1. TELECOMMUNICATIONS Dr. Hugh Blanton ENTC 4307/ENTC 5307

2. Complex Numbers

3. Dr. Blanton - ENTC 4307 - Complex Numbers 3 Complex numbers M = A + jB Where j 2 = -1 or j = √-1 | M | = √(A 2 + B 2 ) and tan  = B/A B A M  Real ImaginaryImaginary ARGAND diagram

4. Dr. Blanton - ENTC 4307 - Complex Numbers 4 j notation Refers to the expression Z = R + jX X is not imaginary Physically the j term refers to +j = 90 o lead and -j = 90 o lag

5. Dr. Blanton - ENTC 4307 - Complex Numbers 5 Complex Number Definitions Rectangular Coordinate System: Real (x) and Imaginary (y) components, A = x +jy Complex Conjugate (A  A*) refers to the same real part but the negative of the imaginary part. If A = x + jy, then A* = x  jy. xx +jy  jy +x 1 2 11 22 12

6. Dr. Blanton - ENTC 4307 - Complex Numbers 6 Complex Number Definitions Polar Coordinates: Magnitude and Angle Complex conjugate has the same magnitude but the negative of the angle. If A =M  90 , then A*=M  -90 

7. Dr. Blanton - ENTC 4307 - Complex Numbers 7 Rectangular to Polar Conversion By trigonometry, the phase angle “  ” is, Polar to Rectangular Conversion y = imaginary part= M(sin  ) x = real part = M(cos  )

8. Dr. Blanton - ENTC 4307 - Complex Numbers 8 +jy  jy +x 1 2 11 22 12

9. Dr. Blanton - ENTC 4307 - Complex Numbers 9 Vector Addition & Subtraction Vector addition and subtraction of complex numbers are conveniently done in the rectangular coordinate system, by adding or subtracting their corresponding real and imaginary parts. If A = 2 + j1 and B = 1 – j2: Then their sum is: A + B = (2+1) + j(1 – 2) = 3 – j1 and the difference is: A - B = (2  1) + j(1  (– 2)) = 1 + j3

10. Dr. Blanton - ENTC 4307 - Complex Numbers 10 For vector multiplication use polar form. The magnitudes (M A,M B ) are multiplied together while the angles (  ) are added. MuItiplying “A” and “B”: AB = (2.24  26.60  )(2.24  63.40  ) = 5  36.8 

11. Dr. Blanton - ENTC 4307 - Complex Numbers 11 Vector division requires the ratio of magnitudes and the differences of the angles:

12. Dr. Blanton - ENTC 4307 - Complex Numbers 12 +jy  jy +x 1 2 11 22 12 22 11 A-B A+B

13. Dr. Blanton - ENTC 4307 - Complex Numbers 13 Complex Impedance System RF components are frequently defined by their terminal impedances or admittances in the complex rectangular coordinate system. Complex impedance is the vector sum of resistance and reactance. Impedance = Resistance ± j Reactance RR +jX  jX +R inductive capacitive

14. Dr. Blanton - ENTC 4307 - Complex Numbers 14 Series connections are handled most conveniently in the impedance system. 

15. Dr. Blanton - ENTC 4307 - Complex Numbers 15 Complex Admittance System Parallel circuit descriptions may be viewed in the complex admittance system Complex impedance is the vector sum of conductance and susceptance. Admittance = Conductance ± j Susceptance where and GG +jB  jB +G inductive capacitive

16. Dr. Blanton - ENTC 4307 - Complex Numbers 16 Parallel connections are handled most conveniently in the admittance system. 

17. Dr. Blanton - ENTC 4307 - Complex Numbers 17 Z dependence on  (RCL ) 12510.20.50.100. frequency 1 5 10 50 100 500 1000 Impedance oo parallel series

18. Dr. Blanton - ENTC 4307 - Complex Numbers 18 Currrent dependence on  12510.20.50.100. frequency 100 500 1000 Current (ma)  o oo I min I max x √2 parallel series

19. Dr. Blanton - ENTC 4307 - Complex Numbers 19 At RF, particularly at high power levels, it is very important to maximize power transfer through careful impedance matching. Improperly matched component connection leads to “mismatch loss.”

20. Dr. Blanton - ENTC 4307 - Complex Numbers 20 RF Components & Related Issues Unique component problems at RF: Parasitics change behavior Primary and secondary resonances Distributed vs. lumped models Limited range of practical values Tolerance effects Measurements and test fixtures Grounding and coupling effects PC-board effects

21. Dr. Blanton - ENTC 4307 - Complex Numbers 21 V and I Phase relationships VLVL I VRVR VSVS  VCVC V L -V C

22. Dr. Blanton - ENTC 4307 - Complex Numbers 22 R, X C and Z relationships XLXL I R Z  XCXC X L -X C

23. Dr. Blanton - ENTC 4307 - Complex Numbers 23 Example 1 Consider this circuit with  = 10 5 rad s -1 1 k  0.01  F

24. Dr. Blanton - ENTC 4307 - Complex Numbers 24 Example 2 5  20  10 

25. Dr. Blanton - ENTC 4307 - Complex Numbers 25 Example 2 Cont’d 5  20  10  ~ 200 V

26. Dr. Blanton - ENTC 4307 - Complex Numbers 26 Example 2 Cont’d VLVL I VRVR VSVS  VCVC V L -V C XLXL I R Z  XCXC X L -X C Z = 15 - 10jI = 8/13(15 + 10j)

27. Dr. Blanton - ENTC 4307 - Complex Numbers 27 General procedures convert all reactances to ohms express impedance in j notation determine Z using absolute value determine I using complex conjugate draw phasor diagram Note: j = -1/j so R + (1/j  C) = R - j/  C

28. Dr. Blanton - ENTC 4307 - Complex Numbers 28 Example 3 5  10  20  15  - express the impedance in j notation - determine Z (in  s) and  - determine I for a voltage of 24V

29. Dr. Blanton - ENTC 4307 - Complex Numbers 29 Example 3 Cont’d 5  10  20  5 

30. Dr. Blanton - ENTC 4307 - Complex Numbers 30 Example 3 Cont’d 5  20  10  ~ 24V

31. Dr. Blanton - ENTC 4307 - Complex Numbers 31 Example 4 Construct a circuit which contains at least one L and one C components which could be represented by: Z = 10 - 30j

32. Dr. Blanton - ENTC 4307 - Complex Numbers 32 Parallel circuits 10 - 30j 20 - 10j * *

33. Dr. Blanton - ENTC 4307 - Complex Numbers 33 Parallel circuits 20 - 30j * *