Basic ElectronicsNinth Edition
Basic ElectronicsNinth Edition
©2002 The McGraw-Hill Companies
GrobSchultz
GrobSchultz
Basic ElectronicsNinth Edition
Basic ElectronicsNinth Edition
©2003 The McGraw-Hill Companies
25CHAPTER
Complex Numbers for AC Circuits
Topics Covered in Chapter 25
Positive and Negative Numbers
The j Operator
Definition of a Complex Number
Complex Numbers and AC Circuits
Impedance in Complex Form
Topics Covered in Chapter 25(continued)
Operations with Complex Numbers
Magnitude and Angle of a Complex Number
Polar Form
Converting Polar to Rectangular Form
Complex Numbers in Series AC Circuits
Topics Covered in Chapter 25 (continued)
Complex Numbers in Parallel AC Circuits
Combining Two Complex Branch Impedances
Combining Complex Branch Currents
Parallel Circuit with Three Complex Branches
Phasors Expressed in Rectangular Form
6+j0
0+j6
0-j6
6+j6
3-j3
• The j-operator rotates a phasor by 90°.• j0 means no rotation. • +j means CCW rotation.• -j means CW rotation.
Circuit Values Expressedin Rectangular Form
6+j0
6+j6
3-j3
0+j6 XL
0-j6 XC
6 6
3 3
Phasors Expressed in Polar Form
• Magnitude is followed by the angle. 0 means no rotation. • Positive angles provide CCW rotation.• Negative angles provide CW rotation.
6
6
6
8.496
6
4.24
Circuit Values Expressedin Polar Form
6XL
6 6
3 3
6
XC 6
8.49
4.24
Why Different Forms?
• Addition and subtraction are easier in rectangular form.
• Multiplication and division are easier in polar form.
• AC circuit analysis requires all four (addition, subtraction, multiplication, and division).
Rectangular-to-Polar Conversion
• General expression for the conversion:
R±jX = Z
arctangent
X
R• Second Step:
Z R X 2 2• First Step:
Polar-to-Rectangular Conversion
• General expression for the conversion:
ZR±jX
X Z sin• Second Step:
R Z cos• First Step:
Operations with Complex Expressions
• Addition (rectangular form) R1+jX1 + R2+jX2 = (R1+R2)+j(X1+X2)
• Subtraction (rectangular form) R1+jX1 R2+jX2 = (R1R2)+j(X1X2)
• Multiplication (polar form) Z11Z22 = Z1Z21 + 2)
• Division (polar form)
VS 6 4
8 4
Complex Numbers Appliedto a Series-Parallel Circuit
Recall the product over sum methodof combining parallel resistors:
21
21
RR
x RRR
EQ
The product over sum approach canbe used to combine branch impedances: 21
21
ZZ
x ZZZ
EQ
Complex Numbers Appliedto a Series-Parallel Circuit
VS 6 4
8 4 21
21
ZZ
x ZZZ
EQ
Z1 = 6+j0 + 0+j8 = 6+j8 = 1053.1°
Z2 = 4+j0 + 0-j4 = 4-j4 = 5.6645°
Z1 + Z2 = 6+j8 + 4-j4 = 10+j4 = 10.821.8
Z1 x Z2 = 1053.1° x 5.6645° = 56.6
56.610.821.8ZEQ = = 5.24
The Total Current Flowin the Series-Parallel Circuit
56.68.110.821.8ZEQ = = 5.2413.7
245.2413.7
IT = = 4.5813.7ANote: The circuit is capacitive since the current is leading by 13.7°.
4.5813.7A
24 V 6 4
8 4 21
21
ZZ
x ZZZ
EQ
The Total Power Dissipationin the Series-Parallel Circuit
WxxVx I x CosPT
107972.058.424
24 V 6 4
8 4 21
21
ZZ
x ZZZ
EQ
4.5813.7A
The Branch Dissipationsin the Series-Parallel Circuit
WxxV x I x CosPT
107972.058.424
1053.1°I1 =
24= 2.453.1° A
5.6645°I2 =
24= 4.24° A
P1 = I2R1 = 2.42 x 6 = 34.6 W
P2 = I2R2 = 4.242 x 4 = 71.9 W
Power check: PT = P1 + P2 = 34.6 + 71.9 = 107 W
6 4
8 4
24 V
4.5813.7A
Combining the Branch Currents
1053.1°I1 =
24= 2.453.1° A
5.6645°I2 =
24= 4.24° A
Convert branch currents to rectangular form for addition:
2.453.1° A = 1.44-j1.92 A
4.24° A = 3+j3 A
IT = 1.44-j1.92 + 3+j3 = 4.44+j1.08 A
6 4
8 4
24 V
4.5813.7A
KCL check: 4.44+j1.08 A = 4.5813.7A
Branch 1 Voltages
6 4
5.6645°I2 =
24= 4.24° A8
4
1053.1°I1 =
24= 2.453.1° A
24 V
VR1 = 2.453.1° x 6° = 14.453.1° V = 8.65-j11.5 V
VL1 = 2.453.1° x 8° = 19.2° V = 15.4+j11.5 V
KVL check: 8.65-j11.5 + 15.4+j11.5 = 24+j0 V
1
Branch 2 Voltages
1053.1°I1 =
24= 2.453.1° A6 4
5.6645°I2 =
24= 4.24° A8
4 24 V
VR2 = 4.24° x 4° = 17° V = 12+j12 V
VC1 = 4.24° x 4° = 17° V = 12-j12 V
KVL check: 12+j12 + 12-j12 = 24+j0 V
2