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EE334 - Reflection Coefficient & Standing Waves
4 Lecture: pp 53-61 2-
The voltage at the load will be the phasor sum of the wave traveling to the right and
to the left
j z jz V z V e V e
and the phasor current at the load will be:
j z jz e Z
V
e Z
V
I z
β + β
− −
= − 0
0
0
Let z=0 at the load and z=-l at the generator then
- − ( ) − + ( ) + − = = = + = 0 + 0
0 0
0 0
V z V V e V e V V
j j L
β β
( ) ( )
0
0 0
0
0
0
0 0
0
0 0
0
Z
V V
Z
V
Z
V
e Z
V
e Z
V
I z I
j j L
− −
β β
finally the load impedance is the ratio of the phasor voltage to the phasor current.
0 0 0
0 0 ~
Z
V V
V V
I
V
Z
L
L L (^) ⎟
solve for the voltage traveling in the negative direction:
− + ⎟
0
0 0 V Z Z
Z Z
V
L
L
if the plus phasor voltage is the input to the load then the – phasor will be the reflect
voltage phasor and their ratio will be the reflection coefficient.
load L
L
Z Z
Z Z
V
V
−
0
0
0
0 voltage reflection coefficient
Voltage reflected is equal to the voltage in times the reflection coefficient
Vref =Γ V in
the reflection coefficient can be complex.
jr e
θ Γ=Γ
in which case there is a phase shift corresponding to the reflection.
Matched impedanceÎZL = Z 0 Î Γ = 0 no reflection
Other extremes
Open circuitÎZL = infinityÎΓ = 1, Vref = Vin
Short circuitÎZL = 0ÎΓ = -1, Vref = -Vin
EXAMPLE:
Ζο = 100Ω
10pF
f = 100 MHz
( )( )( )
−
j
j
C
j Z R
j C
Z R
L
L L
L
L L
8 11
Gets reflected and creates a returning wave with different amplitude and phase
These interfere to create a standing wave
What is the voltage as a function of position?
( ) ( ) ( )
j z j z j z j j z V z V e e V e e e
V V
− +
0 0
0 0 ~
This is a complex number therefore must take square root of product with complex
conjugate to get the magnitude
( ) { ( ) ( )]}
2
1 2 0
2
1
0 0
1 2 cos 2
β θ β β θ β
V z
V z V e e e V e e e
j z j j z j z j j z
What does this functionality look like?
ZL = infinity (open circuit)
ZL = 0 (short circuit)
j e
- 5 Γ=( 0. 1 , 0. 5 , 0. 9 )
Maxima and minima separated by half a wavelength.
No standing wave without some reflected signal (impedance matched)
EXAMPLE: a slotted-line probe is used to measure the voltage as a function of position
on a 50 ohm line. A standing wave ratio of 3 is found, and successive minima were 30cm
apart and the first minimum was 12cm from the load. Determine the load impedance.
Given: Z (^) 0 = 50 Ω S = 3 l min= 12 cm
Minimum are λ/2 apart then 30 cm ( )( 2. 3 m ) 0. 6 ( m )
( )
1
3
= = = m m
S
S
o
. 12 0. 2 36 3
(^2) min 2 min (^2) ⎟ − =− = ⎠
θ r β l π θ r β l π
36 e e j
j (^) r j Γ =Γ = = −
θ − o
0
0
0 0 0 0 0
0
Z Z
Z Z
Z Z Z Z Z Z Z Z
Z Z
Z Z
L
L
L L L L L
L
0 j j
j
j
j Z (^) L Z
