V(z). +. V 0. +.  V0. . 3/4. /2. /4. V(z). +. . 2 V0. V 0. V(z). V(z). +. 2 V0. . 3/4. /2. /4. +. 1/2. . 3/4. /2. /4. =  V 0  [ 1+   ² + 2 cos(2 z + r )]. 16.360 Lecture 9. Standing Wave. Special cases.
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+
V0
+
V0

3/4
/2
/4
V(z)
+

2V0
V0
V(z)
V(z)
+
2V0

3/4
/2
/4
+
1/2

3/4
/2
/4
= V0 [1+  ² + 2cos(2z + r)]
16.360 Lecture 9
Standing Wave
Special cases
+
V(z)
= V0

ZL
Z0
=
+
ZL
Z0
2. ZL= 0,short circuit, = 1
+
1/2
V(z)
= V0 [2 + 2cos(2z + )]
3. ZL= ,open circuit, = 1
+
1/2
V(z)
= V0 [2 + 2cos(2z )]
V0
Z0
jz
jz
jz
jz
e
(e
e
(e
16.360 Lecture 9
short circuit line
B
Ii
A
Zg
Vg(t)
sc
Z0
VL
Zin
ZL = 0
l
z =  l
z = 0
ZL= 0, = 1, S =
+
V(z) = V0 )

= 2jV0sin(z)
+
+
i(z) =
)
= 2V0cos(z)/Z0
V(l)
Zin
= jZ0tan(l)
=
i(l)
= 2V0cos(z)
+
V0
Z0
jz
jz
jz
jz
e
(e
e
(e
16.360 Lecture 9
open circuit line
B
Ii
A
Zg
Vg(t)
oc
Z0
VL
Zin
ZL =
l
z =  l
z = 0
ZL = , = 1, S =
+
V(z) = V0 )
+

i(z) =
)
= 2jV0sin(z)/Z0
V(l)
oc
Zin
= jZ0cot(l)
=
i(l)
For a line of known length l, measurements of its input impedance, one when terminated in a short and another when terminated in an open, can be used to find its characteristic impedance Z0and electrical length
Line of length l = n/2
tan(l) = tan((2/)(n/2)) = 0,
Zin
= ZL
Any multiple of halfwavelength line doesn’t modify the load impedance.
Z0
(1 + )
j2l
j2l
e
e

+
(1
(1
)
)
Z0
16.360 Lecture 9
Quarterwave transformer l = /4 + n/2
l = (2/)(/4 + n/2) = /2 ,
j
e
+
)
(1
Zin(l)
=
=
Z0
=
j
e

(1
)
= Z0²/ZL
An example:
A 50 lossless tarnsmission is to be matched to a resistive load impedance with
ZL = 100 via a quarterwave section, thereby eliminating reflections along the feed line.
Find the characteristic impedance of the quarterwave tarnsformer.
Z01 = 50
ZL = 100
/4
= Z0²/ZL
Zin
Zin = Z0²/ZL= 50
½
½
Z0 = (ZinZL) = (50*100)
+

V0
V0
V0
Z0
Z0
Z0
jz
jz
jz
(e
e
e
16.360 Lecture 9
jz
+
e
V(z) = V0()
+

i(z) =
)
At load z = 0, the incident and reflected voltages and currents:
i
i
+
V = V0
i =
r

r
V = V0
i =
i
i
i
P(t) = v(t) i(t) = Re[V exp(jt)] Re[ i exp(jt)]
+
+
+
+
= Re[V0exp(j )exp(jt)] Re[V0/Z0 exp(j )exp(jt)]
+
+
= (V0²/Z0) cos²(t + )
r
r
r
P(t) = v(t) i(t) = Re[V exp(jt)] Re[ i exp(jt)]

+

+
= Re[V0exp(j )exp(jt)] Re[V0/Z0 exp(j )exp(jt)]
+
+
=  ²(V0²/Z0) cos²(t + + r)
T
+
+
(V0²/Z0) cos²(t + )dt
16.360 Lecture 9
Timedomain approach:
i
T
T
i
Pav =
P (t)dt
=
2
0
0
+
= (V0²/2Z0)
r
+
Pav
= ² (V0²/2Z0)
Net average power:
i
r
Pav
+ Pav
= Pav
+
= (1²) (V0²/2Z0)
Phasordomain approach
Pav
= (½)Re[V i*]
i
+
+
+
Pav = (1/2) Re[V0 V0*/Z0]
= (V0²/2Z0)
r
+
Pav
= ² (V0²/2Z0)
+
Pav
= (1²) (V0²/2Z0)
Wave Equation
TL effect
Lumped element model
TL Equation
l/>0.01
Wave (Input) Impedance
Reflection coefficient
Standing Wave
Lossless TL
+
Complete Solution
Solving for V0
Power
+