PDF《V2e-j. .l》

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13 Smith Chart Tutorial Part1 To begin with we start with the definition of VSWR, which is the ratio of the reflected voltage over the incident voltage.

The Reflection coefficient Γ is simply the complex (ie has phase) version of VSWR:- Define voltage standing wave ratio (VSWR) V V Voltage reflection coefficient - Complex max min = + ? V V V V

1 2

1 2 Γ ZL V1e+j.β.l T.L V2e-j.β.l L Γ Γ Γ = V V = V V At the load (l = 0) ;

= V V

2 1

2 1

2 1 e e e j j j ? + ? ? . . . . . β β β l l l

2 but this may be complex number if there is an instantaneous phase change which we'll call (φ) on reflection. θ V2 V1 Phase Diagram

0 = L (a) .....

1 2

1 2 ? = j e V V V V Sheet

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13 ( ) ( ) Diagram Crank on d represente = V V = constant is L ith not vary w do V & V line lossless a For

0 > L At .

2 1

2 2

1 2

1 2 . )

0 ( (L) l l β φ β φ φ ? ? ? Γ Γ Γ ∴ = Γ = Γ j j j e e V V e Crank Diagram We use a crack diagram as a way of representing the reflection coefficient phasor. ( ) V = V V V V V V

1 2

1 2

1 e e e e e j j j j j + ? + ? ? + . . . . . . . . β β β β φ β l l l l l

1 1

2 2 Γ φ j.

1 e

1 V V

0 = L At Γ + = φ O A L P |Γ| Between

0 &

1 OP AP At the origin of argand diagram.OP = magnitude of total voltage/incident voltage φ-2β.L O A B P |Γ| P' Vmin = 1-|Γ| C Vmax = 1+|Γ| Sheet

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13 As we saw previously the crack diagram with a circle drawn between points A & C is the beginnings of a Smith chart less the constant resistance and reactance circles/lines. VSWR = V V =

1 + 1- or = VSWR -1 VSWR +1 max min Γ Γ Γ | V |

0 Lmin Vmax Vmin L B B C C Length OP' -crank diagram voltage incident v At B -

2 . = - =

2 . - =

4 . = min min g φ β π φ β π π λ π φ l l l min ? ∴From standing wave pattern measure VSWR ? | Γ | @ lmin ? φ at load. Sheet

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13 SmithChart - Impedance (Z) or Admittance Y chart (1) Crank diagram + constant resistance & constant reactance circles. (2) Graphical solution to the equation complex =

1 1 ) .

2 ( ) ( l β φ? Γ Γ Γ ? Γ + = j o in e Z Z (3) Smith Chart is a reflection coefficient diagram |Γ| =

1 A A = |Γ| =

0 θ θ = φ-2.β.L |Γ| Smith Chart A Const r Open circuit x ? ∞ Short circuit v=0,x=0 X =

0 ∴ pure resistance R=0 pure reactance circle R=1 circle A is the matched point no reflection Const x |Γ| =

0 F O Sheet

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13 Impedance is plotted on the smith chart by first normalising to the characteristic impedance of the system (usually

50 ohms). In a

50 ohm system the centre of the smith chart is a pure

50 ohms. For example say we wanted to plot an impedance of

150 + j75? First normalise ie 150/50 = 3? ;

75/50 = 1.5? normalised impedance = 3+ j1.5? So the real part of the impedance will lie somewhere along the r =

3 constant resistance circle ie:- R=3 circle

3 Next we follow the constant reactance line at 0.75 to find the intersection of the r =

3 circle to get to our impedance point. R=3 circle

3 X=0.75 line 3+j1.5 Sheet

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13 Using the Smith Chart (1) Moving along the T.L = rotating around the Smith Chart. BACKWARDS FORWARDS ZL L FORWARDS (TO LOAD) BACKWARDS (FROM LOAD) (2) Constant |Γ| or VSWR circles For a lossless line |Γ| & VSWR do not vary with L. Constant |Γ| or VSWR circles |Γ| =

0 VSWR =

1 |Γ| =

1 VSWR = ∞ VSWR = 1+ 1- Vmax = Z(max) Zo (real VSWR) Vmin = Z(min) Zo Γ Γ S S S = =

1 Vmin Vmax Sheet

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13 (3) Measure Lmin/λg ..... determines φ (at load). φ B Lmin |V| L FORWARD by Lmin/λg takes us the load. Vmin ZL/Zo Lmin/λg φ π λ π =