จานวนเซิิงซ้อน ccoommpplleexx nnuummbbeerr · 1 1...
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จ านวนเซงซอนจ านวนเซงซอน
วตถประสงคเชงพฤตกรรมวตถประสงคเชงพฤตกรรม หลงจากศกษาบทเรยนนจบแลว ผเรยนมความรความสามารถดงตอไปน
1) อธบายความหมายและสวนประกอบของจ านวนเซงซอนได 2) อธบายวธการเปลยนรปของจ านวนเซงซอนในรปตาง ๆได 3) อธบายวธการคอนจเกทจ านวนเซงซอนได 4) อธบายวธการบวก ลบ คณ และการหารจ านวนเซงซอนได 5) แสดงออกถงพฤตกรรมดานคณธรรม จรยธรรม คานยม และคณลกษณะอนพงประสงค
- มมนษยสมพนธทดตอเพอนรวมชนเรยนและตอคร – อาจารย - มความรบผดชอบตอตนเองและผอน - มความสนใจใฝรในเนอหาวชาทเรยน - แสดงออกถงการมพฤตกรรมการท างานกลมทเหมาะสมได - ตระหนกในการตรงตอเวลาและรรกสามคค
CCoommpplleexx NNuummbbeerr
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จ านวนเซงซอน หมายถง ตวเลขหรอจ านวนทประกอบดวย
เลขจ านวนจรง (Real Number) กบเลขจ านวนจนตภาพ (Imaginary Number)
Z = X + jY
จ านวนเซงซอนจ านวนเซงซอน CCoommpplleexx NNuummbbeerr
X
j
Z1 = 2 + j4
Z2 = -5 + j5
Z3 = -2 – j5 Z4 = 5 – j4
5 -5
-5
4
Z5 = 3
Z6 = j2
Complex Plane
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ปรมาณเซงซอนจะประกอบดวยสวนตาง ๆ ดงน
จ านวนจรง ( Real Number) จ านวนจรง คอ จ านวนทเราทราบคาวามเทาไร ซงจะเปนคาทแนนอนหรอไมแนนอนกได แบงออกเปน 2 ชนด คอ
จ านวนตรรกยะ (Rational Number) คอ จ านวนทเปน
เลขเตมบวก เตมลบ เศษสวน ทศนยม (ทราบคาแนนอน) จ านวนอตรรกยะ (Irrational Number) คอ จ านวน
ทเราไมทราบคาทแนนอนไดหรอคาถกตอง โดยเราจะทราบเพยงคาโดยประมาณเทานน (ทราบคาไมแนนอน) เชน
2 = 1.4142…..
= 3.1428…..
log 3 = 0.47712…..
e = 2.71828…..
จ านวนจนตภาพ (Imaginary Number)
จ านวนจนตภาพ คอ จ านวนทเราไมทราบคา วามคามากนอย
เพยงใด เชน –4 , –9 , –25 เลขจ านวนทกลาวมานเรยกวา Pure Imaginary Number
Question –49 = ?
–100 = ?
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Z = X + jY
โดยท
X เรยกวา Real Part
Y เรยกวา Imaginary Part เมอ
Z = Complex Number ซงอยในรปของ
Rectangular form (ระบบพกดฉาก) X , Y = เลขจ านวนจรง
j = –1 เปนคาทเขยนหนาเลขจ านวนจรงใด ๆ หรอเปนคาทคณกบเลขนน ท าใหเปนคา Imaginary Part
ก าหนดให j = –1 หรอเรยกวา “ j Operator ”
ดงนน j2 = (–1)
2 = –1
j3 = j
2 j = (–1) j = –j
j4 = j
2 j
2 = (–1) (–1) = 1
j5 = j
4 j = ( j ) ( 1 ) = j
Question j32
= ? j32
= 1
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รปแบบของ Complex Number จากรปของ Rectangular form (ระบบพกดฉาก)
Z = X + jY
สามารถเขยนเปนรปบนพนราบระหวางแกน X กบแกน Y จะได Z ในแนวเสนตรง r ท ามมกบแนวแกน X เปนมม ไดดงน
จากรปจะไดวา cos = และ sin =
X = r cos …1
Y = r sin …2
Imaginary ( j )
Real ( X )
Z = X + jY
r
Y
X
sin =
cos =
tan =
ชดฉาก ขามฉาก
ขามมม ขามฉาก
ขามมม ชดฉาก
X r
Y r
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น าสมการ (1) และ (2) แทนในรปของ Rectangular form คอจากสมการ Z = X + jY
ดงนน
Z = r cos + j r sin
หรอ Z = r ( cos + j sin )
จ านวนเซงซอนลกษณะนเราเรยกวา Trigonometric form (ระบบพกดตรโกณมต) ของ Complex Number
เมอ r = X2 + Y
2 = tan
-1 ( Y
X)
คา r เรยกวา Absolute Value ของ Z
คา เรยกวา Amplitude ของ Z หรอเปนมม
ทท ากบแนวระนาบ
เมอทราบคา r และ เราสามารถเขยน Complex Number ไดอกรปหนงคอ Z = r
Z = r = X2+Y2
คาของ Complex Number ในรปนเราเรยกวา “Polar form” (ระบบพกดเชงขว) ของ Complex Number
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สรปสรปรปแบบตาง ๆ ของจ านวนเซงซอน รปแบบตาง ๆ ของจ านวนเซงซอน
Complex Number ซงโดยปกตเราจะเขยนใหอยในรปตาง ๆ ได 4 แบบ คอ
1. Rectangular form (ระบบพกดฉาก) คอ Z = X + jY
2. Polar form (ระบบพกดเชงขว) คอ
Z = r
3. Trigonometric form (ระบบพกดตรโกณมต) คอ
Z = r (cos + j sin )
4. Exponential form (ระบบพกดชก าลง) คอ
จาก Euler’s Formula e
j = cos + j sin
และจาก Trigonometric form Z = r (cos + j sin )
จะได
Z = r e j ; มหนวยเปน เรเดยน
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Example 1. จงเปลยน Polar form ใหเปน Rectangular
form , Trigonometric form และ Exponential form
a) 6.360o
b) 0.33180o
c) 4.845o
d) –50–90o
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วธท า a) จากสตร Pol. Rec.
Z = r cos + j r sin
= 6.36 cos 0o + j 6.36 sin 0
o
= 6.36 + j0
= 6.36
จากสตร Pol. Tri.
Z = r cos + j r sin
= 6.36 cos 0o + j 6.36 sin 0
o
จากสตร Pol. Exp.
Z = r e j
= 6.36 e j0
= 6.36
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Example 2. จงเปลยน Rectangular form ใหเปน Polar
form , Trigonometric form และ Exponential form
a) –12 – j12
b) 4 + j4
c) 90 + j0
d) –j90
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วธท า a) จากสตร Rec. Pol.
Z = r ; r = 2 2
x + y ; = y-1
tanx
r = 22
12 + -12 ; = -12-1
tan12
= 16.97 45
จากสตร Rec. Tri.
Z = r cos + j r sin
= 16.97 cos ( 45) + j 16.97 sin ( 45)
= 16.97 cos 45 j 16.97 sin 45
จากสตร Rec. Exp.
Z = r e j
= 16.97
π-j4e
NOTE การเปลยนมมลบ ( ) เปนมมบวก ( )
1. sin ( ) = sin [ sine ลบ เทากบ ลบ sine]
2. cos ( ) = cos [ cos ลบ เทากบ cos]
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การคอนจเกทจ านวนเซงซอน Conjugate of Complex Number ( Z
)
การคอนจเกทจ านวนเซงซอน หมายถง จ านวนเซงซอนทมเครองหมายตรงขามในสวนของ Imaginary Part
คาของ Conjugate นจะเขยนแทนดวย “ Z ” ดงน
Z = X + jY
Z = X – jY
Polar form คอ
Z = r
Z = r–
Trigonometric form คอ
Z = r ( cos + j sin )
Z = r ( cos – j sin )
Exponential form คอ
Z = r ej
Z = r e-j
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Example 3. จงหาคอนจเกท Complex Number ตอไปน a) 6.8812
o
b) –6.36 – j6.36
c) –126.5 sin (t + 75o)
d) 33
πj32e
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วธท า a) จากสตร Pol. Pol.
Z = r Z = r–
= 6.88 –12
จากสตร Rec. Rec.
Z = x + j y
Z = x – j y
= –6.36 + j6.36
จากสตร Exp. Exp.
Z = r cos + j r sin
Z = r cos – j r sin
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การบวกและการบวกและการการลบลบ CCoommpplleexx NNuummbbeerr
Complex number ทจะท าการบวกหรอลบกนไดนนจะตองอยในรปของ Rectangular form เทานน สวนอยในรปของ Polar form จะน าเอามาบวกหรอลบกนไมได
การบวก/ลบ กระท าไดโดยเอาสวนทเปน Real Part บวก/ลบ
กนกบสวนทเปน Real Part และสวนทเปน Imaginary Part
บวก/ลบกนกบสวนทเปน Imaginary Part
ถาก าหนดให Z1 = a + jb
Z2 = c + jd
ดงนน Z1 + Z2 = (a + jb) + (c + jd)
= (a + c) + j(b + d)
= (a + c) + j(b + d)
และ Z1 – Z2 = (a + jb) – (c + jd)
= (a – c) + j(b – d)
= (a + c) + j(b + d)
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Example 4. ถา Z1 = 5 – j2 และ Z2 = –3 – j8 จงหา
a) Z1 + Z2 c) Z1 + Z2
c) Z1 – Z2 d) Z1 – Z2
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การคณและหารการคณและหาร CCoommpplleexx NNuummbbeerr
การคณ Complex Number 2 จ านวนเขาดวยกน เราอาศย
หลกการทางพชคณต โดยใชหลกทวา
j2 = (–1)
2 = –1
j3 = j
2 j = (–1) j = –j
j4 = j
2 j
2 = (–1) (–1) = 1
j5 = j
4 j = ( j ) ( 1 ) = j
การคณจะมลกษณะดงน Rectangular form
Z1 Z2 = (a + jb)(c + jd)
= (ac + j2bd) + j (ad + bc)
= (ac – bd) + j (ad + bc)
Polar form
Z1 Z2 = r11 r22
= (r1 r2 )1+2
Exponential form
Z1 Z2 = j 1 j 2
1 2r e r e
= j 1 21 2r r e
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Example 5. ถาก าหนดให Z1 = 3 – j5 , Z2 = –4 + j2
Z3 = 230, Z4 = 3.72–143
, Z5 =
j44e
และ
Z6 =
2j
35e
จงหา
a) Z1 Z2 b) Z3 Z4 c) Z5 Z6
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การหาร Complex Number ในรปของ Rectangular
form มล าดบขนตอนในการหารดงตอไปน 1. ใหเขยนอยในรปของเศษสวน 2. คณทงเศษและสวนดวยคาของ Conjugate ( Z*
) 3. เอาเศษคณเศษ และสวนคณสวน
ดงนนเราจะไดวา
1
2
Z a jb
Z c jd
= a jb c jd
c jd c jd
= 2
2 2
ac j bd j bc ad
c d
= 2 2
ac bd j bc ad
c d
NOTE การคณ/หาร Complex Number ในรปของ Rectangular form จะไมคอยนยมกนมากนก เพราะวาไมคอยสะดวกเหมอนกนกบการคณและหารในรปของ Polar form
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การหารในรปของ Polar form และ Exponential form
กระท าไดโดยการน าเอาคา Absolute Value ( r ) ของ Complex Number แตละจ านวนมาหารกน และเอาคามมของแตละจ านวนมาลบกน ดงน
Polar form
1
2
ZZ =
1 1
2 2
r
r
= 1
1 2
2
r
r
Exponential form
1
2
ZZ =
j 11
j 22
r e
r e
= 1 j 1 2
2
re
r
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Example 6. ถาก าหนดให Z1 = 2 – j4 , Z2 = –8 – j2
Z3 = 1090 , Z4 = 0.2–170
, Z5 =
j440e
และ Z6 =
3j
225e
จงหา
a) Z1 Z2
b) Z3 Z4
c) Z5 Z6
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ใบงาน เรอง จ านวนเซงซอน (Complex Number)
ชอ ......................................................................................... เลขท ................ ระดบชน ปวช. 2/...............
1. Complex Plane หมายถงอะไร
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2. จงอธบายความหมายของ Complex number
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3. รปแบบของ Complex number มกรปแบบ อะไรบาง
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4. จงอธบายวธการเปลยน Complex number จากระบบพกดฉากไปเปนระบบพกดเชงขว
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5. จงอธบายวธการเปลยน Complex number จากระบบพกดไปเปนระบบพกดฉาก
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6. จงอธบายวธการคอนจเกทจ านวนเซงซอน
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7. จงอธบายหลกการบวกจ านวนเซงซอนในรปแบบของ Rectangular form
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8. จงอธบายวธการลบจ านวนเซงซอนในรปแบบของ Rectangular form
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9. จงอธบายหลกการคณจ านวนเซงซอนในรปแบบของ Polar form
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10. จงอธบายวธการลบจ านวนเซงซอนในรปแบบของ Polar form
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แบบฝกหด เรอง จ านวนเซงซอน (Complex Number)
ชอ ......................................................................................... เลขท ................ ระดบชน ปวช. 2/...............
1. จงเปลยน Polar form ใหเปน Rectangular form , Trigonometric form และ Exponential form a) 45.50o f) 12.86690o b) 53180o g) 0.0132700 c) 25–45o h) 12.25450 d) 86–30o i) 100600 e) 0.05–90o j) 6600
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2. จงเปลยน Rectangular form ใหเปน Polar form , Trigonometric form และ Exponential form 2.1) –33 + j 33 2.7) 0.17 + j 0.47 2.2) 2 – j 4 2.8) – j4000 2.3) –590 – j 0 2.9) –220 + j 220 2.4) 700 + j 700 2.10) –8 – j 8 2.5) 4 + j 5 2.11) –10 + j 250 2.6) 40 + j 0 2.12) j 55
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3. ถา Z1 = 5 – j2 , Z2 = –3 – j8 , Z3 = –4180 และ Z4 = 3-90 จงหา
a) Z1 + Z2 – Z*3 e) Z*3 + Z4 – Z1 b) Z*2 – Z1 + Z*4 f) Z*1 + Z*2 + Z4 c) (Z*1 Z2 ) / Z4 g) Z*1 ( Z*3 / Z2 )
d) (Z2 – Z1) Z*3 h) (Z*1 / Z*2 ) Z4 ...........................................................................................................................................................................................
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การเปลยนรปของ Complex Number
การเปลยนรปจาก Rectangular form เปน Polar form ( Rec. Pol. ) เครองคดเลขยหอ / รน CASIO fx – 5500 L และ fx - 6300 G และ fx – 4500 P
X + jY r 1. กด SHIFT แลวกด Pol 5. กด ) เครองหมายวงเลบปด 2. ใสคาจ านวนจรง ( X ) 6. กดหาผลลพธ ( EXE ) จะไดคาจรง r 3. กดเครองหมาย , ( เครองหมายลกน า ) 7. กด ALPHA แลว กด J แลวกด EXE จะไดคามม θ 4. ใสคาจ านวนจนตภาพ ( jY )
การเปลยนรปจาก Polar form เปน Rectangular form ( Pol. Rec. )
r X + jY
1. กด SHIFT แลวกด Rec. 5. กด ) ( เครองหมายวงเลบปด ) 2. ใสคาจรง r 6. กดหาผลลพธ ( EXE ) จะไดคาจรง ( X จ านวนจรง) 3. กดเครองหมาย , ( เครองหมายลกน า ) 7. กด ALPHA แลว กด J แลวกด EXE จะไดคาจ านวน 4. ใสคามม θ จนตภาพ ( jY ) หมายเหต การใสเครองหมาย , (เครองหมายลกน า) เครองค านวณ Casio fx – 6300 G ตองกด SHIFT กอนและรน 4500 – P ใหกด ALPHA W EXE การเปลยนรปจาก Rectangular form เปน Polar form ( Rec. Pol. ) เครองคดเลขยหอ / รน CASIO fx - 3600 PV , fx3800P , CANNON F-502
X + jY r 1. กด คาจ านวนจรง X แลวกด a 3. กด 2ndF แลวกด a จะไดคาจรง r ( คาหนามม ) 2. กด คาจ านวนจนตภาพ Y แลวกด b 4. กด b จะไดคามม θ
การเปลยนรปจาก Polar form เปน Rectangular form ( Pol. Rec. )
r X + jY 1. กด คาจรง r ( คาหนามม ) แลวกด a 3. กด 2ndF แลวกด b จะไดคา X : จ านวนจรง 2. กด มม θ แลวกด b 4. กด b จะไดคา Y : จ านวนจนตภาพ ถาคา จานวนจรง จานวนจนตภาพ หรอคามม θ มเครองหมายลบ ( - )ใหใส ตวเลข กอน แลวกดปมทมเครองหมาย +/-
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การเปลยนรปจาก Rectangular form เปน Polar form ( Rec. Pol. ) เครองคดเลขยหอ / รน SHARP EL – 556L
X + jY r 1. กด Mode แลวกด 1 แลวปอนคา เชน 6 + j10 ใหกด 6 + ab/c 10 2. ถาตองการเปน Polar ใหกด 2ndF แลวกด 8 แลวกด = จะไดคา r ( คาหนามม ) 3. กด จะไดคามม θ การเปลยนรปจาก Polar form เปน Rectangular form ( Pol. Rec. )
r X + jY 1. กด Mode แลวกด 1 แลวปอนคา เชน 65 40 ใหกด 65 DMS 40 2. ถาตองการเปน Rec. ใหกด 2ndF แลวกด 9 แลวกด = จะไดคา X : จ านวนจรง 3. กด จะไดคา Y : จ านวนจนตภาพ การเปลยนรปจาก Rectangular form เปน Polar form ( Rec. Pol. ) เครองคดเลขยหอ / รน CASIO fx - 3600 PV , fx3800P
X + jY r 1. กดคา X : จ านวนจรง 4. กด = จะไดคา r ( คาหนามม ) 2. กด 2ndF แลวกด R P 5. กด 2ndF แลวกด X Y จะไดคามม θ 3. กด คา Y : จ านวนจนตภาพ การเปลยนรปจาก Polar form เปน Rectangular form ( Pol. Rec. )
r X + jY 1. กดคา r ( คาหนามม ) 4. กด = จะไดคา X : จ านวนจรง 2. กด 2ndF แลวกด P R 5. กด 2ndF แลวกด X Y จะไดคา Y : จ านวน 3. กด คามม θ จนตภาพ หมายเหต เครองคานวณ CASIO fx – 3600 PV และ 3800 P รนเดยวกน แตกดตางกน (แปลก) ใหพจารณาจากลกษณะของตวเครองวาเปนแบบใด
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การเปลยนรปจาก Rectangular form เปน Polar form (Rec. Pol.)
เครองคดเลขยหอ / รน CASIO fx – 5500 L และ fx - 6300 G และ fx – 4500 P
X + jY r 1. กด Shift แลวกด Pol 5. กด ) : เครองหมายวงเลบปด 2. ใสคาจ านวนจรง ( X ) 6. กดหาผลลพธ ( EXE ) จะไดคาจรง r 3. กดเครองหมาย , ( เครองหมายลกน า ) 7. กด ALPHA แลว กด J แลวกด EXE จะไดคามม θ 4. ใสคาจ านวนจนตภาพ ( jY )
Example Rectangular Polar form 123 + j456 r
1. กด
2. กด
3. กด
4. กด
5. กด
6. กด จะไดคาจรง r
7. กด จะไดคามม θ
หมายเหต การใสเครองหมาย , ( เครองหมายลกน า ) เครองค านวณ Casio fx – 6300 G ตองกด Shift กอนและรน 4500 – P ใหกด ALPHA W EXE
1
Pol(
2 3
,
4 5 6
)
SHIFT
EXE
M-
ALPHA M+
J
EXE