TELECOMMUNICATIONS Dr. Hugh Blanton ENTC 4307/ENTC 5307

Complex Numbers

Dr. Blanton - ENTC 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

Dr. Blanton - ENTC 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

Dr. Blanton - ENTC 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

Dr. Blanton - ENTC 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 

Dr. Blanton - ENTC 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  )

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

Dr. Blanton - ENTC 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

Dr. Blanton - ENTC 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   )(2.24   ) = 5  36.8 

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

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

Dr. Blanton - ENTC 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

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

Dr. Blanton - ENTC 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

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

Dr. Blanton - ENTC Complex Numbers 17 Z dependence on  (RCL ) frequency Impedance oo parallel series

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

Dr. Blanton - ENTC 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.”

Dr. Blanton - ENTC 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

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

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

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

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

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

Dr. Blanton - ENTC 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 = jI = 8/13( j)

Dr. Blanton - ENTC 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

Dr. Blanton - ENTC 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

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

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

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

Dr. Blanton - ENTC Complex Numbers 32 Parallel circuits j j * *

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