1. Basic Electronics Ninth Edition Grob Schultz
©2002 The McGraw-Hill Companies

2. Complex Numbers for AC Circuits
Basic Electronics
Ninth Edition 25
CHAPTER
Complex Numbers for AC Circuits
©2003 The McGraw-Hill Companies

3. 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

4. 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

5. 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

6. Phasors Expressed in Rectangular Form
6+j6
0+j6
6+j0
3-j3
0-j6
The j-operator rotates a phasor by 90°.
j0 means no rotation.
+j means CCW rotation.
-j means CW rotation.

7. Circuit Values Expressed
in Rectangular Form 6W
6+j6 W
0+j6 W XL
3-j3 W
0-j6 W XC
6+j0 W 3W

8. Phasors Expressed in Polar Form
6Ð90
6Ð0
6Ð-90
8.49Ð45 6
4.24Ð-45
Magnitude is followed by the angle.
Ð0 means no rotation.
Positive angles provide CCW rotation.
Negative angles provide CW rotation.

9. Circuit Values Expressed
in Polar Form 6W
8.49Ð45 W
6Ð90 W XL
4.24Ð-45 W XC
6Ð-90 W
6Ð0 W 3W

10. 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).

11. Rectangular-to-Polar Conversion
General expression for the conversion:
R±jX = ZÐq Z R X = + 2
First Step: q = æ è ç ö ø ÷
arctan
gent X R
Second Step:

12. Polar-to-Rectangular Conversion
General expression for the conversion:
ZÐq = R±jX R Z =
cos q
First Step: X Z =
sin q
Second Step:

13. Operations with Complex Expressions
Addition (rectangular form)
R1+jX1 + R2+jX2 = (R1+R2)+j(X1+X2)
Subtraction (rectangular form)
R1+jX1 - R2+jX2 = (R1-R2)+j(X1-X2)
Multiplication (polar form)
Z1Ðq1 ´ Z2Ðq2 = Z1Z2Ð(q1 + q2)
Division (polar form)

14. Complex Numbers Applied to a Series-Parallel CircuitVS
6 W
4 W
8 W
Recall the product over sum method
of combining parallel resistors: 2 1 R
x R EQ + =
The product over sum approach can
be used to combine branch impedances: 2 1 Z
x Z EQ + =

15. Complex Numbers Applied to a Series-Parallel CircuitVS
6 W
4 W
8 W 2 1 Z
x Z EQ + =
Z1 = 6+j0 + 0+j8 = 6+j8 W = 10Ð53.1° W
Z2 = 4+j0 + 0-j4 = 4-j4 W = 5.66Ð-45° W
Z1 + Z2 = 6+j8 + 4-j4 = 10+j4 = 10.8Ð21.8 W
Z1 x Z2 = 10Ð53.1° x 5.66Ð-45° = 56.6Ð8.1 W
56.6Ð8.1 W
10.8Ð21.8 W
ZEQ =
= 5.24Ð-13.7 W

16. in the Series-Parallel Circuit
The Total Current Flow
in the Series-Parallel Circuit
24 V
6 W
4 W
8 W 2 1 Z
x Z EQ + =
4.58Ð13.7 A
56.6Ð8.1 W
10.8Ð21.8 W
ZEQ =
= 5.24Ð-13.7 W 24
5.24Ð-13.7
IT =
= 4.58Ð13.7 A
Note:
The circuit is capacitive since the current is leading by 13.7°.

17. The Total Power Dissipation in the Series-Parallel Circuit
24 V
6 W
4 W
8 W 2 1 Z
x Z EQ + =
4.58Ð13.7 A W x
Vx I x Cos P T
107
972 . 58 4 24 = q

18. The Branch Dissipations in the Series-Parallel Circuit
6 W
4 W
8 W
24 V
4.58Ð13.7 A
10Ð53.1°
I1 = 24
= 2.4Ð-53.1° A
5.66Ð-45°
I2 = 24
= 4.24Ð45° A
P1 = I2R1 = 2.42 x 6 = 34.6 W
P2 = I2R2 = x 4 = 71.9 W W x
V x I x Cos P T
107
972 . 58 4 24 = q
Power check: PT = P1 + P2 = = 107 W

19. Combining the Branch Currents
6 W
4 W
8 W
24 V
4.58Ð13.7 A
10Ð53.1°
I1 = 24
= 2.4Ð-53.1° A
5.66Ð-45°
I2 = 24
= 4.24Ð45° A
Convert branch currents to rectangular form for addition:
2.4Ð-53.1° A = 1.44-j1.92 A
4.24Ð45° A = 3+j3 A
IT = 1.44-j j3 = 4.44+j1.08 A
KCL check: 4.44+j1.08 A = 4.58Ð13.7 A

20. Branch 1 Voltages KVL check: 8.65-j11.5 + 15.4+j11.5 = 24+j0 V 1 I1 =
6 W
4 W
5.66Ð-45°
I2 = 24
= 4.24Ð45° A
8 W
10Ð53.1°
I1 =
= 2.4Ð-53.1° A
24 V
VR1 = 2.4Ð-53.1° x 6Ð0° = 14.4Ð-53.1° V = 8.65-j11.5 V
VL1 = 2.4Ð-53.1° x 8Ð90° = 19.2Ð36.9° V = 15.4+j11.5 V
KVL check: 8.65-j j11.5 = 24+j0 V

21. Branch 2 Voltages KVL check: 12+j12 + 12-j12 = 24+j0 V 2 24 I1 =
10Ð53.1°
I1 = 24
= 2.4Ð-53.1° A
6 W
4 W
5.66Ð-45°
I2 =
= 4.24Ð45° A
8 W
24 V
VR2 = 4.24Ð45° x 4Ð0° = 17Ð45° V = 12+j12 V
VC1 = 4.24Ð45° x 4Ð-90° = 17Ð-45° V = 12-j12 V
KVL check: 12+j j12 = 24+j0 V