EEE445_Lectr20_binomial

EEE445_Lectr20_binomial - 5.5 Theory of Small Reflections...

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Unformatted text preview: 5.5 Theory of Small Reflections • Rational – The λ /4 transformer can match real load impedance with any line impedance – The bandwidth is narrow – For more bandwidth, multi-section transformer • Review – Single-section transformer – Multi-section transformer • Device approximate results for total reflection EEE 591/445 1 Lecture 20 Single-section transformer (N=1) A single section transformer in Fig. 5.13 EEE 591/445 Lecture 20 2 2 1 2 12 12 1 2 1 2 1 1 21 12 21 2 2 1 2 3 2 2 , 1 2 , 1 L L Z Z Z T Z Z Z Z Z T Z Z Z Z Z Z- Γ = = +Γ = + + Γ = - Γ = +Γ = +- Γ = + (5.37) (5.38) The total reflection, by the lattice diagram In EM theory, generalized reflection of multi-layer is by recursive equation (EM Theory by J. Kong) EEE 591/445 Lecture 20 3 ( 29 2 2 4 1 12 21 3 12 21 3 2 2 2 1 12 21 3 3 2 2 1 12 21 3 2 3 2 2 1 3 2 1 3 1 1 1 j j n j j n j j j j T T e T T e T T e e T T e e e e θ θ θ θ θ θ θ θ-- ∞-- =---- Γ = Γ + Γ + Γ Γ + ⋅⋅⋅ = Γ + Γ Γ Γ = Γ + Γ- Γ Γ Γ + Γ = + Γ Γ ∑ 1, 1 1, 1 2 ( ) , 1 1, 2 , 1 2 ( ) , 1 1, 2 1 i z i i i z i i ik d d i i i i i i ik d d i i i i R R e R R R e + + + +- + + + +- + + + + = + % % % (5.39) (5.40) (5.41) Fig. 5.13 If reflections are small, i.e., Total reflection is dominated by the reflection at initial discontinuity (the next reflection with corresponding phase delay). Multi-section transformer (Fig 5.14 and 5.13 are inconsistent in numbering) EEE 591/445 Lecture 20 4 1 3 2 1 3 1 j e θ- Γ Γ << ⇒ Γ ≈ Γ +Γ 2 1 3 and j e θ- Γ Γ (5.42) Fig. 5.14 Partial Reflection Coefficients (N=1) 1 Z L Z Z = 2 Z θ Γ 1 Γ 1 1 1 1 1 , 1 L L Z Z Z Z Z Z N Z Z- Γ = +- Γ = = + EEE 591/445 5 Lecture 20 (5.43a) (5.43c) Single section (re-numbered) Partial Reflection Coefficients • In general, • Assume Γ n << 1, for all n . • For the circuit above, Pozar shows from the multiple reflection point of view (using a bounce diagram) 1 1 2 0 1 , 0 ,1, 2 , 3, ..., 1 n n n n n L N N L N j Z Z n N Z Z Z Z Z Z e θ + +-- Γ = = - +- Γ = + Γ ≈ Γ +Γ EEE 591/445 6 Lecture 20 (5.43b) (5.43c) Re-numbered (5.42) N-Section Transformer...
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EEE445_Lectr20_binomial - 5.5 Theory of Small Reflections...

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