Formula sheet: Transmission Lines | Electromagnetics - Electronics and Communication Engineering (ECE) PDF Download

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Electromagnetics: T ransmission Lines F orm ula Sheet
for GA TE
T ransmission Line P arameters
• Characteristic Imp edance (Z
0
) : Ratio of v oltage to curren t for a w a v e tra v eling
in one direction.
Z
0
=
v
R+j?L
G+j?C
whereR is resistance (?m
-1
),L is inductance (Hm
-1
),G is conductance (Sm
-1
),C is
capacitance (Fm
-1
), ? is angular frequency (rads
-1
). F or lossless lines (R = G = 0 ):
Z
0
=
v
L
C
• Propagation Constan t (? ) :
? = a+jß =
v
(R+j?L)(G+j?C)
where a is att en uation constan t (Npm
-1
), ß is phase constan t (radm
-1
). F or lossless
lines:
? = jß = j?
v
LC
• Phase V elo cit y (v
p
) :
v
p
=
?
ß
F or lossless lines:
v
p
=
1
v
LC
W a v e Propagation
• V oltage and Curren t on Line :
V(z) = V
+
e
-?z
+V
-
e
?z
I(z) =
V
+
Z
0
e
-?z
-
V
-
Z
0
e
?z
where V
+
and V
-
are forw ard and bac kw ard v oltage w a v es, z is distance along the
line.
• Reflection Co e?icien t ( G ) : A t load (z = 0 ):
G
L
=
Z
L
-Z
0
Z
L
+Z
0
where Z
L
is load imp edance. A t distance l from load:
G(l) = G
L
e
-2?l
• V oltage Standing W a v e Ratio (VSWR) :
VSWR =
1+|G
L
|
1-|G
L
|
1
Page 2


Electromagnetics: T ransmission Lines F orm ula Sheet
for GA TE
T ransmission Line P arameters
• Characteristic Imp edance (Z
0
) : Ratio of v oltage to curren t for a w a v e tra v eling
in one direction.
Z
0
=
v
R+j?L
G+j?C
whereR is resistance (?m
-1
),L is inductance (Hm
-1
),G is conductance (Sm
-1
),C is
capacitance (Fm
-1
), ? is angular frequency (rads
-1
). F or lossless lines (R = G = 0 ):
Z
0
=
v
L
C
• Propagation Constan t (? ) :
? = a+jß =
v
(R+j?L)(G+j?C)
where a is att en uation constan t (Npm
-1
), ß is phase constan t (radm
-1
). F or lossless
lines:
? = jß = j?
v
LC
• Phase V elo cit y (v
p
) :
v
p
=
?
ß
F or lossless lines:
v
p
=
1
v
LC
W a v e Propagation
• V oltage and Curren t on Line :
V(z) = V
+
e
-?z
+V
-
e
?z
I(z) =
V
+
Z
0
e
-?z
-
V
-
Z
0
e
?z
where V
+
and V
-
are forw ard and bac kw ard v oltage w a v es, z is distance along the
line.
• Reflection Co e?icien t ( G ) : A t load (z = 0 ):
G
L
=
Z
L
-Z
0
Z
L
+Z
0
where Z
L
is load imp edance. A t distance l from load:
G(l) = G
L
e
-2?l
• V oltage Standing W a v e Ratio (VSWR) :
VSWR =
1+|G
L
|
1-|G
L
|
1
Imp edance Along the Line
• Input Imp edance (Z
in
) : A t distance l from load:
Z
in
(l) = Z
0
Z
L
+jZ
0
tan(ßl)
Z
0
+jZ
L
tan(ßl)
• Short-Circuited Line (Z
L
= 0 ) :
Z
in
= jZ
0
tan(ßl)
• Op en-Circuited Line (Z
L
=8 ) :
Z
in
=-jZ
0
cot(ßl)
Imp edance Matc hing
• Quarter-W a v e T ransformer : Matc hes load Z
L
to line Z
0
using a line of length ?/4
and imp edance:
Z
quarter
=
v
Z
0
Z
L
• Stub Matc hing : Uses short- or op en-circuited stubs to cancel reactance. F or a stub
at distance d from load:
Y
in
= Y
0
Y
L
+jY
0
tan(ßd)
Y
0
+jY
L
tan(ßd)
where Y
L
= 1/Z
L
, Y
0
= 1/Z
0
.
Key Notes
• Common GA TE F o cus : Characteristic imp edance, reflection co e?icien t, VSWR,
and quarter-w a v e transformer.
• Lossless Lines : Assume R = G = 0 unless sp ecified.
• W a v elength : ? =
2p
ß
=
vp
f
, where f is frequency .
• Units : Ensure consistency (e.g., ? in rads
-1
, Z
0
in ? , ß in radm
-1
).
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