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EE4101 Smith Chart and Impedance Matching

EE4101 Smith Chart and Impedance Matching - EE4101 RF...

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ECE NUS EE4101 Smith Chart / 1 EE4101 RF Communications EE4101 RF Communications Smith Chart and Impedance Matching - additional lecture material by Marek E Bialkowski
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ECE NUS EE4101 Smith Chart / 2 Lossless Transmission Line Lossless Transmission Line Terminated in Z Terminated in Z L L We recall some results, which we obtained for a lossless transmission line terminated in an arbitrary impedance Z L . Z 0 , β Z L l ' tan tan ) ( 0 0 0 l jZ Z l jZ Z Z l Z L L i β β + + = ( ) ] 1 [ 2 ) ( 2 0 l j l j L L e e Z Z I l V β β - Γ + + = 1 1 1 ) / ( 1 ) / ( 0 0 0 0 + - = + - = + - = Γ = Γ = Γ Γ L L L L L L j L z z Z Z Z Z Z Z Z Z e ( ) ] 1 [ 2 ) ( 2 0 0 l j l j L L e e Z Z Z I l I β β - Γ - + = ) ( l Z i l j e l β 2 ) ( - Γ = Γ
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ECE NUS EE4101 Smith Chart / 3 The Smith Chart The Smith Chart For the loaded transmission line the relationship between the complex valued Г and the complex valued normalized load impedance z L is given by a bilinear transformation: This transformation can be visualized in the form known as the Smith Chart . The particular interest is in visualizing constant reflection coefficient, constant resistance and constant reactance. 1 1 + - = Γ = Γ Γ L L j z z e Γ - Γ + = 1 1 L z or
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ECE NUS EE4101 Smith Chart / 4 The Smith Chart The Smith Chart It is worthwhile knowing that the bilinear transformations transform circles into circles. This means that the constant resistance and constant reactance lines in the Z plane appear as circles in the complex Γ plane . 1 1 + + - + = Γ + Γ = Γ = Γ Γ jx r jx r j e i r j i r i r L L j j jx r z Γ - Γ - Γ + Γ + = Γ - Γ + = + = 1 1 1 1 2 2 2 2 2 1 1 i r r i r r Γ + Γ - Γ + Γ - Γ - = 2 2 2 1 2 i r r i x Γ + Γ - Γ + Γ = To demonstrate that, we consider each of the following equations (neglecting the other one makes x or r arbitrary):
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ECE NUS EE4101 Smith Chart / 5 The Smith Chart The Smith Chart - - constant resistance circles constant resistance circles 1 1 + + - + = Γ + Γ = Γ = Γ Γ jx r jx r j e i r j jx r z + = 2 2 2 1 1 1 + = Γ + + - Γ r r r i r 2 2 2 2 2 1 1 i r r i r r Γ + Γ - Γ + Γ - Γ - = Can be rewritten as: Circles in the Γ plane:
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ECE NUS EE4101 Smith Chart / 6 The Smith Chart The Smith Chart - - constant reactance circles constant reactance circles 1 1 + + - + = Γ + Γ = Γ = Γ Γ jx r jx r j e i r j jx r z + = 2 2 2 1 1 ) 1 ( x x i r = - Γ + - Γ 2 2 2 1 2 i r r i x Γ + Γ - Γ + Γ = Can be rewritten as: Circles in the Γ plane:
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ECE NUS EE4101 Smith Chart / 7 The Smith Chart The Smith Chart - - constant reflection coefficient, constant reflection coefficient, constant resistance and constant reactance circles constant resistance and constant reactance circles 1 1 + + - + = Γ + Γ = Γ = Γ Γ jx r jx r j e i r j θ arbitrary x const r - = ; arbitrary r const x - = ; arbitrary const - = Γ = Γ Γ θ r >0 to have passive loads
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ECE NUS EE4101 Smith Chart / 8 Reflection coefficient Reflection coefficient Γ ∠θ The reflection coefficient is proportional to the length of the radial vector on the chart.
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