1. 1 Complex Algebra and Load Power EE341 Ali Keyhani Circuit Theory Lecture #3

2. 2 Complex Numbers A complex number is a quantity of the form of Where a and b are real numbers, and a : real part b : imaginary partir0 a b  r0 a b  is called the conjugate of z (1)

3. 3 Complex Numbers A complex number can also be written in phasor form: - Modulus (or norm) - Argument (or phase) where (2) (3) (4)

4. 4 Complex Numbers i r0 a b  Conversion between two forms: (5) (6) (7) (8)

5. 5 Complex Numbers Operation Addition / Subtraction: Multiplication: A complex number times its conjugate  the square of its modulus. (9) (10) (11)

6. 6 Complex Numbers Operation Division: Addition and subtraction can be easily done in regular form. While multiplication and division are a little bit complicated. (12)

7. 7 Complex Numbers Operation Multiplication: Multiplication and division are much easier to be done in phasor form. Division: (13) (14)

8. 8 Electric Power For DC circuit: For single-phase AC circuit: Let Impedance angle So (15) (16) (17)

9. 9 Electric Power For three-phase AC circuit: Since We have (19) (18) (20)

10. 10 Electric Power For AC circuit: Complex power: VA, kVA, MVA Active power: W, kW, MW W, kW, MW Reactive power: Var, kVar, MVar (21) (22) (23) (24) (25)

11. 11 Power Factor Lagging: Q > 0,  > 0, inductive Leading: Q < 0,  < 0, capacitive Power factor is defined as: (26)

12. 12 Electric Load Example 1 Consider a three-phase 480V, 300kVA load with p.f. = 0.9 lagging, what is the active, reactive, and complex power of the load? Solution: Known: To compute: P, Q, from S lagging

13. 13 Electric Load Example 1 According to eqn (24), According to eqn (25), Q > 0 because p.f. is lagging.

14. 14 Electric Load Example 2 lagging

15. 15 Electric Load Example 2 According to eqn (24), According to eqn (25), Q > 0 because p.f. is lagging.

16. 16 Electric Load Example 3 leading

17. 17 Electric Load Example 3 According to eqn (24), According to eqn (25), Q < 0 because p.f. is leading.