGRJw~ELUWQ

#xz#x

T_GSINsYsYGIFHJKGJYL_ZH^lL|Nh[G

hKUWju|NhHT_QOGIiDdUxL_Z

^tQ_SbUWGIFL_UxaSSI^iWSIhKiW^`LONTnSI^jH^`VHiGN`DjXGITOD{NT_sYUFKp^`T_UxL_ZKsYGbLOU\SD{hKFHSL_UNFHQnG^QOUWi|YNFSbNsYjKiWGbcFhKsVaGT_Qw~EL!dUWii

sc^`vGo|NhKTQ]LOhHJK|N`DSbUWT$SbhKULshHS$Z0sYNT_GojKiWG^QO^FLLOZH^FUD|Nh

T_GgD{NT$SbGRJqLONuJPN^iiSI^iWSIhKi\^lLOUWNFaQLOZHG

iWNFKpGITd!^z|w

JHJPUxL_UNF^FHJoQOhKVPLOT$^SbLOUWNFodULOZYSbNsYjKiGIgFhHstVXGIT$QUWFtT_GSbL_^`FHphKi\^`TD{NT_sU\QG^Q]|wKNT^JKJPULOUWNFf`QOUWsBjHi|

^JHJghKjgL_ZKGT_G^iSbNsYjaNFKGIFL_QN`DPL_ZKGnSINsYjKiWGbFhKstVXGIT$QL_NJPGILOGTOsYUWFKG}LOZHGnT_G^iSbNsYjaNFKGIFLNDPLOZKGQ]hHsufl^FHJ

^JHJhHjqLOZKGoUWsc^`pUFH^TO|SbNsYjXNFKGFL_QNDL_ZKGoSbNsYjKiGIuFhKsVaGT_QLON[JPGILOGIT_sYUFHGgLOZKGgUWsc^`pUFH^TO|SbNsYjXNFKGFL

N`DLOZHGoQ]hKsq

2 + j584 - j3*+96 + j2:

175 - j3480 - j15+9

255 - j49

-36 + j1020 + j828+9-16 + j92

ZKGFuQOhKVPL_T_^SLOUWFKpYSbNsBjHiGI[FhHstVXGIT$QnUWFuTOGRSL_^FKphHiW^TD{NTOsqfKQOUsYjKiW|[Q]hHVPLOT$^SbL!L_ZKGTOGR^`iSbNsYjaNFKGIFLN`D

LOZHG[Q]GRSbNFaJeSINsYjKiWGbeFhHstVXGITD{T_NsL_ZKG[T_G^iSbNsYjXNFKGFLNDnL_ZKGcHT$Q]LgLON0^`T_T_UkGB^`LgLOZKGcT_G^iSbNsYjXNFKGFL

N`D!LOZHG[JPUxGIT_GIFaSbGf^`FaJMQOhKVPL_T_^SLgLOZHGcUsc^`pUFa^`T_|SbNsBjXNFHGIFLgN`D!LOZKGQOGSINFHJeSbNsYjKiGIFhKstVXGIToD{TONsLOZHG

UWsY^pUWFH^`T_|[SbNsYjaNFKGIFLN`DLOZKGaT_Q]LLON[^`T_T_UkGLOZHGgUsc^`pUFa^`T_|[SbNsYjXNFKGFLN`DL_ZKGoJPUXGTOGFHSbG

2 + j584 - j3*

175 - j3480 - j15

-36 + j1020 + j828- - -

-2 + j8 95 - j19; -56 - j72KNTYiNFKpZH^FHJshKixL_UjHiU\SI^`LOUWNF^`FHJyJPUWkU\QOUNFfjXNi\^`TYUWQL_ZKGuD^zkNT_GJFKN`L$^lLOUWNFLONmd!NT_vedULOZwZKGF

sthHixL_UjKiW|UFHpuSINsYjKiWGb0FhKstVXGIT$QUFejaNiW^TD{NTOsqfQOUsYjKiW|

b

L_ZKGBjaNiW^Tsc^`pFKULOhHJPGRQrN`DL_ZKGcSbNsYjKiWGb

FhKstVXGIT$QLONJKGbLOGTOsYUWFKG0L_ZKGjXNi\^`Tusc^`pFKUxL_hHJPGMN`DgLOZHGjKT_NPJPhHSLRf^FHJ

`[

L_ZKGm^`FKpiGRQ[N`DgL_ZKGmSbNsYjKiWGb

FhKstVXGIT$Q!LON[JKGbLOGTOsYUWFKGrLOZKGo^FKpiWGN`DLOZKGgjHTONPJPhHSbL

K= 77o

11.273 150o

0.6 ? 60o

NNVPL$^`UWFgLOZHGT_GSbUWjKT_NPSI^`ifRNTUFkGITOL_@R

fl^SbNsYjKiWGbFhKstVXGITRflQ]UWsYjKiW|JKUkU\JPGL_ZKGFhKsVaGTSUFjaNiW^T

D{NT_s

UWFL_N0^0Q_SI^`i\^`Tgkl^ihKGcNDgfdZKUWS$ZU\QgFKNLOZKUWFKp0sYNTOGYL_ZH^`F^SINsYjKiWGbMFhKsVaGTtdULOZFKNUsc^`pUFa^`T_|

SbNsYjaNFKGIFL#^FKpiWGA0

1= =

1= =

1= =

1 0o

1 0o

1 0o

35 65o 35 65o

10 -12o 10 -12o

0.0032 10o 0.0032 10o

0.02857 ? -65o

0.1 ? 12o

312.5 > -10o

ZKGRQ]Gt^TOGoLOZKGVH^QOUWSoNjXGIT$^lLOUWNFaQ|NhdUiWiFKGIGRJqLONvFHNldUWF0NT_JPGTL_N[sc^`FKUWjKhKi\^lL_GSbNsYjKiGIFhHstVXGIT$Q

UWFL_ZKG^`FH^i|PQOUWQBNDrSbUWT_SIhKUxL$QIwjXGIT$^lLOUWNFaQdULOZSINsYjKiWGbFhKsVaGT_QY^`T_GqV|FKNmsYG^`FaQiWUWsBULOGRJM~hHQL

LONu^JKJKUxL_UNFfXQOhKVPL_T_^SL_UNFfashKiLOUWjKiU\SI^`LOUWNFfJPUWkUWQOUNFf^`FHJUWFkGT_QOUNFfHZHNldnGkGITRwUWT]L_hH^`iWiW|q^F|u^TOULOZKsYGILOU\S

NjXGIT$^lL_UNFyLOZH^`LSI^`FVXGJPNFKGdULOZQ_SI^`i\^`TcFhKsVaGT_QSI^`FVXGJPNFKG0dULOZSbNsYjKiGIFhKsVaGT_QfUFHSIihaJPUFHp

jXNldnGT_QfnT_NN`L$QIfQONiWkUWFKpQOUshKiL_^`FHGINhaQcG

hH^`LOUWNFHQdUxL_ZSbNsYjKiWGbSbNGBcSIUGFL_Qf^FHJGIkGIFL_TOUWpNFKNsYGbL_TOU\S

D{hKFHSbLOUWNFHQ^`iLOZKNhKpZLOZHUWQUWFkNiWkGRQ^dZKNiGFKGdjXGIT$Q]jXGSbLOUWkG!UFBLOT_UWpNFHNsYGbL_TO|gSI^iiWGJ

#+H\I$-

PiP`F

dZKU\S$ZUWQ[dnGiiVXGI|NFHJyL_ZKG0Q_SbNjXG0NDL_ZKUWQJPU\QOSIhHQOQOUWNF

w!GQOhKTOGqL_ZH^lL[|Nh

T_GqD^`sYUiWU\^`T[dULOZL_ZKG0VH^Q]U\S

^`T_ULOZKsYGbL_UWSuNjaGT_^`LOUWNFHQBN`D^JKJKUxL_UNFf!QOhKVPL_T_^SL_UNFfshKixL_UjHiU\SI^`LOUWNFf!JPUkU\Q]UWNFf!^FHJUWFkGT_QOUNFf^FHJ|Nh

iWi

ZH^zkGiWUxLOLOiWGgLOT_NhHVKiGdULOZSbUWT_SIhKUxL^FH^`iW|PQ]U\QIw

r nH0n0 n

b&%

No^JKJYSbNsYjKiWGbBFhKstVXGIT$Q}UWFBT_GSbL_^FKphKi\^`TD{NTOsqf^JKJLOZHGTOGR^`iKSINsYjaNFKGIFL$Q^`FHJc^JKJLOZKGUsc^`pUFa^`T_|

SbNsYjXNFKGFL_Qw%PhKVPLOT$^SbLOUWNFuUWQQ]UWsYUi\^`TRw

NBsthKiLOUWjKiW|SbNsYjKiGI[FhKstVXGIT$Q!UFujaNiW^T!D{NT_sufshKixL_UjHi|cL_ZKGsc^`pFHUxL_hHJPGQ^FHJ^JKJcL_ZKGg^`FKpiGRQIwN

JPUkU\JPGfKJKUkU\JPGLOZKGgsc^pFKULOhHJKGQ^`FaJuQOhKVPL_T_^SLNFHG^FKpiWGrD{TONsLOZHGgN`LOZHGITRw

?VDC B do3P$A#6 \%CA

GIL

QtSINFKFKGRSLoL_ZKT_GIGkNiL_^`pGcQ]NhKT_SIGQgUWFQOGIT_UGRQo^`FaJmhHQ]GSbNsYjKiWGbmFhKstVXGIT$QgL_NJPGILOGIT_sYUFHG^JKJKUxL_UkG

kNixL$^`pGQwiWiL_ZKG[T_hKiGRQo^`FaJMi\^zdQgiWG^TOFKGRJeUFeLOZHGQL_hHJP|eN`DgSIUT$SbhHUxL$Qo^`jKjHi|LONSbUWT_SIhKUL_Qo^QgdnGii

ZHs

Q%^zdgf

UWT$S$ZKZKN`

Q~^zdQflFKGbLd!NT_vo^`FH^i|PQOUWQsYGbL_ZKNPJKQ

fdULOZLOZKGGIPSIGIjPL_UNFNDajXNldnGTS^`i\SbhKi\^lL_UNFHQw

ZKGoNFKiW|

hH^`iWUaSI^`LOUWNFqUWQLOZH^`L^`iWikl^TOU\^`VKiWGQ

V:

VXGoGbPjKT_GQ_QOGJuUF0SINsYjKiWGbD{NT_sufHL_^vUWFKpcUWFLON^SISbNhKFL

jKZH^Q]G^Qd!GIiWi^Qsc^`pFHUxL_hHJPGf^`FaJ^`iWikNixL$^`pGRQ^`FaJSbhKT_TOGFL_QsthaQLrVaGoNDLOZKGBQO^sBGD{T_G

hKGFHSb|{UWFNT$JPGT

LOZa^lLLOZHGIUWTjKZH^QOGgT_GIi\^lLOUWNFaQ]ZKUWjHQ!T_GIsc^`UWFuSINFHQ]L_^FL

w

load

+

-

-

+

-

+

E1

E2

E3

22 V 8 -64o

12 V 35o

15 V 0o

ZKGjXNi\^`T_UL|tsc^`T_vPQD{NT!^`iWiKLOZHTOGGkNixL$^`pGQONhKT$SbGRQ^`T_GNT_UWGIFLOGRJYUF[QOhHS$Z[^gd^z|oL_ZH^lLnLOZKGUT!QL$^lLOGRJYkNixLO

^`pGQ!Q]ZHNhKi\J^JKJcL_NYsY^vGLOZKGrL_N`L_^ikNixL$^`pGr^SITONQOQLOZKGiWN^JTOGRQ]U\Q]LONTRwN`L_UWSIGrLOZH^`L^ixL_ZKNhHpZsc^pFKULOhHJKG

^`FaJ0jKZH^Q]GB^`FKpiGUWQrpUkGIFD{NTGR^S$ZkNiL_^pGtQONhKT$SbGfaFHND{T_G

hKGFHSb|qkl^`iWhKGBUWQQOjaGRSbUHGJwEDLOZKU\QrUWQLOZHG

SI^Q]GfUxLgU\Q^Q_Q]hKsYGRJLOZa^lLo^`iWiD{TOG

hHGIFHSIUGRQr^TOGBG

hH^ifL_ZhHQsYGIGbL_UFHpNhKT

ha^`iWUxaS^lL_UNFHQD{NTg^`jKjHi|UWFKpug

T_hKiGRQL_N^`FMSbUWT_SIhKUxLV^iiHphHTOGRQpUWkGFUFMSbNsYjKiWGbqD{NTOsqf^`iWiN`D}L_ZKGYQO^sBGoD{TOG

hHGIFHSI|

wZKGBQ]GILOhKjN`D

NhHTG

hH^lL_UNFL_NBHFHJLONL_^`ikNiL_^pG^`jKjXG^T_Q^QQ]hHS$Z

EE total = E1 + E2 + E3

(22 V -64o) + (12 V 35o) + (15 V 0o)EE total =rT$^`jKZKU\SI^iiW|fL_ZKGgkGRSLONT_Q^JKJuhKjqUFLOZKU\Qsc^`FHFKGITR

KGFPy0r0n!0P2orP} r

22 -64o

12 35o

15 0o

ZKGYQ]hHs N`DL_ZKGQOGkGSbLONT$QdUWiWi}VaGY^T_GQOhKiL_^FLrkGRSLONTNTOUWpUWFH^`LOUWFKp^lLrLOZHGYQ]L_^T]L_UFHpjXNUWFLrD{NTrL_ZKG

[

kNixL!kGRSLONTJPN`L^lLnhHjKjaGT]EiWGbDLN`DJPU\^`pT_^s

^`FHJYL_GIT_sYUFH^`LOUWFKp^`LL_ZKGGFHJPUWFKptjXNUWFLD{NTL_ZKGt

kNixLnkGSbLONT

^TOT_NldL_Uj^lLL_ZKGgsYUWJKJKiGIT_UpZLNDL_ZKGoJPU\^`pT_^s

resultant vectorH

22 -64o

12 35o

15 0o

FqNT$JPGIT!L_N[JPGbL_GIT_sBUWFKGgdZH^`LLOZKGgT_GQOhKiL_^FLkGSL_NT

Q!sc^`pFKULOhHJPGo^FHJq^`FKpiGg^TOGdULOZKNhPLTOGRQ]NT]L_UFHptL_N

pT$^`jHZKUWSUWsc^`pGQf`dnGSI^`FcSINFkGT]LGR^S$ZYNFKGNDLOZKGRQ]GjXNi\^`TOD{NT_sSbNsBjHiGIBFhKsVaGT_Q}UWFLONoT_GSbL_^FKphKi\^`TD{NTOs

^`FaJ^JHJwoGIsYGstVXGITRfdnG

T_G

[`l

LOZHGQOGtHphKT_GQLONpGbL_ZKGITrVXGSI^hHQOGtL_ZKGjXNi\^`T_UL|usc^TOvPQD{NTrLOZKGLOZHTOGG

kNixL$^`pGrQONhKT$SbGRQ^`T_GrNTOUWGIFL_GJ[UWF^`F^JHJPUxL_UkGsY^FKFKGT

nH0n0 n

15 9.8298

9.6442+ j6.8829 V- j19.7735 V

+ j0 V

+

34.4740 - j12.8906 V>

15 V 0o = 15 + j0 V

12 V 35o = 9.8298 + j6.8829 V

22 V 8 -64o = 9.6442 - j19.7735 V

FjXNi\^`TD{NTOsqfKL_ZKU\QG

hH^`LOGRQLON

w

`

kNiL_QI

Kw

H

IwZH^`LLOZHUWQsYG^`FaQUFT_G^iLOGTOscQUWQLOZH^`L

LOZHGkNixL$^`pGsBGR^QOhKT_GJY^SITONQOQL_ZKGQOGL_ZKTOGGkNixL$^`pGQ]NhKT_SIGQ}dUWiWiaVXG

w

[

kNiL_Qfi\^`ppUWFKpLOZKG

kNiL

jKZH^Q]GTOGID{GIT_GIFHSIG

V|

Hw

K

wkNixL_sYGbLOGTnSINFKFKGRSL_GJ[^SbT_NQ_QLOZKGRQ]GjaNUFL$QUWF^gT_G^iaSbUWT$SbhKULnd!NhKi\JYNFHi|

UWFHJPU\SI^lL_GYLOZKGjaNiW^TgsY^pFKULOhaJPG[N`D!LOZHG[kNixL$^`pG

w

`

kNiL_Q

fFKN`LtLOZKG^FKpiWGwFNQ_SbUWiWiNQOSINjXGBSINhKi\J

VXGBhaQ]GRJ0L_NqJKUWQOjKi\^z|qLdnNukNixL$^`pGd!^zkGbD{NTOscQr^`FHJLOZhHQjKT_NlkUWJKG^ujKZH^Q]GYQ]ZHUxDLosYG^Q]hHTOGsBGFLfVKhPLoFHN`Lg^

kNixL_sYGbLOGTwoZHGBQ_^`sYGtjHTOUWFHSbUWjKiWGZHNi\JKQLOT_hKGtD{NTg^`sYsYGbL_GIT$QIL_ZKGI|UWFHJPU\SI^`LOGLOZKGYjXNi\^`Trsc^`pFHUxL_hHJPGBN`D

LOZHGoSbhKT_TOGFLfPFKNL!L_ZKGgjKZH^Q]Gg^FKpiWGw

ZKU\Q!UWQGIL_TOGsYGIiW|cUsYjXNTOL_^`FLUWFT_GIi\^lL_UFKpYSI^iWSIhKi\^lLOGRJcHphKT_GQ!N`DkNiL_^pGr^`FHJuSbhKT_T_GIFLnL_NT_G^iSbUWT_SIhKUxL$QIw

iLOZHNhKpZuT_GSbL_^FKphKi\^`TFKNL_^`LOUWNFqUWQSbNFkGIFKUWGIFLD{NT^JKJPULOUWNF^`FaJuQOhKVPL_T_^SL_UNFfK^`FHJud^QUWFHJPGGJuLOZHGHFH^`i

Q]LOGIjUF0NhKTQ_^`sYjKiWGojKT_NVHiGsZKGTOGfaULU\QFKN`LkGIT_|u^jKjKiWUWS^`VKiWGgL_NjKT$^SbLOU\SI^`isBGR^QOhKT_GIsYGIFL_QwGRSL_^FKphHiW^T

HphKTOGRQsthHQ]LVXGtSbNFkGITOLOGRJcL_N[jaNiW^THphHTOGRQ#QOjaGRSbUaSI^iiW|jXNi\^`T

HIHMn+

VXGbD{NTOGL_ZKGI|qSI^FuVXGoT_GIi\^lL_GJ

LONc^SLOha^`iSIUT$SbhKULsYG^QOhKT_GIsYGIFL$QIw

eGSI^FhHQOG*PO!LON[kGIT_UxD{|LOZHG^SSbhKT$^SI|NDNhKTTOGRQ]hKiL_QwFL_ZKU\QL_GQ]LrSbUWT$SbhKULfaLOZKGcvKJT_GQOUWQ]LONT

kl^`iWhKGtU\Q

hHUxL_G^`T_VKULOT$^`T_|wEL

QLOZHGIT_GQONcLOZH^`LPO!JPNGRQFKNLrJPGRSbi\^`T_Gt^`F0NjaGFPSbUWT_SIhKUxLrGIT_TONT^`FaJ0^`VXNTOL

^`Fa^`iW|QOU\QIwiWQONHfRL_ZKGnS$ZKNUWSIGN`DPD{T_G

hKGFHSbUWGQD{NTLOZKG!Q]UWsthHiW^`LOUWNFp

U\Q

hHUxL_Gn^`T_VKULOT$^`T_|fRVaGRSI^hHQ]GTOGRQ]U\QL_NT$Q

T_GQOjaNFHJhKFHUxD{NTOsYiW|D{NT^`iWirD{TOG

hHGIFHSIUGRQNDkNiL_^`pGM^FHJSIhKTOT_GIFLRwZKGIT_GM^TOGeN`L_ZKGITSINsYjaNFKGIFL$Q

{FHN`L_^VKiW|SI^jH^SIUxL_NT$Q^`FHJBUWFHJPhHSbLONT$Q

dZKUWS$ZcJPNFKNLTOGRQ]jXNFaJhHFKUxD{NTOsYiW|oL_NoJPUxGIT_GIFL}D{T_G

hKGFHSbUWGQf`VKhPLLOZH^`L

U\Q^`FKNLOZKGTQ]hKVK~GSLR

KGFPy0r0n!0P2orP}

+

-

-

+

-

+

3

2

1

0

3

0

VL 1

VL 2

VL 3

RM 1 10 k

22 V 8 -64o

12 V 35o

15 V 0o

NPO/QKRSUTNWVKXYN[Z[Z!\]T^\_R_`

Qbaca/dcNPOea]fcdhg\i`

QKjcjkalNPOea]jcmnfg\o`

QKmcmpjcNPOYjnjpq[rWshg\o`

tbalmpdua]dWv

w

NPOYSP\i`xalrndcrnd y{z}|~^g\o`nVN[tKXWW~PXU`^O]R[crnd

w

t!\o`TcNPO/QDm#dQ

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w

XU`Z

[tKXW QDmP Q

Dm

r

w

d[dndWd!a m

w

r[^aid!aq[j

w

dnfndWnd^a

hKT_GrGIFKNhKpZfPd!GpGIL^tLONL_^ikNiL_^pGrN`D7

w KkNixL$Q

Hw

A

dUxL_ZqTOGID{GIT_GIFHSIGL_NL_ZKGY

kNixLQONhKT$SbGf

dZKNQ]GjKZH^Q]Gg^FKpiWGd!^Q!^TOVHUxL_T_^TOUWi|[Q]L_^`LOGRJu^`LIGIT_NcJPGIpTOGGQQONY^Q!LONYVXGLOZKGoTOGID{GIT_GIFHSIGd!^zkGbD{NT_sw

!LHT$QLpiW^FHSbGfPLOZHUWQU\QSINhKFLOGT]EUWFL_hKULOUWkGwNldUWQULjXNQ_Q]UWVKiWGrLON[NVPL_^UFq^BLONL_^`ikNiL_^pGN`DX~hHQ]LNlkGT

kNiL_QgdULOZ

kNiLfn

kNixLRf^FHJ

`

kNixLtQOhKjKjKiWUWGQoSINFKFKGRSL_GJeUFQOGIT_UWGQ$UxL_ZgfLOZKU\Qod!NhHiWJeVXG

UWsBjXNQ_QOUVKiWGf^QkNixL$^`pGaphKT_GQdUiWiGUxL_ZKGITrJPUWTOGRSLOiW|^JKJqNTQOhKVPL_T_^SLfJPGjaGFHJPUWFKpNFjaNiW^TOUL|w!hPLrdUxL_Z

fNhKTt`jXNi\^`T_UL|

jHZH^QOG[Q]ZHUxDLgSI^Fekl^TO|^F|dZHGIT_GBUWFmVaGILdnGGIFeD{hKiWix^`U\JPUWFKp0^`FaJD{hHiiNjKjXNQOUFKpaf^FHJ

LOZHUWQ^iiWNldQnD{NTQOhHS$ZujH^`T$^JPNzPU\SI^iXQOhKsYsYUFKpaw

ZH^`L!UDd!GrLONNvYL_ZKGgQO^sBGSbUWT$SbhKUL^FHJ[T_GIkGIT$Q]GRJcNFHGNDLOZHGQOhKjKjHi|^ Q!SINFKFHGSL_UNFHQ$1EL_QSbNFLOT_UVKhKLOUWNF

LONBL_ZKGLONL_^`ikNiL_^pGrdnNhKi\J[L_ZKGIFqVXGLOZKGgNjKjaNQ]ULOGNDdZH^`LUxLd^QVXGbD{NTOG

7Pe

load

+

-

-

+

-

+

E1

E2

E3

Polarity reversed onsource E2 !

22 V -64o

12 V 35o

15 V 0o

NLOGZKNldLOZKGc

kNiLQ]hKjHjKi|^ QjKZH^Q]Gg^FKpiWGUWQQ]LOUWiiTOGID{GIT_TOGRJcLON[^Q

zfKGIkGIFLOZKNhKpZLOZHGgiGR^JKQZH^zkG

VXGIGIFcT_GIkGIT$Q]GRJwGsYGIstVXGITLOZH^`L}LOZKGjKZH^QOG^`FHpiWG!ND^`F|okNixL$^`pGJKTONjBU\QQ]L_^`LOGJBUWFcTOGID{GIT_GIFHSIG!LONoUL_Q}FHN`LOGRJ

jXNi\^`T_UxL|w[kGFL_ZKNhHpZeLOZKG^`FKpiGYU\QoQL_UiWidT_UxLOLOGFm^Q

zfL_ZKGckGRSL_NTgdUWiiVXGJPT$^zdFyioNjKjXNQOUxL_GcN`D

dZH^`LUxLd^Q!VXGbD{NT_G

22 -64o

12 35o (reversed) = 12 215oor

-12 35o

15 0o

ZKGT_GQOhKiL_^`FL

Q]hKskGSbLONTQ]ZKNhKi\JYVaGpUWFY^`LLOZHGhKjKjXGITOiWGbDLnjaNUFL

NT_UpUFYN`DL_ZKG

`

kNiLkGRSLONTD^FHJ

LOGTOsYUWFH^lL_Gg^lLLOZHGgTOUWpZL^`T_TONldL_UjqNDL_ZKGc

kNixLkGSL_NTR

KGFy02!P7horP}

resultant vector

22 -64o

12 35o (reversed) = 12 215oor

-12 35

15 0oZKGoSINFKFKGRSL_UNFuT_GIkGIT$QO^iNFuLOZKGc

kNixLQOhKjKjKiW|uS^`FqVXGgTOGjKTOGRQ]GFLOGJUWFqLdnNcJPUXGTOGFLd!^z|PQ!UWFjaNiW^T

D{NT_sqV|[^F^JKJPULOUWNFNDiLONBUxL$QkGSbLONT!^FKpiWG

sc^`vUFHptUL

kNiL_Q

fPNT!^T_GIkGT_Q_^`iaN`DQ]UWpF[NF

LOZHGcsY^pFKULOhaJPG

sc^`vUWFKpULg$

kNiL_Q

bwcULOZKGTrd^z|fSbNFkGIT$Q]UWNFL_NqTOGRSL_^FKphHiW^TD{NTOs|UWGIi\JKQrLOZHG

Q_^`sYGTOGRQ]hKiL

(reversed) =or

=

=

-9.8298 - j6.8829 V

-9.8298 - j6.8829 V

12 V 215o

-12 V 35o

12 V 35o

ZKGT_GQOhKixL_UFHpY^JKJPULOUWNFqN`DkNixL$^`pGRQ!UFqT_GSL$^`FKphKi\^`TnD{NT_sufKLOZKGF

15

9.6442 - j19.7735 V

+ j0 V9

+9-9.8298 - j6.8829 V

14.8143 - j26.6564 VF0jXNi\^`TD{NT_sufaLOZHUWQrG

hH^`LOGRQLON1Hw

`

Hw

wFHSIG^p^`UWFfXdnGtdUiWihHQ]G

O!LONkGIT_UxD{|

LOZHGgTOGRQ]hKiL_QNDNhKTSI^iWSIhKiW^`LOUWNFHQ

NPO/QKRSUTNWVKXYN[Z[Z!\]T^\_R_`

Qbaca/dcNPOea]fcdhg\i`

QKjua/jcNPOea]jcmnfg\o` KRWTKXYT[PXtKXWQXWt!gUNS/R[`KR[ZX`n~U|PKXWt!gljpNU`KZua

QKmcmpjcNPOYjnjpq[rWshg\o` TKRhgn\ |P~S[NUTKXT[PXg]PN

n

\o`VR[O_RU`n`PXPO]T^\_RU`^g

tbalmpdua]dWv

w

NPOYSP\i`xalrndcrnd

w

t!\o`TcNPO/QDm#dQ

Dm7d

w

XU`Z

7Pe

[tKXW QDmP Q

Dm

r

w

d[dndWd!a m

w

d[fndWd!aq[r

w

dnWsnnd^a

b

iWiLOZKGi\^zdQ!^FHJTOhKiWGQ!NDgSIUT$SbhKUL_Q^jKjKiW|cLONcSIUT$SbhHUxL$QIfPdULOZLOZKGgGIKSbGIjKLOUWNFNDjXNldnGTSI^`i\SbhK

iW^`LOUWNFHQfPQ]NYiWNFKpY^Q^iikl^`iWhKGRQ!^TOGGIjHTOGRQOQOGJ^`FHJsc^`FKUWjKhKi\^lL_GJUFqSbNsBjHiGI[D{NT_sqfK^`FHJu^`iWikNixL$^`pGRQ

^`FHJuSbhKT_T_GIFL_Q^`T_Gg^lLL_ZKGoQO^sYGD{T_G

hKGFHSb|w

ZKGIFcT_GIkGIT$Q]UWFKprLOZKGrJPUWTOGRSLOUWNFYND^okGSbLONT G

hHUkl^`iWGIFLL_NoT_GIkGT_QOUWFKpL_ZKGjXNi\^`T_UL|tN`D^F[kNiL_^pG

Q]NhKT_SIGUFTOGiW^`LOUWNFYL_NtN`L_ZKGIT!kNixL$^`pGQ]NhKT_SIGQDfUxLSI^F[VaGGbPjKT_GQ_Q]GRJYUFGIULOZKGTnNDLdnNJPUxGIT_GIFLd!^z|PQ

^JKJPUWFKpqiLONBLOZHGo^`FKpiGfPNTT_GIkGT_QOUWFKpoL_ZKGoQ]UWpFuN`DLOZHGgsY^pFKULOhaJPGw

GbL_GITsYG^Q]hHTOGsBGFL_Q}UWF[^`F[SbUWT$SbhKULnSbNTOT_GQOjXNFHJtLONgL_ZKG

H@_BH}P

N`DSI^iWSIhKiW^`LOGRJkl^ihKGRQIw

GSbL_^`FHphKi\^`TGbPjKT_GQ_Q]UWNFaQNDaSbNsBjHiGI

hH^`FLOULOUWGQUWFB^FBSbUWT_SIhKUxL}ZH^zkGFKNgJPUWT_GSLRf`GIsYjKUWT_UWS^`iG

hKUWk

^`iWGIFLf^`iLOZKNhKpZmL_ZKGI|^TOG[SINFkGFKUGFLoD{NTojXGITOD{NT_sYUFHp^JKJKUxL_UNF^`FHJQOhKVPL_T_^SL_UNFf^Q

UWT_S$ZHZKN` Q

}NiL_^`pGr^FHJ0nhKT_TOGFL^zdQnT_G

hKUWT_Gw

5h

=

i #Bb7AB#D7

R

biAPUP PB[UPUWD_WUAUiK D K WPKU{AUKD UP

Io UnUP n W PKW K

Time

+

-

e =i =

ii_PKi UIPUP ii DKP@h_ioWP_ U7!BK_I P

]U W WKWU_W UYUB[APo UIKUlYKPW1K]UBU UP

PB[IKUKWl BUln @U[!U[lIUWKKKUBU[D_IUP1PUUP

WP_ K[@UKoU#BKiWUPUWD_W2 iUDUKi

U[UPUPP_In _PA@_Ki_

]UU ii1 U

n@@

!o_KKKA@_Ko_P_

D

o!"Ki

PP_[D_KiUP]UPU PB[ iB#WK2PD_[UKBWPWUUUUB[Ko U_ Bi$

%b&Ui!@KUiWPBKI KB[KUPU1Ko U _[U!i'P

UWD_WU WPAiW2_ _[U!i'^_W(U[1UUiW[UWK

)*

+-, .70/21Kl3/.P .0/"4h657/84 .0/:94@/

P]WiBAP1 C%{]pKU KK WKPUWD_KiUP@_Ko_7UUUU _P1 PB[

i_ _ BUBK_K!@i2KP_iEDnK iW _ K_2PWI@UKo2U K_

UD_P(

Time F

+

-

e =i =

p =

GUP_P!@i KiUiPBW_U@UK0KBK Ki[!WW D@WK2K]H[P

UWD_W U!WWUiKK @B 'Io7Ub[U@UK8UnUiB@KiP KB[

KB[@W

Di@cKAKo!PUWD_U2 _ KiW_UPD1oPD!WW ]UP U!@i

KB[@W nKD!BIAPKiJDnPBW_UnK DiK[P_ [U nKD!BHMLK IBUP B[

_!U _NU@BB P@KAi U U] P@K @BKD-nP2U KIWKB

_ioUP PAW _Pi@ BPBU"K KB2KIBKiW2!WWAUKB[@W iW

{KP_iiKBU

HO QP RSAB#D7

PWK_7DiP]UK_U#i UDTKBiW#o UD#KAWK[KCP@_ ioU

PWKUKi

DnIWKKK WUUUi KU!UUPUU2KBKB[H_PP WUP[

UK

IBKB[KUKWPBDnUKPK2WUUUiPWWWP_WK

_

6V

UK

#_BKB[

UBBUDK_ XWT!ZY$[

W

\

UK 7P oAWUUU#U]{U!PD2!U _W{22_WD_

PB[@Koi[@UK7LP_K Ki[ P iWK1UUKPKKPPBi1UWD_U

_PPUW_PPBiW]P@_^K#BKiW o iWK#nK!U _ bBUi_PUKP1$

PBiWJP@ UP[1PiBoULPWK!WWl BKiWDPUKU B_i

`B

!

D@PB_1o D_PB

`aKii_Ki_ BUUKB @WPKD! BBKUWD_WKWKoU WKPKPUUP

D@ U PB[DP_PU2KUPUKPP UWPDb

e = c did

dtdLe

LKKoU

f

@

WKUBUBK!i_PKKI_UDP_PU_PUWD_KiUP KB[

@UiK_P!BiBUP(0LP2PUP

%iK_P K P_[D_KiUPUWD_W

]H

_^ WKU7UUTLK@ P_P_K!B7UBP@P BKiW@Bi __[i2IPPP U

D K

1 gK/."4X;h.Cij.k.l;Pam +Bn

Le

ABPUPIBKB[UPWU_W UK UBKI BK! WK nU KK

WK

Time

+

-

e =i =

oiIiD!B^KWU_WUPo UB[2_ PU iWU _WU2K

UK

KB[

PWKU7LPB UnKP_[D_KiUPWU_W iBKBKiUiKPUWD_KiUP Ki[ _

!i-

Bi1DP_KW!U2iUi7U!^U K KB[2P]U]H _PKIPD_[UKBWPUUUU @

!i-KiiUiPP_[D_KiUP KB[ _-nAP D_KW

K2W[_BB!iU!WP

PB[]W[KB WoK BiIK]HpLK oPDAWU_W]UP_ C*

,rq

W_PPU

K PB[]U@%{nnPI_K UD_K(WP2WUUU]U2iBIP]U2 UKiW UAWP

PB[]W^[PUUUUk_iWKsAP Ki[K_K KB[3_ _WD!BPP1KUWD_W

Time

+

-

current slope = max. (-)tvoltage = max. (-)u

current slope = 0tvoltage = 0

voltage = 0u

e =i =

current slope = t

current slope = max. (+)tvoltage = max. (+)

0

LKKWW BWB1U2[ioKKi K_K@BWK D K

+` .70/21Kl3/.P .0/"4h657/84 .0/:94@/

Time F

+

-

e =i =

p =

ii_PPUWD_KiUP!@B KPP _KPUWD_KiUPWUUU _P PPU[_PBU

PB[ 'I]H[P!@iKW_ BiAKBKiUiPP_[D_KiUP Ki[l

WU_W iB#@KBZv

BWBK P_[D_KiUP PB[_P UUUU2_"D!_ !WW -D!@UK K]HK @B !WU

K2o U!K_KWP!@i _ !WU2KBK2PU[_PBUBKB[UPWUUU

__D!_PBW_U Di@PAK]H@BWBBD!iBUP KI Ki[_PWU_W ]Wi_*

, q

U_

KPW[Ki _oPB1UK2!WWPK _KB PBW_UWiKK1K[P_K[Ki[

BBKB iU=

[@

D

'

nS

\

P_ni_

[@

@Bo_$wi_P_KPW ioUK!@iCDPUP

D KK I!WW @Bi_P_ -DPUDKKA!@iWxKBD K

PB2K !WW

_KB[@W@B i_KW_!UUKPK_KD@W@Ui#[K2PP UBiUiyxP

UAD@B=DPWAAK2BD PU-DPUDP@UBK P_UABUK 2BnB0P@K i_

1IKDU B_ BP2P@PoUD_PBU_1PP UKP_iIKBiKBUUiB#ZK[K KKW

PiUP U{iWWKDKP_iBPBUAKW_bPi@azUW(UK #W!Bi

PP UDUPKDP]WP o D_PB

1PP Uo UK!WW DP_PU2 PB[D_P_o_1UK!WU _iP_K KB[

UiKBD_{PPD lDnK!|PKU _]D_KWK PUWD_KiUP_UPPUP Ki U(lLK

UP[U1UBP@P Ki[ Ui D_PBBDKK}!iiW@_]iK

KPWKDBiB Ki[UP UUUUb_P AKP_i iB !@Bii_PI_K}!iiP o

APUK~}^BB[APU

[

1Z gB.6"4X;[email protected];TPam +

BKiW# KPUBD P #K[PUn KUWD_UkUWDnKiDnPPKPW

iU UP ] WK!NxPU W_keBKiW 1KIiUI BK K[P_ PIWUUU

UWDno]D[KiUP UPH

Le 10 mH10 V60 Hz

XL = 3.7699 (inductive reactance of 10 mH inductor at 60 Hz)

I = E

X

I = 10 V3.7699

I = 2.6526 A@BWB_KiiUiBP_WU_W_ Ki[_KUKPWKB7WK@

iUB7K1UWD_U PUI KPUK_kl*

,q

ioI K KB[1iKoi[APi

KPWI_KWoUUWD_U_P Ki[@PB@i_PAWU UK!nK Di! |PPP_

_1PP U UK!WW KB[PUIKPWUKUn[$

Opposition =CurrentVoltage

Opposition = 10 V 90o

2.6526 A 0

Opposition = 3.7699 90o

or

0 + j3.7699

+)

.0"0/721B3/7.#" .0/:"4657/843" .0/:9 64@/

Opposition

For an inductor:

90 o

0 o

90 o

(XL)

E

I

z_Ki_ BUo]P_bKKPWUKU_KUPKPU {UK!WW2BKB[{ b*

,q

]iUKK

@#_PP W UK!WW PB[{ 7!WW_UPUkK[P_[@LK PPUUKUUKoUW

UP[U BKiWDo WIo i_ I!UU[I/BD K_PU#io _ WBUK!

D KKBiWD_PB_PAo _ [BDU T[K@UpD!BP!|BUWBKiB[C# U!UKi[

UP[U Ki[2B2U WK[PD!BDD@KB2P_ i_ _kKW_[i2_o D_PBUP

iU UP

X"

+

.0"0/721B3/7.#" .0/:"4657/843" .0/:9 64@/

BW 1KKPiPUBD P

Time

+

-

e =i =

phase shift =37.016o

KWKo U _PIPW^KKPWIi_UPKP"DBiB WUUUUP BKiW P]WB(

DPUKUoy W7UUUUUB[Ko U# 7KPU_

,rq

K#KBKB[7PWKU^n_PP

UWD_WAUB[KIPKPU2l*

,

q UK_KPUA KI KB[WUKPWKUk Ii_WB

P_Ki_ BU

E = IZ

ER = IRZR

E R = (1.597 A -37.016o)(5 0o)

ER = 7.9847 V -37.016o

Notice that the phase angle of ER is equal tothe phase angle of the current.LK UWD_WUB[Po UPUK!KWUI KU2UKUUKBKiWPWKU iP

PP@3`UPX_2KPWUKo UWKB!

E = IZ

EL = ILZL

E L = (1.597 A -37.016o)(3.7699 90o)

EL = 6.0203 V

52.984o

Notice that the phase angle of EL is exactly90o more than the phase angle of the current.LKUWD_W UB[AP1PWPU KPW_PU U

+

B *-)

q

K1K Ki[KWKU/P

PP UUKPWUKUU7v

Z

,#n

q

~}^BB _!KW*

,rq

DBiB K#LK 2i P

@`Y_P

+

.0"0/721B3/7.#" .0/:"4657/843" .0/:9 64@/

_Kih|PUPo_UiBIDUA_!UiUBPD K BKWK WUK D!3D!BK| _

UP PK

, .0"0/721B3/7.#" .0/:"4657/843" .0/:9 64@/

E

I

Volts

Amps

OhmsZ

R L Total

5 + j05 0o

0 + j3.76993.7699 90o

10 + j010 0o

5 + j3.76996.262 37.016o

1.597 -37.016o1.597 -37.016o1.597 -37.016o

OhmsLaw

OhmsLaw

6.3756 - j4.80717.9847 -37.016o

3.6244 + j4.80716.0203 52.984o

1.2751 - j0.96141.2751 - j0.96141.2751 - j0.9614

E = IZ E = IZ P@WK DK WKBLK!KU AUIKi _KKi K_PU Uh_

D KUKKu D KWi{{ KB]Wi@ P@AU8K[PU[i AD!BKiB[o UP

BUBK @o1BUK! D@KBP_i_ _UaIUPIWPPU2KKU!BiKoi[i 1UP

BUBK @UUKiKPPK_iWD_^}!iiP P@UKKWWDDUBD P_PU#WBD

_"

G@ Un BnBxKB @UKD! BilKiBUBK_o|PWKo_P/iUKK[

UUiJDn U PUPKB[IoUKiB[_WUUU UP BKB[JLKX|UKiKBP@o

B @1o_v}iWPiIi[_1K[

-

$KUoD_KWKU Y_KBUDK

UWB WKKo WU B U WKIo U2 K BK UPP B_WU

KUB

+

.0"0/721B3/7.#" .0/:"4657/843" .0/:9 64@/

E

I

Volts

Amps

OhmsZ

R L Total

5 + j05 0o

0 + j3.76993.7699 90o

10 + j010 0o

10 + j010 0o

10 + j010 0o

Rule of parallel circuits:

Etotal = ER = ELG@lBU_PKAK 0%{]sIl`=y{UB BU222BUKP_PKDKUBUBK @K KB[

PWKU Ki U_P1 PB[KUPU KPP W

E

I

Volts

Amps

OhmsZ

R L Total

5 + j05 0o

0 + j3.76993.7699 90o

10 + j010 0o

10 + j010 0o

10 + j010 0o

0 - j2.65262.6526 -90o

2 + j02 0o

OhmsLaw

OhmsLaw

I = EZ

I = EZ

WPWkpBD KBDKD_DBKiWDPUUBbpBD KUP U KUU{ KB[

DDKK-}C KiW3%]pbPU KK2UcU W3kh!

E

I

Volts

Amps

OhmsZ

R L Total

5 + j05 0o

0 + j3.76993.7699 90o

10 + j010 0o

10 + j010 0o

10 + j010 0o

0 - j2.65262.6526 -90o

2 + j02 0o

2 - j2.6526

Rule of parallelcircuits:

Itotal = IR + IL

3.3221 -52.984o

_P_UoK_ i_D!i_ K @iDn1PKAK "%{]{(Il`h?WB BU K L{v

U& WK(_ B[U^PUUBI!iP_PBIi_ UXD!Ii_ K @iDnPK o Kni_{UAK

-)

.0"0/721B3/7.#" .0/:"4657/843" .0/:9 64@/

87 9(RS7A;: B1

.0"0/721B3/7.#" .0/:"4657/843" .0/:9 64@/

LKKWKBK2@_KW=M Kl_KKB KPP W aii_P oUIWKWKUPU

iUP ^PBiKiy[ KWZv-MYPP UaD! i_ @hv-MYPW#UKiWIPK!Wi7Ki_

PP UUP P]UOMeUZ|PPKBB!}^i U o _

ii_PPWiU UP l@Ui KWPBPB#PM @@BUiPBK o W

!}^i UPP UDs[ !}^ibDU @W Wo^iP PB[bUPnBi _ 1@_

K[KB 1M K[o K_@UPWWWP_ iWD_PB UDB WM @_KP]UKiB

iUKP#AP3D!i |Po1__ K _iKWPBPB

[] o D_PB ( KAWK1PW"K[KsKiiXD!]__6KKA WP_P

APKKPUbBUKKPP UDUiP_D@iIU iWD_PB8Dn_ PK @KAW_E _

@_K (UUU_KBpnPI_^Bi i_KPK_U$!UP]UKUiWUUB_U UP BWBU

D! BB PPhBUPUBD_PBaD!KUi KK!nDP_KByDKAQ i]@ K

!UW7P@#0D!B]UiUi}^BB[U PUP U_PB UKKBa__[KoUW

KK_7o_ PP W

!R S A#R UTWVX< R PY XV

{KBnWPIi[UKoyoKn!}^i{KB_i@KBKiW BPPb]UWD]Ui@KUPU P

i[BUW BUPWnKDi WPP U Pi_KK UBaLK !}^iWBP

Wvi UPU^BUPKPUUoI]@U -DK2i_ UBP@PBiW]P@_KP iWKIKAo D_PB

_@ WPP UD@W@WK KW_D Wo KB[

Cross-sectional area of a roundconductor available for conductingtDC current

"DC resistance"

Cross-sectional area of the sameconductor available for conductingtlow-frequency AC

"AC resistance"

Cross-sectional area of the sameconductor available for conductingthigh-frequency AC

"AC resistance"LK Bi i_ iUP 2_P UPU_b Wvi UPU!UoAP CnK@UP

_o D_PB KUho _ _PUIBUPWBBK1KUPBk|PWKiKK

U P sn !}^i WU1B_1i@KU K[KBPBoKYBKB[]UWKD]UiKUKWW

1Z[Z5ih3/ih0/]\mFk62/+^U^0/"._\

N

_P WPP U W6v oWP_7_i#XKW KPK!W_ WPP K KB[{KIU[AW

ByDKW@`

AUDUUKKi@U8 U[BPPU]I[K_D-DKB^K!}^i !KUi(@nP W-v}KWPBPB

k KB[UPb ]WB7KWKU K K_BUPWA_nn]bn K_3xAPU@

D_!KiU_{U D_^o_]W3D!_ iWW_P Ww]z[_[iKPP Ki_

U UKW@cB_^BD!iWK iUUb

LK2BWiK DK[KB }^i K2!}^i U2o D_PB_{2 UPUIUiDn

PWUU _7@KP _Uv}[_Wo!K&DKUKUKWKP on!}^i DPUKU

iUP W khP_Uv}[_Wi_UnUUB K[KBPBaLK K[P_U U_KP?_K

s[1!}^i2@KU KWPBPBoWo@iPU

n

z UW@

R AC = (RDC)(k) fa

Where,

R AC =

R DC =

k =b

f =

AC resistance at given frequency "f"c

Resistance at zero frequency (DC)Wire gage factor (see table below)=

Frequency of AC in MHz (MegaHertz)LK2U@KID-DK UUiUKK?@@UKo_0-Z UUW]UUUPP &Bi

d

eZ

g

Z fgZ#B

hhhNhhhhhNhhhhhNhhhNhNhhNhhh

i

NrNNNNrNN

?

=

i

NrNNNNrNN

Np

i

NrNNNNrNN

Np

NNNNrNNNNrN

Np

NNNNrNNNNrN

Np

NNNNrNNNNrN

Bp

NNNNrNNNNrN

7

?

NNNrNNNNrN

p

NNNrNNNNrN

p

NNNrNNNNrN

?

N

NNNrNNNNrN

86N

KWK_K{ iK_ UnKD!B

n,

[_UI ]khBPZv=-vBP o D_PBAU=

+]

UK

P]W2U }!oU?o _ _aZ

n

_AK[KiP _

n,

z3

R AC = (RDC)(k) fa

R AC = (25 )(27.6) 10

R AC = 2.182 kj

.0"0/721B3/7.#" .0/:"4657/843" .0/:9 64@/

iWiBD!B@K =|PWK2 "

I!iP_PBK_K[ol'

BUP B_niU U!}^iD

PP UWBUPUBW0LKKIUi_o|PUKU^PKi D_PBUKBUPW"P_

UP[U K2@u_Bi UP2K D H P@I!@B KUUKo@HHBUoi

Usn }!o]oiWD_PBUP K BU DKPi !}^iDUiWUP _ i D_PB !iKUP ?H7_

B[B }!ii[_BD

k lnmpoWqsrut v

w x y z { y | z x y | }

~ x } y | z x z y y z ~0{ ~ x

, i QP #7 { #

R

biAPUP PB[UPUWD_WUAUiK D K WPKU{AUKD UP

Io UnUP n W PKW K

Time

+

-

e =i =

ii_PPiW_@_ UUK[U KB[oKU!UUPUAKUWD_UU W#_

_U!BnP{UI]K]W WcUbKBKiW!KU KUP]U WcU{KUWD_U

B_n __n!UW_UKIKKU&BU[D_b _KPU_P WP_2K[@_Ko_

PB[_P UWD_U21iUD1_KB_=_nXUPUPP_An @P2@UKo_7]U _2BBoW

B

-'N

#!HWo_KK P@_Ki@P@#

-

HaPBPD_[UKBWP#@_KW

UWD_W=iB#WK2D_[UKBWPBKiWKUPUKi U _ Bi$@%{WB [_KWIi[

KBK UWD_WUB[K iW @!WW!i'[K Ki[KWKU K iW

_ A@!WWo-n_PUb7_nIUUB!U[_UP K]WirAK =%{]KWK

*

, .0"0/7'Z/8.$ .0/"4 658/84 .0/9.0y.@P@?>0/

K2UKP_[D_KiUP@_KiUWUUU _P1 KB[

i_ _ BUBK_K!@i2KP_iEDnK iW _ K_2PWI@UKo2U K_

UD_P(

Time

+

-

e =i =

p =

GUP_P!@i KiUiPBW_U@UK0KBK Ki[!WW D@WK2K?H[P

UWD_W U!WWUiKK @B"'I7Ub[U@UK8UnUiB@KiP KB[

KB[@WDi@cKAK!PUWD_U2 _ KiW_UPD1oPD!WW ]UP U!@i

KB[@W nKD!BIAPKiJDnPBW_UnK DiK[P_ [U nKD!BHHMLK IBUP B[

_!U _NU@BB P@KAi U U] P@K @BKD-nP2U KIWKB

_ioUP PAW _Pi@ BPBU"K KB2KIBKiW2!WWAUKB[@W iW

{KP_iiKBU

, HO QP P/3/A #D7

_U WIK_CD!BP]WKU2Wi UDTKBiWi UDU@ P@_BiWPKUPU

PBo KU!UUPUK1UWD_UKW(BUPUBUDUK!W

U

-NrH UWD_WDn D]K

UIPKK[K KB[UAPBDP_U UKDP_UK1KBxWU_WBUiMLKXP@x_ioU

@KUKW#i_PW U oPWWWP_!Ks

6V

U

-Nr_7UWD_W2W WKBUPUBU

LK UK!WWUWD_UDP_KW UK_KBU _8

!

!NDKUPP@KiBBUK!W

P_[P1!K&DKo

'Z gB..0"y.@PCih.@6_.l;bPam

n

ABPUPIBKB[UPWU_W UK UBKI BK! WK nU KK

WK

Time F

+

-

e =i =

oiIiD!B[K PB[KUKW Ai_PW W AiWU _WUK

U

1UUUUW W

LKiBUKKIPU[_PBU PB[ iBKiKBWBKAPUWD_KiUPUWD_UA2@ o-iB

DPUKU@U#BUiKU!UUKWU_WK]WHW_PAKPUWD_KiUP7BKB[# #@2o-APBBUi

PD_[UKBWPUWD_U @-nAP DPUKUK!U[U#iB!iU!U1KUWD_W2]U

KiB[i KiBK!LK AiKlUWD_W]WP_Av*

,q

WA_KUP

PB[]U=%bnnK@KWUK(KKABKB[]UBBP]W_KoUD_W KAWUUU

]U^[K KB[3_iUPsAKWUUUUPKUWD_WlUUWD!BKP P Ki[

Time

+

-

e =i =

voltage slope = 0ucurrent = 0t

voltage slope = max. (-)current = max. (-)t

voltage slope = 0current = 0t

voltage slope = max. (+)ucurrent = max. (+)t

WU U[P]U2UKoi(UKUI2KnPP_^!@i]WP_ ]c KK2PP U

D KPoiW1KKBUPUBU D Kn[$

-

.0"0/7'Z/8.$ .0/"4 658/84 .0/9.0y.@P@?>0/

Time F

+

-

e =i =

p =

7IKKPKPU D K@K3*

,

iUBKPWKpD! BiUUUU_PI PB[#oPD

!@i]UP@_iP_oK[PUDBiB!WUUPKiW_U0LK o_PP_B_U W

niKU7KP_@B7U7iWDUW_PDPUKUo{AWUUU^@#BB-DPUDP{UPioUib!@B

_i@i

i_PW U UK!WU11DP_KWWUUU D_ @i _WK[U _i@KWUUU

WBKiU{KD _Dn |PKU_]n DPUKUK PUWD_KiUP UUKPKUP oUbXKW

_n UUB1_WKPU7cWU_W@UUi1KWPBPBPi_PW UUWWBiTW UP

iD_ _WKP UBKB[UWPWKBKB[KUKW IiW APPW U{P2WUUU

UB[Ki UUPIKo _ -}^BiDnKo U_K PB[#PWKUi_PW W#

KP U_KUUUU UB[P_Ks

-

-}^BiXDnKi_PW W7PP UD

PiWUP _B_U W KoiWK_PnD!UiiDnIP Bh W7 D2U

o |!

nP i_PW WkUBUPKP Ki[ KU!UU K_UWU_WID_KWbKB U

UA PB[W WBsvDPUKUKUWD_Wi=UKiD_U_ DPUW KIUIAUUUU!i

i2H_P oBKiW2U@iv DP_PUKUWD_UoP@ Po_P2 2P_ iWD_PB

UPW_n i_PW U _

!2KU!UUPUK K[KiP _{K_i@K PB[

XL C = 2j pifC1

B?NZ

#N?B=

NB#?jsBrZ' BN#?BH-

g

NNrNNNNrNNNNrNNrrNrNNrrNrNNNNrNrrNN

8

N

NrNNNNrNNNNrNNrrNrNNrrNrNNNNrNrrN

?N

8

NrNNNNrNNNNrNNrrNrNNrrNrNNNNrNrrN

N

rNN

NNrNNNNrNNNNrNNrrNrNNrrNrNNNNrNrrNN

iWIKUIP_KB @WPK _BUPUBWiWUP 1K[KiP WWK[IU

@U^PUiWD_PBp_PW UoU _ UP7o iUi7 PBoUP K[KB

UnWBDiPP UiWD_PB UPPBoUi PBoUK hK[KiP PKPUDUBv

!W UiD_KWK Ki[pDnKP KUi@BWUUUU^nBUPUBUDUP[ UBDP_PUK

UWD_W2UhD[ _@KIUi@B PB[

KB[{AKi_PW U D K bKW__KWU_W3UWDbn iD[PBUPUBW

iU UP ] WK!NxPU W_keBKiW 1KIiUI BK K[P_ PIWUUU

UWDno]D[KiUP UPH

'Z gB..0"y.@PCih.@6_.l;bPam

10 V60 Hz C

100 F

XC = 26.5258

I = EXL

I = 10 V26.5258

I = 0.3770 A@BWB_KiiUiBP_WU_W_ Ki[_KUKPWKB7WK@

iUBUK2BKiWPUKPUK_@l*

,

ioAK UWD_WbiKiB[KiPPU

_PUiUUWD_UUP BKB[I_Ki@ BU7 i_ BUBK @ K KPW1_KWUK1PP U

iU U2UK!WU PB[

Opposition = VoltageCurrent

Opposition = 10 V 0o

0.3770 A 90o

Opposition = 26.5258 j -90o

I

E Opposition

For a capacitor:

90 o

0 o

-90o

(XC)z_Ki_ BUi]P@bKKU_KW#Uni_PW U ^UK!WW2 PB[b v*

,q

]iUKK

@#BUPUBU {UK!WU PB[ 7KiW_U_UPUkK[P_[@LK PPUUKUUKoUW

UP[U BKiWDo WIo i_ I!UU[I/BD K_PU#io _ WBUK!

?)

.0"0/7'Z/8.$ .0/"4 658/84 .0/9.0y.@P@?>0/

D KKBiWD_PB_PAo _ [BDU T[K@UpD!BP!|BUWBKiB[C# U!UKi[

UP[U PB[1BU7BUK! nK DiK_ KUaxPi_ _CK[P_[iUi D_PB2UP

iU UP

X"

.0"0/7'Z/8.$ .0/"4 658/84 .0/9.0y.@P@?>0/

E

I

Volts

Amps

OhmsZ

R TotalC10 + j010 0o

5 + j05 0o

0 - j26.525826.5258 -90o

5 - j26.525826.993 -79.325o

370.5m 79.325o68.623m + j364.06m

Ki[2ii BK 7PUoK[PU_Dn_ U!UKi[_K=|PWKo{PW iAAKCL{U_

WPI1W Ki[i_DKDPo _ _PB WPIUB

E

I

Volts

Amps

OhmsZ

R TotalC10 + j010 0o

5 + j05 0o

0 - j26.525826.5258 -90o

5 - j26.525826.993 -79.325o

370.5m 79.325o68.623m + j364.06m68.623m + j364.06m

370.5m 79.325o68.623m + j364.06m

370.5m 79.325o

Rule of seriescircuits:

Itotal = IR = ICU[nKKUP_PUKBUUKKAP] 3%]`=I"6oWB BU BIKWUUU

UB[Ko UUP i_PW W

E

I

Volts

Amps

OhmsZ

R TotalC10 + j010 0o

5 + j05 0o

0 - j26.525826.5258 -90o

5 - j26.525826.993 -79.325o

370.5m 79.325o68.623m + j364.06m68.623m + j364.06m

370.5m 79.325o68.623m + j364.06m

370.5m 79.325o

343.11m + j1.82031.8523 79.325o

OhmsLaw

OhmsLaw

9.6569 - j1.82039.8269 -10.675o

E = IZ E = IZGU P@lKWU_WUB[KiW#PUK!KU UKPWUKUUK Ki[KUPU

KBKP_"`_

.0"0/7'Z/8.$ .0/"4 658/84 .0/9.0y.@P@?>0/

I!iP_PBipy _7_PUUi8xP{W#o D_PBiao^ii BD K{__n Bo !iKUP o

UK U K _D_!iKUP UD1K iW_i_ K @WPlBUK!:KU

i_ _H#W8

q!

I " $ CC7 %

iUKUP@1!iKUP o U]UKcBiB[_Di_P_ n _ABUWKB[D

UK KI!iKUP o_LP_2 ^iU!iKUP ^PKPWoK_PBbUPBUPUBW

!iKUP _2D! i@i PUI ]@PB@i_

pPKi o U2!iKUP _]P]UIKPW_PU2UW

,rq

{&:Io

,q

!

eKKB BUPUBW!iKUP _]nP]WKPWUKU _#!KU v *

,q

I

v*

,q

H

AK C%]U BKp`:Is^ZCI `=J^$MI`h?

KBiW_1i_PW W_2Ii WUBKi D KP _D_boK_ b]U

KPU_KW UBKikD! Bi

,rq

UP

'Z('Zy3*)#)T/+)M3/7mZmCih.0y.@PCih.@6_.l;TPam

-*

E

I

Volts

Amps

OhmsZ

R TotalC10 + j010 0o

5 + j05 0o

0 - j26.525826.5258 -90o

10 + j010 0o

10 + j010 0o

Rule of parallelcircuits:

Etotal = ER = ECG@lBU_PK AP] a%{] sIl`=ybWB BU222 WKP7KDKUBUBK @K KB[

PWKU Ki U_P1 PB[KUPU Ki_PW U

E

I

Volts

Amps

OhmsZ

R TotalC10 + j010 0o

5 + j05 0o

0 - j26.525826.5258 -90o

10 + j010 0o

10 + j010 0o

2 + j02 0o

0 + j376.99m376.99m 90o

OhmsLaw

OhmsLaw

I = EZ

I = EZ

WP#Uk BKDKUPDBKiWD{PUUB BD PUPK2UeP_D_ KB[

DDKK-}C KiW3%]p_[_$H

E

I

Volts

Amps

OhmsZ

R TotalC10 + j010 0o

5 + j05 0o

0 - j26.525826.5258 -90o

10 + j010 0o

10 + j010 0o

2 + j02 0o

0 + j376.99m376.99m 90o

2 + j376.99m2.0352 10.675o

Rule of parallelcircuits:

Itotal = IR + IC_@U_noK_PBB_D!i_ K @i DnPKAK a%{](Il`= Uii_K3L{U_

WPIb]K PP UP DPUi@PUUB[I!iP_PBi_I_ "DBUBK @oDnPP

, .0"0/7'Z/8.$ .0/"4 658/84 .0/9.0y.@P@?>0/

i KB_UAK KB[ BU#P@Po i_ K @PP_D_iiUP o

'ZCBr .0y.@PCihD;b6_Fm

n

, 87 P/3/D: B1

.0"0/7'Z/8.$ .0/"4 658/84 .0/9.0y.@P@?>0/

Idealcapacitort

Equivalent circuit for a real capacitor

Rseries

Rparallel

KWKP_B[Ki]iUP oU2PP__oIU @cBio _ _P KW

PUUB^iUP ?HAD1iWK~|!B_[PU1KD]o _ iPoiWU]UBD_WPP U

`i U[ BUPUBUDrnK@WKBi_UiKUBUPUBD_PB_@WsnKAWU_Wn_

_ [P@UKBKUWUP[PiPKKDUKDPUWB BUK W UPi_K

Bi 7|PUPAKioU[PWUGKio _WU in UBiW[BUPUBUD

KUK KBWB_D!Pi hPKo ~o:DK Ui#2BUPUW_uUUUUIKBUi[K

P2i_PW UWxBWB8D!BK#DZxi iIiUiUWD_WT`UiKi(nPBi WDP_DU i i

]D! niUB KWD WIKUK_KP B_U_nn]

k lnmpoWqsrut

w x y z { y | z x y | }

~ x } y | z x w y | } z

7i BWs/RS P

%bB UK W@K!K_K BKUP1_PUNB

120 V60 Hz

250

R

L

C650 mH

1.5 F

LKl|PDi K iK PoUD_PBikUKUKPUUPKBUPUBU

-)

.0"0/8]Br/8.$ .0/"4 658/84 .0/:9)#"4 .

XL L = 2pifL

XL L = (2)(pi)(60 Hz)(650 mH)

XL = 245.04

XC = 2pifC1

XL C = (2)(pi)(60 Hz)(1.5 F)1

XC = 1.7684 kLKP!nIi A !KiA_i D_PBiUPiU UP o/@PB@i_ WIW U]

I!iP_PBCoBiD!BP@UPWiU UP 2D_ @i[!WWUUP_I!iP_PB

W UoK_ /_Ml*

,rq

HKli_PW UiWUP UP_i[pKB[@W_W_

I!iP_PBoK_PB_pv*

,

q

!0oiUP U_bBUKDU iWUoW KPi_iU&2I!iP_PB

!U _UKU _

,q

!

ZR = 250 + j0 or 250 0o

Z L = 0 + j245.04 or 245.04 90o

Z C = 0 - j1.7684k or 1.7684 k -90o

120 V60 Hz

ZR

ZLZ C

250 0o

245.04 90o

1.7684 k -90oG@_]_rK[PU[i_PWK[U2Bi BKiW!KioBUU(_BUK!nKD!B

U@_ W!iKUP oKUP P_UiUP oWoUD_PBiH[PB i_]D PUPi K_2]

U7PUiUP o7__lBD P@LK # _ iUD]lP__PUD-DKU#K BK

_Pi_K_WWB#|PUKi_D_7UWD_U!UPK!iKUP o2_Ki U!PW UP

BUPUBUHH

.0"0/8]Br/8.$ .0/"4 658/84 .0/:9)#"4 .

E

I

Volts

Amps

OhmsZ

R L TotalC

250 + j0250 0o

0 + j245.04 254.04 90o

0 - j1.7684k1.7684k -90o

120 + j0120 0o

250 - j1.5233k1.5437k -80.680o

12.589m + 76.708m77.734m 80.680o

OhmsLaw

I = EZ

BK BiBD K#BKB[APD K[PUPWKU U U!UKi[L[7 BU -WP

|PWK DUKoWU_ Ki[UP&DKoUD1_{K_KBBUKP

E

I

Volts

Amps

OhmsZ

R L TotalC

250 + j0250 0o

0 + j245.04 254.04 90o

0 - j1.7684k1.7684k -90o

120 + j0120 0o

250 - j1.5233k1.5437k -80.680o

12.589m + 76.708m77.734m 80.680o

12.589m + 76.708m77.734m 80.680o

12.589m + 76.708m77.734m 80.680o

12.589m + 76.708m77.734m 80.680o

Rule of seriescircuits:

I total = IR = IL = ICG@ PiP_i _KKAK "%{]`hI"syoUDUKAPnKP_ U!UKi[ WKP

K DKnK iKUWD_W2U

E

I

Volts

Amps

OhmsZ

R

L

TotalC

250 + j0250 0o

0 + j245.04 254.04 90o

0 - j1.7684k1.7684k -90o

120 + j0120 0o

250 - j1.5233k1.5437k -80.680o

12.589m + 76.708m77.734m 80.680o

12.589m + 76.708m77.734m 80.680o

12.589m + 76.708m77.734m 80.680o

12.589m + 76.708m77.734m 80.680o

OhmsLaw

OhmsLaw

OhmsLaw

3.1472 + j19.17719.434 80.680o

-18.797 + j3.084819.048 170.68o

135.65 - j22.262137.46 -9.3199o

E = IZ

E = IZ

E = IZ

GU W PKD_KWKB#UKUKW WKKKPWUUU UK

n

,

UWD[KWUUUW W

Pi_PW W

n

Z )rUWD4@ i_ K D?wQLK_BIiIlK[BDUWJD! BB/P

PP UUPB_U UiWUP o@`aKioW#I!iP_PBiWBUBP@KPWUP[o

PB[ _KKiKiBB WK[P_A_P BUPUBU]`pnPoi i UKUK _ WP

PP U oK_ PUI [U1UUP_ B UPKB_U WUI KiW_U1_W_

iKi Ko U[UI!iP_PBiUUKKiiiH KB iP BUP i#iWD UKi

UKAKUKW KB 3

U

!1KnKP K P@P W_!_ iKB

_K1P[ PU i_PW U1UIPKPW!iKUP o_UKJUP_UWUPUKKK WUBKi

"U8]Br58

$

"5 ++

)#*

N

#

#

'Z#

?ys$

a

s$:(

'Z#

?$

a

#:

Z

rB

?$

a

(

8

rNZ'

B

Z'

Z

Nr

ZN

86NN

N

rB

y$

a

8

rNZ'

8

N

Z'

ZN

8

NZN

8r

Z'

E R = 19.43 V 80.68o

EL = 19.05 V 170.7o

E C = 137.5 V -9.320o

I = 77.73 mA -99.32o (actual phase angle = 80.68o)

Interpreted SPICE results

Z8?

?[?_P44??[3!

5H4H?4#4O[-?C444O44H

4[!C-3

4[!C-

44P5[#NN?[?PX[4?4N?-

8[

-5`[

[4!4+?41E-4N-[

-45[+[

`

`

48?34[![5POP5

[4

444[

??--?4+?[O#4P!?N*?CP?4+!44

?5?

85

X"

Br By3*)#)+)M)#5

120 V60 Hz

R L C250 650 mH 1.5 F

58??*4!??[![-?#-?+-[4

_?[P4

[?4P-?`?44

?`[8[344C[8!4P[_!_[

120 V60 Hz

ZR ZL ZC

250 0o

245.04 90o1.7684 k -90o

[?H4[`4??? 3[

{(

XOH?HH[?!

4 ?s-O[J[444?JP[ s[[4_4[![

U?4[4

E

I

Volts

Amps

OhmsZ

R L TotalC

250 + j0250 0o

0 + j245.04 254.04 90o

0 - j1.7684k1.7684k -90o

120 + j0120 0o

P[3-?JN[[U4N* O![![4J?!

?C!*?O[4_4P[

E

I

Volts

Amps

OhmsZ

R L TotalC

250 + j0250 0o

0 + j245.04 254.04 90o

0 - j1.7684k1.7684k -90o

120 + j0120 0o

120 + j0120 0o

120 + j0120 0o

120 + j0120 0o

Rule of parallelcircuits:

Etotal = ER = EL = EC

`! *

#"

hy

4??4[[![?4?4[4?4N?

!4

"U8]Br58

$

"5 ++

)#*

E

I

Volts

Amps

OhmsZ

R L TotalC

250 + j0250 0o

0 + j245.04 254.04 90o

0 - j1.7684k1.7684k -90o

120 + j0120 0o

120 + j0120 0o

120 + j0120 0o

120 + j0120 0o

OhmsLaw

OhmsLaw

OhmsLaw

480m + j0$480 $ 0o

0 - j489.71m489.71m $ -90o

0 + j67.858m67.858m % 90o

I = E

Z

I = E

Z

I = E

Z

??s-?!4[44?H!*4?![

-_4

44?*?? ?N[![

!

B

&J

&

?4?!X?N_NN?!`[O?#4

hy

44'N???[43?JP[?s44!__?!(

P[?U?4[4?

y

?J?C

???4 ?

4?[44!4H? COP

????

N

@

[ ? 4-*? X?8!*;`C[44?

)**?UN4!-#4 [-

?J?4?3!14N

O[?J[!?N?[*4N

??N

![4[!C#*O

`

*??`? ?!!?4N

N4? *

+"#

4?4[?!4?E?[4N

{

h?

E

I

Volts

Amps

OhmsZ

R

L

TotalC

250 + j0250 0o

0 + j245.04 254.04 90o

0 - j1.7684k1.7684k -90o

120 + j0120 0o

120 + j0120 0o

120 + j0120 0o

120 + j0120 0o

480m + j0,480 0o

0 - j489.71m489.71m -90o

0 + j67.858m67.858m - 90o

480m - j421.85m$639.03m % -41.311o

141.05 + j123.96187.79 41.311o

[?43U-? ??XP[+??83-4!!"4

`? [

![_PC

Zp?

?4P_44

Br By3*)#)+)M)#5

battery symbols are "dummy"voltage sources for SPICE touse as current measurementpoints.. All are set to 0 volts.

2

4/

56

1

2

3

0 0 0 0

2 2

120 V60 Hz

R L0 C250 650 mH 1.5 F

V ic V

il V

ic

V iR bogus

13254

NN

2628794:29;79=?=5@?132A=9BC@

7ED

7F=5B?132G@

794HBHIH132J@

7

B

132J@

4LK3M

e

;

FN

=9OCP:=9B

7L2JBHQH132J@

4>=RIH@?B

N

@

=

N

@?Q

N

@ES

23=RQH@T=VU

N

;

UW132

7XDY=RQC@?QC@

U

4:7ED8

gfU`h]BLiRh5`f

jflk

Gfgk5nm++lflk

oh)3flk*

I total = 639.0 mA -41.31o

I R = 480 mA 0o

IL = 489.7 mA -90o

IC = 67.86 mA 90o

Interpreted SPICE results

p

5]JOC??C?)??4rq8s

p?

(5JtJ?4[ou-[[

v

v

C!?4 !!!? ! 4N? [-[?

v

v

?-?3[H?O8?! ??4Ns[?? ? C!?![#XP!Rq8s

p?

41?4!Ew_41?5?[N4

4[-[?*

C!5-!gu4?4N4?)w1O*?8[54*[5?[N-4-

q8s

p?

[?4N+5Xx

?

?4N+?5[N45

?48[O!P44?

y)z {|}e~}>:gG?

?] ?44?4[![

4[-![OC44N?

`

![-[_-+???-5?_-?- ![[?O5`?

4A

-5?N4![-?U??4?4NPC!4N

*?_?4- [4![PP4[

120 V60 Hz

C 1

4.7 F L 650 mH

R 470 C 2 1.5 F

!!-? ??4u\w+_[44N!

- N4O

4`?!

JX ?!-4

uwU[[8!+4[?4O4-8!4uw?-!4ulw1?

?P[u\w

Li8iJ`h}m\flh}fgV`hrj3fgk5

Reactances and Resistances:

XC1 = 2pifC11

XC1 = (2)(pi)(60 Hz)(4.7 F)1

XC1 = 564.38

XL = 2pifL

XL = (2)(pi)(60 Hz)(650 mH)

XL = 245.04

XC2 =1

2pifC2

XC2 = (2)(pi)(60 Hz)(1.5 F)1

XC2 = 1.7684 k

R = 470

ZC1 = 0 - j564.38 or 564.38 -90o

Z L = 0 + j245.04 or 245.04 90o

Z C2 = 0 - j1.7684k or 1.7684 k -90o

ZR = 470 + j0 or 470 0o

g`-4_?`[?_[!

E

I

Volts

Amps

OhmsZ

TotalC 1 L C 2 R

470 + j0470 0o

120 + j0120 0o

0 - j564.38564.38 -90o

0 + j245.04245.04 90o

0 - j1.7684k1.7684k -90o

4?4[ !4

4

:

4[!CN-P?*C?4*?5?

-?4#

!4 5?"

#

?454[?4s

?4_[P4?

4?543P[J[4P[?34

5???_?44[?[4X[PNNCP[

?`[`

+

???!

p

O++!5 [?-?N[P[4???OC[[

?C`??!s-???-N?55[[5?4NO?!?P

P_[!?4+-[`--

p

P[*4 414?

!4?[NC!?-?`4N5?

flU`hTLiRh5`f

jflk

Gfgk5nm++lflk

ohrje3flk*

E

I

Volts

Amps

OhmsZ

L -- C 2 R // (L

-- C2) C

1 -- [R // (L -- C2)]Total

??u4[?:w5P?N[ [? ?4`

[?

v

4[?4

v

?*4[5s[4

_[4_

4*#?`??X`?N[?43?_?P

E

I

Volts

Amps

OhmsZ

L -- C2 R // (L -- C2) C

1 -- [R // (L -- C2)]Total

0 - j1.5233k1.5233k -90o

429.15 - j132.41449.11 -17.147o

429.15 - j696.79818.34 -58.371o

120 + j0120 0o

Rule of seriescircuits:

Rule of parallelcircuits:

Rule of seriescircuits:

ZL--C2 = ZL + ZC2

ZR//(L--C2) =

ZR ZL--C211

+

1

Ztotal = ZC1 + ZR//(L--C2)

qCX]`G]`(X]3o)l]L']t3`t

v]

)

v

#lX'']'o'3]

']'#3'W']t

3tX3]3)+'o]e]33o](X']t+t(('+]3

A'3r#CA'o()t:X]3uxXx3

]! d

x

L

E)wg3(o(]']]t']]|uWE|])

w\X]g gV9u

p(

\w+]((]t|'

v

]']

v

]('(]j]'

(]']>'(XLX

L8J8dgV `rj

E

I

Volts

Amps

OhmsZ

L -- C 2 R // (L

-- C2) C

1 -- [R // (L -- C2)]Total

0 - j1.5233k1.5233k -90o

429.15 - j132.41449.11 -17.147o

429.15 - j696.79818.34 -58.371o

120 + j0120 0o

76.899m + j124.86m146.64m 58.371o

OhmsLaw

I = E

Z

#>'VjtLV#`Wg])WXtX:'+LXW(]V>'(X(`]CgjL'>]V]jL ]tj(]j

+)R(3't(3(|()#])]|]'!()#'(XL

p

(W:'

3 '|j)tX

3(]|

u

wWj'}(3r'ou (]']w#3('LXC(t3l'(]':o:X33o:(X

(3d()t:X]3E ('XX

LjX()3Wr'3'(]']`''LtL(:'

t3oj

E

I

Volts

Amps

OhmsZ

C 1 L C 2 R

470 + j0!470 0o

0 - j564.38"564.38 # -90o

0 + j245.04"245.04 90o

0 - j1.7684k"1.7684k -90o

76.899m + j124.86m$146.64m 58.371o

Rule of seriescircuits:

I% total = IC1 = IR//(L--C2)

E

I

Volts

Amps

OhmsZ

L -- C2 R // (L -- C2) C

1 -- [R // (L -- C2)]Total

0 - j1.5233k"1.5233k -90o

429.15 - j132.41449.11 -17.147o

429.15 - j696.79818.34 -58.371o

120 + j0120 0o

76.899m + j124.86m&146.64m 58.371o

76.899m + j124.86m$146.64m 58.371o

Rule of seriescircuits:

I%

total = IC1 = IR//(L--C2)

XeXX']t']]8']3>'''

3g'`('tE

3)3'tr8

]

u

w

3(t3g3 } 9Tu

gp

\w]((]t(LW']r]3X

)(

lTL*l++,-./10`-324rje5

E

I

Volts

Amps

OhmsZ

C 1 L C

2 R

470 + j0!470 0o

0 - j564.38"564.38 # -90o

0 + j245.04"245.04 90o

0 - j1.7684k"1.7684k -90o

76.899m + j124.86m&146.64m 58.371o

OhmsLaw

70.467 - j43.400$82.760 6 -31.629o

E = IZ7

E

I

Volts8

Amps9

OhmsZ

L -- C 2 R // (L

-- C2) C

1 -- [R // (L -- C2)]Total

0 - j1.5233k"1.5233k -90o

429.15 - j132.41449.11 ! -17.147o

429.15 - j696.79818.34 -58.371o

120 + j0120 0o

76.899m + j124.86m&146.64m 58.371o

76.899m + j124.86m&146.64m 58.371o

OhmsLaw

49.533 + j43.40065.857 : 41.225o

E = IZ

L)8]3t

)3X)>#]'`':]L##]3:g((+(X]`'g]t']]l8']j

](L(

]3(l('X

3]']tC3(]o\

]

u

wt3X(r'(]';l])8

(=

EB total should be equal to EC1 + ER//(L--C2)70.467 - j43.400 VC

49.533 + j43.400 V+120 + j0 V Indeed, it is!

3]W}d']'|o'X]8(]V

p

o']+'C|W(j+]#(3+r3]XC(3'

3)]:]((3d#X(']EeW]jj(L(

)X)8e]g(3r(9gjX'(]+] #]'

j|3XX'(]('3W()'t|C|8L'C('t(o(rj3VW(X!] '']

+ '39Ct(]]Eo]]]t']]+8']3](L(

3o'l3(]

u

wC#lLW

]3'(]Ee'`3XW(]>+3](X+]:]XL'Wj'l'(]o])l]+('LD

p

|'+X](]L(3

'X(W(gugw`]3(3g3(]V(gt33']3'rWE]jj]t(gu

wWj'+(r'

])]:XX3(+(3(lW'`>o:X33X`](+3]']t3+'X)o](XX

X(]']]#lX()jdX

(3]]t']]g3]3(FEd3WlWtX|]+L(o(]3#]}A]3

L8J8dgV `rj

E

I

Volts

Amps9

OhmsZ

C 1 L C

2 R

470 + j0470 ! 0o

0 - j564.38"564.38 # -90o

0 + j245.04"245.04 G 90o

0 - j1.7684k"1.7684k -90o

76.899m + j124.86m&146.64m 58.371o

70.467 - j43.400&82.760 -31.629o

49.533 + j43.400H65.857 : 41.225o

Rule of parallelcircuits:

E7 R//(L--C2) = ER = EL--C2

E

I

Volts8

Amps9

OhmsZ

L -- C 2 R // (L

-- C2) C

1 -- [R // (L -- C2)]Total

0 - j1.5233k"1.5233k -90o

429.15 - j132.41449.11 ! -17.147o

429.15 - j696.79818.34 -58.371o

120 + j0120 0o

76.899m + j124.86m&146.64m 58.371o

76.899m + j124.86m&146.64m 58.371o

49.533 + j43.4006