engineering lessons in electronic circuits ii ac
Post on 29-Mar-2016
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DESCRIPTION
TRANSCRIPT
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length = 5.66angle = 45 degrees
length = 9.43
(-57.99 degrees)angle = 302.01 degrees
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AmplitudeLength
Waveform Vector representation
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K Kronn r
A
B
Phase shift = 90 degrees A is ahead of B
(A "leads" B)
B APhase shift = 90 degrees
B is ahead of A(B "leads" A)
A
BPhase shift = 180 degreesA and B waveforms are
mirror-images of each other
A
B
Phase shift = 0 degreesA and B waveforms are
in perfect step with each other
(of "A" waveform withreference to "B" waveform)
B
A
B
A
BA
A
B
90 degrees
-90 degrees
180 degrees
Waveforms Phase relations Vector representations
BA
B
A
phase shift
angle
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LOZHGIFqNFKGgNDL_ZKNQOGgQ]NhKT$SbGQ!U\Q^`T_VKULOT$^`T_UiW|[S$ZKNQ]GFL_NBVXGgL_ZKGgjKZH^Q]GT_GbD{GIT_GIFaSbGrD{NT^iiNLOZKGTsYG^QOhKT_GIsYGIFL$Q
UWFL_ZKGoSbUWT$SbhKULw
ZKU\QoSbNFHSbGjPLtND^qT_GbD{GTOGFHSbGcjXNUWFLtU\QgFKN`LhKFKiWUvGcLOZH^`LtN`DL_ZKG`pTONhKFHJKjXNUWFLUF^0SIUT$SbhHUxLgD{NToLOZHG
VXGIFKGIKLNDkNiL_^pGgT_GbD{GIT_GIFaSbGw!ULOZM^cSIiGR^`T_i|uJPGbaFKGJjXNUWFLUWFLOZKGSbUWT_SIhKUxLrJPGSIiW^TOGRJuLON[VXGt`pTONhKFHJf YUL
VXGSbNsYGQjXNQ_QOUVKiWGLONoL_^`iWv^VaNhPLkNiL_^`pG`NFHrNT^lL_gQOUWFKpiWGjaNUFL$QUWF[^oSbUWT_SIhKULfVXGIUWFKpohHFHJPGIT$Q]LONNPJBLOZH^`L
LOZHNQOGkNixL$^`pGQp^`iWd!^z|PQgT_GIi\^lL_UkG[VaGILdnGGIF
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SbhHTOT_GIFL_QUF[^FYSIUT$SbhKULZH^zkUWFKpoJPGbaFKUxL_GjKZH^Q]G^`FKpiGRQIw}K^`sYjKiWGlLOZHGSbhKT_T_GIFL}LOZKT_NhHpZYTOGRQ]U\Q]LONTUWQ
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-
- nH0n0 n
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d!^zkGdZKUWiGLOZHG^FKpiWGrN`D^kGRSL_NT!TOGjKTOGRQ]GFL_QLOZKGjKZH^Q]G^`FHpiWGNDLOZHGrd^zkGTOGiW^`LOUWkGLONcQ]NsYGNLOZKGT
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GsBGstVXGITYL_ZH^lLckGRSL_NT$Q^TOGusc^lLOZHGIsc^lL_UWS^`iNVK~GSL$Q~hHQLciWUvGqFhHstVXGIT$QNFy^MFhKstVXGITciWUWFKGLOZKG|SI^F
VXGq^JKJPGRJfnQOhKVPL_T_^SLOGRJfshKiLOUWjKiUWGJf!^FHJJKUkU\JPGJwJHJPUxL_UNFUWQYjXGIT_ZH^`jaQtL_ZKGqG^Q]UWGQ]LckGSbLONTNjaGT_^`LOUWNF
LONukUWQOhH^iUIGfQ]NudnG
iWiVaGpUWFdULOZMLOZH^`Lw(EDnkGRSL_NT$QdULOZmSbNsYsYNFe^`FKpiGRQ^TOGY^JHJPGJfL_ZKGIUWTsc^`pFHUxL_hHJPGQ
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length = 6angle = 0 degrees
length = 8angle = 0 degrees
total length = 6 + 8 = 14
angle = 0 degreesUWsYUi\^`T_i|fUDkNixL$^`pGQONhHT_SIGQ}dULOZcL_ZKGQ_^`sYGjKZa^QOG^`FKpiG^`T_GSbNFHFKGSbLOGJcLONpGbL_ZKGITUWFQOGIT_UGRQIfLOZKGUT
kNixL$^`pGQ^JKJt~hHQ]L^Q|NhusBUWpZLGIjXGSbLdUxL_ZVH^lLOLOGTOUWGQ
0 deg 0 deg
0 deg
- + - +
- +
- + - +
- +
6 V 8 V
14 V 14 V
6 V 8 V
iWG^Q]GFKNLOGL_ZKG E
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LrZH^zkGojaNiW^TOUL|PYUFL_ZKGQ_^`sYGtQOGIFaQ]GgL_ZH^lLrJPNGRQIfaLOZKGRQ]Gsc^`T_vQ^`T_GoGQ_Q]GFLOU\^`iL_N
vFKNldUWFKp[ZKNldL_NT_GbD{GIT_GIFaSbGoL_ZKGBpUWkGIFjHZH^QOG^FKpiWGQN`D}L_ZKGkNiL_^pGQwZKU\QrdUiWiVaGRSbNsYGBsYNT_Gt^`jKja^`T_GIFL
UWFL_ZKGgFKGILGIK^`sYjKiWGw
EDkGSL_NT$QnJKUT_GSbLOiW|cNjKjXNQOUFHptG^S$Z[N`L_ZKGIT]
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ha^`FLOULOUWGQQOhKVPL_T_^SLdZKGIF^JKJPGJ
length = 6 angle = 0 degrees
length = 8
total length = 6 - 8 = -2 at 0 degrees
angle = 180 degrees
or 2 at 180 degreesUWsYUi\^`T_i|fKUDNjKjXNQOUFKp[kNiL_^`pGgQ]NhKT_SIGQ^TOGoSINFKFKGRSL_GJuUF0QOGIT_UWGQfPLOZKGUTkNixL$^`pGRQQ]hHVPLOT$^SbL^Q|Nh
sYUpZLGbPjaGRSLdULOZ0gVH^`L]L_GIT_UGRQSbNFHFKGSbLOGJUWFq^FNjKjXNQOUFKpBD^Q]ZKUWNF
-
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0 deg- + - +
- +
- + -+
-+
180 deg
180 deg
6 V 8 V6 V 8 V
2 V 2 V
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JPU\^`pT_^sQ]GGIsLONcUWFHJPU\SI^lL_Go^JKJKUxL_UkGgkNiL_^pGRQD{T_NsiWGbDLLONcT_UpZLRfKd!GoQ]GGn^`FHJ0NFuLOZHG
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f
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^YjKZH^Q]G^`FKpiGN`D^`FHJuL_ZKGoN`L_ZKGIT^cjKZH^Q]Go^FKpiWGgN`D
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0 deg- + - +
- +
- +
-+
180 deg
6 V8 V
6 V8 V
2 V 2 V
0 deg
- +
NLOGZKNld,LOZKGujaNiW^TOULOUWGQ^`jKjXG^TtL_NVXGNjKjaNQ]GRJmLONMG^S$ZN`LOZHGITBFKNldgf}JPhKGLON0L_ZKGT_GIkGIT$QO^iN`DdUT_G
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LOZHGI|BLOT_hKiW|Y^TOGNjKjXNQOGJBL_NtNFHG^`FHN`LOZHGITRf^`FaJYLOZKGNlkGT_^iiKGIXGRSL!UWQLOZKGrQO^sYG^QLOZKGD{NT_sYGITnQ_SbGFH^`T_UWNodUxL_Z
^JKJPULOUWkGRtjXNi\^`T_UxL_UGRQ^`FHJuJPUXGTOUWFKpcjKZa^QOG^FKpiWGQ}^LON`L$^`ikNiL_^`pGNDNFKiW|
kNiL_Qw
-
nH0n0 n
0 deg- + -+
- +
180 deg
0 deg
0 deg+ -
6 V 8 V
2 V
2 V
Just as there are two ways toexpress the phase of the sources,there are two ways to expresstheir resultant sum.ZKGT_GQOhKiL_^FLnkNiL_^pGS^`FcVXGGbPjKT_GQ_QOGJYUWFcLdnNoJKUxGIT_GIFLnd^z|PQI
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8 V180 deg- +
8 V-+
0 degThese voltage sourcesare equivalent!
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^BjaNLOGFL_UW^iiW|SINFPD{haQ]UWFKpcQ]hHVP~GSbLwD{GIdUWiiWhHQ]LOT$^lLOUWNFaQnsc^z|[ZHGIiWjL_N[Sbi\^`T_UxD{|[LOZKGgU\QOQOhKGw
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jKZH^Q]GwELU\QNFKiW|dZHGIFqdnGoSINsYjH^TOGULdUxL_ZN`LOZHGITd^zkGbD{NTOscQnLOZa^lLd!GgSI^`FQ_^`sYGo^`F|LOZHUFKpYsYG^FKUWFKp`D{hKi
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QOUsYjKiW|cLONBjKT_NlkUWJPG^tD{T_^sBGNDTOGID{GIT_GIFHSIGD{NTjKZH^QOG^`FKpiGRQIfT$^lLOZHGIT!LOZH^FL_NB^SL_hH^`iWi|[JKGIFKNLOGjaNiW^TOUL|cN`D^
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GIiWGSbLOT_NFKU\SIQnd!NT_vXfT_GJcL|jKU\SI^iiW|cT_GIjKT_GQOGIFL_Qn`jXNQOUxL_UkG^`FHJVKiW^S$vYL|jKUWS^`iWi|YT_GIjKT_GQOGIFL$QnFKGIp^lLOUWkGw tNGQ
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-
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V
V A
AOFF
6 V
COMA
V
V A
AOFF
6 V
Test lead colors provide a frame of referencefor interpreting the sign (+ or -) of the metersindication.
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D{T$^`sYGtNDTOGID{GIT_GIFHSIGgD{NTrUFL_GIT_jKTOGILOUWFKpcL_ZH^lLgQ]NhKT$SbG
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QOGIT_UGRQIf^`FaJ0d!Gd^`FLL_NL_TO|qL_NuJKGbLOGTOsYUWFKGYdZH^lLrLOZHGtL_N`L_^iTOGRQ]hKiL_^FLrkNiL_^`pGoU\QV|T_G^JPUFHp[LOZHGBkNiL_^pG
^SITONQOQG^S$Z0NFKG{UWFHJPUWkUWJKhH^`iWi|
dUxL_Ze^kNixL_sBGILOGTfXLOZHGIFeGIULOZKGT^JKJPUWFKpNTrQOhKVPL_T_^SL_UFKpusc^lL_ZKGIsc^lL_UWS^`iWi|
LONBHFaJuLOZKGg^`FaQ]d!GITRw}UWT$QLRfPdnGgd!NhKi\JsBGR^QOhKT_GL_ZKGgkNixL$^`pGrN`DNFHGQONhHT_SIG
-
[ nH0n0 n
COMA
V
V A
AOFF
Source 1 Source 2
Total voltage?
The meter tells us +24 volts
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T_G^`iWiW|[UWQ!L_NhHS$ZHUFKpBL_ZKGgjaNQ]ULOUWkGw}ZhaQIfKd!GvFKNldQONhKT$SbGgUWQ^BVH^`L]L_GIT_|cD^SIUFKpcUWFLOZKU\QNT_UWGIFL_^`LOUWNF
Source 1 Source 2
Total voltage?
24 V
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-
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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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LOZHGQ_^`sYG!jKhKT_jaNQ]G^QL_ZKGTOGRJt^FHJtVHiW^S$voSbNiWNT$QN`DX^rgmkNixL_sYGbLOGT
QLOGRQL}iWG^JHQIwEDHL_ZKGIT_Gnd^QQOhHS$ZY^L_ZKUFHp
^Q^u`jKZa^QOGIsYGbL_GIT$cLOZa^lLoSbNhKiWJeJPUWTOGRSL_i|UWFHJPU\SI^`LOGYjKZH^Q]GY^`FKpiG{ULod!NhKi\J0T_G
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QONhKT$SbGrLON[^SL^Q^YTOGID{GIT_GIFHSIGrUWFuLOUWsYG
faUxLrSbNhHiWJujKT_NlkUWJPGrLd!NcJPUxGIT_GIFLUWFHJPU\SI^`LOUWNFHQNDjHZH^QOGo^`FKpiGdZKUWiWG
sYG^Q]hKT_UWFKptL_ZKGgd^zkGbD{NTOsNDL_ZKGoQO^sYGQONhKT$SbGfJKGIjXGIFHJPUWFKpcNFuZHNldd!GSINFKFHGSL_GJLOZKGL_GQ]LiGR^JKQ
-
` nH0n0 n
180 deg
- +0 deg
+ -
redblack
blackred
"phasemeter"
"phasemeter"NafPdZKGIFq|NhuQOGIGg^FukNixL$^`pGJPGRQ]UWpFH^`LOGRJ^Q!QOhHS$Zmwww
+-0 deg8 V
www}UxLsYG^FHQ!LOZa^lLNhKTZ|jXN`LOZHGbLOU\SI^i`jKZH^Q]GsYGbLOGT_d!NhKi\J[T_G^JjKZa^QOGQOZKUDLdUxL_ZuUL_QFKGIp^lL_UkG
{VHiW^S$v
LOGRQLiGR^JBL_NhHS$ZKUWFKptLOZKGriGIDL!dUWT_G^`FHJ[UxL$Q!jaNQ]ULOUWkGRp{T_GJ
LOGRQLiWG^JcLONhaS$ZKUFHpgLOZKGrT_UpZLndUT_GfLOZHG
^FHJ
Q]|sVaNiWQsYGTOGi|[jHTONlkU\JPUFHpY^D{T$^`sYGgN`DT_GbD{GIT_GIFaSbGrD{NTL_ZKGgjKZH^Q]Go^FKpiWGrHphHTOGw%ELdnNhKi\J^iWQON
VXGojaGT]D{GRSL_i|Nvl^z|cLONcT_GIjHTOGRQ]GFLL_ZH^lLkGIT_|QO^sYGoQ]NhKT_SIG
QjKZH^Q]GiWUvGLOZKU\Q FKN`L_UWSIGLOZKGtQOdUL_S$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
TOGrQ]GGIUWFKpZHGIT_GdULOZLdnNYykNixL$^`pGQQOiWUpZL_i|cNhPLN`D
jKZH^Q]GwgkNiL_^pGRQS^`FYNFHi|tJKUT_GSbLOiW|^`U\JBNTJPUWT_GSL_i|tNjKjXNQOGf`dULOZcFHN`LOZHUFKpgUWFYVaGILdnGGIFwULOZ[flLd!N
kNixL$^`pGQS^`FoVXG^`U\JPUFHpNTNjKjXNQOUWFKpNFKGn^FKN`L_ZKGIT
E`H#E`O$
VaGILdnGGIFgD{hKiWiW|^`U\JPUWFKp^FHJgD{hHiiW|NjKjaNQ]UWFKpaf
UWFHSbiWhHQOUkGwUxL_ZKNhKLnL_ZKGhHQOGNDkGRSLONTSbNsYjKiWGbcFhKsVaGT
FKN`L$^lLOUWNF[LONJKGQ_SbT_UVXG
hH^FL_UxL_UGRQIfUL!d!NhKi\J
VXG
zP
JKU[SbhHixLL_NYjaGT]D{NTOssc^`LOZKGsY^`LOU\SI^iSI^iWSIhKiW^`LOUWNFHQnD{NTSIUT$SbhKUL^`FH^i|PQOUWQw
FYLOZHGFKGILnQ]GRSL_UNFfd!G
iWiaiWG^TOFcZKNldyL_NtT_GIjKT_GQOGIFLkGRSL_NT
hH^FL_UxL_UGRQUWFQO|stVXNiWUWST$^lL_ZKGITLOZH^F[pT$^`jHZP
U\SI^`iD{NTOsqw}GSL_NTg^`FaJ0L_TOU\^`FKpiGYJPU\^`pT_^sYQrQOh[SbGBLONqUWiiWhHQ]LOT$^lLOGBL_ZKGYpGIFHGIT$^`iSbNFaSbGIjKLfVKhKLosBNTOGBjKT_GSIUWQOG
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-__
JPGIjXGIFHJKUFKpcNFL_ZKGgjKZH^Q]GgQOZKUDLVaGILdnGGIFuLOZKGsqw
?V (P# %!A#BC\oArA%\(
F0NT_JPGTLONdnNTOvdUxL_ZLOZHGQOGSbNsYjKiGIuFhKsVaGT_QdUxL_ZKNhKLrJPT$^zdUFHpckGSbLONT_QfHd!GgHT$QLFKGIGRJ0Q]NsYGovUFHJ0N`D
Q]L_^`FaJK^`T$JYsc^lLOZHGIsc^lL_UWS^`iHFHN`L_^`LOUWNFw}ZKGTOG^`T_GLdnNtVH^QOUWSD{NT_scQN`DSINsYjKiWGbYFhKstVXGIT!FKN`L$^lL_UNF
H
^FHJ
OP `bo
w
Ni\^`TD{NT_s UWQrdZKGTOGB^SINsYjKiWGbFhKstVXGITU\QrJKGIFKNLOGJV|L_ZKG
b-{#
{NLOZKGTOdU\Q]GvFKNldF^QLOZHG
HI
HMn
f
:IbE Z- H
fHNT
cb 0
^`FaJL_ZKG
l-W
N`DUxL$QkGRSL_NT#{haQ]hH^iiW|JKGIFKNLOGJV|^Fq^FKpiWGgQ]|sVaNi
LOZa^lLiNNvQiWUWvGLOZKU\QI
wNYhHQOGL_ZKGgsc^`jq^`FH^iNp|fjXNi\^`TFKNL_^`LOUWNF[D{NTLOZKGgkGSbLONT!D{T_NsGIdNT_vnUxL|
LONtK^`FmUWGIpNdnNhKiWJMVXGQ]NsYGbLOZHUFKpqiWUvG[
`
sYUiWGQfQONhPL_ZdnGRQLRw GTOG[^TOGLd!NGbK^`sYjKiWGQND!kGSL_NT$Q
^`FaJ[L_ZKGIUWTjXNi\^`TFKNL_^lL_UNFHQ
-
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
{ZHNT_UoNFL_^i
fsc^vUWFKp
cQ]LOT$^`UWpZLohKjf
BLON0LOZKGiGIDLf^FHJ
[
YQ]LOT$^`UWpZLJPNldFwiWG^Q]GcFKNLOGLOZH^`L
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
[
HQL_T_^UpZLJPNldF
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$^`jKZqSbNTOT_GQOjXNFHJKQnL_N[^`FKNLOZKGT
FhKstVXGITiWUWFKGoQ]ULOha^lLOGRJq^lL
L_NYLOZKGoTOGR^`i\tNFKGw#}GRSL_NT$QVaGUFKpBLd!N`JPUWsBGFHQOUNFH^`iXLOZHUFKpQIfHdnGsthaQLZH^zkG
^LdnNEJKUsYGIFaQ]UWNFH^i`sc^`jHBhKjXNFqdZKU\S$ZuLONYGbPjKT_GQ_Q!LOZHGIsqfPLOZhHQLOZKGLd!NYFhHstVXGITiWUFKGRQjXGIT_jaGFHJPU\SbhKi\^`TL_N
G^S$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
ULOZHGITtsYGILOZKNPJNDFKN`L$^lLOUWNFUWQkl^`iWUWJmD{NTSINsYjKiWGbFhKstVXGIT$QIwMZKGjKT_UWsY^TO|eT_G^QONFmD{NTtZH^zkUWFKpLd!N
sYGbL_ZKNPJKQND}FKN`L$^lLOUWNFU\QD{NTGR^QOGgN`DiWNFKpZH^`FaJuS^`i\SbhKi\^lL_UNFfXTOGRSL_^FKphHiW^TD{NT_siWGIFHJKUFKpUxL$Q]GixD}L_N^JHJPUxL_UNF
^`FaJQOhKVPL_T_^SLOUWNFfK^`FHJujXNi\^`T!D{NT_siWGIFHJKUFKpYUL_QOGIiDL_NcsthKiLOUWjKiWUWS^lL_UNFq^FHJuJPUkU\Q]UWNFw
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
LOZHGZ|jaNLOGIFhHQOGNDXL_ZKGL_TOU\^`FHpiWGTOGjKTOGRQ]GFLOUWFKpLOZKGkGRSLONTUL_QOGIiD%UxL$QiWGIFHp`LOZ^`FHJc^FKpiWGdUxL_Z[T_GQOjaGRSLL_NoLOZHG
ZKNTOUINFL$^`iHSINFHQ]LOULOhKLOUWFKpgLOZHGjXNi\^`TD{NTOs
fL_ZKGZKNTOUINFL_^`iK^FHJckGITOLOU\SI^`iaQ]U\JPGQTOGjKT_GQOGIFLOUWFKpLOZHG`T_G^iW^FHJ
`UWsc^`pUFH^TO|PtTOGRSL_^FKphHiW^TSINsYjXNFKGFL$QIfPT_GQOjXGSL_UkGIiW|
-
KaY2aI
+j3
+4
length = 5
angle =36.87o
(polar form)
(real component)(imaginary component)
4 + j3 (rectangular form)
(5)(cos 36.87o) = 4(5)(sin 36.87o) = 3
5 ) 36.87o
NSbNFkGITOLaD{T_NsTOGRSL_^FKphHiW^TLONjXNi\^`TRfbHFHJrL_ZKGjXNi\^`Tsc^pFKULOhHJKGLOZKT_NhKpZL_ZKGhHQOG}N`DL_ZKG|L_ZH^`pNT_G^`F
ZKGNT_GIsL_ZKGjaNiW^Tsc^pFKULOhHJKG!U\QLOZKGZ|jaNLOGFhaQ]G!N`Da^T_UpZLLOT_U\^`FKpiGf`^`FHJoLOZKGT_G^`iP^FHJoUWsc^`pUWFH^TO|gSbNsB
jXNFKGFL$Q^TOGoLOZKGB^Jz~]^SbGIFL^`FHJqNjKjaNQ]ULOGBQ]U\JPGRQIfXTOGRQ]jXGSbLOUWkGi|
f^`FaJqLOZKGB^`FKpiGtV|uL$^`vUFHpcLOZKGY^`T$SL$^`FKpGIFL
N`DLOZHGgUsc^`pUFa^`T_|[SbNsYjXNFKGFLJKUkU\JPGJuV|cLOZHGgTOGR^`iSbNsYjaNFKGIFL
4 + j3* (rectangular form)
c =+ a, 2 + b2 (pythagorean theorem)
polar magnitude = 4* 2 + 32
polar magnitude = 5)
polar angle = arctan, 34*
polar angle =
(polar form)
36.87o
5 ) 36.87o
b&%
-
o-
FKNL_^lL_UNF[JPGIFHN`LOGRQn^tSbNsYjKiWGbYFhKstVXGIT!UWFcLOGTOscQnN`DUxL$QkGSbLONT
QiWGIFKpLOZ^FHJc^`FKphKi\^`TnJKUT_GSbLOUWNF
D{TONsLOZKGoQ]L_^T]L_UFHpBjXNUWFLwP^sYjKiGH|
sYUWiGRQ
{eGQ]LV|dNhKLOZdnGRQL
w
/.
\lb
FKNL_^lL_UNFJPGFKN`L_GQ!^SINsYjKiWGbcFhKstVXGIT!UFLOGTOscQnN`DUL_Q!ZKNTOUINFL$^`iX^`FHJckGITOLOU\SI^iXJPUWsYGIFP
Q]UWNFHQw}}K^`sYjKiWG}JPT_UWkG
rsYUiWGQeGQ]LfKLOZKGFL_hKT_Fq^FHJuJPTOUWkGY
sYUWiGRQNhKLOZw
-
nH0n0 n
FeT_GSbL_^FKphKi\^`TgFHN`L_^`LOUWNFfLOZKG[HT_Q]L
hH^`FL_UxL|MUWQL_ZKGTOGR^`i\uSINsYjXNFKGFLZKNT_UINFL$^`iJPUWsYGIFHQOUNFmN`D
kGSbLONT
^`FHJLOZKGgQOGSINFHJ
ha^`FLOUL|[U\Q!LOZKGg`UWsc^`pUFH^TO|PoSINsYjXNFKGFL#kGITOLOU\SI^iXJKUsYGIFaQ]UWNFuN`DkGSbLONT
w
ZKGUsc^`pUFa^`T_|YSINsYjXNFKGFLU\QjKT_GSbGRJPGJV|^BiNld!GITOSI^QOGb~bf YQONsYGbL_UsYGQS^`iWiGRJ[L_ZKGg~!NjXGIT$^lLONTw
nNLOZYjXNi\^`Tn^FHJBTOGRSL$^`FKphKiW^T}D{NT_scQ}N`DFKNL_^`LOUWNFYD{NTn^oSbNsBjHiGIFhKstVXGIT!SI^FcVaGT_GIi\^lL_GJYpT_^jKZKU\SI^`iWiW|
UFLOZKGD{NTOsN`D^BT_UpZL!L_TOU\^`FHpiWGfPdULOZLOZKGgZ|jaNLOGFhaQ]GT_GIjKT_GQOGIFLOUWFKpLOZKGgkGSbLONTUL_QOGIiDDjaNiW^T!D{NT_su
Z|jXN`L_GIFhHQOGiGFKp`L_Z10sY^pFKULOhaJPGP^`FHpiWGdUxL_Z[TOGRQ]jXGSbLLONZKNT_UINFL$^`iHQOUWJKG20^`FKpiG
fLOZKGZKNT_UoNFL_^i
Q]U\JPG0T_GIjKT_GQOGIFL_UFKpmL_ZKG0T_GSbL_^`FHphKi\^`TTOGR^`i\mSbNsYjXNFKGFLf^`FaJLOZKGkGITOLOU\SI^iQOUWJKG0TOGjKT_GQOGIFLOUWFKpmLOZHG
TOGRSL_^FKphHiW^T!Usc^`pUFa^`T_|SbNsBjXNFHGIFLw
?V3 3P C54UA76 A~\
UWFHSbGMSbNsYjKiWGbFhKstVXGIT$Q[^TOGiWGIpUxL_Usc^lL_G0sc^lL_ZKGIsc^lL_UWS^`iGFLOULOUWGQf~hHQLuiUWvGMQOS^`i\^`TcFhKstVXGIT$QfnL_ZKGI|SI^F
VXGY^JKJPGRJfQ]hHVPLOT$^SbLOGJfsthKiLOUWjKiWUGRJfJPUWkUWJKGJfQ
hH^`T_GJfXUWFkGT]L_GJf^FHJQOhHS$Zf~hHQ]LoiWUvGB^F|qNLOZKGTrvUWFHJMN`D
FhKstVXGITRwNsYGQOSIUGFLOUaSSI^`i\SbhHiW^`LONT$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
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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$ZUWQ[dnGiiVXGI|NFHJyL_ZKG0Q_SbNjXG0NDL_ZKUWQJPU\QOSIhHQOQOUWNF
w!GQOhKTOGqL_ZH^lL[|Nh
T_GqD^`sYUiWU\^`T[dULOZL_ZKG0VH^Q]U\S
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iWi
ZH^zkGiWUxLOLOiWGgLOT_NhHVKiGdULOZSbUWT_SIhKUxL^FH^`iW|PQ]U\QIw
-
r nH0n0 n
b&%
No^JKJYSbNsYjKiWGbBFhKstVXGIT$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$SbGRQ^`T_GrNTOUWGIFL_GJ[UWF^`F^JHJPUxL_UkGsY^FKFKGT
-
nH0n0 n
15 9.8298
9.6442+ j6.8829 V- j19.7735 V
+ j0 V
+
34.4740 - j12.8906 V>
15 V 0o = 15 + j0 V
12 V 35o = 9.8298 + j6.8829 V
22 V 8 -64o = 9.6442 - j19.7735 V
FjXNi\^`TD{NTOsqfKL_ZKU\QG
hH^`LOGRQLON
w
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kNiL_QI
Kw
H
IwZH^`LLOZHUWQsYG^`FaQUFT_G^iLOGTOscQUWQLOZH^`L
LOZHGkNixL$^`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
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HIHMn+
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LONc^SLOha^`iSIUT$SbhKULsYG^QOhKT_GIsYGIFL$QIw
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kl^`iWhKGtU\Q
hHUxL_G^`T_VKULOT$^`T_|wEL
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^`Fa^`iW|QOU\QIwiWQONHfRL_ZKGnS$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
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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
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Dm
r
w
d[dndWd!a m
w
r[^aid!aq[j
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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
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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$1EL_QSbNFLOT_UVKhKLOUWNF
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-
7Pe
load
+
-
-
+
-
+
E1
E2
E3
Polarity reversed onsource E2 !
22 V -64o
12 V 35o
15 V 0o
NLOGZKNldLOZKGc
kNiLQ]hKjHjKi|^ QjKZH^Q]Gg^FKpiWGUWQQ]LOUWiiTOGID{GIT_TOGRJcLON[^Q
zfKGIkGIFLOZKNhKpZLOZHGgiGR^JKQZH^zkG
VXGIGIFcT_GIkGIT$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_$wi_P_KPW ioUK!@iCDPUP
D KK I!WW @Bi_P_ -DPUDKKA!@iWxKBD K
PB2K !WW
_KB[@W@B i_KW_!UUKPK_KD@W@Ui#[K2PP UBiUiyxP
UAD@B=DPWAAK2BD PU-DPUDP@UBK P_UABUK 2BnB0P@K i_
1IKDU B_ BP2P@PoUD_PBU_1PP UKP_iIKBiKBUUiB#ZK[K KKW
PiUP U{iWWKDKP_iBPBUAKW_bPi@azUW(UK #W!Bi
PP UDUPKDP]WP o D_PB
1PP Uo UK!WW DP_PU2 PB[D_P_o_1UK!WU _iP_K KB[
UiKBD_{PPD lDnK!|PKU _]D_KWK PUWD_KiUP_UPPUP Ki U(lLK
UP[U1UBP@P Ki[ Ui D_PBBDKK}!iiW@_]iK
KPWKDBiB Ki[UP UUUUb_P AKP_i iB !@Bii_PI_K}!iiP o
APUK~}^BB[APU
[
-
1Z gB.6"4X;h.Cihj.@6k.l;TPam +
BKiW# KPUBD P #K[PUn KUWD_UkUWDnKiDnPPKPW
iU UP ] WK!NxPU W_keBKiW 1KIiUI BK K[P_ PIWUUU
UWDno]D[KiUP UPH
Le 10 mH10 V60 Hz
XL = 3.7699 (inductive reactance of 10 mH inductor at 60 Hz)
I = E
X
I = 10 V3.7699
I = 2.6526 A@BWB_KiiUiBP_WU_W_ Ki[_KUKPWKB7WK@
iUB7K1UWD_U PUI KPUK_kl*
,q
ioI K KB[1iKoi[APi
KPWI_KWoUUWD_U_P Ki[@PB@i_PAWU UK!nK Di! |PPP_
_1PP U UK!WW KB[PUIKPWUKUn[$
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"
- 1Z1ZmZ/836/7m
-
+
.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
- 1Z1ZmZ/836/7m
-
+
.0"0/721B3/7.#" .0/:"4657/843" .0/:9 64@/
_Kih|PUPo_UiBIDUA_!UiUBPD K BKWK WUK D!3D!BK| _
UP PK
- 1Z1ZmZ/836/7m
-
, .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[
-
$KUoD_KWKU Y_KBUDK
UWB WKKo WU B U WKIo U2 K BK UPP B_WU
KUB
+
- 1Z('Zy3*)#)T/+)M3/7mZmCih64
-
.0"0/721B3/7.#" .0/:"4657/843" .0/:9 64@/
E
I
Volts
Amps
OhmsZ
R L Total
5 + j05 0o
0 + j3.76993.7699 90o
10 + j010 0o
10 + j010 0o
10 + j010 0o
Rule of parallel circuits:
Etotal = ER = ELG@lBU_PKAK 0%{]sIl`=y{UB BU222BUKP_PKDKUBUBK @K KB[
PWKU Ki U_P1 PB[KUPU KPP W
E
I
Volts
Amps
OhmsZ
R L Total
5 + j05 0o
0 + j3.76993.7699 90o
10 + j010 0o
10 + j010 0o
10 + j010 0o
0 - j2.65262.6526 -90o
2 + j02 0o
OhmsLaw
OhmsLaw
I = EZ
I = EZ
WPWkpBD KBDKD_DBKiWDPUUBbpBD KUP U KUU{ KB[
DDKK-}C KiW3%]pbPU KK2UcU W3kh!
E
I
Volts
Amps
OhmsZ
R L Total
5 + j05 0o
0 + j3.76993.7699 90o
10 + j010 0o
10 + j010 0o
10 + j010 0o
0 - j2.65262.6526 -90o
2 + j02 0o
2 - j2.6526
Rule of parallelcircuits:
Itotal = IR + IL
3.3221 -52.984o
_P_UoK_ i_D!i_ K @iDn1PKAK "%{]{(Il`h?WB BU K L{v
U& WK(_ B[U^PUUBI!iP_PBIi_ UXD!Ii_ K @iDnPK o Kni_{UAK
- 1Z('Zy3*)#)T/+)M3/7mZmCih64
-
-)
.0"0/721B3/7.#" .0/:"4657/843" .0/:9 64@/
87 9(RS7A;: B1
- 1ZCBr64
-
.0"0/721B3/7.#" .0/:"4657/843" .0/:9 64@/
LKKWKBK2@_KW=M Kl_KKB KPP W aii_P oUIWKWKUPU
iUP ^PBiKiy[ KWZv-MYPP UaD! i_ @hv-MYPW#UKiWIPK!Wi7Ki_
PP UUP P]UOMeUZ|PPKBB!}^i U o _
ii_PPWiU UP l@Ui KWPBPB#PM @@BUiPBK o W
!}^i UPP UDs[ !}^ibDU @W Wo^iP PB[bUPnBi _ 1@_
K[KB 1M K[o K_@UPWWWP_ iWD_PB UDB WM @_KP]UKiB
iUKP#AP3D!i |Po1__ K _iKWPBPB
[] o D_PB ( KAWK1PW"K[KsKiiXD!]__6KKA WP_P
APKKPUbBUKKPP UDUiP_D@iIU iWD_PB8Dn_ PK @KAW_E _
@_K (UUU_KBpnPI_^Bi i_KPK_U$!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"
- 'Z1ZmZ/836/7m
-
.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_"`_
- 'Z1ZmZ/836/7m
-
.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
- BrmZ/836/7m
-
.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
- BrmZ/836/7m
-
"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
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