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ECE 391 supplemental notes - #11
168 Oregon State University ECE391– Transmission Lines Spring Term 2014
Adding a Lumped Series Element Consider the following T-line circuit:
ZL
Z0,1! Z0,2!R
zL =ZL
Z0,2zin,2 = rin,2 + jxin,2
zL,1 = rin,2Z0,2Z0,1
+ RZ0,1
+ jxin,2Z0,2Z0,1
zin,1 = rin,1 + jxin,1Zin,1 = zin,1Z0,1
è All calculations can be done on Smith chart
renormalization
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169 Oregon State University ECE391– Transmission Lines Spring Term 2014
Smith Chart as Admittance Chart
For circuits involving parallel (shunt) connections of circuit elements, it is more convenient to work with admittances.
How can the Smith chart be used with admittances?
è How is Y=1/Z = G + jB related to reflection coefficient Γ ?
Consider normalized admittance y
y = YY0
= G + jBY0
= g + jb = Y Z0 =Z0Z
= 1z= 1r + jx
170 Oregon State University ECE391– Transmission Lines Spring Term 2014
Reflection coefficient Γ
Smith chart can directly be used with y=g+jb after rotating reflection coefficient by 180°.
Smith Chart as Admittance Chart
Γ = z −1z +1
= 1 y −11 y +1
= − y −1y +1
y −1y +1
= −Γ = Γ e jπ g = const
b = const circles
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171 Oregon State University ECE391– Transmission Lines Spring Term 2014
Impedance vs. Admittance Coordinates
r + jx
r - jx SC OC
Re(Γ)
Im(Γ)
inductive
capacitive
0
g + jb
g - jb OC SC
Re(-Γ)
Im(-Γ)
inductive
capacitive
0
Impedance Chart Admittance Chart
172 Oregon State University ECE391– Transmission Lines Spring Term 2014
Example 4 Given the normalized load admittance
yL = 0.5 + j2.0
Determine the normalized admittance at distance
d = λ/16 = 0.0625λ
Solution: use Smith chart with admittance coordinates è see web applet (example 4)
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173 Oregon State University ECE391– Transmission Lines Spring Term 2014
Example 5 Given a 50Ω T-line that is terminated in ZL = ZT= 25Ω and has a shunt resistance Rsh = 50Ω connected at distance d = 1m from the termination. The wavelength on the line is λ = 3m.
Determine the input impedance at the location of the shunt resistance.
Solution: see web applet, example 5
174 Oregon State University ECE391– Transmission Lines Spring Term 2014
Additional Example
( )Ω+= 2525 jZLΩ= 500Z
λ1025.0=l
Results
6.2SWR ≈
45.0≈ΓL5.116≈Lθ
1.15.1zin j+≈j−=1yL
5.05.0z jL +=
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175 Oregon State University ECE391– Transmission Lines Spring Term 2014
Impedance Matching
Implementation Issues § Types of Networks
- lumped elements - distributed elements - mixed - reactive vs. resistive
§ Design Criteria - bandwidth - lossless/lossy - complexity - parasitics
176 Oregon State University ECE391– Transmission Lines Spring Term 2014
Impedance Matching
è Matching network needs to have at least
two degrees of freedom (two knobs)
to match a complex load impedance
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177 Oregon State University ECE391– Transmission Lines Spring Term 2014
Transmission-Line Matching Idea: Use impedance transformation property of transmission line
transformation circle
2βz’ load zL (or yL)
178 Oregon State University ECE391– Transmission Lines Spring Term 2014
Transmission-Line Matching Transformation to real z or y
transformation circle
2βz’
real z (or y) real z (or y)
load zL (or yL)
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179 Oregon State University ECE391– Transmission Lines Spring Term 2014
Transmission-Line Matching Transformation to r=1 (or g=1) circle
transformation circle
2βz’
r=1 or g=1
load zL (or yL) z=1+jx (y=1+jb)
z=1-jx (y=1-jb)
180 Oregon State University ECE391– Transmission Lines Spring Term 2014
Single Stub Matching Stub tuners are useful for matching complex load impedances. The two design parameters in single stub matching networks are
§ distance from load at which stub is connected
§ length of stub
Other design considerations are
§ open or shorted stub
§ series or shunt connected stub
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181 Oregon State University ECE391– Transmission Lines Spring Term 2014
Basic Design Principle
Objective: Match load admittance by 1. transforming along T-line to Yin = Y0+jB 2. tuning out remaining jB with a parallel shunt element of
admittance –jB 3. shunt element can be realized with a T-line stub
Yin = Y0
182 Oregon State University ECE391– Transmission Lines Spring Term 2014
Design Steps
1. Determine distance d/λ from the load YL (or ZL) at which the § input admittance is Y0 +jB (for shunt stub matching) § input impedance is Z0+jX (for series stub matching)
2. Realize the stub with a length l/λ of open- or short-circuited transmission line (stub).
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183 Oregon State University ECE391– Transmission Lines Spring Term 2014
Example 6
Using a short-circuited stub, match the load impedance ZL=(80 + j70)Ω to a transmission line with Z0 = 50Ω.
Z0 ZL
d
Solution: use Smith chart with admittance coordinates è see web applet (example 6)
184 Oregon State University ECE391– Transmission Lines Spring Term 2014
Additional Example
Ω=125LRpF54.2=LC
GHz10 =fΩ= 500Z
jL −= 5.0z
58.11yin_1,2 j±≈
mS6.31/58.1 0sh_1,2 jZjY ==
Need shunt element with
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185 Oregon State University ECE391– Transmission Lines Spring Term 2014
Quarter-Wave Transformer
LZZZ 01 =
increased mismatch
NOTE: for complex load, insert λ/4 transformer where Zin=R
4/λ at design frequency f0
186 Oregon State University ECE391– Transmission Lines Spring Term 2014
Example: Quarter-Wave Transformer
Ω=125LRpF54.2=LC
GHz10 =fΩ= 500Z
jL −= 5.0z
Ω≈ 5.24Z T0,
24.0zin,1 ≈ Ω≈12Zin,1
in,1
2,0
0in,2ZZZ
Z T==From
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187 Oregon State University ECE391– Transmission Lines Spring Term 2014
Bandwidth Performance
GHz1at 0 =f
188 Oregon State University ECE391– Transmission Lines Spring Term 2014
Lumped Element Matching Reactive L-Section Matching Networks
0ZRL <0ZRL >
220
220
LL
LLLLL
XRZRXRZRX
B+
−+±=
LL
L
RBZ
RZX
BX 001 −+=
( )0
0
ZRRZ
B LL−±=
( ) LLL XRZRX −−±= 0
different solutions may result in different bandwidths
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189 Oregon State University ECE391– Transmission Lines Spring Term 2014
Example
Design criteria § bandwidth § realizability of components § parameter sensitivity § … cost … ?
190 Oregon State University ECE391– Transmission Lines Spring Term 2014
Lumped-Distributed and Distributed Matching Networks
Series Reactance Matching Shunt Reactance Matching
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191 Oregon State University ECE391– Transmission Lines Spring Term 2014
Examples
192 Oregon State University ECE391– Transmission Lines Spring Term 2014
Double-Stub Tuner • Double-stub (and multi-stub) tuners use fixed
stub locations (typical spacing: λ/8)
• Advantages: - easier to accommodate different loads - better bandwidth performance
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193 Oregon State University ECE391– Transmission Lines Spring Term 2014
Smith Chart with Lossy T-Lines Recall for a lossless T-line
For low-loss T-lines, Z0 ≈ real and γ = α + jβ
Then
zin (z ') =Zin (z ')Z0
= 1+ ΓLe−2γ z '
1− ΓLe−2γ z ' =
1+ ΓL e−2αz 'e jφ z '( )
1− ΓL e−2αz 'e jφ z '( )
zin (z ') =Zin (z ')Z0
=1+ ΓL e
jφ z '( )
1− ΓL ejφ z '( ) φ(z ') = θL − 2βz '
194 Oregon State University ECE391– Transmission Lines Spring Term 2014
Smith Chart with Lossy T-Lines Γ z '( ) = ΓL e
−2αz '
Γre
Γim As z '→∞zin →1(Zin → Z0 )
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195 Oregon State University ECE391– Transmission Lines Spring Term 2014
Approach
Γre
Γim
zL
Γ inlossy = Γ in
losslesse−2αz '
Γ inlossless = ΓL e
jφ
1. Assume lossless line and rotate by -2βz’ 2. Change |Γ| by factor
e−2αz '
196 Oregon State University ECE391– Transmission Lines Spring Term 2014
Example 7
Given a lossy T-line with characteristic impedance Z0 = 100Ω, length l = 1.5m (l < λ/2), and ZSC(z’=l) = 40 – j280 Ω.
(a) Determine α and β.
(b) Determine Zin for ZL = 50 + j50 Ω an l = 1.5m.
Solution using Smith chart
è see web applet (example 7)