Transcript Solid State III, Lecture 23

Lecture 5 - Impedance mismatch
Reflection and transmission at the
boundary between two media.
 Aims:
 Derivation reflection/transmission
coefficients:
boundary conditions.
 Energy reflection/transmission:
Impedance matching.
 Eliminating reflection.
l/4 coupling;
gradual impedance changes.
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Reflection and transmission
 Section 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:
 Transverse displacement is continuous across
the boundary.
Self-evident that the following fig. is
impossible:
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Boundary conditions
 Cont……
 The transverse force is continuous across the
boundary.
Tx 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
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Reflection coefficient
 Continuity of displacement, , at x=0
A1eiwt  B1eiwt  A2eiwt
 must be true at all times
hence w must be same on both sides.
 Thus
[4.1]
A1  B1  A2
 Continuity of trans. force, Tx at x=0
T  ik1 A1  T ik1 B1  T  ik2 A2
 substitute k=w/v and hence get an expression
involving Z (=T/v).

T
T
T
wA1  wB1   wA2
v1
v1
v2
 Z1A1  Z1B1  Z2 A2
[4.2]
 Multiply [4.1] by Z2 and add [4.2]
Z2  Z1 A1  Z2  Z1 B1  0
 Amplitude reflection coefficient, (r).
reflected amplitude B1 Z1  Z 2 
r
 
incident amplitude
A1 Z1  Z 2 
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Transmission coefficient
 Amplitude transmission coefficient (t).
 Multiply [4.1 by Z1 and subtract from [4.2].
2Z1A1  Z1  Z2 A2
transmitte d amplitude A2
2Z1
t


incident amplitude
A1 Z1  Z 2 
 Special cases:
 Z2 =  (end of the string is clamped) r = -1
 Z2 = 0 (end of the string is free) r = +1
 Z1 = Z2 (impedance match) r = 0, t = 1
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Summary: reflection
 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” Zt ( 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.
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Energy transmission
 Reflection/transmission of energy
 Recall from 3.2, rate at which energy is carried
by a harmonic wave is
2 2
1
2
Zw A
 Energy input
 Energy reflected
 Energy transmitted
1 Z w 2 A2
1
2 1
1 Z w 2B2
1
2 1
1 Z 2w 2 A22
2
Combining the above and using amplitude
reflection coefficients to eliminate the A’s and B.
 Power reflection/transmission coefficients
2
2
reflected power Z1B1  Z1  Z 2 
R



2
incident power Z1 A1  Z1  Z 2 
transmitte d power Z 2 A22
4 Z1Z 2
T


incident power
Z1 A12 Z1  Z 2 2
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Conservation of energy
 Equations are consistent with conservation
of energy:
 Refected power + transmitted power
= incident power.
2
2
 Z  Z2 
 Z1  Z 2 
4Z1Z 2
R T   1

 


2
 Z1  Z 2  Z1  Z 2 
 Z1  Z 2 
1
 Special cases:
 Z2 = : R = 1, T = 0
All energy reflected.
 Z2 = 0: R = 1, T = 0
All energy reflected.
 Z1 = Z2: R = 0, T = 1
All energy transmitted.
 Note:
 For complex Z1 and Z2: power =
 e.g. power reflected
1 2 Z1 w 2 B 12
R

2 2
1 2 Z1 w A 1
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B1
A1
1 Z w 2 A2
2
2

B1
 Z1  Z 2
 rr 

Z1  Z 2
A1
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Impedance matching
 Lots of examples
 2 common approaches to reduce reflection:
 l/4 coupler
 examples: anti-reflection coatings on lenses.
Wave of zero amplitude
 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|, Z C  Z A Z B
(gives the same reflection coefficient at each
surface)
out-of-phase by l/2. i.e. l = l/4 (hence name).
 Note: for an optical system l = l/4 is chosen in
the middle of the visible spectrum. Hence the
“purple” bloom on coated lenses.
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Impedance matching, continued..
 Gradual impedance changes
 change impedance over a distance >> l.
 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….
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