Post on 27-Mar-2020
Impedance Matching
Impedance MatchingMicrowave Seminar
J. Richie
February 22, 2013
Impedance Matching
Outline
Introduction
Narrow-Band MethodsLumped Element MatchingStub TunersQuarter-Wave Transformer
Varying Bandwidth MethodsLumped Element MethodsMultiple Quarter-wave SectionsBode-Fano CriterionTheory of Small Reflections
Conclusions
Impedance Matching
Introduction
T-lines, Loads, and Input Impedance
Transmission Line Types (many more than listed here) Coaxial Line Twin Lead Microstrip, Stripline, etc.
The Load: ZL
Input Impedance:
Zin = RoZL + jRo tanβℓ
Ro + jZL tanβℓ
Impedance Matching
Narrow-Band Methods
Narrow-Band Methods
Matching can easily be accomplished at one specificfrequency.
The design depends on component values or lengths
Then, the bandwidth is generally narrow and depends tosome extent on how far apart Ro and ZL are from eachother.
Impedance Matching
Narrow-Band Methods
Lumped Element Matching
Outline
Introduction
Narrow-Band MethodsLumped Element MatchingStub TunersQuarter-Wave Transformer
Varying Bandwidth MethodsLumped Element MethodsMultiple Quarter-wave SectionsBode-Fano CriterionTheory of Small Reflections
Conclusions
Impedance Matching
Narrow-Band Methods
Lumped Element Matching
Lumped Element Matching
Adjusting impedances to get maximum power transfer
Can be used at higher frequencies now due tosurface-mount technology
Component losses can limit usefulness of matchingnetwork
Impedance Matching
Narrow-Band Methods
Lumped Element Matching
L-Nets: Analytic Considerations
jX p
jX s
VAC
Z left
50
1000
Note how jXp pulls 1kΩ down since in parallel Choose Xp so that Rleft = 50Ω (to match). Then, jXs used to cancel jXleft
Impedance Matching
Narrow-Band Methods
Lumped Element Matching
Answers to Prev. Problem
VAC 50
1000−j218
j229 VAC 50
1000j218
−j229
On left, 0 output at DC On right, 0 output at infinite frequency
Impedance Matching
Narrow-Band Methods
Lumped Element Matching
Analysis
Let us define
QEL =
√
Rhigh
Rlow− 1
Then, we haveXs
Rlow=
Rhigh
Xp= QEL
However, QEL is not the Q = fo/∆f but it can be shown that
1Q
=2
QEL
Note that as Rhigh/Rlow increases, the Q increases.
Impedance Matching
Narrow-Band Methods
Lumped Element Matching
L-Nets on a Smith Chart
Need impedance Smith chart with g = 1 circle added
Example
ZL = 200− j100, 100Ω line, f = 500MHz.
zL = 2− j1
inside r = 1 circle→ high impedance → Xp first.
(see chart)
Impedance Matching
Narrow-Band Methods
Stub Tuners
Outline
Introduction
Narrow-Band MethodsLumped Element MatchingStub TunersQuarter-Wave Transformer
Varying Bandwidth MethodsLumped Element MethodsMultiple Quarter-wave SectionsBode-Fano CriterionTheory of Small Reflections
Conclusions
Impedance Matching
Narrow-Band Methods
Stub Tuners
Single Stub Tuner
Z s
Z L
d
l
There are two variables, d and ℓ. Zs is either an open circuit or a short circuit The stub adds only reactance Principle
Find d so that yin = 1± jx Find ℓ so that yin,stub = 1∓ jx
Impedance Matching
Narrow-Band Methods
Stub Tuners
Example
ZL = 60− j80 (R = 60Ω, C=0.995pF at 2 GHz)
Zo = 50Ω
zL = 1.2− j1.6
see Smith chart
Impedance Matching
Narrow-Band Methods
Quarter-Wave Transformer
Outline
Introduction
Narrow-Band MethodsLumped Element MatchingStub TunersQuarter-Wave Transformer
Varying Bandwidth MethodsLumped Element MethodsMultiple Quarter-wave SectionsBode-Fano CriterionTheory of Small Reflections
Conclusions
Impedance Matching
Narrow-Band Methods
Quarter-Wave Transformer
Quarter-Wave Transformer
Recall the input impedance relation:
Zin = RoZL + jRo tanβℓ
Ro + jZL tanβℓ
if ZL, Ro are real, and ℓ = λ/4, then,βℓ = (2π/λ)(λ/4) = π/2 and tan(π/2) → ∞.
Therefore,
Zin =R2
o
ZL
Impedance Matching
Narrow-Band Methods
Quarter-Wave Transformer
Quarter-Wave Transformers, Part II
Recall:
Zin =R2
o
ZL
This relationship can be used to define a section oftransmission line with impedance R′ which is λ/4 long andhas characteristic impedance:
R′ =√
RoRL
and there will be no reflections at the center frequency.
Impedance Matching
Varying Bandwidth Methods
Bandwidth Considerations
In all of the methods discussed, the match is “perfect” at asingle frequency.
Sometimes, the bandwidth of the match is important. There are instances where a narrower bandwidth or a
wider bandwidth is desired. For the rest of the presentation, we will investigate
techniques (mostly based on previous methods) thatprovide either a wider or a narrower bandwidth.
The Bode-Fano Criterion also helps us understand someof the fundamental limitations of wide-band matchingnetworks.
Impedance Matching
Varying Bandwidth Methods
Lumped Element Methods
Outline
Introduction
Narrow-Band MethodsLumped Element MatchingStub TunersQuarter-Wave Transformer
Varying Bandwidth MethodsLumped Element MethodsMultiple Quarter-wave SectionsBode-Fano CriterionTheory of Small Reflections
Conclusions
Impedance Matching
Varying Bandwidth Methods
Lumped Element Methods
Lumped Element Methods: Narrower Bandwidth
Z=10+j0 Z=10+j0
501000
−j100.5
j99.5 j20
−j25
Pi net will have narrower BW Intermediate impedance is additional degree of freedom Can also choose Z > Rhigh, then, L-nets flip and have T-net
Impedance Matching
Varying Bandwidth Methods
Lumped Element Methods
Lumped Element Methods: Wider Bandwidth
Z1
1000 50
50 < Z < 10001
Multiple Sections can be used Many sections and structure begins to look like tapered
t-line
Impedance Matching
Varying Bandwidth Methods
Multiple Quarter-wave Sections
Outline
Introduction
Narrow-Band MethodsLumped Element MatchingStub TunersQuarter-Wave Transformer
Varying Bandwidth MethodsLumped Element MethodsMultiple Quarter-wave SectionsBode-Fano CriterionTheory of Small Reflections
Conclusions
Impedance Matching
Varying Bandwidth Methods
Multiple Quarter-wave Sections
Multiple Quarter-wave Section
Can use multiple sections to create wider bandwidth match
Each section has length λ/4 Each section has impedance between Ro and RL
Structure will take more space (length) More t-line will also mean more loss in structure
Impedance Matching
Varying Bandwidth Methods
Multiple Quarter-wave Sections
Example
Z =Rin oAZ
BZ
oR 3R 2R 1R
LR
For example, suppose have 3 sections. Let
r =RL
ZB=
ZB
ZA=
ZA
Roor
RL
Ro=
ZA
Ro
ZB
ZA
RL
ZB= r3
Therefore, use
r = 3
√
RL
Ro
Impedance Matching
Varying Bandwidth Methods
Bode-Fano Criterion
Outline
Introduction
Narrow-Band MethodsLumped Element MatchingStub TunersQuarter-Wave Transformer
Varying Bandwidth MethodsLumped Element MethodsMultiple Quarter-wave SectionsBode-Fano CriterionTheory of Small Reflections
Conclusions
Impedance Matching
Varying Bandwidth Methods
Bode-Fano Criterion
Bode-Fano Criterion: Introduction
Bode-Fano Criterion answers: Can we achieve perfect
match (Γ = 0) over abandwidth (BW)?
If not, how well can we do? What is tradeoff between
| Γ | and BW? How complex must
matching net be?
Bode-Fano gives theoreticallimit on |Γ|min
Ro
Z L
Matching
Net
Impedance Matching
Varying Bandwidth Methods
Bode-Fano Criterion
Bode-Fano Criterion
Criterion related to
∞∫
0
ln1
|Γ(ω)|dω
For example, with a parallel RC load (ICBST):
∞∫
0
ln1
|Γ(ω)|dω ≤ π
RC
if |Γ(ω)| = 1, have complete reflection and contribution tointegral is zero.
Thus, criterion is concerned with pass band
Impedance Matching
Varying Bandwidth Methods
Bode-Fano Criterion
Simple Example
| |Γ
minΓ
ω
1
∆ω
Using |Γ| as shown,
∞∫
0
ln1|Γ|dω =
∫
∆ω
ln1
Γmindω = ∆ω ln
1Γmin
≤ π
RC
Conclusions: For a given load, as ∆ω increases, Γmin increases Γ in passband cannot be zero unless ∆ω = 0. as R or C increase, ∆ω or 1/Γmin must decrease (higher Q
implies harder to match)
Impedance Matching
Varying Bandwidth Methods
Theory of Small Reflections
Outline
Introduction
Narrow-Band MethodsLumped Element MatchingStub TunersQuarter-Wave Transformer
Varying Bandwidth MethodsLumped Element MethodsMultiple Quarter-wave SectionsBode-Fano CriterionTheory of Small Reflections
Conclusions
Impedance Matching
Conclusions
Conclusions
Many methods available for impedance matching narrow-band methods wider-band methods
Bode-Fano Criterion helps us understand the fundamentallimits of wide-band matching
(not covered) Theory of small reflections can be used tocreate filter-like designs that both match the load to the lineand provide filtering.