impedance matching - microwave seminarrichiej/seminar/matching.pdf · impedance matching...

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.