TELECOMMUNICATIONS

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

2. Background • An AT&T Bell Lab engineer, Philip Smith, developed a graphical tool in 1933 to simplify the task of plotting impedance variation in passive transmission line circuits. • Although Smith’s original paper has been rejected by the IRE (predecessor of the IEEE) it has become one of the most popular design aids for RF and microwave engineers. • It is estimated that over 70,000,000 copies have been distributed throughout the world during the past sixty years. Dr. Blanton - ENTC 4307 - Smith Chart 2

3. Recognizing that passive transmission circuit impedances may vary through a very wide range (zero to infinity), • Smith decided to plot reflection coefficient that has a limited magnitude range (zero to one). • To “translate” reflection coefficient to impedance, he created a unique overlay that became the Impedance Smith Chart. • Later, a second chart was created to provide conversion between reflection coefficient and admittance, and finally the two charts were superimposed to form the Impedance-Admittance (Immittance) Chart. Dr. Blanton - ENTC 4307 - Smith Chart 3

4. Although initially the charts were used for passive circuit impedance manipulation only, later additional applications were developed for active circuit design. • Constant-gain, constant-noise, constant power output, and RF stability plots are now traditionally shown on the Smith Chart. • Modern RF/MW test equipment and CAE software can also display their output on the Chart. • Therefore, anyone involved with development, production, or test of RF/MW components, circuits and systems will benefit from a thorough understanding of this powerful graphical tool. Dr. Blanton - ENTC 4307 - Smith Chart 4

5. Smith Chart • Slide rule of the RF engineer • Circuit matching • Impedance –Admittance transformation • Conversion between  and ZL • What is it •  is complex • Phasor diagram or Argand diagram of  Dr. Blanton - ENTC 4307 - Smith Chart 5

6. The Impedance Smith Chart is a result of a mathematical transformation of the rectangular impedance Z, to a polar reflection coefficient G, where Dr. Blanton - ENTC 4307 - Smith Chart 6

7. This transformation places all impedances with positive real parts (R  0) inside of a circle. • The center of the circle refers to Zo, which is called the characteristic impedance. • Zo is generally resistive and equal to 50W. Dr. Blanton - ENTC 4307 - Smith Chart 7

8. 0 -j25 -j50 -j75 Ideal Inductors Pure Positive Reactance No Resistive Component +90° jwXL j75 Ideal Resistors No Reactive Component j50 j25 25 50 75 100 0° 180° jwXC -90° Ideal Capacitors Pure Negative Reactance No Resistive Component Dr. Blanton - ENTC 4307 - Smith Chart 8

9. Impedances of passive circuits may vary from zero to infinity and are difficult to plot due to their wide range. • Converting impedances to reflection coefficients limits the magnitudes to be between 0 and 1. • Referencing the impedances to Z0 and plotting in a polar coordinate system, a small manageable chart is created that includes all points of the right-hand side of the rectangular impedance system (0  R  ). • At the center of the chart, the reflection coefficient is zero (G=0), and the impedance is the characteristic impedance (Z = Z0). Dr. Blanton - ENTC 4307 - Smith Chart 9

10.  • Sample transformations using Z0 = 50 W. • Infinite impedance has three possible combinations: •   jX • R + j • R  j Dr. Blanton - ENTC 4307 - Smith Chart 10

11. 10 W 25W 50W 100W 250W 0  Ideal Inductor • The top half of the Impedance Smith Chart represents inductive terminations, while the lower half represents capacitive terminations. • Ideal resistors (X = 0) are located on the horizontal centerline, • ideal inductors (R = 0) on the upper half of the chart’s circumference, and • ideal capacitors on the lower half of the circumference. j50 j50 Ideal Capacitor Dr. Blanton - ENTC 4307 - Smith Chart 11

12. Formation of Smith Chart 125º ||=0.73 =0.2+0.5j For a passive system || < 1 so  must be within the unit circle. The area marked on diagram. We know what the axis are so we can miss them out Plot  as either a phasor or complex number Dr. Blanton - ENTC 4307 - Smith Chart 12

13. j75 j50 j25 25 50 75 100 0 -j25 -j50 -j75 • In the rectangular impedance system, • Vertical lines represent constant resistances with varying reactances. • Horizontal lines are the loci of impedances with constant reactance and varying resistance. Dr. Blanton - ENTC 4307 - Smith Chart 13

14. The Smith Chart transformation changes both vertical and horizontal lines to circles. • The families of constant resistance and constant reactance circles form the impedance Smith Chart. Dr. Blanton - ENTC 4307 - Smith Chart 14

15. Dr. Blanton - ENTC 4307 - Smith Chart 15

16. Now plot constant reactance contours. This finally gives the standard Smith chart. Formation of Smith Chart Mark on the diagram contours of constant normalized resistance. The resistance has been normalized against the characteristic impedance of the transmission line Dr. Blanton - ENTC 4307 - Smith Chart 16

17. The lower part of the commercially available 50 W Smith Chart includes several scales, including |G|, Return Loss, Mismatch Loss, and VSWR. Dr. Blanton - ENTC 4307 - Smith Chart 17

18. Generally, Zo is 50 W and the center of the chart refers to that impedance. • However at times the characteristic impedance is other than 50 W and the Smith Chart must be labeled accordingly. • Instead of creating a different Smith Chart for every characteristic impedance, the transformation equation may be normalized by dividing all terms by Zo. Letting Dr. Blanton - ENTC 4307 - Smith Chart 18

19. Lossless Series Inductors • Series additions of components are most conveniently handled in the impedance system. • Beginning with a component of reflection coefficient G1, we first convert to impedance z1. • A lossless series inductor adds inductive reactance, keeping the real part (resistance) constant. Dr. Blanton - ENTC 4307 - Smith Chart 19

20. The sum of the two impedances, zT, is computed as: Dr. Blanton - ENTC 4307 - Smith Chart 20

21. On the impedance Smith Chart, addition of a series lossless inductor represents an upward movement on the constant resistance circle, toward x = +j. Dr. Blanton - ENTC 4307 - Smith Chart 21

22. Insert a 80 nH series inductor to a one port of G1 = 0.45-116. • Find the new combined input impedance, zT, at 0.1 GHz. Use Zo = 50 W for normalization. • Locating G1 on the normalized Impedance Smith Chart, convert • The reactance of the 80 nH inductor jXL = j0.126FGHzLnH j1 • Move from z1 on the constant resistance circle of r = 0.5, upward +j1 unit. • Read zT = 0.5 + j0.5. Dr. Blanton - ENTC 4307 - Smith Chart 22

23. The transformation moves on the appropriate constant resistance circle that represents the resistance of the termination. Dr. Blanton - ENTC 4307 - Smith Chart 23

24. Series capacitors are also handled most conveniently in the impedance system. • Beginning with an arbitrary impedance z1, a lossless series capacitor adds capacitive reactance, keeping the real part (resistance) constant. • The sum of the impedances, zT, is computed as: On the impedance Smith Chart, addition of a series lossless capacitor represents a downward movement on the constant resistance circle, toward x = -. Dr. Blanton - ENTC 4307 - Smith Chart 24

25. Insert a 32 pF series inductor to a one port of G1 = 0.45-116. • Find the new combined input impedance, zT, at 0.1 GHz. Use Zo = 50 W for normalization. • Locating G1 on the normalized Impedance Smith Chart, convert • The reactance of the 32 pF capacitor -jXC = 3.18/(jFGHzCpF)  -j1 • Move from z1 on the constant resistance circle of r = 0.5, upward -j1 unit. • Read zT = 0.5 – j1.5. Dr. Blanton - ENTC 4307 - Smith Chart 25

26. The transformation moves on the appropriate constant resistance circle that represents the resistance of the termination. Dr. Blanton - ENTC 4307 - Smith Chart 26