TELECOMMUNICATIONS Dr. Hugh Blanton ENTC 4307/ENTC 5307

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

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

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

Dr. Blanton - ENTC Smith Chart 8 0 -j25 -j50 -j75 j75 j50 j jXLjXL jXCjXC Ideal Inductors Pure Positive Reactance No Resistive Component +90° Ideal Capacitors Pure Negative Reactance No Resistive Component -90° Ideal Resistors No Reactive Component 180° 0°0°

Dr. Blanton - ENTC Smith Chart 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 Z 0 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 (  =0), and the impedance is the characteristic impedance (Z = Z 0 ).

Dr. Blanton - ENTC Smith Chart 10 Sample transformations using Z 0 = 50 . Infinite impedance has three possible combinations: 1.   jX 2.R + j  3.R  j  

Dr. Blanton - ENTC Smith Chart 11 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  25  50  100  250   j50  j50 Ideal Inductor Ideal Capacitor

Dr. Blanton - ENTC Smith Chart 12 Formation of Smith Chart 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  = j 125º |  |=0.73

Dr. Blanton - ENTC Smith Chart 13 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. 0 -j25 -j50 -j75 j75 j50 j

Dr. Blanton - ENTC Smith Chart 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 Smith Chart 15

Dr. Blanton - ENTC Smith Chart 16 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 Now plot constant reactance contours. This finally gives the standard Smith chart.

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

Dr. Blanton - ENTC Smith Chart 18 Generally, Z o is 50  and the center of the chart refers to that impedance. However at times the characteristic impedance is other than 50  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 Z o. Letting

Dr. Blanton - ENTC Smith Chart 19 Lossless Series Inductors Series additions of components are most conveniently handled in the impedance system. Beginning with a component of reflection coefficient  1, we first convert to impedance z 1. A lossless series inductor adds inductive reactance, keeping the real part (resistance) constant.

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

Dr. Blanton - ENTC Smith Chart 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 Smith Chart 22 Insert a 80 nH series inductor to a one port of  1 = 0.45  . Find the new combined input impedance, z T, at 0.1 GHz. Use Z o = 50  for normalization. 1.Locating G1 on the normalized Impedance Smith Chart, convert 2.The reactance of the 80 nH inductor jX L = j0.126F GHz L nH  j1 3.Move from z1 on the constant resistance circle of r = 0.5, upward +j1 unit. 4.Read z T = j0.5.

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

Dr. Blanton - ENTC Smith Chart 24 Series capacitors are also handled most conveniently in the impedance system. Beginning with an arbitrary impedance z 1, a lossless series capacitor adds capacitive reactance, keeping the real part (resistance) constant. The sum of the impedances, z T, 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 Smith Chart 25 Insert a 32 pF series inductor to a one port of  1 = 0.45  -116 . Find the new combined input impedance, z T, at 0.1 GHz. Use Z o = 50  for normalization. 1.Locating  1 on the normalized Impedance Smith Chart, convert 2.The reactance of the 32 pF capacitor -jX C = 3.18/(jF GHz C pF )  -j1 3.Move from z1 on the constant resistance circle of r = 0.5, upward -j1 unit. 4.Read z T = 0.5 – j1.5.

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