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For extra copies of the Smith Chart go to the LEARN page for my part of the course and click on the Smith Chart link

For extra copies of the Smith Chart go to the LEARN page for my part of the course and click on the Smith Chart link. Voltage amplitude |V|. l /4. V max. V min. x . x = 0 . Z o Z L. The Voltage Standing Wave Ratio (VSWR) is defined as:

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For extra copies of the Smith Chart go to the LEARN page for my part of the course and click on the Smith Chart link

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  1. For extra copies of the Smith Chart go to the LEARN page for my part of the course and click on the Smith Chart link

  2. Voltage amplitude |V| l/4 Vmax Vmin x x = 0 Zo ZL The Voltage Standing Wave Ratio (VSWR) is defined as: VSWR = Vmax / Vmin Like ρ, the VSWR is a measure of the mismatch between ZL and Zo.

  3. Hence the distance between neighbouring maxima and minima is lmax - lmin = p/2b But b = 2p/l, hence the distance between a voltage maximum and the adjacent voltage minimum is l/4 l/4 lmax Vmax Vmin lmin x N.B. l is the wavelength of the travelling waves that are giving rise to the standing wave pattern – it is NOT the wavelength of the standing wave.

  4. Voltage amplitude |V| l/4 Vmax Vmin Current amplitude | I | x x = 0 l/4 Imax Imin Impedance magnitude |Z| |Z|=|V| / | I | x x = 0 l/4 Zmax Zmin x x = 0

  5. l/4 lmax Vmax Vmin lmin x VOLTAGE STANDING WAVE MEASUREMENT The voltage standing wave pattern on a transmission line can be measured using a "slotted line": V voltage probe ZL V E x = 0 The probe enables the voltage (strictly the E field) to be measured at any point along the slot cut in the outer skin of the line. If the slot is narrow it will not affect the properties of the line or cause reflections.

  6. From measurement of Vmax / Vmin we can obtain the magnitude of r: VSWR =Vmax / Vmin = => |r| = (VSWR - 1) (VSWR + 1) The phase of r can be found from the positions of the voltage maxima and minima, lmax and lmin. At the first voltage maximum: f - 2blmax = 0 => f = 2blmax = 2(2p/l) lmax radians or f = 2blmax = 2(lmax /l) x 360 degrees

  7. The wavelength l can be determined since the distance between a neighbouring maximum and minimum is l/4: |lmax - lmin| = l/4 Knowing r, the load impedance ZL can be found:

  8. Part 3 - The Smith Chart and its Applications Lec. 7 Introduction to the Smith Chart Principle of Operation Construction of the Smith Chart Key Points on the Smith Chart Using Smith Chart with Load and Line Combinations

  9. THE SMITH CHART • Devised in 1944 by Philip H. Smith, Bell Labs, U.S.A. • A graphical aid for transmission line calculations. • Used nowadays for designing “matching” circuits and displaying RF data. Philip Smith: 1905-1987

  10. l ZL Load Transmission line, Zo Generator The Smith Chart basically enables you to convert between  and Z graphically, either at the load or at an arbitrary point down the line.

  11. Simplified Smith Chart 1 In this simplified version of the Smith Chart most of the grid of lines has been removed. For clarity, in these lectures this version will be used to illustrate the properties of the Smith Chart. 0.5 2 3 0.2 5 10 0 0.2 0.5 1 2 3 5 10 0 -10 -5 -0.2 -3 -2 -0.5 -1

  12. THE SMITH CHART

  13. Simplified Smith Chart 1 In this simplified version of the Smith Chart most of the grid of lines has been removed. For clarity, in these lectures this version will be used to illustrate the properties of the Smith Chart. 0.5 2 3 0.2 5 10 0 0.2 0.5 1 2 3 5 10 0 -10 -5 -0.2 -3 -2 -0.5 -1

  14. SMITH CHART BASICS •  is a complex quantity:  = ||e j • For all ZL, 0  ||  1 Circle of UNIT RADIUS in complex number plane Imaginary +jX R+jX  || (1,0) 0 R Real -jX Complex Impedance Plane Complex Reflection Coefficient Plane

  15.  ZL To avoid having to use a different Smith Chart for every value of Zo, the normalised impedance, z , is used: z = ZL/ Zo The normalised impedance z can then be “denormalised” to obtain the load impedance ZL by multiplying by Zo: ZL = z.Zo In terms of its real and imaginary components, the normalised impedance can be written as: z = r + jx

  16. CONSTRUCTION OF THE SMITH CHART (pages 3-5) is for information only IT IS NOT EXAMINABLE

  17. MAPPING BETWEEN IMPEDANCE PLANE AND THE SMITH CHART r is the normalised resistance axis x is the normalised reactance axis Lines of constant normalised resistance Lines of constant normalised reactance r = 0 1 0.5 2 0.4 0.2 0 -0.2 -0.4 x = X/Zo 3 0.2 5 x = 0 10 x r = R/Zo 0 0.2 0.5 1 2 3 5 10 r 0.2 0.4 0.6 0.8 0 -10 -5 -0.2 -3 -2 -0.5 Complex (Normalised) Impedance Plane -1 Smith Chart

  18. 1 0.5 2 3 0.2 5 10 x 0 0.2 0.5 1 2 3 5 10 0 r -10 -5 -0.2 -3 -2 -0.5 -1 normalised reactance, x normalised resistance, r

  19. 0.70745º 0.707 45º 10º Step 1: Plot  Example (page 2): Measurements on a slotted line with Zo = 50Ω give: || = 0.707 and  = 45º. Find ZL ρ Unit Circle in Reflection Coefficient Plane

  20. 1 0.5 2 3 0.70745º 0.2 5 0.707 10 x 0 0.2 0.5 1 2 3 5 10 0 45º r 10º -10 -5 -0.2 -3 -2 -0.5 -1 Step 2: Superimpose Smith Chart grid Example (page 2): Measurements on a slotted line with Zo = 50Ω give: || = 0.707 and  = 45º. Find ZL Unit Circle in Reflection Coefficient Plane

  21. Step 3: Read off normalised values for r and x 1 0.5 2 3 0.70745º 0.2 5 0.707 10 x 0 0.2 0.5 1 2 3 5 10 0 45º r 10º -10 -5 -0.2 -3 -2 -0.5 -1 Example (page 2): Measurements on a slotted line with Zo = 50Ω give: || = 0.707 and  = 45º. Find ZL , x = 2 r = 1 Unit Circle in Reflection Coefficient Plane

  22. 1 0.5 2 3 0.70745º 0.2 5 0.707 10 x 0 0.2 0.5 1 2 3 5 10 0 45º r 10º -10 -5 -0.2 -3 -2 -0.5 -1 Step 4: Calculate R and X by “denormalizing”: ZL = Zo(r + jx) = 50(1+j2) = 50 + j100Ω Example (page 2): Measurements on a slotted line with Zo = 50Ω give: || = 0.707 and  = 45º. Find ZL , x=2 r=1 Unit Circle in Reflection Coefficient Plane

  23. KEY POINTS ON THE SMITH CHART Open Circuit (ZL = , z = ZL/Zo = ) z = r + jx =  1 0.5 2  = 10º  = 10º 3 0.2 5 10 x 10º 0 0.2 0.5 1 2 3 5 10 r 0 -10 -5 -0.2 -3 -2 -0.5 -1 Reflection Coefficient Plane Smith Chart

  24. Short Circuit (ZL = 0, z = 0) 1 0.5 2 3 0.2 5 10 x 10 0 0.2 0.5 1 2 3 5 10 r 0 -10 -5 -0.2 -3  = 1180º -2 -0.5  = 1180º z = r + jx = 0 + j0 -1 Reflection Coefficient Plane Smith Chart

  25. Matched Load (ZL = Zo, z = 1) 1 0.5 2 3 0.2 5 10 x 10 0 0.2 0.5 1 2 3 5 10 r 0 -10 -5 -0.2 -3  = 0 -2 -0.5  = 0 z = r + jx = 1 + j0 -1 Reflection Coefficient Plane Smith Chart

  26. Resistive Load (ZL = R + j0 , z = r) R and Zo (for a lossless line) are REAL, therefore  is real. R > Zo  +ve,  = 0º R < Zo  -ve,  = 180º 1 0.5 2 3 0.2 5 10 x R < Zo R = Zo R > Zo 10 0 0.2 0.5 1 2 3 5 10 R = 0 r 0 -10 R =  -5 -0.2 -3 -2 -0.5 -1 Reflection Coefficient Plane Smith Chart

  27. Purely Reactive Load (ZL = 0 + jX , z = 0 + jx) 1 0.5 2 3 z = jx (inductive) 0.2 5 10 x 10 0 0.2 0.5 1 2 3 5 10 z = j0 r 0 -10 z = j -5 -0.2 -3 z = jx (capacitive) -2 -0.5 -1 Reflection Coefficient Plane Smith Chart

  28. Purely Reactive Load (ZL = 0 + jX , z = 0 + jx) 1 0.5 2 3 z = jx (inductive) 0.2 5 10 x 10 0 0.2 0.5 1 2 3 5 10 z = j0 r 0 -10 z = j -5 -0.2 -3 z = jx (capacitive) -2 -0.5 -1 Reflection Coefficient Plane Smith Chart

  29. General Case (ZL = R + jX) 1 0.5 2 3 0.2 5 || 10  x 10 0 0.2 0.5 1 2 3 5 10 r 0 -10 -5 -0.2 -3 -2 -0.5 -1 Reflection Coefficient Plane Smith Chart

  30. Pure Reactance 1 Pure Resistance 0.5 2 3 0.2 5 Matched Load 10 x 0 0.2 0.5 1 2 3 5 10 0 r -10 Short Circuit Open Circuit -5 -0.2 -3 -2 -0.5 -1

  31. RATIO OF V-/V+ AT AN ARBITRARY DISTANCE l FROM THE LOAD ZL Zo ZL x = -l x = 0 If the line is losslessgl = jbl , hence: r-l= re-2gl= re-j2bl= |r|ejf e-j2bl= |r|ej(f- 2bl) i.e. the magnitude of the reflection coefficient at x = -lis the same as at x = 0 but its phase changes fromfto f - 2bl . An additional phase change of -2blhas been added by the introduction of the length of line,l .

  32. l ZL Load Transmission line Generator 1 0.5 2 3 z = r+jx  of ZL only 0.2 5 ||  10 x 2βl 10 0 0.2 0.5 1 2 3 5 10 2βl r 0 -10 -5  of ZL plus line of length l -0.2 z of load and line of length l -3 -2 -0.5 Point rotates clockwise by 2βlradians (2x360xl/λ degrees) at a constant radius -1 Reflection Coefficient Plane Smith Chart

  33. In the case of a lossless line a = 0, so g = a + jb = jb and is purely imaginary, so the hyperbolic functions reduce to trigonometric functions: Zin becomes: or: Original expression:

  34. l ZL “Clockwise towards generator” Load Transmission line Generator 1 0.5 2 3 z = r+jx  of ZL only 0.2 5 ||  10 x 2βl 10 0 0.2 0.5 1 2 3 5 10 2βl r 0 -10 -5  of ZL plus line of length l -0.2 z of load and line of length l -3 -2 -0.5 -1 Point rotates clockwise by 2βl radians (2x360 l/λ degrees) Reflection Coefficient Plane Smith Chart

  35. Tutorial C, Question 2 Solution by Smith Chart To find Zin: • zL = ZL/Zo • = (100+j50)/75 • = 1.33+j0.67 zL 3. 2l = 2x2l/λ = 2x2fl/v = 1.26 radians = 72º 4. Read z-l from chart: z-l = 1.75-j0.45 72º 2. Plot zL on Smith Chart z-l 5. Denormalise to find Zin: Zin = Zo x z-l = 75(1.75-j0.45) = 131-j34 Ω (cf. 126-j36 Ω by exact calculation) Zo = 75 Ω ZL = 100 + j50 Ω l = 2.2 m, f = 100 MHz v = 2 x 108 ms-1

  36. For extra copies of the Smith Chart go to the WebCT page for my part of the course and click on the Smith Chart link

  37. The Smith Chart & General Transmission Lines r-l = re-2gl = re-2αle-j2bl = |r|ejf e-2αl e-j2bl = |r|e-2αlej(f-2bl)  =  +j  - propagation constant  - attenuation constant  - phase constant 1 0.5 2 z = r+jx  3 0.2 5 10 x 10 0 0.2 0.5 1 2 3 5 10 r 0 -l -10 z of load and line of length l -5 -0.2 -3 -2 -0.5 -1 Reflection Coefficient Plane Smith Chart

  38. The Smith Chart and Variation of Frequency r-l = |r|ej(-2bl)for a lossless line rotation angle is -2l= -2.(2/λ).l = -2.(2.f/v).l = -4fl/v or constant x f 1 0.5 2 dc zdc 3 zf1 f1 0.2 5 10 x 10 0 0.2 0.5 1 2 3 5 10 f2 r 0 zf2 -10 -5 -0.2 f3 zf3 -3 -2 -0.5 frequency f3 > f2 > f1 -1 Reflection Coefficient Plane Smith Chart

  39. Summary The Smith Chart is used as a graphical aid for converting between a load impedance, Z, and a reflection coefficient,. (This can be done with or without sections of line being present.)  ZL

  40. To avoid having to use a different Smith Chart for every value of Zo, the normalised impedance, z, is used: • z = ZL/Zo • (z = r + jx) • z can then be denormalised to obtain the load impedance ZL, by multiplying by Zo: • ZL = z.Zo

  41. The circles correspond to lines of constant normalised resistance, r. • The arcs correspond to lines of constant normalised reactance, x. r = 0 1 0.5 2 0.4 0.2 0 -0.2 -0.4 x = X/Zo 3 0.2 5 x = 0 10 x r = R/Zo 0 0.2 0.5 1 2 3 5 10 r 0.2 0.4 0.6 0.8 0 -10 -5 -0.2 -3 -2 -0.5 -1 Complex Impedance Plane Smith Chart

  42. Adding a length, l, of lossless line to a load, ZL, corresponds on the Smith Chart to rotating at constant radius from zLCLOCKWISE through an angle 2l. (“CLOCKWISE TOWARDS GENERATOR”) • If the line is not lossless, the radius decreases as we rotate around the centre. • Increasing the signal frequency causes z-l to rotate clockwise around Smith Chart at constant radius.

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