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TELECOMMUNICATIONS

TELECOMMUNICATIONS. Dr. Hugh Blanton ENTC 4307/ENTC 5307. Complex Numbers. Complex numbers. ARGAND diagram. M = A + jB Where j 2 = -1 or j = √-1 | M | = √(A 2 + B 2 ) and tan  = B/A. Imaginary. B. M. . A. Real. j notation. Refers to the expression Z = R + jX

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TELECOMMUNICATIONS

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

  2. Complex Numbers

  3. Complex numbers ARGAND diagram M = A + jB Where j2 = -1 or j = √-1 |M| = √(A2 + B2) and tan  = B/A Imaginary B M  A Real Dr. Blanton - ENTC 4307 - Complex Numbers 3

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

  5. Complex Number Definitions +jy • 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. 2 1 x +x 1 2 1 2 jy Dr. Blanton - ENTC 4307 - Complex Numbers 5

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

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

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

  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 Dr. Blanton - ENTC 4307 - Complex Numbers 9

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

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

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

  13. Complex Impedance System inductive +jX • 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 capacitive Dr. Blanton - ENTC 4307 - Complex Numbers 13

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

  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 capacitive +jB G +G jB inductive Dr. Blanton - ENTC 4307 - Complex Numbers 15

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

  17. Z dependence on (RCL ) Impedance 1000 500 series 100 50 parallel o 10 5 frequency 1 1 2 5 10. 20. 50. 100. Dr. Blanton - ENTC 4307 - Complex Numbers 17

  18. Currrent dependence on  Current (ma) 1000 parallel o 500 Imaxx√2 series Imin 100 frequency 1 2 5 10. 20. 50. 100. o Dr. Blanton - ENTC 4307 - Complex Numbers 18

  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.” Dr. Blanton - ENTC 4307 - Complex Numbers 19

  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 Dr. Blanton - ENTC 4307 - Complex Numbers 20

  21. VL VS VL-VC  I VR VC V and I Phase relationships Dr. Blanton - ENTC 4307 - Complex Numbers 21

  22. R, XC and Z relationships XL Z XL-XC  I R XC Dr. Blanton - ENTC 4307 - Complex Numbers 22

  23. Example 1 Consider this circuit with  = 105 rad s-1 1 k 0.01 F Dr. Blanton - ENTC 4307 - Complex Numbers 23

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

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

  26. Example 2 Cont’d VL XL VR R  VL-VC  XL-XC I I VS Z VC XC I = 8/13(15 + 10j) Z = 15 - 10j Dr. Blanton - ENTC 4307 - Complex Numbers 26

  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 Dr. Blanton - ENTC 4307 - Complex Numbers 27

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

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

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

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

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

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

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