1 Waves 5

Lecture 5 - Impedance mismatchLecture 5 - Impedance mismatch

Reflection and transmission at the Reflection and transmission at the boundary between two media.boundary between two media.

Aims:Aims: Derivation reflection/transmission

coefficients: boundary conditions.

Energy reflection/transmission: Impedance matching.

Eliminating reflection. /4 coupling; gradual impedance changes.

2 Waves 5

Reflection and transmissionReflection and transmission

Section 4Section 4 What happens when a wave approaches a

boundary between wave-media? Some energy is transmitted, Some energy is reflected.

With a knowledge of the boundary conditions, we can calculate what fraction is reflected or transmitted.

Boundary conditions:Boundary conditions: Transverse displacement is continuous across

the boundary. Self-evident that the following fig. is

impossible:

3 Waves 5

Boundary conditionsBoundary conditions

Cont……Cont…… The transverse force is continuous across the

boundary. Tx is continuous ( hence so is x).

Reason: at the boundary, an element of infinitesimal mass cannot be subject to a finite net-force since that would give an infinite acceleration. e.g.

Occasional exceptions are seen (e.g. tripos questions with a finite mass element at the boundary)

So, we have

4 Waves 5

Reflection coefficientReflection coefficient

Continuity of displacement, Continuity of displacement, , at x=0, at x=0

must be true at all times hence must be same on both sides.

Thus

Continuity of trans. force, Continuity of trans. force, TTxx at x=0 at x=0

substitute k=/v and hence get an expression involving Z (=T/v).

Multiply [4.1] by Z2 and add [4.2]

Amplitude reflection coefficient, (Amplitude reflection coefficient, (rr).).

tititi eAeBeA 211

211 ABA ]1.4[

221111 AikTBikTAikT

22

11

11

AvT

BvT

AvT

221111 AZBZAZ ]2.4[

0112112 BZZAZZ

21

21

1

1ZZZZ

AB

r

amplitude incidentamplitude reflected

5 Waves 5

Transmission coefficientTransmission coefficient

Amplitude transmission coefficient (Amplitude transmission coefficient (tt).). Multiply [4.1 by Z1 and subtract from [4.2].

Special cases:Special cases: Z2 = (end of the string is clamped) r = -1

Z2 = (end of the string is free)r = +1

Z1 = Z2 (impedance match) r = 0, t = 1

221112 AZZAZ

21

1

1

2 2ZZ

ZAA

t

amplitude incident

amplitude dtransmitte

6 Waves 5

Summary: reflectionSummary: reflection

Notes and summary:Notes and summary: Results are general and apply to a wide variety

of wave systems. Results for r and t obtained using:

continuity of displacement; continuity of “force” Zt ( slope).

Easy to derive (no need to remember), especially as coefficients for the “force”

(rather than “displacement”) are often required. Eg: voltage in a transmission line; electric field in an E.M. wave; pressure in a sound wave.

In these latter cases 1/Z replaces Z. If Z is complex, there are phase differences

between incident, reflected and transmitted waves.

7 Waves 5

Energy transmissionEnergy transmission

Reflection/transmission of energyReflection/transmission of energy Recall from 3.2, rate at which energy is carried

by a harmonic wave is

Energy input

Energy reflected

Energy transmitted

Combining the above and using amplitude reflection coefficients to eliminate the A’s and B.

Power reflection/transmission coefficientsPower reflection/transmission coefficients

222

1 AZ

22

22

21

21

21

21

21

21

21

AZ

BZ

AZ

221

21211

222

2

21

21211

211

4

ZZ

ZZ

AZ

AZT

ZZZZ

AZ

BZR

power incidentpower dtransmitte

power incidentpower reflected

8 Waves 5

Conservation of energyConservation of energy

Equations are consistent with conservation Equations are consistent with conservation of energy:of energy: Refected power + transmitted power

= incident power.

Special cases:Special cases: Z2 = : R = 1, T = 0

All energy reflected. Z2 = : R = 1, T = 0

All energy reflected. Z1 = Z2: R = 0, T = 1

All energy transmitted.

Note:Note: For complex Z1 and Z2: power =

e.g. power reflected

1

42

21

212

21

212

21

21

ZZZZ

ZZ

ZZZZZZ

TR

222

1 AZ

2

21

21

1

1

1

121

21

21

21

21

21

ZZZZ

rrA

BAB

AZ

BZR

9 Waves 5

Lots of examples 2 common approaches to reduce reflection:

/4 coupler/4 coupler examples: anti-reflection coatings on lenses.

Destructive interference of the reflected waves gives a total reflected wave of zero amplitude.

Energy conservation gives 100% transmission. Conditions for destructive interference:

same amplitude, |r| =|rtr|,

(gives the same reflection coefficient at each surface)

out-of-phase by /2. i.e. l = /4 (hence name). Note: for an optical system l = /4 is chosen in

the middle of the visible spectrum. Hence the “purple” bloom on coated lenses.

Impedance matchingImpedance matching

Wave of zero amplitudeWave of zero amplitude

BAC ZZZ

10 Waves 5

Impedance matching, continued..Impedance matching, continued..

Gradual impedance changesGradual impedance changes change impedance over a distance >> . infinitesimal reflections at each infinitesimal

change in impedance. Lots of small reflections with a large range of

phase give a small net resultant reflected wave. Thus, most energy is transmitted.

Examples: Bell on trumpet or horn. Microwave horn …etc….