Electricity Energy Conversion
168
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Transcript of Electricity Energy Conversion
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אאאאאא،אאאאאא
אאאK
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אאאא
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אאאKאאאאאא
אאאאאא،אאאאאאאאאKאאא
אא،אאאאאא،אאאאאאאאאאאK
אאJ٢אאאאאאאאאK
אא،אאאאאא،אאאא،אK
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א١٤١ א א
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١J١אאאאאאא
א א א א א א אאאאאאK
١J٢אEnergy ConversionאW
١K אאWאאאאאאאאFKE
٢K Wאאאאאאאאאאאאאא
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١J٢J١אאאPrimary Energy Resources אאאאאאא
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אאWF١E אא(Fossil Energy Resources)F٢E אא(Renewable Energy sources)F٣E אא (Nuclear Energy sources)
א١J٢J١J١ אאFossil Energy Sources
אאאאאאאאאאא
אאאאאאא
א١٤١ א א
אאא -٢
- ٣ -
אאאאאאאאW
FIE א(Petroleum Oil)FIIE אא(Natural Gas)FIIIE א(Coal)
אאאאאאאא א א א
א א אא א א א אא אא א א
אאאאאאאאKאאאא
א א א א אא א אא א אא א א
אאאאאאאאאאאאאאאKא
אאאאאאאאא،אא
א א א א אאא
אאWאאאFאאE
אאא
א
٤٠٦٠٢٥٠
F١J١Eאאאאא
א١٤١ א א
אאא -٢
- ٤ -
אאאאאאאאאאאאאאא
אאאאK١J٢J١J٢ אאNuclear Energy
אאFאEאאאאאאאאאאאאאאא
א(Radioactive Material)Kאאאאאאאאאאא
אאאKאאאאאאאאאא
אאאאאאאאאאKאאאJאא–א
אKאאאאאFfissionEאא
(Fusion)אאאאאאאאאאאאK
אאאאאאFאEאאאFאEא
אאאאא،אאאאאאאK
אאאאאאאאאאאאאאאאאK
١J٢J١J٣ אאRenewable Energyאאאאאאא
אאאאאאאאאאאאK
א١٤١ א א
אאא -٢
- ٥ -
אאאאאאאאW
FIE אאאאאאKFIIE אאאאאאאאא
(Wind Energy)אFאאKE
FIIIE אאאא(Water falls)אאFאאKE
FIVE אאא(Sea waves)אאאאאK
FVE אאאאא(Geophysics)אאאFאאKE
FVIE אאאאאא(Ocean Energy)אאאאא
KFVIIE אאאאאא(Solar Lakes)FאאאKEFVIIIE אאאאאא
א(Bio gas)K
אאאאאא،אאאא
א א א א א א א א ،א אא
אאאאאאאא א K א אא א
Kאאאאא
א١٤١ א א
אאא -٢
- ٦ -
אאאאאאאאאא
אK
١J٣אאGeneration of Electrical Energy
אאא،אאאאאא
אאאאאאאאKאא
אאאאאKאאא
אאKאאW
١J٣J١ אאאאHydro- Power Station
אאאאאFאאאEKאאאא
אאF١J١EK
א١٤١ א א
אאא -٢
- ٧ -
F١J١Eאא
אאאאאאאFאאKEאאאא
אאאאאאאאאא
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אאאאאאאאFאEאא
F١J٢EK
F١J٢Eאאא
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אאאאאאא
אאאאאאFאEאאאאאאאKאאא
אאאאא،אאאאאאאFאא
אאאאאEאאאאאאאFאא
EF٣EK
F١J٣Eאאאאא
١J٣J٢ אאאאThermal Power Stationאאאאאא
אאאאאאאאא
א
P
G
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א١٤١ א א
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אאאאאאאאא،אאא
אאאאאאאאאאKF١J٤E
אאאאאאאאאאאאאFאEאא
אאאאאK
F١J٤Eאאאא
אאאאאאאאא א אא א F ،א
א،אEאא K F١J٥ E، א א
אאאאאאא،אאאאאאאאאK
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P
T
א١٤١ א א
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F١J٥Eאא
١J٣J٣ אאאאGas Turbine Power Station
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F١J٦Eאא،אא
אK
مكثف مضخة
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א
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א١٤١ א א
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١J٣J٤ אאאDiesel Power Station
אאאאאאא،אאK
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F١J٧EאW
F١J٧Eא
١J٣J٥ אאאWאאאאאא
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١K אאאאK ٢K אאאאK ٣K אאאאK ٤K אאאK ٥K אאאאK ٦K אאאאK ٧K אאאאאK ٨K אאאאK ٩K אאאאאאאאK
١٠K אאאאאאאאK ١١K אאאאאאK ١٢K אאאאאאK ١٣K אאאאאK ١٤K אאאאאאאK ١٥K אאאאאאאK ١٦K אאאאאK
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٢J١Introduction אאאאאאא
אאאאאKאאאאאאאא
،אאאאאאאאאK
٢J٢אאאMagnetic Effects of Electrical Current ٢J٢J١אאGeneration & Concentration of Magnetic Field אאא،אא
א(Magnetic Field)אאאFאאFאאE،φEאאא،
F٢J١KE
F٢J١EאאאK
אאא،אאאאWאאאא،אאאאא،אאאא
אאאאאKאאFאאφEאא،FאEאאאא،
אאאא،אF٢J٢KE
Iϕ
אא ١٤١ א
אא -٢
- ١٧ -
F٢J٢EאK
אWF١E אאאאאא
( )×א( )•אא،אFאאאEאאא،
F٢J٢KEF٢E ،אא،אאאא
א،א،אאא،אאK
F٣E אאאאאאאא،אK
F٤E אאאאF٢J٣KE
F٢J٣EאאאK
אאFφEא،אאאאא،אאא،אא
ϕ١
אא ١٤١ א
אא -٢
- ١٨ -
אאאאFEאא،אאאאאאK
F٢J٤EאאאאK
אא،אאאאאאאFאאE
אאאאאאא،אאאאאאאאאKא
אאא(φ٢)אאFφ١EאאF٢J٤KE
א،אאאאאאאF٢J٥KE
F٢J٥EKW
ϕ١<ϕ٢<ϕ٣אאא،אאאאאאא
אא،אאאאK
ϕ
A B
אא ١٤١ א
אא -٢
- ١٩ -
٢J٢J٢אאFaraday's LawאאאN
אF٢J٦Eא،EFEאאאא،אא
Nאא،אF٢J١WE
dtdNE φ
−= )١-٢(
F٢J٦EאאאאאאK
٢J٣אאBasic Concepts of Electrical Generator F٢J٥Eאאאא،
אאאא،אאאאאאאאאאאא
אאא،،אF٢J٧KE
F٢J٧EאאאאאK
E = - N dϕ/dt
ϕ = f(t)
N
S
אא ١٤١ א
אא -٢
- ٢٠ -
אאאאאWאאא،אאאאא
אאW
dtdNE ϕ
−= )٢-٢(
אאאBW
BA
=ϕ
)٣-٢(
W
BA=ϕ )٤-٢( BdAd =ϕ )٥-٢(
אlאאאאϕ،אאEWאא
dxאאאא،=ϕd،WBdAd =ϕ )٦-٢(
dxdA l= )٧-٢( dxBd l=ϕ )٨-٢(
א،אאF٢J٢E،WN=١،W
ν== ll BdtdxBE
)٩-٢(
WBZאאFאאאElZאνZאאאאאאאK
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- ٢١ -
אאאאEאFWEאא،W ϕ νאאא،
א،W،אאאאאאאאאאאאאאאאאאא
אאאK٢J٣J١אאGeneration of Sine Wave
אאאאא،אאאאF٢J٨E،
אאאאאאאאאאאFEאאK
F٢J٨EאאK
אאאF٢J٨E،אאABCDAא،WF١E אAאאא
אKF٢E אBאאאא
אאFאאEאאאF٢J٨KE
ω
N
S
xφ
A
B
C
D
אא ١٤١ א
אא -٢
- ٢٢ -
F٣E אCאאאאK )٤( אDאאאא
אאFאאEאאאאK אאאאאאאאא
א،אאאאאF٢J٩KE
1
05
0
05
1
0 90 180 270 360
אא
א
אL
א
F٢J٩EאאK
٢J٣J٢אאאMathematical Analysis of Sine Wave
אאאאאאאאאאאאK
אא(V)WאאאאאאאKאאא(ω)Wאאאאאאא،אאא
אאradiansא(f)WאאאאאאאאK
א،אdאאאאDW
2D2d π= )١٠-٢( אאאאθW
( )( ) π=π
=ϑ 22D
2D2
)١١-٢(
אא ١٤١ א
אא -٢
- ٢٣ -
אאאאא،fא،אאVW
( ) f2D2V ⋅π= )١٢-٢ ( אאωW
f2 π=ω )١٣-٢( אVωW
( )2DV ⋅ω= )١٤-٢ ( tאW
tVd ⋅= )١٥-٢( אW
t⋅ω=θ )١٦-٢( אאאאאθ
V،F٢J١٠E،W
F٢J١٠Eאא
υ = V sinθ )١٧-٢(
υאאאאϕKW( )tsinVBsinVBBE ω=θ=ν= lll )١٨-٢(
θ
θ
θV
N
S
ω
السرعة الخطية
السرعة العمودية
אא
θ
אא ١٤١ א
אא -٢
- ٢٤ -
( ) ( )tsinEtsinVBE max ω=ω=∴ l )١٩-٢( ( )2DBVBE max ω== llQ )٢٠-٢ (
( )tsin2
DBE ω⎟
⎠⎞
⎜⎝⎛ ω
=∴l
)٢١-٢( אאאאאאאאאK
٢J٣J٣אאאEvaluation of Sine Wave
0 90 180 270 360
אא
א
F٢J١١Eאא
אאאאF٢J١١E،
אאא،אWאאא؟
אאאאWאאאאKאאאאאאאא
٢J٣J٣J١אאאAverage value of sine wave אאWאאאאאאא
אאאא(T)א،אZKאאאT/٢אאאא
אאאא،אאאT/٢
אא ١٤١ א
אא -٢
- ٢٥ -
אאאF٢J١٢E،אאא٢/TאאאאWא
אאK
( ) ( )[ ] mm0
mav 2I0coscosIdsinII =−π=θθ=π ∫π
)٢٢-٢ (
mav I2Iπ
= )٢٣-٢ (
אאZ
[ ] 0)2cos()0cos(IsinI)21(I mmav =π−=θπ
= ∫ )٢٤-٢(
0
0.25
0.5
0.75
1
0 30 60 90 120 150 180
F٢J١٢Eאאאא
٢J٣J٣J٢אאאEffective value of sine wave אאWאאאאא،
אאא،אא،אאאאאאאאאא
אאאאאאאאאKאr.m.s אroot
Iav
Im
אא ١٤١ א
אא -٢
- ٢٦ -
mean squareאאאאאאאאאWθ= sinIi mW
F١E א،אW2m
2 )sinI(i θ=(square)،F٢J١٣KE
F٢J١٣Eאא
F٢Eאאאאא
אאאא،אא (mean):
( ) ( )2
I2
2cos1IsinI1 2
m2m22
m =θ−
⎟⎟⎠
⎞⎜⎜⎝
⎛
π=θ−
π ∫∫ )٢٥-٢(
F٣Eאאאא(root)W
2I
2I
I m2m
eff == )٢٦-٢(
אא(root mean square) אא(rms)W
⎟⎠
⎞⎜⎝
⎛⎟⎠⎞
⎜⎝⎛= ∫ θsin I π1 I 22
meff )٢٧-٢(
2mI
mI
π ٢π
אא ١٤١ א
אא -٢
- ٢٧ -
٢J٣J٤אאPhase angle
F٢J١٠Eאאאאאאאא
אF٢J١٤EאאאאאאK
F٢J١٤EK
אאאWα & ψ،F٢J١٥Eא،W
F٢J١٥E
φ
N
S
شمالي قطب
جنوبي قطب
θ = ωt
ψ
α
C
AB
N
S
אא ١٤١ א
אא -٢
- ٢٨ -
JאאאאאAW
eA = Em sin ωt )٢٨-٢( JאBאאאאW
eB = Em sin (ωt+ψ) )٢٩-٢(
J אCאאאאW
eC = Em sin (ωt - α) )٣٠-٢ (
אאאאאאBאאאψאAאא
אαאCאF٢J١٦KE
F٢J١٦EאאאאK
אאאאאא
אאא،אאאאא(reference)אאK
אאאAאאB،Cאאאאאאאאא،אאאאW
A B C
ψ α
אא ١٤١ א
אא -٢
- ٢٩ -
♦ אAאא،אZW )0tsin(Ee mA +ω= )٣١-٢ (
♦ אאאאW )tsin(Ee mB ψ+ω= )٣٢-٢(
♦ אאאאW )tsin(Ee mC α−ω= )٣٣-٢(
٢J٣J٥אאאRepresentation of Sine Wave
אאאאאאEmאωאאאאאאא
אאωtאאאאאאאF٢J١٧KE
-1
-0.5
0
0.5
1
F٢J١٧EאאאK
٢J٣J٥J١אאMathematical Representation of Vectors
אאW١J אאאK٢J אאK ٣J אאK ٤J אאK
ωtωt π ٢π
אא ١٤١ א
אא -٢
- ٣٠ -
WF١E אאאWA
W 21 jaaA += )٣٤-٢(
jאWjWאאאאZ1−
אאאWאאאא
F٢J١٨KE
F٢J١٨EאאK
אX،אאj
XjjX −=⋅⋅אאאFF–EE
אאjאאאא،٩٠א،jאאאא٩٠Kאא
אאאאאאאאאאאאF٢J١٩KE
to - ∞ to ∞ ٠
אא ١٤١ א
אא -٢
- ٣١ -
F٢J١٩EאאאאK)aa(A 2
221 += )٣٥-٢(
⎟⎟⎠
⎞⎜⎜⎝
⎛=θ −
1
21
aa
tan )٣٦-٢(
F٢E אאWאאאW
θ= cosAa1 )٣٧-٢( θ= sinAa 2 )٣٨-٢(
אW21 jaaA +=W( )θ+θ=θ+θ= sinjcosAsinjAcosAA )٣٩-٢(
F٣E אאWאאאW
θ+θ=θ sinjcose j )٤٠-٢(
אאW( )θ+θ= sinjcosAAאWθ= jeA A )٤١-٢(
F٤E אאWאאאW
θ A A ∠= )٤٢-٢(
j
⎟A⎢
a٢
a١ θ
אא ١٤١ א
אא -٢
- ٣٢ -
٢J٣J٥J٢אאאBasic Calculations of Vectors
F١E אאB,AאCאDאW
21 jaaA += )٤٣-٢( 21 jbbB += )٤٤-٢(
21 jccC += )٤٥-٢( 21 jddD += )٤٦-٢(
BAC += )٤٧-٢( BAD −= )٤٨-٢(
( ) ( )2211212121 bajbajbbjaajcc +++=+++=+ )٤٩-٢( )ba(j)ba()jbb()jaa(jdd 2211212121 −+−=+−+=+ )٥٠-٢(
111 bac += )٥١-٢( 222 bac += )٥٢-٢(
111 bad −= )٥٣-٢( 222 bad −= )٥٤-٢(
])ba()ba[(C 222
211 +++= )٥٥-٢(
( )( )11
221C ba
batan
++
=θ −
)٥٦-٢( ])ba()ba[(D 2
222
11 −+−= )٥٧-٢( ( )( )11
221d ba
batan
−−
=θ −
)٥٨-٢(
אא ١٤١ א
אא -٢
- ٣٣ -
F٢E אאאאאאא،אא
אאאאKCAB: ⋅
( ) ( )2121 jbbjaaBAC +⋅+=⋅= )٥٩-٢( ( ) ( )12212211 babajbaba ++−= )٦٠-٢(
( ) ( ) ]babababa[C 21221
22211 ++−= )٦١-٢(
( )( )2211
12211C baba
babatan
−+
=θ −
)٦٢-٢( אאאאאאאW
CBA CBABAC θ∠=θ∠⋅θ∠=⋅= )٦٣-٢( BAC ⋅= )٦٤-٢(
baC θ∠+θ∠=θ∠ )٦٥-٢( אW
( ) cbaba jθθθjjθjθ eCeBAeBeAC =⋅=⋅= + )٦٦-٢(
אW
BAD =
)٦٧-٢(
BA
D = )٦٨-٢(
bad θ∠−θ∠=θ∠ )٦٩-٢(
אא ١٤١ א
אא -٢
- ٣٤ -
אא
WאWFE אאאאא FE אאאא FE אאאא FE אא FE אאאאאאאαK FE אאאאאאאא FE אאאאאאא FE אאאא FE אאאאK
WאW١K אאאW
FE FE 6j8B,8j6A −=−=FE 6j8B,8j6A +=−= FE 6j8B,8j6A −=+= FE 6j8B,8j6A +=+= FE 6j8L,8j6K +−=−= FE 6j8L,8j6K +−=−−= FE 6j8L,8j6K −−=−−= FE 6j8L,6j8K +−=+−= FE 6j8L,8j6K +=−−=
٢K אאBA −W FE 6j8B,8j6A −=−= FE 6j8B,8j6A +=−= FE 6j8B,8j6A −=+= FE 6j8B,8j6A +=+=
אא ١٤١ א
אא -٢
- ٣٥ -
FE 6j8B,8j6A +−=−= FE 6j8B,8j6A +−=−−= FE 6j8B,8j6A −−=−−= FE 6j8B,6j8A +−=+−= FE 6j8B,8j6A +=−−=
٣K אאאאW FE 8j6A −= FE 6j8B += FE 8j6A +=6j8B −= FE 8j6A −−= FE 6j8B −−= FE 6j8A +−= FE ,8j6A +−=
٤K אאאאאW FE °∠= 3010A FE °−∠= 3010A FE °∠= 6010A FE °−∠= 6010A FE °∠= 87.3610A FE °−∠= 87.3610A FE °∠= 13.5310A FE °−∠= 13.5310A
٥K אאאW FE °−∠=°∠= 3010B,3010A FE °∠=°∠= 6010B,3010A FE °∠=°∠= 010B,3010A FE °∠=°∠= 010B,010A FE °−∠=°∠= 6010B,12010A
אא ١٤١ א
אא -٢
- ٣٦ -
٦K אאBAW
FE °−∠=°∠= 3010B,3010A FE °∠=°∠= 6010B,3010A FE °∠=°∠= 010B,3010A FE °∠=°∠= 010B,010A FE °−∠=°∠= 6010B,12010A
٧K א،אABW
FE °−∠=°∠= 3010B,3010A FE °∠=°∠= 6010B,3010A FE °∠=°∠= 010B,3010A FE °∠=°∠= 010B,010A FE °−∠=°∠= 6010B,12010A
- ١٤ -
אאאאאא
JJ٢٢
אאא<Üé×Ãj×Ö<íÚ^ÃÖ]<퉉ö¹]ã¹]<gè…‚jÖ]æ<ËÖ]
אאאא
א
אא
٣
אא ١٤١ א
אאא -٢
- ٣٧ -
אאאא
אאא،אאW• אאK • אאאאK • אאK • אאאאK • אאאK • אאK • אאאאאאאאjK • אאאK • אאאK • אאאאK • אאאאאK
אא ١٤١ א
אאא -٢
- ٣٨ -
٣J١Introduction אאאאאאאאאאאאאאאא،אאא
אאK٣J٢אאBasic Elements of A.C Circuits
٣J٢J١אאThe Resistance
F٣J١EאאאאאK
אאאF٣J١E،Wאאא
tsinEE m ω= )١-٣( W
RitsinEE m ⋅=ω= )٢-٣(
tsinItsinR
Ei m
m ω=ω⎟⎠⎞
⎜⎝⎛=
)٣-٣( tsinIi m ω= )٤-٣(
אF٣J١EF٣J٤Eאאאא،אאאאK
RE
i
אא ١٤١ א
אאא -٢
- ٣٩ -
٣J٢J٢אאThe Inductive Reactance
،אF٣J٢E،ΦאאאאΦ
אN אi،W
F٣J٢EאאאאK
iN ∝φ )٥-٣( LiN =φ )٦-٣(
אאאאLא(Henry)KאאW
dtdNE φ
−= )٧-٣(
LiN =φQ )٨-٣(
dtdiL
dtdN =
φ∴
)٩-٣(
dtdiL
dtdNE −=
φ−=∴
)١٠-٣( אאאאW)tsin(Ii m ω=
W
( ))t(sinmIdtdLE ω⋅⋅−=
)١١-٣( ( )tcosILE m ω⋅ω⋅⋅−= )١٢-٣( ( ) ( )90tsinmE90tsinLmIE +ω⋅=+ω⋅⋅ω⋅−= )١٣-٣(
Φ
N i
אא ١٤١ א
אאא -٢
- ٤٠ -
LIE
m
m ⋅ω= )١٤-٣(
אאאאFEωLאאא(Inductive Reactance)א،(Ω)،
WL f 2 L π=ωא،אאאEאiא
٩٠KאאאW
Lf2L ⋅π=ω )١٥-٣(
٣J٢J٣אאThe Capacitive Reactance
،אאאאF٣J٣E،
אא،אKWVQ ∝ )١٦-٣( CVQ = )١٧-٣(
אאאCאא،אאא FaradK
F٣J٣EאאאאK
E
i
++++- - - -
Q
אא ١٤١ א
אאא -٢
- ٤١ -
٣J٢J٣J١אאאאאCאאR،
F٣J٤E،אאאאאאא
אא،א،אאאא
אאKאאאW
F٣J٤Eאא
אF٣J١٧EאאאW
Idt
dVC
dtdQ C ==
)١٨-٣( אאאFאJ١E،
אW
dtdV
RCERdt
dVCERIEV CC
C −=⋅−=⋅−= )١٩-٣(
Edt
dVRCV C
C =+∴ )٢٠-٣(
אF٣J٢٠Eאא،אא،אאW
R C
CERE
I
E
אא ١٤١ א
אאא -٢
- ٤٢ -
⎟⎟⎠
⎞⎜⎜⎝
⎛−⋅=
−RC
t
C e1EV )٢١-٣(
F٣J٤Eא،W
( )R
e1EE
RVE
I
RCt
CC
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−⋅−
=−
=
−
)٢٢-٣(
⎟⎟⎠
⎞⎜⎜⎝
⎛=∴ τ
− t
C eREI
)٢٣-٣( W
ICWאRC=τWא
٣J٢J٣J٢א،אאאאאאאאאא
F٣J٥Eאאאא،אאK،אאאא
אאאKאאW
F٣J٥Eאא
R C
CERE
I
+ −
אא ١٤١ א
אאא -٢
- ٤٣ -
0RIV dC =⋅+ )٢٤-٣(
אIdאא،אW
dtdV
CI Cd −=
)٢٥-٣( אF٣J٢٤Eא،W
0dt
dVRCV C
C =− )٢٦-٣(
אF٣J٢٦E،אא،אאאאW
⎟⎟⎠
⎞⎜⎜⎝
⎛⋅=
−RC
t
CiC eVV )٢٧-٣(
VCiאאאאאאאF٣J٢٤EאIdW
⎟⎟⎠
⎞⎜⎜⎝
⎛⋅−=−=
−RC
tCiC
d eR
VR
VI
)٢٨-٣( אאאאאK
F٣J٦Eאאאאא،FאאEאאאא،FאאKE
א
א
F٣J٦Eאאאאא
F٣J١Eא٥٠ µFאא،١٢ Vא
א٢א kΩא،Wאאτאא،٢٠ msecKא
אא ١٤١ א
אאא -٢
- ٤٤ -
אאא،אאאאאא،אאא١٠א msecאK
אאאτ،W
sec1.0CR =⋅=τ א٢٠ msec،W
V175.2e1V12e1EV 1.0020.0
RCt
C =⎟⎠⎞
⎜⎝⎛ −⋅=⎟⎟
⎠
⎞⎜⎜⎝
⎛−⋅=
−−
א٢٠ msec،W
mA91.4e2000
V12e
REI 1.0
020.0t
C =⎟⎠⎞
⎜⎝⎛
Ω=⎟⎟
⎠
⎞⎜⎜⎝
⎛=∴
−τ−
א١٠ msecא،אW
V97.1eV175.2eVV 1.0010.0
RCt
CiC =⎟⎟⎠
⎞⎜⎜⎝
⎛⋅=⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅=
−−
mA98.0e2000
V175.2e
RV
I 1.0010.0
RCt
Cid −=⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅−=⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅−=
−−
٣J٢J٣J٣אאאאאא،אאאאאאאאא،אאא
אאW
dtdVC
dtdQi ==
) ٢٩-٣( אאאאאW tsinVv m ω=אא،
אW( ) tsin
dtd VC
dtdvC i m ω⋅⋅==
)٣٠-٣( ( )90tsinVC tcosVC i mm +ω⋅⋅⋅ω=ω⋅ω⋅⋅= )٣١-٣(
אא ١٤١ א
אאא -٢
- ٤٥ -
אאW( )90tsinIi m +ω⋅= )٣٢-٣(
WCVI mm ω⋅= )٣٣-٣(
W
C1
IV
m
m
ω=
)٣٤-٣( אFE،
C1
ωא
א(Capacitive Reactance)א،(Ω)،W
C f 21
C 1
π=
ω )٣٥-٣( אאאאא٩٠°K
٣J٣אאא٣J٣J١אאאjאאאj Operator in Electrical Circuits
אאiRאאאLC،F٣J٧KE
F٣J٧ER، L، CאאאK
WאRWRiE r ⋅=א،(volt) KאL WLiE L ω⋅=א،(volt)
i
R
E
L C
ECER EL
אא ١٤١ א
אאא -٢
- ٤٦ -
אCWCiEC ω
=א،(volt)
אאאאאאFאאE،W)L(iE L ω=،א٩٠°אiF٣J٨Eאאא،j
אW
F٣J٨EאאאאאאאKiLjE L ⋅ω= )٣٦-٣(
אijωLאאא٩٠°ELאi–j/ωCאאא٩٠°ECK
אאאאW
⎟⎠⎞
⎜⎝⎛
ω−ω+=
ω−ω+
C1LjR
CjLjR
)٣٧-٣( אZ
XRKjXRZ += )٣٨-٣(
אאאZW
)XR(Z 22 += )٣٩-٣(
RXtan 1
Z−=θ
)٤٠-٣(
i ER
EL
EC
אא ١٤١ א
אאא -٢
- ٤٧ -
٣J٣J٢אאא٣J٣J٢J١ אאא
אאאF٣J٩E،אאאאXLW
F٣J٩Eאאאא)LL(jLjLjLjX 2121L +ω=ω+ω=ω= )٤١-٣(
)LL(L 21 +=∴ )٤٢-٣(
،אאאאאאאאאK
F٣J٢E
אאאאאאאא٥ mHא٧ mHK
א
F٣J٤٢Eאאאא،WmH12mH7mH5)LL(L 21 =+=+=∴
٣J٣J٢J٢ אאאאאאF٣J١٠E،אא
אאXCW
E
L٢
1LE2LE
i
L١
E
L
i
≡
אא ١٤١ א
אאא -٢
- ٤٨ -
F٣J١٠Eאאאא
⎟⎟⎠
⎞⎜⎜⎝
⎛ω
+ω
−=⎟⎟⎠
⎞⎜⎜⎝
⎛ω
−+⎟⎟⎠
⎞⎜⎜⎝
⎛ω
−=⎟⎠⎞
⎜⎝⎛
ω−=
2121C C
1C1j
C1j
C1j
C1jX
) ٤٣-٣( W
⎟⎟⎠
⎞⎜⎜⎝
⎛+=⎟
⎠⎞
⎜⎝⎛
21 C1
C1
C1
)٤٤-٣(
⎟⎟⎠
⎞⎜⎜⎝
⎛+⋅
=21
21
CCCC
C )٤٥-٣(
אF٣J٤٥EאאאאאאK
F٣J٣E
אאאאא١٠٠א µFא٢٥ µFK
א
F٣J٤٥Eאא،W
F20F25F100
F25F100CCCC
C21
21 µ=µ+µ
µ⋅µ=⎟⎟
⎠
⎞⎜⎜⎝
⎛+⋅
=
E
C٢
2CE1CE
i
C١
E
C٢
i
≡
אא ١٤١ א
אאא -٢
- ٤٩ -
٣J٣J٢J٣ אאRL Series Circuit אאאF٣J١١E،אא
אאZW
F٣J١١EאאאKLjRZ ω+= )٤٦-٣(
))L(R(Z 22 ω+= )٤٧-٣(
RLtan 1
Zω
=θ∠ −
)٤٨-٣( אF٣J٤٨E אאZθאKאiW
iZ
E iZE
ZEi θ∠=
θ∠
θ∠==
)٤٩-٣( W
)LR(
EZE
i222 ω+
==
, )٥٠-٣( ZEi θ∠−θ∠=θ∠ )٥١-٣(
אFEאאW0EE ∠= )٥٢-٣(
אאWϕ∠=θ−∠=θ∠−∠=θ∠ ZZi 0 )٥٣-٣(
R
E
L
RE LE
i
אא ١٤١ א
אאא -٢
- ٥٠ -
ϕאאאאאאאאאFEא،אא،אK
אאאF٣J١٢KE
F٣J١٢EאאאאאאK
WLR EEE += )٥٤-٣(
( ) ZiLjRiLjiRiE ⋅=ω+⋅=ω⋅+⋅= )٥٥-٣( F٣J٤E
א٥ ΩאאאL = ٠١٠ Henry،Wvolts200e =Wf = ٥٠ HzK
אWF١E אאKF٢E אK
אF١EאWאאאωW
f2π=ω sec/rad 314 50 14.3 2 =××=ω
אאXLWLX L ×ω=
Ω=×= 14.3 01.0 314 X L
RiER ⋅=
ZiE ⋅=LjiEL ω⋅=
i
θi= ϕ
אא ١٤١ א
אאא -٢
- ٥١ -
אאZW( ) Ω=+= 9.5 14.3 5 Z 22
°=⎟⎠⎞
⎜⎝⎛=θ − 13.32
514.3tan 1
Z
אiW
Zei =
אFEאא،٠° אאא
W
13.32A9.33 13.329.5
0V200 i °−∠=
°∠Ω°∠
=
אאא١٣٣٢°אא،٠°אא١٣٣٢° א،אאאK
F٣E אW♦ אאW
°−∠=Ω⋅°−∠=⋅= 13.32V5.169 513.32A9.33 Ri ER ♦ אאW
°∠=°∠Ω⋅°−∠=⋅= 87.57V45.106 9014.313.32A9.33 Xi E LL
٣J٣J٢J٤ אאRC series Circuit F٣J١٣Eא،אאW
אא ١٤١ א
אאא -٢
- ٥٢ -
F٣J١٣EאאאK
C1jRZ
ω−=
)٥٦-٣( 2
2
C1RZ ⎟
⎠⎞
⎜⎝⎛
ω+=
)٥٧-٣(
⎟⎠⎞
⎜⎝⎛
ω−
=θ∠ −
CR1tan 1
Z ⇒ א )٥٨-٣(
אiW
i
Z
22
E
Z
E i
C1R
EZE
ZEi θ∠=
θ∠⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎠⎞
⎜⎝⎛
ω+
θ∠=
θ∠
θ∠==
)٥٩-٣( W
( )
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎠⎞
⎜⎝⎛
ω+
==2
2
C1R
EZE
i
, )٦٠-٣( ZEi θ−θ=θ )٦١-٣(
אאאFאEא٠° ،Wϕ=θ−°=θ Zi 0 )٦٢-٣ (
R
E
C
CERE
i
אא ١٤١ א
אאא -٢
- ٥٣ -
אאZθאאא،אא،KאאF٣J١٤KE
F٣J١٤EאאאאאאK
WCR EEE += )٦٣-٣(
RiER ⋅= )٦٤-٣( )
Cj(iEC ω
−⋅=
)٦٥-٣( )
Cj(iRiE
ω⋅−⋅=
)٦٦-٣(
ZiC
1jRiE ⋅=⎟⎠⎞
⎜⎝⎛
ω−⋅=
)٦٧-٣( ZiE •= )٦٨-٣(
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
ω+•=
22
C1RiE
)٦٩-٣( 2
C2
R
2222 EE
C1iRiE +=⎟
⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
ω•+=
)٧٠-٣(
E
i
CE
RE
ϕ
אא ١٤١ א
אאא -٢
- ٥٤ -
F٣J٥Eא١٢٥ volts٥٠ Hzא
،אאF٣J١٥Eאאאאא،٢٢ Amp.אאאא٨٩٦ Wattא،אאא
אאאK
F٣J١٥EאאאF٣J٥KEא
RiP 2=
Ω=== 20 2.2
8.96 iP R
22
volts 44 202.2 iR E R =×==
( ) ( ) volts 117 44125 EE E 222R
2C =−=−=
CC XiE •=
Ω== 2.53 2.2
117 XC
C5014.321
Cf21
C12.53XC ×××
=⋅π
=ω
==
Farad 1060 C 6−×=
Watts 8.96P = C
A 2.2١٢٥ volts ٥٠ Hz
אא ١٤١ א
אאא -٢
- ٥٥ -
F٣J١٦EאאאK
F٣J١٦EאאאF٣J٥KE
٣J٣J٢J٥ אאאאRLC Series Circuit
אאאאאRאאאLאאC
F٣J١٧E،iאאאא،אאFאEאאאאFאאEאF
KEאאאZW
F٣J١٧ER، L، CאאאK
⎟⎠⎞
⎜⎝⎛
ω−ω+=⎟
⎠⎞
⎜⎝⎛
ω−ω+=
C1LjR
C1jLjRZ
)٧١-٣(
( )2
2
C1LRZ ⎟
⎠⎞
⎜⎝⎛
ω−ω+=
)٧٢-٣(
R
E
L C
EC ER EL
i
volts117EC =volts125E =
i volts44E R =
ϕ
אא ١٤١ א
אאא -٢
- ٥٦ -
⎟⎟⎠
⎞⎜⎜⎝
⎛
ω−ω
=⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛⎟⎠⎞
⎜⎝⎛
ω−ω
=θ∠ −−
CR1LCtan
RC
1Ltan
211
Z
)٧٣-٣(
אW
°θ−∠=θ∠
°∠= Z
Z ZE
Z0E
i )٧٤-٣(
( )⎟⎟⎠
⎞⎜⎜⎝
⎛
ω−ω
−∠
⎟⎠⎞
⎜⎝⎛
ω−ω+
= −
CR1LCtan
C1LR
Ei
21
22
)٧٥-٣(
אאאW
( )⎟⎟⎠
⎞⎜⎜⎝
⎛
ω−ω
−∠
⎟⎠⎞
⎜⎝⎛
ω−ω+
⋅= −
CR1LCtan
C1LR
REE
21
22
R
)٧٦-٣( LL XiLjiE ⋅=ω⋅= ( )٧٧-٣
( )⎟⎟⎠
⎞⎜⎜⎝
⎛
ω−ω
−°∠
⎟⎠⎞
⎜⎝⎛
ω−ω+
ω⋅= −
CR1LCtan90
C1LR
LEE
21
22
L
)٧٨-٣(
CC XiC
1jiE ⋅−=ω
⋅−= )٧٩-٣(
( )
( )⎟⎟⎠
⎞⎜⎜⎝
⎛
ω−ω
−°−∠
⎟⎠⎞
⎜⎝⎛
ω−ω+
ω= −
CR1LCtan90
C1LR
CEE2
12
2C
)٨٠-٣(
אא ١٤١ א
אאא -٢
- ٥٧ -
F٣J١٨Eאאא(XL < XC)(XC < XL)
F٣J١٨JWE(XL < XC)F٣J١٨JWE(XC < XL)
F٣J٦E ١٠٠ µFאאא ٢٠ Ωא
H05.0L =،F٣J١٩KEאאאאאא٥٠ Hzאא،٢٠٠ voltsKאא
אKא
F٣J١٩EאאאF٣J٦KE
sec/rad314 5014.32 f2 =××=π=ω Ω=×=ω= 7.15 05.0314 L X L
( ) Ω=××
=ω
=−
84.31 10100314
1 C
1 X6C
i LE
CE
E
RE
iθ
i
LE
CE
E
RE
iθ
Ω20
Hz50V200
H05.0F100µ
i
אא ١٤١ א
אאא -٢
- ٥٨ -
( ) 14.16j2084.317.15j20Z −=−+=
( ) ( ) Ω=+= 7.25 14.1620 Z 22
°−=⎟⎠⎞
⎜⎝⎛ −
=θ − 9.38 20
14.16tan 1Z
.Amp 9.3878.7 9.387.25
0V200
ZE i °∠=
°−∠Ω°∠
== 9.38V6.155209.38A78.7RiER ∠=Ω•°∠=⋅=
9.128V146.122 907.159.38A78.7 Xi E LL °∠=∠Ω•∠=⋅= 1.51V72.247 9084.319.38A78.7 Xi E CC °−∠=−∠Ω•∠=⋅=
F٣J٢٠Eאאא
F٣J٢٠EאאF٣J٦E
E = ٢٠٠ V
i = ٧٫٧٨ A
i⋅R = ١٥٫٦ V
i⋅XL= ١٢٢٫١٤٦ V
i⋅XC= ٢٤٧٫٧٢ ٣٨٫٩°
١٢٨٫٩°
-٥١٫١°
אא ١٤١ א
אאא -٢
- ٥٩ -
٣J٣J٣אאאParallel Circuits ٣J٣J٣J١אאא
אאאF٣J٢١E،אאאאXLW
F٣J٢١Eאאאא
א،אאאאW
LjEiω
= )٨١-٣(
א،אאאאאW
⎟⎟⎠
⎞⎜⎜⎝
⎛ω
+ω
⋅=ω
+ω
=+=2121
21 Lj1
Lj1E
LjE
LjEiii
)٨٢-٣(
א،אאW
LjiEX L ω==
)٨٣-٣(
⎟⎟⎠
⎞⎜⎜⎝
⎛ω
+ω
=
⎟⎟⎠
⎞⎜⎜⎝
⎛ω
+ω
⋅
==
2121
L
Lj1
Lj1
1
Lj1
Lj1E
EiEX
)٨٤-٣(
≡EL٢
i
L١
1i 2i
E
L
i
אא ١٤١ א
אאא -٢
- ٦٠ -
א،אאאW
21 L1
L1
L1
+= )٨٥-٣(
)LLLL
(L21
21
+⋅
=∴ )٨٦-٣(
،אאאאאאאאאK
F٣J٧E
אאאאאאאא٥ mHא٧ mHK
א
F٣J٨٦E،אאאא،W
mH92.2mH7mH5mH7mH5
)LLLL
(L21
21 =+×
=+⋅
=∴
٣J٣J٣J٢אאאאאאF٣J٢٢E،אא
אאXCW
F٣J٢٢Eאא
E
C٢
i
≡EC٢
i
C١
1i 2i
אא ١٤١ א
אאא -٢
- ٦١ -
א،אאאאW
( ) CjECj1
Ei ω⋅=ω
= )٨٧-٣(
א،אאאאאW
( ) ( ) 2121
21 CjECjECj1
ECj1
Eiii ω⋅+ω⋅=ω
+ω
=+= )٨٨-٣(
א،אאאWCjECjECjEi 21 ω⋅=ω⋅+ω⋅= )٨٩-٣(
אEj ωא،W21 CCC += )٩٠-٣(
אF٣J٩٠EאאאאאאK
F٣J٨Eאאאאא١٠٠א µFא
٢٥ µFKא
F٣J٩٠Eאא،WF125F25F100CCC 21eq µ=µ+µ=+=
٣J٣J٣J٣אאRL Parallel Circuit
אאאF٣J٢٣E،אאאאZW
F٣J٢٣Eאא
E L
Ri Li
Ti
V
אא ١٤١ א
אאא -٢
- ٦٢ -
ZWאא( )
( ) ( )( )
( )222
222
LRLRjLR
LjRLjRLjRLjR
LjRLjR
Zω+
ω+ω=
ω−⋅ω+ω−⋅ω⋅
=ω+
ω⋅=
) ٩١-٣( ( )
( )( )
( )222
2
222
22
LRLR
jLR
LRZω+
ω+
ω+
ω=
)٩٢-٣(
( )( )
( )( )
2
222
22
222
22
LRLR
LRLRZ ⎟
⎟⎠
⎞⎜⎜⎝
⎛
ω+
ω+⎟
⎟⎠
⎞⎜⎜⎝
⎛
ω+
ω=
)٩٣-٣(
⎟⎠⎞
⎜⎝⎛
ω=⎟
⎟⎠
⎞⎜⎜⎝
⎛
ω
ω=θ∠ −−
LRtan
LRLRtan 122
21
Z )٩٤-٣(
אאW°∠=
°∠°∠
= 0RE
0R0E
iR )٩٥-٣(
°−∠ω
=°∠ω
°∠= 90
LE
90L0E
iL )٩٦-٣(
°−∠ω
+°∠=+= 90L
E0REiii LRT )٩٧-٣(
( ) ( ) ⎟⎠⎞
⎜⎝⎛
ω−=
ω−=−
ω++=
L1j
R1E
LEj
RE1j0
LE0j1
REiT
) ٩٨-٣(
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
ω+⎟
⎠⎞
⎜⎝⎛⋅=
22
T L1
R1Ei
)٩٩-٣(
⎟⎠⎞
⎜⎝⎛
ω−=θ −
LRtan 1
i) ١٠٠-٣ (
אאF٣J٢٤KE
אא ١٤١ א
אאא -٢
- ٦٣ -
F٣J٢٤EאאאאK
F٣J٩EF٣J٢٥Eא٥ Ωאאא
L = ٠١٠ Henry،Wvolts 200e =Wf = ٥٠ HzKאאא،אאאK
F٣J٢٥EאאאF٣J٩KEא
°∠=°∠Ω
°∠=°∠=
°∠°∠
= 0A4005
0V2000
RE
0R0E
iR
°−∠π
=°−∠ω
= 90Lf2
E90L
EiL
°−∠=°−∠Ω×⋅⋅π
=−
90A7.6390101502
V200i
2L
E
Ri
LiTi
iθ
Ω= 5R
Hz50fV200E
== H01.0L =
Ri Li
Ti
V
אא ١٤١ א
אאא -٢
- ٦٤ -
A2.7514.31
51200
L1
R1Ei
2222
T =⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛×=⎟
⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
ω+⎟
⎠⎞
⎜⎝⎛⋅=
°−=⎟⎠⎞
⎜⎝⎛−=⎟
⎠⎞
⎜⎝⎛
ω−=θ −− 9.57
14.35tan
LRtan 11
i
٣J٣J٣J٤אאRC Parallel Circuit
אאאF٣J٢٦E،אאאאZW
F٣J٢٦EאאאאאK
( )⎟⎠⎞
⎜⎝⎛
ω−ω
⎟⎠⎞
⎜⎝⎛
ω−
=⎟⎠
⎞⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
ω−
⎟⎠
⎞⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
ω−⋅
=
CjCR
CRj
C1jR
C1jR
Z ) ٣-
١٠١(
( )( )
( ) ( )jCRjCRjCRRj
jCRRj
Z+ω⋅−ω
+ω⋅−=
−ω−
= ) ٣-
١٠٢(
( )( ) ( )
( )( )1CR
CjRRjCRjCR
jCRRjZ
222
2
+ω
ω−=
+ω⋅−ω+ω⋅−
= ) ١٠٣-٣ (
C R E
Ri Ci
Ti
V
אא ١٤١ א
אאא -٢
- ٦٥ -
( ) ( )1CRCRj
1CRRZ
222
2
222 +ω
ω−
+ω= )١٠٤-٣(
( ) ( )2
222
22
222 1CRCR
1CRRZ ⎟
⎟⎠
⎞⎜⎜⎝
⎛
+ω
ω+⎟
⎟⎠
⎞⎜⎜⎝
⎛
+ω= ١٠٥( )٣-
( )( )( )( ) ⎟
⎟⎠
⎞⎜⎜⎝
⎛ ω−=
+ω
+ωω−=θ∠ −
RCRtan
1CRR1CRCR 2
1222
2222
Z ) ١٠٦-٣ (
אאW
°∠=°∠°∠
= 0RE
0R0E
iR F٣J١٠٧E
°∠ω=°−∠
ω
°∠= 90CE
90C
10E
iC F٣J١٠٨E
°∠ω+°∠=+= 90CE0REiii CRT F٣J١٠٩E
( ) ⎟⎠⎞
⎜⎝⎛ ω+=ω++⎟
⎠⎞
⎜⎝⎛ += CEj
RECEj00j
REiT F٣J١١٠E
( )22
T CEREi ω+⎟
⎠⎞
⎜⎝⎛= F٣J١١١E
( )CRtanRECEtan 11
i ω=⎟⎠
⎞⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ω=θ∠ −− F٣J١١٢E
אאF٣J٢٧EK
אא ١٤١ א
אאא -٢
- ٦٦ -
F٣J٢٧EאאאאK
F٣J١٠Eא٥ ΩאאC = ٠١٠ µF،
Wvolts 200e =Wf = ٥٠ HzF٣J٢٨KEאאאאאאK
F٣J٢٨EאאאF٣J١٠KE
א
°∠=°∠Ω
°∠=°∠=
°∠°∠
= 0A4005
0V2000
RE
0R0E
iR
°∠×=°∠×××π×=°∠ω= −− 90A108.62901001.050220090CEi 36C
( ) ( ) ( )262
22
22
T 1014.351200C
R1ECE
REi −×+⎟
⎠⎞
⎜⎝⎛×=ω+⎟
⎠⎞
⎜⎝⎛=ω+⎟
⎠⎞
⎜⎝⎛=
E
Ri
CiTi
iθ
F01.0C µ=Ω= 5R
Hz50fV200E
==
Ri Ci
Ti
V
אא ١٤١ א
אאא -٢
- ٦٧ -
( ) A401014.351200i
262
T =×+⎟⎠⎞
⎜⎝⎛×= −
( ) ( ) °=××××π×=ω=θ∠ −−− 051001.0502tanCRtan 611i
٣J٣J٣J٥אאאאRLC Parallel Circuit
RאאאLאאCאאאEF٣J
٢٩KE
F٣J٢٩EאאאאאK
אאאiWאאא(iR)Hאאא(iL)Hאאא(iC)K
CLR iiii ++= ) ١١٣-٣ (
אE،אאאאאFאZE
Cj
E Lj
E RE i
⎟⎠⎞
⎜⎝⎛
ω−
+ω
+=
) ١١٤-٣ (
RE L C
Ti
Ci Li Ri
אא ١٤١ א
אאא -٢
- ٦٨ -
⎟⎠
⎞⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
ω−ω+⋅=⎟⎟
⎠
⎞⎜⎜⎝
⎛ ω−
ω+⋅=
L1 C j
R1E
jC
Lj1
R1E i
) ١١٥-٣ ( 22
L1C
R1E i ⎟
⎠⎞
⎜⎝⎛
ω−ω+⎟
⎠⎞
⎜⎝⎛⋅=
) ١١٦-٣ (
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
ω−ω
=⎟⎠
⎞⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
ω−ω=θ∠ −−
L1LCRtan
R1
L1Ctan
211
i
) ١١٧-٣ (
אאF٣J٣٠E
F٣J٣٠EאאאאK
E iR
iC
iL
iL
iC
i
אא ١٤١ א
אאא -٢
- ٦٩ -
F٣J١١EאאאF٣J٣١EאאאאאאK
א
F٣J٣١EאאאF٣J١١KE
°∠=°∠Ω°∠
= 0A5.001000V50
iR
( ) °−∠=°∠Ω××π×
°∠= 90mA145.3
9053.2100020V50
iL
( ) °∠=°−∠Ω×××π×
°∠=
−90mA145.3
901001.01000210V50
i6C
°∠+°−∠+°∠=++= 90mA145.390mA145.30A5.0iiii CLRT
°∠= 0A5.0iT
אאאאאאא،אK
R = ١٠٠ ΩE = ٥٠ V f = ١ kHz
L = ٢٫٥٣ H C= ٠٫٠١ µf
Ti
Ci Li Ri
אא ١٤١ א
אאא -٢
- ٧٠ -
F٣J١٢EאאאF٣J٣٢E،אא،W
6 j - 8 Z& 8 j 6 Z 21 =+=אאא،volts 100 E =(f =٦٠ Hz)KאאאK
א
F٣J٣٢EאאאF٣J١٢KE
°∠Ω=+= 3.5110 8j6 Z1
°−∠Ω=−= 9.3610 6j8 Z2
°∠= 0V100E
°−∠=°∠
°∠= 3.51A10
3.51100100 i1
°∠=°−∠
°∠= 9.36A10
9.36100100 i2
( )( )( )( ) 6j88j6
6j88j6 ZZ
ZZ Z
21
21t −+
−+=
+=
13.81.7 13.814.1426.16100
2j1428j96 Zt °∠Ω=
°∠°∠
=++
=
°−∠=∠
∠== 13.8A1.14
13.81.70100
ZE i
t
Z١
Z٢
i
i١
i٢
١٠٠ V
אא ١٤١ א
אאא -٢
- ٧١ -
٣J٤אאאPower in A.C Circuits
٣J٤J١אPower Triangle
א،אאאאאאאאא،(Voltage lead)
אא (Current lead)אאאאאא(in phase)אא،
אאאϕF٣J٣٣KE
F٣J٣٣EאאאאK
אIWאא=ϕ⋅ cosIאא= ϕ⋅ sinIF٣J٣٤KE
F٣J٣٤EאאאאK
V
i
ϕ
V
I
ϕ
ϕ⋅ sinI
ϕ⋅ cosI
אא ١٤١ א
אאא -٢
- ٧٢ -
אא،אאאאאאאאאK
אאVאאF٣J٣٥KE
F٣J٣٥Eא
אאWא،אאאW
F١Eאא(S)WApparent PowerWIVS ⋅= ) ١١٨-٣(
F٢Eאא(P)WActive PowerW ϕ⋅⋅= cosIVP ) ١١٩-٣ (
F٣Eאא(Q)WReactive PowerWϕ⋅⋅= sinIVQ ) ١٢٠-٣ (
٣J٤J٢אPower Factorאאאאאאאϕcos،
אאϕאאאאK
SP)(cos =ϕ
) ١٢١-٣ (
V
IVS ⋅=
ϕ
ϕ⋅⋅= cosIVP
ϕ⋅⋅= sinIVQ
אא ١٤١ א
אאא -٢
- ٧٣ -
٣J٤J٣אPower Factor Correction
אאאאאאאא(Current lag.)F٣J٣٦Eאא،
אSאאאאPאאKאאQאא،SP
אאא،אKאאאϕ ϕcosאא
אKSPאאאK
F٣J٣٦EאאאאK
،אאאאאF٣J٣٦E،אאאאI١אאϕ١،
אP.F١ = cos ϕ١KאאאאאFאאאאE،אאא
F٣J٣٧Eאא،ϕאK
V
1I
1ϕ
1I
مصدر التغذيةV א
אא ١٤١ א
אאא -٢
- ٧٤ -
F٣J٣٧EאאK
F٣J١٣EF٣J٣٨Eאא،אאא
אאאK
F٣J٣٨EאאאF٣J١٣KE
אאZLאW
( ) ( ) Ω=+= 1086Z 22L
אאθZLW
°==θ − 3.5168tan 1
ZL
1I
8j6ZL +=0V380V ∠=
2I
CI
1I
LZCV
V
CI
1I
2I
2ϕ1ϕ
CI
CI
1I
V
אא ١٤١ א
אאא -٢
- ٧٥ -
א1IW
°−∠=°∠°∠
= 3.51A383.51100380
I1
א،אאאאא،אאאϕ٢KאאW
A66.29)3.51(sin38sinII 11C −=°−⋅=ϕ⋅=
אאא°=θ 90CW
°∠−=∴ 90A66.29IC
אאW
90CV90)C1(
0V)C1(j
0VIC ∠ω=
−∠ω∠
=ω−
∠=
F42.248502380
66.29C µ−=×π×
−=
אאF٣J٣٩KE
F٣J٣٩EאאK
V
11 sinI ϕ⋅
CI
1I
1ϕ
22211 IcosIcosI =ϕ⋅=ϕ⋅
אא ١٤١ א
אאא -٢
- ٧٦ -
אא
WאW
١K אאאאאאאK
٢K אאאא٩٠°אK
٣K אאאא٩٠°אK
٤K אאא.XL
٥K אאא.XC
٦K אאאאאאאאK
٧K אאאאאאאאK
٨K אאאאאאאאK
٩K אאאאאאאאK
١٠K אאאאאאאאK
١١K אאאאאאאאK
WאW
١K אאאא(٥Ω) אא(٥ mH)אאK
٢K אאאא(١٥Ω)אא(١µf)אאK
٣K אאאא(٢٠Ω) אא(٥ mH)אא(١µf)אאK
אא ١٤١ א
אאא -٢
- ٧٧ -
٤K אאאא(٥Ω) אא(٥ mH)אאK
٥K אאאא(١٥Ω)אא(١µf)אאK
٦K אאאא(٢٠Ω) אא(٥ mH)אא(١µf)אאK
٧K אאאאאאאא٥٠ mHא٧٠ mHK
٨K אאאאאאאא٣٠ mHא٢٠ mHK
٩K אאאאא١٠٠א µF٢٥א µFK
١٠K אאאאא١٠٠א µF٢٥א µFK
١١K ،אאאא،אאW8 j - 6 Z& 6 j 8 Z 21 =+=אאא،V 200 E =(f =٥٠ Hz)K
אאאK
Z١
Z٢
i
i١
i٢
٢٠٠ V
אא ١٤١ א
אאא -٢
- ٧٨ -
١٢K אא،אאאאאאאאK
١٣K א١٠ µFאא،١٢ Vאא١א kΩא،Wאאτאא،٥٠
msecKאא،אאאא٢ kΩאאא،אאאא
٣٠ msecאK
1I
8j6ZL +=0V380V ∠=
2I
CI
1I
LZCV
- ٣٦ -
אאאאאאאאאאאא
JJ٢٢
אאאã¹]<gè…‚jÖ]æ<ËÖ]<Üé×Ãj×Ö<íÚ^ÃÖ]<퉉ö¹]
אאאא
א
אא
א
א
א
٤
אאא ١٤١ א
אאאאאא -٢
- ٧٩ -
אאאאא
אא،אאאW• אאאK • אאאאK • אאאאאאK • אאאאאאK • אאאאאאK • אאאאאאK • אאאאאאK • אאאאK • אאאאאאK • אאאאאK
אאא ١٤١ א
אאאאאא -٢
- ٨٠ -
٤J١Introduction א،אאאאאאאא
אאאאאאאאK
٤J٢אאאאאBasic Theorems of A.C Circuits
٤J٢J١ Ohm's Lawאאאאא
אאאאVאIאZאא،FאE
אW
(a) Z
Vi Z
Vi,
ZVi
θ∠θ∠
=θ∠= )١-٤(
(b) ZiV ZiV,ZiV θ∠•θ∠=θ∠•= )٢-٤(
(c) i
VZ i
VZ,
iVZ
θ∠θ∠
=θ∠= )٣-٤(
F٤J١E
אאWF١EאאZKF٢EאאiK
F٤J١EאאאF٤J١KE
8j6Z +=
i
١٠٠ V٥٠ Hz
אאא ١٤١ א
אאאאאא -٢
- ٨١ -
אF١E אZ،W
W٦אאΩ6KWאאFאאEΩ=⋅ω 8LW
Hm465.251008
f288
L =π
Ω=
πΩ
=ωΩ
=
F٤J٢EאZF٤J١KE F٢Eאiא،Z،W
°∠Ω=∠+= − 13.531068tan86Z 122
אi،W
°−∠=°∠Ω
°∠== 13.53A10
13.53100V100
ZVi
F٤J٢EאאW
F١E אאZF٢E
F٤J٣EאאאF٤J٢KE
Ω= 6R Hm465.25L =
Z
°∠= 30A20i
١٠٠ V
אאא ١٤١ א
אאאאאא -٢
- ٨٢ -
אF١E אZאW
i
VZ i
VZ,
iVZ
θ∠θ∠
=θ∠=
°−∠Ω=°∠°∠
= 30530A200V100
Z
F٢E W
)30(sinj)30(cos5Z −+−Ω=
4.2j33.4)5.0j866.0(5Z −=−Ω=
אאWאZW
W٣٣٤אאΩ33.4
WאאFאאEΩ=⋅ω
4.2C
1
W
mF326.11004.21
4.21C =
π××=
ω=
F٤J٤EאZF٤J٢KE
Ω= 33.4R mF326.1C =
אאא ١٤١ א
אאאאאא -٢
- ٨٣ -
٤J٢J٢ Kirchhoff's Laws
٤J٢J٢J١ Kirchhoff's Current Law ?אאאאאאא
א،W
0ink
1kk =∑
=
= )٤-٤(
nאאאאא،אאאאאאא،אאאאאאא
אKF٤J٥EאאK
F٤J٥Eאאאאאא
F٤J٣EאאF٤J٦Eאאא،ITK
F٤J٦EאאאF٤J٣KE
1i
2i 3i
ni
Z١ Z٢
I١٠ =١ A ∠٣٠° I٢٠=٢ A ∠-٢ °
IT A
B
אאא ١٤١ א
אאאאאא -٢
- ٨٤ -
אאאITאא،ABW
• AW FE אאאאא=FITEFE אאאא=FI١ + I٢E
W (IT) = (I١ + I٢)
• BW FE אאאאא=FI١ + I٢EFE אאאא=FITE
W(I١ + I٢) =(IT)
אאABאK
W
( ) =°−∠+°∠=+= 20A2030A10III 21T
( ) ( ) °−∠=−=−++= 84.3A5.2784.1j46.2784.6j8.185j66.8IT
F٤J٤EF٤J٣EאאZ١, Z٢אא،W
E=١١٠ V, f=٦٠HZ
Z١ Z٢
I١٠ =١ A∠٣٠° I٢٠=٢ A∠-٢٠°
IT A
B
E=١١٠ V f = ٦٠ HZ
אאא ١٤١ א
אאאאאא -٢
- ٨٥ -
F٤J٧EאאאF٤J٤KE
אאWW
°−∠Ω=°∠°∠
= 301130A10
0V110Z1
°∠Ω=°−∠
°∠= 205.5
20A200V110
Z2
אW
( ) ( ) 5.5j52.930sin11j30cos113011Z1 −=°−⋅+°−⋅=°−∠Ω=
٥٢٩אR٥٢٩ =١ Ωאא،אאאXC= ٥٥ Ωא،C١W
C1XC ⋅ω
=
F28.4825.5602
1X1C
C1 µ=
××π=
⋅ω=
W
( ) ( ) 88.1j17.520sin5.5j20cos5.5205.5Z2 +=°+°=°∠Ω=
١٧٥אR١٧٥ =٢ Ωאא،אאאXL= ٨٨١ Ωאאא،LW
2L LX ⋅ω=
mH98.4602
88.1XL L
2 =×π
Ω=
ω=
אאא ١٤١ א
אאאאאא -٢
- ٨٦ -
F٤J٨Eא21 Z,ZF٤J٤KE
F٤J٥EאאF٤J٩Eאאא،I١K
F٤J٩EאאאF٤J٥KE
אאאI١אא،AW
• AW אאאאאFITE=אאאאFI١ + I٢E
W
( ) ( )21T III +=
( ) ( )2T1 III −=
( ) ( )5j66.884.6j8.18301020A20III 2T1 +−+=°∠−°∠=−=
°∠=+= 28.10A3.1084.1j14.10I1
Z١ Z٢
I١
IT =٢٠ A∠-٢ °
I١٠ =٢ A∠٣٠°
A
B
Ω= 52.9R1 F28.482C1 µ= Ω= 17.5R 2 mH98.4L2 =
Z١ Z٢
אאא ١٤١ א
אאאאאא -٢
- ٨٧ -
אאאאK
٤J٢J٢J١J١ אאאParallel Connection F٤J٧EF٤J٤Eא،Z١, Z٢
אאFאJ١Eאאא،אא،אאא
א V אאKKאאאאאZ
F٤J١٠E،W
F٤J١٠Eאאא
אאאWZIZIZIV T2211 === )٥-٤(
W( ) ( )21T III += )٦-٤(
I٢אEF،W
⎟⎟⎠
⎞⎜⎜⎝
⎛+=⎟⎟
⎠
⎞⎜⎜⎝
⎛
21 ZV
ZV
ZV
)٧-٤( אZא،W
⎟⎟⎠
⎞⎜⎜⎝
⎛+=⎟
⎠⎞
⎜⎝⎛
21 Z1
Z1
Z1
)٨-٤(
Z١ Z٢
I١ I٢
IT A
B
V Z
IT
IT A
B
V≡
אאא ١٤١ א
אאאאאא -٢
- ٨٨ -
⎟⎟⎠
⎞⎜⎜⎝
⎛ +=⎟
⎠⎞
⎜⎝⎛
21
21
ZZZZ
Z1
)٩-٤(
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛+
=21
21
ZZZZ
Z )١٠-٤(
אאאאאאאאאK
F٤J٦EF٤J٤EאאאZ١ & Z٢K
F٤J١١EאאאF٤J٦E
אאZ١ & Z٢אF٤J٤Eא،W
°−∠Ω= 3011Z1
°∠Ω= 205.5Z2
אאZאF٤J١٠EW
( ) ( ) ( )88.1j17.55.5j52.920305.511
205.53011205.53011
ZZZZ
Z21
21
++−°+°−∠Ω×Ω
=°∠Ω+°−∠Ω°∠Ω×°−∠Ω
=⎟⎟⎠
⎞⎜⎜⎝
⎛+
=
Z١١ = ١ Ω ∠-٣٠° Z٥٫٥ = ٢ Ω ∠٢٠°
I١ I٢
IT A
B
Z
IT
IT A
B
V ≡
אאא ١٤١ א
אאאאאא -٢
- ٨٩ -
( ) ( ) °−∠Ω°−∠Ω
=−
°−∠=
84.1313.15105.60
62.3j69.14105.60
Z2
( ) ( ) °∠Ω=°+°−∠Ω
Ω= 84.3484.1310
13.155.60
Z2
אאא،אFF٤J٣EE،W
°−∠=°∠Ω
°∠= 84.3A5.27
84.340V110
IT
אאאF٤J٣KE
٤J٢J٢J١J٢ אCurrent Divider F٤J١٢EאאITI١, I٢K
F٤J١٢Eאא21 Z,Z
אאאאITאאאI١, I٢W( ) ( )21T III +=Q )١١-٤( ( ) ( )1T2 III −=∴ )١٢-٤(
ZIZIZIV T2211 ===Q )١٣-٤( ( ) 21T11 ZIIZIV −==∴ )١٤-٤(
2T2111 ZIZIZI =+ )١٥-٤( ( ) 2T211 ZIZZI =+ )١٦-٤(
1Z 2Z
I١ I٢
IT A
B
V
אאא ١٤١ א
אאאאאא -٢
- ٩٠ -
( )21
2T1 ZZ
ZII
+=∴
)١٧-٤( W
( )21
1T2 ZZ
ZII
+=∴
)١٨-٤( אאא،אאאאאאא
،אאאאאK
F٤J٧EאאF٤J١٣Eאאאא،2i, 1iK
F٤J١٣EאאאF٤J٧E
אאF٤J١٧EF٤J١٨Eאאא،I١, I٢
W
( ) ( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛°∠Ω+°−∠Ω
°∠Ω×°−∠=
+=
205.53011205.5
84.3A5.27ZZ
Zii
21
2T1
( ) ( ) ( )⎟⎟⎠
⎞⎜⎜⎝
⎛++−
°∠Ω×°−∠=
88.1j17.55.5j53.9205.5
84.3A5.27i1
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛−
°∠Ω×°−∠=
62.3j7.14205.5
84.3A5.27i1
1Z = ١١ Ω ∠-٣٠° 1Z = ٥٫٥ Ω
1i 2i
Ti = ٢٧٫٥ A ∠- A
B
אאא ١٤١ א
אאאאאא -٢
- ٩١ -
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛°−∠Ω
°∠Ω×°−∠=
84.1314.15205.5
84.3A5.27i1
( ) ( ) °∠=°∠×°−∠= 30A1084.33363.084.3A5.27i1
( ) ( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛°∠Ω+°−∠Ω
°−∠Ω×−∠=
+=
205.530113011
84.3A5.27ZZ
Zii
21
1T2
( ) °−∠=⎟⎟⎠
⎞⎜⎜⎝
⎛°−∠Ω
°−∠Ω×−∠= 20A20
84.1314.153011
84.3A5.27i2
אאאאאאאאF٤J٣EאאK
٤J٢J٢J٢ Kirchhoff's Voltage Law ?אאא
אאאאא،W
0vnk
1kk =∑
=
= )١٩-٤(
nא،אאאאאאאאאאאאאKא
אKF٤J١٤EאאאK
אאא ١٤١ א
אאאאאא -٢
- ٩٢ -
F٤J١٤Eאאא
אאW
0VVVE 321 =−−− )٢٠-٤(
W
321 VVVE ++= )٢١-٤(
אאאאאאאאK
1Z
2Z
3Z
iE
1V
i
i
i
3V
2V
אאא ١٤١ א
אאאאאא -٢
- ٩٣ -
F٤J٨EF٤J١٥EאאאA, BK
F٤J١٥EאאאF٤J٨E
א،אאW
321AB VVVV ++=
אאW
( ) ( ) °−∠=°−∠Ω⋅°−∠=⋅= 50V5020530A10ZiV 11
( ) ( ) °−∠=°−∠Ω⋅°−∠=⋅= 40V7010730A10ZiV 22
( ) ( ) °−∠=°∠Ω⋅°−∠=⋅= 15V8015830A10ZiV 33
אVABא،W
°−∠+°−∠+°−∠=++=− 15V8040V7050V50VVVV 321BA
( ) ( ) ( ) =−+−+−= 7.20j27.7745j62.533.38j14.32VAB °−∠=−= 54.32V4.193104j163VAB
אאאאK
°−∠Ω= 205Z1
°−∠= 30A10i
°−∠Ω= 107Z2
°∠Ω= 158Z3
A
B
אאא ١٤١ א
אאאאאא -٢
- ٩٤ -
٤J٢J٢J٢J١ אאאSeries ConnectionF٤J١٥EF٤J٨Eא،321 Z,Z,Z
אאFאJ١Eא،1Zא2אZא2Zאא3Zא
אאאאאאiאאK
אאאאאZF٤J١٦E،W
F٤J١٦Eאאא
אVאאW
321 VVVV ++=
אאאA, B،אW
ZiZiZiZiV 321 =++= )٢٢-٤(
אאiא،W
ZZZZ 321 =++∴ )٢٣-٤(
אאאאאאאאK
1Zi
2Z
3Z
A
B
1V
2V3V
V Z
i
A
B
V≡
אאא ١٤١ א
אאאאאא -٢
- ٩٥ -
F٤J٩EאF٤J٨EאאאאK
F٤J١٧EאאאF٤J٩E
אF٤J١٧EאZ١אאZ٢
אZ٢אאZ٣אאאאאאאiאאK
אא،אאאאW
°∠Ω+°−∠Ω+°−∠Ω=++= 158107205ZZZZ 321eq
( ) ( ) ( ) Ω++Ω−+Ω−= 1.2j7.722.1j9.671.1j7.4Zeq
( ) °−∠Ω=−= 46.232.1983.0j3.19Zeq
٤J٢J٢J٢J٢ אPotential Divider F٤J١٨EאאאאאZ١, Z٢& Z٣
אEאא،V،אאW
°−∠Ω= 205Z1
°−∠Ω= 107Z2
°∠Ω= 158Z3
A
B
i
Z
A
B
≡i
אאא ١٤١ א
אאאאאא -٢
- ٩٦ -
F٤J١٨Eאאאאאא
W
11 ZiV ⋅= )٢٤-٤(
22 ZiV ⋅= )٢٥-٤(
33 ZiV ⋅= )٢٦-٤(
W
( )321321 ZZZiVVVE ++⋅=++= )٢٧-٤(
1VE،W
( )321
11
ZZZZ
EV
++=
)٢٨-٤(
אE،W
( )⎟⎟⎠
⎞⎜⎜⎝
⎛++
⋅=321
11 ZZZ
ZEV
)٢٩-٤(
1Zi
2Z
3Z
1V
2V3V
A
B
E
אאא ١٤١ א
אאאאאא -٢
- ٩٧ -
א2VE،W
( )321
22
ZZZZ
EV
++=
)٣٠-٤(
א×E،W
( )⎟⎟⎠
⎞⎜⎜⎝
⎛++
⋅=321
22 ZZZ
ZEV
)٣١-٤(
א3VE،W
( )321
33
ZZZZ
EV
++=
)٣٢-٤(
אE،W
( )⎟⎟⎠
⎞⎜⎜⎝
⎛++
⋅=321
33 ZZZ
ZEV
)٣٣-٤(
אW
( )⎟⎟⎠
⎞⎜⎜⎝
⎛⋅=
T
xx Z
ZEV
F٤J١٠EF٤J١٩Eאאאאא
אK
אאא ١٤١ א
אאאאאא -٢
- ٩٨ -
F٤J١٩EאאאF٤J١٠E
א
אאאאאאFKEאאאW
( )⎟⎟⎠
⎞⎜⎜⎝
⎛++
⋅=321
11 ZZZ
ZEV
( ) ( ) ( ) ( )( )⎟⎟⎠
⎞⎜⎜⎝
⎛°∠Ω+°−∠Ω+°−∠Ω
°−∠Ω⋅°∠=
158107205205
0V110V1
( ) ( ) ( ) ( )( )⎟⎟⎠
⎞⎜⎜⎝
⎛°∠Ω+°−∠Ω+°−∠Ω
°−∠Ω⋅°∠=
158107205205
0V110V1
( ) ( ) ( ) ( )( )⎟⎟⎠
⎞⎜⎜⎝
⎛Ω++Ω−+Ω−
°−∠Ω⋅°∠=
1.2j7.722.1j9.671.1j7.4205
0V110V1
( ) ( ) °−∠=⎟⎟⎠
⎞⎜⎜⎝
⎛°−∠Ω
°−∠Ω⋅°∠= 54.17V47.28
46.232.19205
0V110V1
( ) ( ) ( )⎟⎟⎠
⎞⎜⎜⎝
⎛°−∠Ω
°−∠Ω⋅°∠=⎟⎟
⎠
⎞⎜⎜⎝
⎛++
⋅=46.232.19
1070V110
ZZZZ
EV321
22
°−∠= 54.7V86.39V2
°−∠Ω= 205Z1
°−∠Ω= 107Z2
°∠Ω= 158Z3
A
B
E= ١١٠ V f = ٦٠ Hz
אאא ١٤١ א
אאאאאא -٢
- ٩٩ -
( ) ( ) ( )⎟⎟⎠
⎞⎜⎜⎝
⎛°−∠Ω
°∠Ω⋅°∠=⎟⎟
⎠
⎞⎜⎜⎝
⎛++
⋅=46.232.19
1580V110
ZZZZ
EV321
33
°∠= 46.17V55.45V3
אאV١, V٢, V٣W
°∠+°−∠+°−∠=++= 46.1755.4554.786.3954.1747.28VVVE 321
( ) ( ) ( ) ( ) °∠=−=++−+−= 0V1100j1107.13j4.432.5j5.39V5.8j1.27E
٤J٢J٣ אאאCompound Circuitsאאאאאאאא،אא
אאאאKF٤J٢٠Eאאאא،
אאא،אאZ٢, Z٣אאFאאאEא،Z١, Z٤אאאFא
אאKE،אאאאאאא
אK
F٤J٢٠Eא
٤J٢J٣J١ אאSimplification Methodאאאאאאאא
א،אאאאK
1Z
1i
2Z
4Z
A
B
V 3Z
2i 3i
V١
C
D
אאא ١٤١ א
אאאאאא -٢
- ١٠٠ -
F٤J١١EF٤J٢١EאWא321א i&i,iK
F٤J٢١EאאאF٤J١١E
אWאאאא
KWא
،אאאאאאאאKא
אW• אCאW
321 iii += • אאACDBW
( ) 22411412211 ZiZZiZiZiZiE ++=++= ( ) ( ) 231411 ZiiZZiE −++=
• אאCEFDW3322 ZiZi =
אאFאאאאEא3iW
3
223 Z
Zii =
°∠Ω= 105Z11i
°∠Ω= 105Z2
A
B
Hz60fV110E
==
°−∠Ω= 1010Z3
2i 3iC
D°−∠Ω= 1010Z4
E
F
אאא ١٤١ א
אאאאאא -٢
- ١٠١ -
אאW( )
3
232
3
2221 Z
ZZi
ZZi
ii+
=+=
אא2iW
( )23
312 ZZ
Zii
+=
אאW
( ) ( ) ( )⎟⎟⎠⎞
⎜⎜⎝
⎛
+++=⎟
⎟⎠
⎞⎜⎜⎝
⎛
+++=
23
23411
23
231411 ZZ
ZZZZi
ZZZZ
iZZiE
אאiW
( )⎟⎟⎠⎞
⎜⎜⎝
⎛
+++
=
23
2341
1
ZZZZ
ZZ
Ei
( ) ( )( ) ( )⎟⎟
⎠
⎞⎜⎜⎝
⎛°∠Ω+°−∠Ω°∠Ω⋅°−∠Ω
+°−∠Ω+°∠Ω
°∠=
10510101051010
1010105
0V110i1
( )( ) ( ) ⎟
⎟⎠
⎞⎜⎜⎝
⎛
Ω++Ω−°∠Ω
+°−∠Ω+°∠Ω
°∠=
87.0j92.474.1j85.9050
1010105
0V110i
21
( )( ) ⎟
⎟⎠
⎞⎜⎜⎝
⎛
Ω−°∠Ω
+°−∠Ω+°∠Ω
°∠=
87.0j77.14050
1010105
0V110i
21
( )( ) ⎟
⎟⎠
⎞⎜⎜⎝
⎛
°−∠Ω°∠Ω
+°−∠Ω+°∠Ω
°∠=
37.38.14050
1010105
0V110i
21
אאא ١٤١ א
אאאאאא -٢
- ١٠٢ -
( )°∠Ω+°−∠Ω+°∠Ω°∠
=37.338.31010105
0V110i1
( ) ( ) ( )( )Ω++Ω−+Ω+°∠
=2.0j37.374.1j85.987.0j92.4
0V110i1
( ) °∠=°−∠Ω
°∠=
Ω−°∠
= 12.2A05.612.215.18
0V11067.0j14.180V110
i1
א2iW
( ) ( ) ( )( ) ( )( )⎟⎟
⎠
⎞⎜⎜⎝
⎛°∠Ω+°−∠Ω
°−∠Ω⋅°∠=
+=
10510101010
12.2A05.6ZZ
Zii
23
312
( ) ( )( ) ( )( ) =⎟⎟
⎠
⎞⎜⎜⎝
⎛Ω++Ω−
°−∠Ω⋅°∠=
87.0j92.474.1j85.91010
12.2A05.6i2
( ) ( )( ) =⎟⎟
⎠
⎞⎜⎜⎝
⎛−
°−∠Ω⋅°∠=
87.0j77.14101012.2A05.6i2
( ) °−∠=⎟⎟⎠
⎞⎜⎜⎝
⎛°−∠Ω
°−∠Ω⋅°∠= 5.4A09.4
4.38.141010
12.2A05.6i2
א3iW
( ) ( )°∠=
°−∠Ω°∠Ω⋅°−∠
== 5.15A05.21010
1055.4A09.4Z
Zii
3
223
אאא ١٤١ א
אאאאאא -٢
- ١٠٣ -
Wאא
F٤J٢٢Eאא
F٤J٢٢Eאאאאאא،Z٢, Z٣אאאאאZ١, Z٤א
אאZW
32
3241 ZZ
ZZZZZ
+++=
אאאאא1iא،W
( )⎟⎟⎠⎞
⎜⎜⎝
⎛
+++
=
23
2341
1
ZZZZ
ZZ
Ei
אא1i،א1אi،W
°∠= 12.2A05.6i1
א2iא،W
( )23
312 ZZ
Zii
+=
1Z1i
2Z
A
B
Hz60fV110E
==
3Z
4Z
2i 3iC
D
E
F
Z
A
B
≡
1i
E, f
אאא ١٤١ א
אאאאאא -٢
- ١٠٤ -
אא2i،א2אi،W
°−∠= 5.4A09.4i2
א3iאא،W• אW
( )32
213 ZZ
Zii
+=
( ) ( ) °∠=°−∠Ω
°∠Ω∠= 5.15A05.2
4.38.14105
12.2A05.6i3
אא32 ZZ +א• אW
321 iii +=Q
213 iii −=∴
( ) ( ) =°−∠−∠=∴ 5.4A09.412.2A05.6i3
( ) ( ) °∠=+=−−+=∴ 5.15A05.254.0j97.132.0j08.422.0j05.6i3
• אאאאאW
2233 ZiZi =
א3Z،W
3
223 Z
Zii =∴
אאאאאאW
°∠=∴ 5.15A05.2i3
אאא ١٤١ א
אאאאאא -٢
- ١٠٥ -
אאאא2iK
٤J٢J٣J٢ אJאJStar-Delta, Delta-Star ConversionאF٤J٢٣Eאאא،
אאאאKאאאKאאאא،אאא
אאאK
F٤J٢٣Eאא
אJאJאאאאאאK،אאF٤J٢٤EאאA, B,
CאאאאKאאאא(∆)אאאא(Y)K
אK
b c
1Z 2Z
3Z4Z
E, f
a
d
1i 2i
3i 4i
5Z5i
Ti
אאא ١٤١ א
אאאאאא -٢
- ١٠٦ -
F٤J٢٤Eאא
אאאאאאאאאאאאאאאא
אאאאKאKא
F٤J٢٥Eאא،A, BאאאA, B∆אא
אA, BYKאW
F٤J٢٥EאאA, B
( ) ( )( )⎟⎟
⎠
⎞⎜⎜⎝
⎛+++⋅
=+CABCAB
CABCABBA ZZZ
ZZZZZ )٣٤-٤(
A
B C
O
ZA
ZB ZC
A
B C
ZAB
ZBC
ZCA ≡
A
B C
A
B C
O
ZA
ZB ZC
ZAB
ZBC
ZCA
אאא ١٤١ א
אאאאאא -٢
- ١٠٧ -
( ) ( )( ) ( ) ⎟⎟
⎠
⎞⎜⎜⎝
⎛++
+=⎟⎟
⎠
⎞⎜⎜⎝
⎛+++⋅
=+CABCAB
CAABBCAB
CABCAB
CABCABBA ZZZ
ZZZZZZZZZZ
ZZ ) ٣٥-٤(
F٤J٢٦Eאא، B, C
אאאB, C∆אאאB, CYKאW
( ) ( )( )⎟⎟
⎠
⎞⎜⎜⎝
⎛+++⋅
=+CAABBC
CAABBCCB ZZZ
ZZZZZ
)٣٦-٤(
( ) ( )( ) ( )CABCAB
CABCABBC
CAABBC
CAABBCCB ZZZ
ZZZZZZZZZZ
ZZ++
+=⎟⎟
⎠
⎞⎜⎜⎝
⎛+++⋅
=+ ) ٣٧-٤(
F٤J٢٦EאאB, C
F٤J٢٧Eאא،A,
C אאאA, C∆אאאA, CYKאW
A
B C
A
B C
O
ZA
ZB ZC
ZAB
ZBC
ZCA
אאא ١٤١ א
אאאאאא -٢
- ١٠٨ -
F٤J٢٧EאאC, A
( ) ( )( )⎟⎟
⎠
⎞⎜⎜⎝
⎛+++⋅
=+BCABCA
BCABCAAC ZZZ
ZZZZZ )٣٨-٤(
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛++
+=+
BCABCA
BCCAABCAAC ZZZ
ZZZZZZ
)٣٩-٤(
אאF٤J٣٥EF٤J٣٧EF٤J٣٩EאW
⎟⎟⎠
⎞⎜⎜⎝
⎛++
⋅=
BCABCA
ABCAA ZZZ
ZZZ
)٤٠-٤(
⎟⎟⎠
⎞⎜⎜⎝
⎛++
⋅=
CABCAB
ABBCB ZZZ
ZZZ
)٤١-٤(
⎟⎟⎠
⎞⎜⎜⎝
⎛++
⋅=
BCABCA
ABCAC ZZZ
ZZZ
)٤٢-٤(
אW
C
BABAAB Z
ZZZZZ
⋅++=
)٤٣-٤(
A
CBCBBC Z
ZZZZZ
⋅++=
)٤٤-٤(
A
B C
A
B C
O
ZA
ZB ZC
ZAB
ZBC
ZCA
אאא ١٤١ א
אאאאאא -٢
- ١٠٩ -
B
ACACCA Z
ZZZZZ
⋅++=
)٤٥-٤(
F٤J١٢EאאאF٤J٢٨Eאאאאא،
(∆-Y)(Y-∆)K
F٤J٢٨EאאאF٤J١٢Eא
אאאאאאאאa,
b, c∆א،b, c, d∆،א a, c, dYbא،
a, b, dYcKאאאאאאKאא∆א
אa, b, cYאKא،אאF٤J٤٠EF٤J٤٢E،
W
( ) ( )⎟⎟⎠
⎞⎜⎜⎝
⎛°∠Ω+∠Ω+°−∠Ω
°∠Ω⋅°−∠Ω=⎟⎟
⎠
⎞⎜⎜⎝
⎛++
⋅=
02087.361013.531087.361013.5310
ZZZZZ
Z521
21A
b c
°−∠Ω= 13.5310Z1 °∠Ω= 87.3610Z2
°∠Ω= 13.5310Z3 °−∠Ω= 87.3610Z4
E=٢٢٠ Vf= ٥٠ Hz
a
d
1i 2i
3i 4i °∠Ω= 020Z5
5i
Ti
אאא ١٤١ א
אאאאאא -٢
- ١١٠ -
F٤J٢٩E
( )( ) ( ) ( )
( )( )Ω−
°−∠Ω=⎟
⎟⎠
⎞⎜⎜⎝
⎛
Ω++Ω++Ω−°−∠Ω
=2j34
26.160100j206j88j6
26.16010Z
22
A
( )( ) °−∠Ω=
°−∠Ω°−∠Ω
= 86.1294.24.334
26.16010Z
2
A
( ) ( )( )°−∠Ω
°∠Ω⋅°−∠Ω=⎟⎟
⎠
⎞⎜⎜⎝
⎛++
⋅=
4.33402013.5310
ZZZZZ
Z521
51B
( )( ) °−∠Ω=
°−∠Ω°−∠Ω
= 73.4988.54.33413.53200
Z2
B
( ) ( )( )°−∠Ω
°∠Ω⋅°∠Ω=⎟⎟
⎠
⎞⎜⎜⎝
⎛++
⋅=
4.33402087.3610
ZZZZZ
Z521
52C
°∠Ω=°−∠Ω°∠Ω
= 27.4088.54.33487.36200
ZC
אאאאאאאאאF٤J٣٠KE
a
b c
°−∠Ω= 13.5310Z1 °∠Ω= 87.3610Z2
°∠Ω= 020Z5
a
b c
o
ZA
ZB ZC
אאא ١٤١ א
אאאאאא -٢
- ١١١ -
F٤J٣٠E
F٤J٣٠EאZ٣, ZB،אאאZ٤, ZCא،אא
אאאאאZAKאאאאאאZeq،W
( ) ( )( ) ( ) A
C4B3
C4B3eq Z
ZZZZZZZZ
Z +⎟⎟⎠
⎞⎜⎜⎝
⎛++++⋅+
=
( ) ( ) ( )( ) ( ) ( )5.4j8.38j673.4988.513.5310ZZ B3 −++=°−∠Ω+°∠Ω=+
( ) °∠Ω=+=+ 65.1941.105.3j8.9ZZ B3 ( ) ( ) ( )( ) ( ) ( )8.3j5.46j827.4088.587.3610ZZ C4 ++−=°∠Ω+°−∠Ω=+
( ) °−∠Ω=−=+ 107.122.2j5.12ZZ C4
( ) ( ) ( ) ( )2.2j5.125.3j8.9ZZZZ C4B3 −++=+++
( ) ( ) °∠Ω=+=+++ 34.334.223.1j3.22ZZZZ C4B3
( ) ( )( ) ( )°−∠Ω+⎟⎟
⎠
⎞⎜⎜⎝
⎛°∠Ω
°−∠Ω⋅°∠Ω= 86.1294.2
34.334.22107.1265.1941.10
Zeq
E=٢٢٠ Vf= ٥٠ Hz
°−∠Ω= 86.1294.2ZA
°∠Ω= 13.5310Z3 °−∠Ω= 87.3610Z4
a
d
Ti o
b c
°∠Ω= 27.4088.5ZC°−∠Ω= 73.4988.5ZB
bi ci
אאא ١٤١ א
אאאאאא -٢
- ١١٢ -
( ) ( )°−∠Ω+°∠Ω= 86.1294.231.692.5Zeq
( ) ( ) ( ) °∠Ω=Ω+=Ω−+Ω+= 075.80j75.865.0j87.265.0j88.5Zeq
אאZeqאאא،KאאאאW
°∠=°∠Ω°∠
= 0A1.25075.80V220
iT
F٤J١٣EאאF٤J١٢Eאא،5iא(∆-Y)(Y-∆)K
אF٤J٢٨Eא،5iאאb, c
אאVbcא5Zאא5iKF٤J٣٠EאVbcאb, c
אאobcF٤J٣١EW
0VVV cobcob =++
אאאobcאאאo, bאאo, cא،Vobא،Vcoאא،אVbcא
אאאVobא،W
0VVV obcobc =+−
אאW
obcobc VVV −=
BbCcbc ZiZiV ⋅−⋅=
אאא ١٤١ א
אאאאאא -٢
- ١١٣ -
F٤J٣١E
אאcb i,iא،W
( )( ) ( )⎟⎟
⎠
⎞⎜⎜⎝
⎛+++
+⋅=
3B4C
4CTb ZZZZ
ZZii
( ) ( )( ) °−∠=⎟⎟
⎠
⎞⎜⎜⎝
⎛°∠Ω°−∠Ω
⋅°∠= 34.13A29.1434.334.22107.12
0A14.25ib
W
( )( ) ( )⎟⎟
⎠
⎞⎜⎜⎝
⎛+++
+⋅=
3B4C
3BTc ZZZZ
ZZii
( ) ( )( ) °∠=⎟⎟
⎠
⎞⎜⎜⎝
⎛°∠Ω°∠Ω
⋅°∠= 31.16A72.1134.334.2265.1941.10
0A14.25ic
אbcVW
BbCcbc ZiZiV ⋅−⋅=
( ) ( ) ( ) ( )°−∠Ω⋅°−∠−°∠Ω⋅°∠= 73.4988.534.13A29.1427.4088.531.16A72.11Vbc
°∠Ω= 13.5310Z3 °−∠Ω= 87.3610Z4
d
Tio
b c
°∠Ω= 27.4088.5ZC°−∠Ω= 73.4988.5ZB
bi ci
אאא ١٤١ א
אאאאאא -٢
- ١١٤ -
( ) ( )°−∠−°∠= 1.63V03.8458.56V91.68Vbc
( ) ( ) °∠=+=−−−+= 90V5.132V5.132j0V75j38V5.57j38Vbc
°∠=°∠Ω
°∠== 90A63.6
02090V5.132
ZV
i5
bc5
٤J٢J٣J٣ Thevinen's TheoremאאאאK
א XZאאאFF٤J٣٢EEאאאאא،Xiאא
אאאא1אVאאeqZאאאאFאאKE
F٤J٣٢E
אאאאאאאאKאאאW
• אאאאאאאאF٤J٣٣JE،אאאאאאאאא
F٤J٣٣JE. • אאאאאאאא
אא(a, b)אאThVF٤J٣٣JE. • אאאאאאאF
אאEF٤J٣٣JE،
E 1V
1Z
2Z XZ1V XZ
eqZXi Xi
אאא ١٤١ א
אאאאאא -٢
- ١١٥ -
• אאאאא(a, b)אאThZK
• אאXiאאאF٤J٣٣JEאא،אאXZאאאThVאThZ
W
( )XTh
ThX ZZ
Vi
+= )٤٦-٤(
FEFE
FE FEF٤J٣٣E
F٤J١٤EF٤J١٢Eא5אi
rאK
אF٤J٢٨Eא،5i
rW
E
1Z
2Z XZ
Xia
b
ThVE
1Z
2Z XZ
a
b
1Z
2Z ThZ
a
b
≡ ThV
ThZ
XZ
a
b
Xi
אאא ١٤١ א
אאאאאא -٢
- ١١٦ -
• Wא5ir
אא،אאאF٤J٣٤KE
F٤J٣٤E • WאThVF٤J٣٤E،
W
0VVV caThab =++
111ab ZiV ⋅=
221ca ZiV ⋅−=
0ZiZiV 221111Th =⋅−⋅+
111221Th ZiZiV ⋅−⋅=
b c
°−∠Ω= 13.5310Z1 °∠Ω= 87.3610Z2
°∠Ω= 13.5310Z3 °−∠Ω= 87.3610Z4
E=٢٢٠ Vf= ٥٠ Hz
a
d
11i 21i
1Ti
ThV
אאא ١٤١ א
אאאאאא -٢
- ١١٧ -
אא2111 i,iW
( ) ( ) ( )°∠Ω+°−∠Ω°∠
=+
=13.531013.5310
0V220ZZ
Ei31
11
( ) ( ) ( ) °∠=°∠Ω°∠
=Ω+°∠
=Ω++Ω−
°∠= 0A33.18
0120V220
0j120V220
8j68j60V220
i11
( ) ( ) ( )°−∠Ω+°∠Ω°∠
=+
=87.361087.3610
0V220ZZ
Ei42
21
( ) ( ) ( ) °∠=°∠Ω°∠
=Ω+°∠
=Ω−+Ω+
°∠= 0A75.13
0160V220
0j160V220
6j86j80V220
i21
( ) ( ) ( ) ( )°−∠Ω⋅°∠−°∠Ω⋅°∠= 13.53100A33.1887.36100A75.13VTh
( ) ( )°−∠−°∠= 13.53V3.18387.36V5.137VTh
( ) ( ) V64.146j110V5.82j110VTh −−+=
( ) °∠=+= 90V14.229V14.229j0VTh
• אאThZאאאאאFאאEאאאאb, c،
F٤J٣٥KEF٤J٣٦EF٤J٣٧Eאאאאאa, dא31 Z,Z
אאא،42 Z,Zאאאאא،אאK
אאא ١٤١ א
אאאאאא -٢
- ١١٨ -
F٤J٣٥E
F٤J٣٦E
b c
°−∠Ω= 13.5310Z1 °∠Ω= 87.3610Z2
°∠Ω= 13.5310Z3 °−∠Ω= 87.3610Z4
a
d
11i 21i
b c
°−∠Ω= 13.5310Z1 °∠Ω= 87.3610Z2
°∠Ω= 13.5310Z3 °−∠Ω= 87.3610Z4
a
d
11i 21i
אאא ١٤١ א
אאאאאא -٢
- ١١٩ -
F٤J٣٧E
אאThZW
=⎟⎟⎠
⎞⎜⎜⎝
⎛+⋅
+⎟⎟⎠
⎞⎜⎜⎝
⎛+⋅
=42
42
31
31Th ZZ
ZZZZZZ
Z
( ) ( )( ) ( )
( ) ( )( ) ( )⎟⎟
⎠
⎞⎜⎜⎝
⎛°−∠+°∠Ω°−∠⋅°∠Ω
+⎟⎟⎠
⎞⎜⎜⎝
⎛°∠+°−∠Ω°∠⋅°−∠Ω
=87.361087.361087.361087.3610
13.531013.531013.531013.5310
ZTh
( ) ( ) ( ) ( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛
Ω−+Ω+°∠Ω
+⎟⎟⎠
⎞⎜⎜⎝
⎛
Ω++Ω−°∠Ω
=6j86j8
01008j68j6
0100Z
22
Th
( ) ( ) ( ) ( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛
°∠Ω°∠Ω
+⎟⎟⎠
⎞⎜⎜⎝
⎛
°∠Ω°∠Ω
=⎟⎟⎠
⎞⎜⎜⎝
⎛
Ω+°∠Ω
+⎟⎟⎠
⎞⎜⎜⎝
⎛
Ω−°∠Ω
=016
0100012
01000j16
01000j12
0100Z
2222
Th
( ) ( ) Ω°∠Ω+Ω°∠Ω= 025.6033.8ZTh
( ) ( ) °∠Ω=Ω++Ω+= 0583.140j25.60j33.8ZTh
• אא5ir
W
( ) ( ) ( ) ( ) Ω++Ω+°∠
=°∠Ω+°∠Ω
°∠=
+=
0j200j583.1490V14.229
0200583.1490V14.229
ZZV
i5Th
Th5
b c
°−∠Ω= 13.5310Z1 °∠Ω= 87.3610Z2
°∠Ω= 13.5310Z3 °−∠Ω= 87.3610Z4
a
d
11i 21i
אאא ١٤١ א
אאאאאא -٢
- ١٢٠ -
( ) °∠=°∠Ω°∠
=Ω+°∠
= 90A63.60583.3490V14.229
0j583.3490V14.229
i5
א5ir
אאא(∆-Y)(Y-∆)אאK
٤J٢J٣J٤ אאFEClosed Loop (Mesh) method אאא،
אאKאא،אאאא
אK
F٤J٣٨EF٤J٣٨Eאאאא1iאאאאabcFאא
١Eאא2iאאאאbdecFאא٢KEאאא41 Z,Zא1iאאא3Z
2אiא2Zא21 ii −אW• אabcFאא١WE
( ) ( ) 221411 ZiiZZiE ⋅−++⋅= )٤٧-٤(
( ) 222411 ZiZZZiE ⋅−++⋅= )٤٨-٤(
• אbdecFאא٢WE
( ) 22132 ZiiZi0 ⋅−−⋅= )٤٩-٤(
E, f
1Z
2Z 3Z
4Z
1i 2i
a
c
d
e
אאא ١٤١ א
אאאאאא -٢
- ١٢١ -
( )32221 ZZiZi0 +⋅+⋅−= )٥٠-٤(
• אאאW
( )( ) ⎥
⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡+−
−++0E
ii
ZZZZZZZ
2
1
322
2241
)٥١-٤(
אאW١K אאאאאאאא
א[ ][ ] [ ]EZV =[ ]Zא[ ]iאא[ ]VאK
٢K אא[ ]iאאאאאאאאאאkאאאאK
٣K אאאא(k × k)אאK
אא11Z22Z،KKKKkkZאא١א٢KKKKKKKkא،F٤J٣٨E
W
( )24111 ZZZZ ++= )٥٢-٤(
( )3222 ZZZ += )٥٣-٤(
k = ٢אKאאאאאmnZ
אאאאmאאn،mnnm ZZ =אאאאnאאmאא
אאmאאnK٤K א[ ]Vאאk21x V,....V,VV =
xVאאאx،W1V٢אאאא١،2Vאאאא
אאא ١٤١ א
אאאאאא -٢
- ١٢٢ -
kVאאאאkKאאxVאאxiK
٥K אאאאאאאא،אW[ ] [ ] [ ]VZi 1−=א،
אאאאאאא[ ]Z[ ] 1Z −Kאא
אאאאKאאW • אא[ ]Z[ ] ZZ =
• אx[ ]Zx
Zאאאאxא[ ]VK
• אxW
Z
Zi xx =
)٥٤-٤(
• אאאאאאאאאאאאאאK
F٤J١٥EאאאF٤J٣٩EאאאabאאFKE
°∠Ω= 3010Z1 °−∠Ω= 2020Z2
Ti a
b
Ιi ΙΙiHz60f
V110E==
1i 2i
°∠Ω= 020Z3
אאא ١٤١ א
אאאאאא -٢
- ١٢٣ -
F٤J٣٩Eא
אאאא،אΙΙΙ i&iF٤J٣٩KEאאא
אKאאW
• א(٢×٢)W
[ ] ( ) ( )( ) ( )⎥⎦
⎤⎢⎣
⎡+−
−+=
211
131
ZZZZZZ
Z
אW
( ) ( ) ( ) °∠Ω=+=°∠Ω+°∠Ω=+ 9.909.295j66.280203010ZZ 31 ( ) ( ) °−∠Ω=−=°−∠Ω+°∠Ω=+ 84.351.2784.1j45.2720203010ZZ 21
⎥⎦
⎤⎢⎣
⎡°−∠Ω∠−
°∠−°∠=∴
84.351.27301030109.909.29
Z
• אאW
[ ] ⎥⎦
⎤⎢⎣
⎡=
ΙΙ
Ι
ii
i
• אW
[ ] ⎥⎦
⎤⎢⎣
⎡°∠°∠
=000110
V
،אאאאאאW
⎥⎦
⎤⎢⎣
⎡°∠°∠
=⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡°−∠Ω∠−
°∠−°∠
ΙΙ
Ι
000110
ii
84.351.27301030109.909.29
אאא ١٤١ א
אאאאאא -٢
- ١٢٤ -
אאאאאאאW
⎥⎦
⎤⎢⎣
⎡°∠°∠
⎥⎦
⎤⎢⎣
⎡°−∠Ω∠−
°∠−°∠=⎥
⎦
⎤⎢⎣
⎡ −
ΙΙ
Ι
000110
84.351.27301030109.909.29
ii 1
• אאW
( ) ( )( ) ( ) ( )( )°∠=−=°∠−°∠=∠−⋅∠−−°−∠Ω⋅°∠=
°−∠Ω∠−°∠−°∠
=
2.08.74512.2j8.7456010006.627.8003010301084.351.279.909.29
84.351.27301030109.909.29
Z
• אאאW
( ) ( )
°−∠=
°∠−°−∠⋅°∠=°−∠°∠
°∠−°∠=
84.31.3026
0084.351.27011084.351.2700
30100110Z 1
( ) ( ) °∠=°∠−°∠−⋅°∠−=°∠°∠−°∠°∠
= 301100003010011000301001109.909.29
Z 2
• ،אאאאW
°−∠=°∠
°−∠==Ι 04.4A06.4
2.08.74584.31.3026
Z
Zi 1
°∠=°∠
°∠==ΙΙ 8.29A475.1
2.08.745301100
Z
Zi 2
• א1iW
( ) ( ) ( )°−∠=−=
°∠−°−∠=−= ΙΙΙ
2.20A95.202.1j77.28.29A475.104.4A06.4iii1
אאא ١٤١ א
אאאאאא -٢
- ١٢٥ -
F٤J١٦EF٤J١٢Eא5אi
rאאאFKE
א،אאאא
אאΙΙΙΙΙΙ i,i,iF٤J٤٠KEאאאאאאK
F٤J٤٠EW
אWΙ= iiTאWΙΙΙ −= iii1אWΙΙ= ii2אWΙΙΙΙ −= iii3אWΙΙΙ= ii4אWΙΙΙΙΙ −= iii5
אא،אאK،א(٣ × ٣) אאk = ٣،
אW
( )°∠Ω=+= 012ZZZ 3111
b c
°−∠Ω= 13.5310Z1 °∠Ω= 87.3610Z2
°∠Ω= 13.5310Z3 °−∠Ω= 87.3610Z4
E=٢٢٠ Vf= ٥٠ Hz
a
d
1i 2i
3i 4i °∠Ω= 020Z5
5i
Ti
Ιi
ΙΙi
ΙΙΙi
אאא ١٤١ א
אאאאאא -٢
- ١٢٦ -
( )°−∠Ω=++= 4.334ZZZZ 52122
( ) ( ) ( )°∠Ω+°−∠Ω+°∠Ω=++= 02087.361013.5310ZZZZ 54333
( ) °∠Ω=Ω+= 8.306.342j34Z33
• אW
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
°∠°∠−°∠−°∠−°−∠°−∠−
°∠−°−∠−°∠=
4.306.3402013.53100204.306.3413.531013.531013.5310012
Z
• אאW
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
ΙΙΙ
ΙΙ
Ι
iii
i
• אW
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
°∠°∠
°∠=
0000
0220V
• ،אאאאאא
W
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
°∠°∠
°∠=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
°∠°∠−°∠−°∠−°−∠°−∠−
°∠−°−∠−°∠
ΙΙΙ
ΙΙ
Ι
0000
0220
iii
4.306.3402013.53100204.306.3413.531013.531013.5310012
אאאאאאאW
אאא ١٤١ א
אאאאאא -٢
- ١٢٧ -
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
°∠°∠
°∠
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
°∠°∠−°∠−°∠−°−∠°−∠−
°∠−°−∠−°∠=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −
ΙΙΙ
ΙΙ
Ι
0000
0220
4.306.3402013.53100204.306.3413.531013.531013.5310012
iii 1
،אאאאאW
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
°∠°∠−°∠−°∠−°−∠°−∠−
°∠−°−∠−°∠
°∠°∠−°∠°∠−°−∠°∠
°∠−°−∠−°∠
=Ι
4.306.3402013.53100204.306.3413.531013.531013.5310012
4.306.34020000204.306.340013.531013.53100220
i
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
°∠°∠−°∠−°∠−°−∠°−∠−
°∠−°−∠−°∠
°∠°∠°∠−°∠−°∠°−∠−
°∠−°∠°∠
=ΙΙ
4.306.3402013.53100204.306.3413.531013.531013.5310012
4.306.340013.53100200013.531013.53100220012
i
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
°∠°∠−°∠−°∠−°−∠°−∠−
°∠−°−∠−°∠
°∠°∠−°∠−°∠°−∠°−∠−
°∠°−∠−°∠
=ΙΙΙ
4.306.3402013.53100204.306.3413.531013.531013.5310012
0002013.5310004.306.3413.5310
022013.5310012
i
• ،אאW
iK אאW
°∠°∠−°∠−°∠−°−∠°−∠−
°∠−°−∠−°∠=
4.306.3402013.53100204.306.3413.531013.531013.5310012
Z
אאא ١٤١ א
אאאאאא -٢
- ١٢٨ -
( ) ( )( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )( )°−∠°∠+°∠°−∠°∠−+
°∠°∠−°∠°−∠−°−∠+
°∠−°∠−−°∠°−∠°∠=
4.306.3413.531002013.531013.531002013.53104.306.3413.531013.5310
0200204.306.344.306.34012Z
( ) ( ) ( )( )( ) ( ) ( )( )( ) ( ) ( )( )
( ) ( ) ( ) ( )( ) ( )
( ) ( ) ( ) ( )( ) ( )
( ) ( ) ( )°∠=
−=°∠−°∠+°∠=°∠°∠−
°∠°−∠+°∠°∠=+°∠−
+−°−∠++°∠=°∠+°−∠°∠−
°∠−°−∠−°−∠+
°∠−°∠°∠=
07.66385.0j7.663853.6935455.1102.354509120
4.165.35413.531063.16352.35413.53100760012
88.99j16.34013.531088.99j16.34013.53100j760012
73.496.34013.5320013.531013.5320073.496.34013.5310
040001160012Z
iiK אאW
°∠°∠−°∠°∠−°−∠°∠
°∠−°−∠−°∠=
4.306.34020000204.306.340013.531013.53100220
Z1
אאאאאW
( ) ( ) ( ) ( ) ( )( )( ) ( ) ( )( ) ( ) ( )( ) ( ) °∠=°∠°∠=
+°∠=°∠−°∠°∠=
°∠−°∠−−°∠°−∠°∠=
0167200076002200j76002200400011600220
0200204.306.344.306.340220Z1
iiiK אאW
°∠°∠°∠−°∠−°∠°−∠−
°∠−°∠°∠=
4.306.340013.53100200013.531013.53100220012
Z2
אאאאאW
אאא ١٤١ א
אאאאאא -٢
- ١٢٩ -
( ) ( ) ( ) ( ) ( )( )( ) ( ) ( )( )( ) ( ) ( ) ( )( ) ( ) °−∠=°∠−=°∠°∠−=
−°∠−=+−°∠−=°∠−°−∠−°∠−=
°∠−°∠−−°∠°−∠−°∠−=
64.16353.7799464.16353.7799464.16352.354022088.99j16.340022088.99j16.3400220
13.5320073.496.3400220
02013.53104.306.3413.53100220Z2
ivK אאW
°∠°∠−°∠−°∠°−∠°−∠−
°∠°−∠−°∠=
0002013.5310004.306.3413.5310
022013.5310012Z
3
אאאאאW
( ) ( ) ( ) ( ) ( )( )( ) ( ) ( )( ) ( ) ( )( ) ( ) °∠=°∠°∠=
+°∠=°∠+°−∠°∠=
°−∠°∠−−°∠−°−∠−°∠=
36.1655.7799436.165.354022088.99j16.340022073.496.34013.532000220
4.306.3413.531002013.53100220Z3
• אאW
°∠=°∠°∠
==Ι 0A16.2507.66380167200
ZZ
i 1
°−∠=°∠
°−∠==ΙΙ 64.163A75.11
07.663864.16353.77994
ZZ
i 2
°∠=°∠
°∠==ΙΙΙ 4.16A75.11
07.66384.1655.77994
ZZ
i 3
Wאאא •
0A16.25iiT ∠== Ι
( ) ( ) °∠=+=°−∠−°∠=−= ΙΙΙ 4.13A3.1431.3j89.1364.16375.11016.25iii1
°−∠== ΙΙ 64.163A75.11ii2
°−∠=−=°∠−°∠=−= ΙΙΙΙ 4.13A3.1432.3j89.134.16A75.110A16.25iii3
אאא ١٤١ א
אאאאאא -٢
- ١٣٠ -
°∠== ΙΙΙ 4.16A75.11ii4
( )°−∠−°∠=−= ΙΙΙΙΙ 64.163A75.114.16A75.11iii5
°∠=+=°∠+°∠= 90A63.663.6j064.163A75.114.16A75.11i5
אאTiאאאאF٤J١٢Eאא5iאאF٤J١٣EF٤J١٤KE
٤J٢J٣J٥ אSuperposition Theorem –١אאא،
אאאאאאאאאאאK
אאא،F٤J٤١Eאא21 E,Eאא،321 i,i,iאא21 E,E
F٤J٤٢EF٤J٤٣KE
אאא ١٤١ א
אאאאאא -٢
- ١٣١ -
F٤J٤١Eאא21 E,E
F٤J٤٢Eא1E
F٤J٤٣Eא2E
W
12111 iii −= )٥٥-٤(
21222 iii −= )٥٦-٤(
32313 iii += )٥٧-٤(
E١, f
1Z 2Z
3Z
1i 2ia b
c
d
E٢, f
3i
E١, f
1Z 2Z
3Z
11i 21ia b
c
d 31i
1Z 2Z
3Z
12i 22ia b
c
d
E٢, f
32i
אאא ١٤١ א
אאאאאא -٢
- ١٣٢ -
F٤J١٧EF٤J٤٤KEא،אאאאאאא
F٤J٤٣EאאאF٤J١٧Eא
אאאאF٤J٤٣EאאאאF٤J٤٤EF٤J٤٥E،אא
אא،אW
( )( )323121
321
32
321
111 ZZZZZZ
ZZE
ZZZZ
Z
Ei
+++⋅
=
++
=
F٤J٤٤EE١
E٢٢٠ = ١ V ∠٠°
°−∠Ω= 3010Z1 °∠Ω= 3010Z2
°∠= 6015Z3
1i 2ia b
c
d
E٢٢٠ =٢ V∠٣٠3i
E٢٢٠ = ١ V ∠٠°
°−∠Ω= 3010Z1 °∠Ω= 3010Z2
°∠= 6015Z3
11i 21i a b
c
d
31i
אאא ١٤١ א
אאאאאא -٢
- ١٣٣ -
F٤J٤٥EE٢
( )( )°∠Ω⋅°∠Ω+°∠Ω⋅°−∠Ω+°∠Ω⋅°−∠Ω
°∠Ω+°∠Ω⋅°∠=
601530106015301030103010601530100V220
i11
( )( ) 225j230
85.3732.290V22090150301500100
18j15.230V220i
22211 +°∠Ω⋅°∠
=°∠Ω+°∠Ω+°∠Ω
Ω+⋅°∠=
°−∠=°∠Ω
°∠Ω⋅°∠= 55.6A05.20
4.4475.32185.3732.290V220
i211
אאW
( ) °−∠=°∠Ω
°∠Ω⋅°−∠=
+= 4.14A84.8
85.3732.293010
55.6A05.20ZZ
Zii
32
21121
( ) °−∠=°∠Ω
°∠Ω⋅°−∠=
+= 6.15A26.10
85.3732.29601555.6A05.20
ZZZii
32
31131
( )
( )323121
312
31
312
222 ZZZZZZ
ZZE
ZZZZ
Z
Ei
+++⋅
=
++
=
( ) ( )
°∠Ω
+⋅°∠=
°∠Ω
°∠Ω+°−∠Ω⋅°∠=
4.4475.3218j16.1630V220
4.4475.3216015301030V220
i2222
( ) ( )
°∠=°∠Ω
°∠Ω⋅°∠= 12A33.12
4.4475.32131.2603.1830V220
i222
°−∠Ω= 3010Z1 °∠Ω= 3010Z2
°∠= 6015Z3
12i 22ia b
c
d
E٢٢٠ =٢ V∠٣٠°
32i
אאא ١٤١ א
אאאאאא -٢
- ١٣٤ -
( ) ( )°∠=
°∠Ω°∠Ω⋅°∠
⋅=+
= 6.45A25.1031.2603.18
601512A33.12ZZ
Zii
31
32212
( ) ( )
°−∠=°∠Ω
°−∠Ω⋅°∠⋅=
+= 4.44A84.6
31.2603.18301012A33.12
ZZZ
ii31
12232
אאW
( ) ( ) 6.9j75.126.45A25.1055.6A05.20iii 12111 −=°∠−°−∠=−= °−∠= 37A97.15i1
( ) ( ) 217.0j18.24.14A84.812A33.12iii 21222 −=°−∠−°∠=−= °−∠= 7.5A2.2i2
( ) ( ) 03.2j77.144.44A84.66.15A26.10iii 32313 −=°−∠+°−∠=+= °−∠= 8.7A9.14i3
אאא ١٤١ א
אאאאאא -٢
- ١٣٥ -
אאא
١K אא،אאאאאZK
٢K אאא،אאאא
°∠= 0V110Eא،٦٠ HzK
٣K אאZאאK
6j8Z −=
i
١٠٠ V٥٠ Hz
Ω= 6R Hm465.25L =
E
Z
°∠= 30A20i
١٠٠ V
אאא ١٤١ א
אאאאאא -٢
- ١٣٦ -
٤K אאא،אאiאא،
°∠= 0V110Eא،٥٠ HzK
٥K אאא،אITK
٦K א،אאZ١, Z٢אא،W
E=١١٠ V, f=٦٠HZ
Ω= 33.4R mF326.1C =
E
i
Z١ Z٢
I٥ =١ A∠-٣٠° I١٠=٢ A∠٢٠°
IT A
B
Z١ Z٢
I١٠ =١ A ∠٣٠° I١٠=٢ A ∠-٣ °
IT A
B
E=١١٠ V f = ٦٠ HZ
אאא ١٤١ א
אאאאאא -٢
- ١٣٧ -
٧K אאא،אאI١K
٨K אאאאK
٩K אF٦E،אאאK
١٠K אאF٤J١٣Eאאאא،I١, I٢،אאK
Z١ Z٢
I١
IT = ٤٠ A ∠-٦٠°
I١٠ = ٢ A ∠-٣٠°
A
B
Z٧ = ١ Ω ∠-٣٠° Z٥ = ٢ Ω ∠-٢٠°
I١ I٢
IT A
B
Z٢٠ = ١ Ω ∠- Z٢ = ١٠ Ω ∠٣٠°
I١ I٢
IT = ١٥ A ∠-٤٥° A
B
אאא ١٤١ א
אאאאאא -٢
- ١٣٨ -
١١K אאא،אאA, Bאאא،K
١٢K אאאא،אאא،אאאאK
١٣K אאא،321א i&i,iאאא،K
°∠Ω= 307Z1
°−∠= 30A8i
°−∠Ω= 108Z2
°∠Ω= 1515Z3
A
B
°∠Ω= 3015Z1
°∠Ω= 1017Z2
°∠Ω= 158Z3
A
B
V ١١٠ V ٦٠ Hz
°∠Ω= 1010Z11i
°∠Ω= 105Z2
A
B
Hz60fV110E
==
°−∠Ω= 1010Z3
2i 3iC
D°−∠Ω= 1010Z4
E
F
אאא ١٤١ א
אאאאאא -٢
- ١٣٩ -
١٤K אאF١٣Eאא،2iאK ١٥K אאאF٤J٢٨Eאאאאא،
(∆-Y)(Y-∆)K
١٦K אאF١٥Eאא،5iא،K
١٧K אאF١٥Eאא،5iאאא،K
١٨K אאאאאא،אאאאK
١٩K אאF١٨Eאא،i٣א(∆-Y)K
b c
°−∠Ω= 13.5320Z1 °−∠Ω= 87.3610Z2
°∠Ω= 13.5310Z3 °∠Ω= 87.3620Z4
E=٢٢٠ V f= ٥٠ Hz
a
d
1i 2i
3i 4i °∠Ω= 020Z5
5i
Ti
E٢٢٠ = ١ V ∠٠°
°−∠Ω= 6020Z1 °∠Ω= 6020Z2
°∠= 010Z3
1i 2ia b
c
d
E٢٢٠ =٢ V∠٣٠3i
אאאאאאאאאא
JJ٢٢
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אאאא
א
א
א
א
א
٥
אא ١٤١ א
אאאאא -٢
- ١٤٠ -
אאאא
אא،אאאW• אאאK • אאאאאK • אאאאאאאאאאאא
אאK • אאK • אאK • אאאאאאK • אאK • אאK
אא ١٤١ א
אאאאא -٢
- ١٤١ -
٥J١ Introductionאאאאאאאאא
אאאאא،אאאאא
KאאאאאאאאאאKא
אאאאאאK
٥J٢ אאMagnetic Induction
٥J٢J١ אאאSelf Inductance of a Coil
אאאאאא(L)אא،אאאאאK
אאאאאאאאאאאאK
אאאאאאאK
٥J٢J١J١ אאאאאא Factors that Affect the Self Inductance
אאאאאא١K א ٢K אFאNא،אא
אNKE ٣K אאאאאאFאאKE ٤K אK
אא ١٤١ א
אאאאא -٢
- ١٤٢ -
٥J٢J١Jא ٢אאCalculation of Self Inductance
F٣J١٠EאאאאאאאאאאאאאiאאאL
אdtdiאאא،W
dtdiL
dtdNE −=φ
−= )١-٥(
אאאLW
( )dtdiEL −
= )٢-٥(
didNL φ
= )٣-٥(
אאאW
אאא(L)
אאאאאאאאאאאאFאאEאאאא
Kאאא(Henry)،F.LKE
א
א א אא א א א א אאאא א א
אאאKאF٥J٢Eאאאאאאא،
אאאאאאאKאאא
אא ١٤١ א
אאאאא -٢
- ١٤٣ -
٥J٢J١J٣ אאאאאאאאאא Effect of Self Inductance on the Relation between induced emf & Current
F٥J١Eאאאאא
אאאאאאאאF
F٥J١EEאאאאאאK
٥J٢J٢ אאMutual Inductance
אFאאאE،אאאאאFאאEא،א
אאאאא،אאאאאאאא
F٥J٢KEאאאאאאאאK
אאWאאאאא
אK
ϕ١
אא
א
אא ١٤١ א
אאאאא -٢
- ١٤٤ -
F٥J٢Eאא
٥J٢J٢J١ אאMutual Inductance Coefficient
♦ Iאאא،אאW
tI
t1
∆∆
α∆ϕ∆ )٤-٥(
♦ אאאאאאξ٢،אאאאאW
t2 ∆ϕ∆
αξ )٥-٥(
♦ אF٥J٤EאF٥J٥EW
tI1
2 ∆∆
αξ )٦-٥(
♦ אF٥J٦EאW
tI
M 12 ∆
∆−=ξ
)٧-٥(
ϕ١
ϕ٢
1I
2I
אא ١٤١ א
אאאאא -٢
- ١٤٥ -
♦ אMאF٥J٧EאאאK ♦ אאאאאאאאאא
אאאאK אא
אF٥J٧EאאMאאW
12
1
2
It
tI
M∆∆
ξ=⎟⎠
⎞⎜⎝
⎛∆∆ξ
=−
)٨-٥( אאאאאאאאא
אאא١אKאא
אF٥J٨EאאF×÷KEאא(Henry)،F.LKEF٥J١E
אאאאא،٥mA/secאא٥٠ V
אאF٥J٨Eא،W
Asec/V100sec/A105
V5.0
tI
M3
1
2 ⋅=×
=⎟⎠
⎞⎜⎝
⎛∆∆ξ
=−−
אא ١٤١ א
אאאאא -٢
- ١٤٦ -
٥J٣ אאTransformer
٥J٣J١ אאFunction of Transformer
אאאאאKאאK٥J٣J٢ אאTransformer Applications
♦ אאאאK ♦ אאK ♦ אאאFאאE
אאאK
٥J٣J٣ אConstruction of Transformer
אאאF٢J٥Eאאאאא،אאאא
1אE،אאאאאא2אEF٥J٣KE
F٥J٣Eאאא
אאאאאאאאאK
ϕ
الملف االبتدائي
الملف1Eالثانوي 2E
אא ١٤١ א
אאאאא -٢
- ١٤٧ -
٥J٣J٤ אTransformer Operation
אאאאW١K ،אאאאאאאאא
אאאאאK ٢K אאאאאאK ٣K אאאאאא
אK ٤K אאאאאאאא
אאאאK ٥K אאאאאא
אאK
٥J٣J٥ אאאאאאemf Relationship
אאאאאאאאW١K אאאN١ = אא.N٢ = ٢K אאאאאא،אאאא
א2EW
tNE 22 ∆
ϕ∆= )٩-٥(
٣K ،אאאאאאאאאאאאאאאאאאאאאאאא
אאאאאאאאאאW
tNE 11 ∆
ϕ∆=
)١٠-٥(
٤K אF٥J٩EאF٥J١٠EאW
אא ١٤١ א
אאאאא -٢
- ١٤٨ -
⎟⎠⎞
⎜⎝⎛
∆ϕ∆
⎟⎠⎞
⎜⎝⎛
∆ϕ∆
=
tN
tN
EE
1
2
1
2
)١١-٥(
1
2
1
2
NN
EE
= )١٢-٥(
٥K אF٥J١٢EאאאאאאאW
1
212 N
NEE ⋅=
)١٣-٥(
٦K אאאאאאאאאאאאK
٥J٣J٦ אאאCurrents Relationship
אאא،אאאאא،אאאאW
21 PP = )١٤-٥(
אW
2211 iEiE ⋅=⋅ )١٥-٥(
אW
1
2
2
1
EE
ii
= )١٦-٥(
אF٥J١٢Eא،W
1
2
1
2
2
1
NN
EE
ii
== )١٧-٥(
אאW
אא ١٤١ א
אאאאא -٢
- ١٤٩ -
2
112 N
Nii ⋅=
)١٨-٥(
אאאאאאאאאK
F٥J٢E
אאאאאאא201אא،
אאאאאאאאאאאא٥٠ Vא١א AK
א
אF٥J١٣Eאאאא،W
( ) V100020V50NN
EE1
212 =⋅=⋅=
אF٥J١٨Eאאא،W
mA50A05.0201A1
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ii2
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⎠⎞
⎜⎝⎛⋅=⋅=
٥J٣J٧ אאTransformer Efficiency،אאאאא
אאWאאאאאאאאאא
אאאאאא،אKאηW
11
22
iViV
⋅⋅
=η )١٩-٥(
אא ١٤١ א
אאאאא -٢
- ١٥٠ -
F٥J٤Eאאא
אאאאF٥J٤KEאאאא LossesTאא Losses١אR١אא
Losses٢אR٢KאאW
1211 RiLosses ⋅= )٢٠-٥(
2222 RiLosses ⋅= )٢١-٥(
2221
2121T RiRiLossesLossesLosses ⋅+⋅=+= )٢٢-٥(
אאאאאאאאאK
٥J٣J٨ אאאאאTransformers in Power Networks אאאאא
אאאאאאאאאאFאאאאEאא
אאאאאKאאאא،F٥J٥KE
1R 1L2R 2L
1E 2E1V 2V
1i 2i
אא
אא ١٤١ א
אאאאא -٢
- ١٥١ -
F٥J٥Eאאאא
F٥J٣Eאאאא٣ kVא،אאא
٢٢٠ Vאא٦٦ kVאא،2
1
NN
אאKא
F٥J٦EאאF٥J٣E
א
א
א
א
א
E٣=١ kV
א
E٦٦=٢ kV E٢٢٠=٣ V المستهلك
אא ١٤١ א
אאאאא -٢
- ١٥٢ -
א
א2
1
NNאא،F٥J١٢EW
22300066000
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EE
11
12
1
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221
NN
12
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א2
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EE
12
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אא ١٤١ א
אאאאא -٢
- ١٥٣ -
אא
١K אאאK٢K אK ٣K אאא(L)K ٤K אK ٥K אאאאאאאK ٦K אאK٧K אאK ٨K אאK ٩K אאאK
١٠K אאאאאאאK ١١K אאאאK ١٢K אאאאאאK ١٣K אאא،אא١אmA/sec
אא٨٠ VK ١٤K אאא،אאא٣mA/sec
אא٥١ VK
١٥K אאאאאאא2001
א،אאאאאאאאאאא
אא١٠٠ Vאא١ AK ١٦K אאאאאאא200א،
אאאאאאאאאאאאא١٠٠ Vאא١ AK
١٧K אאאאאאא20א،אאאאאאאאאאא
אא١٠٠ Vאא١٠ AK
אא ١٤١ א
אאאאא -٢
- ١٥٤ -
١٨K אאאא١ kVאאא،١١٠א Vאא٣٣ kVאא،
12 NNאאK ١٩K אאאא٣ kVאאא،
٢٢٠א Vאא٦٦ kVאא،12 NNאאK
אא ١٤١ א
-٢
- ١٥٥ -
אא(References)
[١] Thomas L. Floyed, Electrical Engineering Fundamentals, Prentice, Inc, sixth
edition, ٢٠٠٠. [٢] B. L. Theraja, A. K. Theraja, " Electrical Technology", published by Ninja
Construction development Co. Ltd. Ram Nagar, New Delhi, ١٩٩٥ ,١١٠٠٥٥. [٣] M. A. PAI, "Introduction to Electric Circuits and Machines", Affiliated east west
presses private limited, ١٩٧٥.
א ١٤١ א
-٢
א
אא
אאWאאא ١אאאא
١J١ ٢
١J٢א٢١J٢J١אאא٢١J٢J١J١אא٢١J٢J١J٢א٤ א١J٢J١J٣אא ٤١J٣אא٦١J٣J١אאאא ٦١J٣J٢אאאא ٨١J٣J٣אאאא ١٠١J٣J٤אאא ١١١J٣J٥אאא ١١
אא١٤אאWאא
אאאא١٥٢J١١٦٢J٢אאא١٦٢J٢J١אא ١٦٢J٢J٢אא ١٩٢J٣אא١٩٢J٣J١אא ٢١
١٤١ א א א
-٢
٢J٣J٢אאא ٢٢٢J٣J٣אאא ٢٤٢J٣J٣J١אאא٢٤٢J٣J٣J٢אאא٢٥٢J٣J٤אא٢٧٢J٣J٥אאא ٢٩٢J٣J٥J١אאW ٢٩٢J٣J٥J٢אאא ٣٢
אא٣٤אאWאאא
אאאא٣٧٣J١٣٨٣J٢אא٣٨٣J٢J١אא ٣٨٣J٢J٢אא ٣٩٣J٢J٣אא ٤٠٣J٢J٣J١א ٤١٣J٢J٣J٢א ٤٢٣J٢J٣J٣אאאא ٤٤٣J٣אאא٤٥٣J٣J١אאאjאאא ٤٥٣J٣J٢אאא ٤٧٣J٣J٢J١אאא ٤٧٣J٣J٢J٢אאא ٤٧٣J٣J٢J٣אא ٤٩٣J٣J٢J٤אא ٥١٣J٣J٢J٥אאאא٥٥
١٤١ א א א
-٢
٣J٣J٣אאא٥٩٣J٣J٣J١אאא٥٩٣J٣J٣J٢אאא٦٠٣J٣J٣J٣אא٦١٣J٣J٣J٤אא٦٤٣J٣J٣J٥אאאא
٦٧
٣J٤אאא٧١٣J٤J١א٧١٣J٤J٢א٧٢٣J٤J٣א٧٣
אא٧٦אאאWאאאאא
אאאאא٧٩٤J١٨٠٤J٢אאאאא٨٠٤J٢J١٨٠٤J٢J٢٨٣٤J٢J٢J١٨٣٤J٢J٢J١Jא١אא٨٧٤J٢J٢J١J٢א٨٩٤J٢J٢J٢٩١٤J٢J٢J٢J١אאא٩٤٤J٢J٢J٢J٢א٩٥٤J٢J٣אאא٩٩٤J٢J٣J١אא٩٩
١٤١ א א א
-٢
٤J٢J٣J٢אJאJ١٠٥٤J٢J٣J٣١١٤٤J٢J٣J٤אאFE١٢٠٤J٢J٣J٥א١٣٠
אאא١٣٥אאWאאאאא
אאאא١٤٠٥J١١٤١٥J٢אא١٤١٥J٢J١אאא١٤١٥J٢J١J١אאאאאא١٤١٥J٢J١J٢אאא١٤٢٥J٢J١J٣אאאאאאאאא١٤٣٥J٢J٢אא١٤٣٥J٢J٢J١אא١٤٤٥J٣אא١٤٦٥J٣J١אא١٤٦٥J٣J٢אא١٤٦٥J٣J٣א١٤٦٥J٣J٤א١٤٧٥J٣J٥אאאאאא١٤٧٥J٣Jא٦אא١٤٨٥J٣J٧אא١٤٩٥J٣J٨אאאאא١٥٠
אא١٥٣אא١٥٥
אאאאאא
אאFאEא
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