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1
Swartz 06/2/15
ECE 488 –
RF Circuits and Systems
– Spring 2009
Lecture 13:
ECE 488
RF Circuits and Systems
Transmission Lines
Reflection Coefficients
Fall 2009 Lecture 13
2009/2/16(18)
4
Swartz 06/2/15
ECE 488 –
RF Circuits and Systems
– Spring 2009
Lecture 13:
Modeling a Transmission Line (1)
•
Transmission line effects become important
–
When their length becomes comparable with a fraction
of a wavelength.
–
When the time to propagate to the other end becomes
comparable to other circuit response times.
•
This model uses lumped elements
–
Series inductances –
slow build up of currents at other end
–
Parallel capacitances –
shunt off currents before reaching
other end
–
No resistances shown here
(usually small, but can be added
to model as a series element if losses need to be modeled)
.
5
Swartz 06/2/15
ECE 488 –
RF Circuits and Systems
– Spring 2009
Lecture 13:
Modeling a Transmission Line (2)
•
Consider a transmission line
–
with a characteristic impedance of Z
0
,
–
attached to of load of impedance Z
0
,
–
then its input impedance is Z
0
.
•
Adding the lumped elements of model
–
Cannot change input impedance
–
L has units of H/m; C has units of F/m.
6
Swartz 06/2/15
ECE 488 –
RF Circuits and Systems
– Spring 2009
Lecture 13:
Modeling a Transmission Line (3)
•
Compute the input impedance:
0
0
0
0
0
1
1
1
xZ
C
j
Z
x
L
j
x
C
j
Z
x
C
j
Z
x
L
j
Z
Z
δ
ω
+
+
=
+
+
=
+
0
0
1
xZ
C
j
Z
+
2
0
0
)
(
0
2
lim
xZ
C
j
Z
x
−
=
→
•
Consider the second term:
=
−
−
•
0
0
1
1
xZ
C
j
xZ
C
j
2
0
2
2
2
0
0
)
(
1
)
1
(
Z
x
C
xZ
C
j
Z
+
−
7
Swartz 06/2/15
ECE 488 –
RF Circuits and Systems
– Spring 2009
Lecture 13:
Modeling a Transmission Line (4)
•
Reducing to
)
(
2
0
0
2
0
0
0
Z
C
L
x
j
Z
xZ
C
j
Z
x
L
j
Z
Z
−
+
=
−
+
=
+
ωδ
•
What happens when
?
C
L
Z
=
0
0
0
Z
Z
Z
=
+
•
What does this mean?
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 Spring '06
 SWARTZ
 Frequency, Gate

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