ECE 6130 Lecture 4: SMITH CHARTSText Section: 2.4Portfolio Question:1) Describe and Demonstrate how to use a Smith Chart to find impedance, Vmin, Vmax, SWR, reflection coefficient, etc. See Chapter 2, Problems 7-12Smith Chart Circles:A Smith chart is a graphical representation of the complex reflection coefficient, Smith Chart for Reflection Coefficient and Load Impedance:Reflection Coefficient and Load (ZL) are directly related: = (ZL / Zo - 1) / (ZL/Zo+ 1) = (zL - 1) / (zL + 1)ORZL / Zo = zL = (1 +  ) / (1 -  )  This is NORMALIZED load impedance zL = rL + j xL The real and imaginary parts of zL are functions of , and these functions can be plotted on the same chart. Remember ||  1.Example: Given ZL , find  using Smith ChartSee transparencies (Copies to be made available in copy room)How to find  : 1) Find Normalized load Impedance, zL = ZL / Zo = rL + j xL2) Find intercept of semicircles for rL and xL and PLOT zL 3) Draw line from center of smith chart to (or through) zL4) Read angle of  from outside of Smith chart5) Measure |  | with a protractor and compare to line on bottom of smith chart labeled “Ref. Coeff. E or I”Zo = 100 ohmsZL = open circuit 1) zL =  =  + j 02) PLOT (far right)3) Draw Line through zL . Read 04) Measure using a protractor (or this one is obviously =1) |  | = 1 =1  0 (which is what we expect for an open circuit)ZL = open circuit 1) zL = 0 = 0+ j 02) PLOT (far left)3) Draw Line through zL . Read 1804) Measure using a protractor (or this one is obviously =1) |  | = 1 =1  180 = -1 (which is what we expect for an short circuit)ZL = 100 + j 0 ohms 1) zL = ZL / Zo = 1 + j 02) PLOT (center of smith chart)3) Draw Line through zL . Not so easy … ?4) Measure using a protractor (or this one is obviously =0) |  | = 0 =0  ? (which is what we expect for a matched load)ZL = 100 + j 100 ohms 1) zL = ZL / Zo = 1 + j 12) PLOT (top right quadrant)3) Draw Line through zL . about 634) Measure using a protractor |  | = 0.45 =0.45 63 = (zL - 1) / (zL + 1) = (0+j1) / (2+j1) =190 / 2.236 26.56  = 0.45 63.43How do you find load impedance if given ?1) Plot 2) Read zL = zL +j zL3) Unnormalize: ZL = zL * ZoAdmittance vs. Impedance:Admittance yL = 1 / zL = (zL - 1) / (zL + 1) = (1/yL - 1) / (1/yL + 1) = - (yL - 1) / (yL + 1) = 180 out of phaseSteps to find  from yL:1) Find normalized yL = Zo / ZL = gL +jbL2) Plot it (Using same curves g=r and b=x)3) “Transform it through the origin” … Rotate 180 degrees = draw a line of equal length through the origin. Now you have found zL4) Read  as beforeEXAMPLE (See transparencies)Input Impedance:Zin = Zo [1 +  e -jl ] / [1 -  e -jl ]zin = Zin / Zo = [1 +  e -j2l ] / [1 -  e -j2l ]Define reflection coefficient at the input (NOT g ) as the reflection coefficient looking into the load frm the input location. l = L -2l This represents moving 2l radians towards the generator. You can convert this distance to degrees, and read it off the outer circles on the Smith Chart (notice DIRECTION to the generator is marked) OR 2l = 2(2 / ) l = 4 (l / ) This has been normalized for you on the outside circle around the Smith Chart. Observe that if l=, this represents 2 complete rotations around the Smith Chart. L = /2 represents one complete rotation. Does this make sense? For a Transmission line of length L = /2, traveling from generatorto the load and back would represent a phase shift of 360 degrees … one complete rotation.Then zin = [1 + l] / [1 - l ]How to find Zin :1) Normalize zL = ZL / Zo2) Plot zL. This also gives you L. 3) Rotate  distance l (given in wavelengths) TOWARDS the generator.4) Read zin and l5) Zin = zin * ZoEXAMPLE (see transparencies)Standing Wave Ratio:To read SWR from the Smith Chart:1) PLOT zL 2) Draw a circle through it.3) Read SWR from real axis to right (SWR  1)EXAMPLE (See transparencies)Voltage Minima and Maxima:To read Voltage maxima off Smith Chart:1) PLOT zL2) First Voltage maximum occurs on right side of real axis. First Voltage minimum occurs on left side of real axis. EXAMPLE (See