1/26/2005
Transmission Line Input Impedance.doc
1/9
Transmission Line
Input Impedance
Consider a lossless line, length A , terminated with a load ZL.
I(z)
IL
+
+
Z0 , β
V (z)
ZL
VL
-
-
A
z = −A
z = 0
Let’s determine the input impedance of this line!
Q: Just what do you mean by input impedance?
A: The input impedance is simply the line impedance seen
at the beginning ( z = −A) of the transmission line, i.e.:
Zin = Z ( z = −A ) =
V ( z = −A )
I ( z = −A )
Note Zin equal to neither the load impedance ZL nor the
characteristic impedance Z0 !
Zin ≠ Z L
Jim Stiles
and
The Univ. of Kansas
Zin ≠ Z 0
Dept. of EECS
1/26/2005
Transmission Line Input Impedance.doc
2/9
To determine exactly what Zin is, we first must determine the
voltage and current at the beginning of the transmission line
( z = −A ).
V ( z = − A ) = V0+ ⎡⎣e + j β A + Γ L e − j β A ⎤⎦
V0+ + j β A
⎡⎣e
− Γ L e − j β A ⎤⎦
I ( z = −A ) =
Z0
Therefore:
⎛ e + j β A + ΓL e − j β A ⎞
V ( z = −A )
Zin =
= Z0 ⎜ + j βA
⎟
I ( z = −A )
− ΓL e − j β A ⎠
⎝e
We can explicitly write Zin in terms of load ZL using the
previously determined relationship:
ΓL =
ZL − Z0
ZL + Z0
Combining these two expressions, we get:
Zin =
ZL + Z 0 ) e + j β A + (ZL − Z 0 ) e − j β A
(
Z0
(ZL + Z 0 ) e + j β A − (ZL − Z 0 ) e − j β A
⎛ Z L (e + j β A + e − j β A ) + Z 0 (e + j β A − e − j β A ) ⎞
= Z0 ⎜
⎟
⎜ Z L (e + j β A + e − j β A ) − Z 0 (e + j β A − e − j β A ) ⎟
⎝
⎠
Now, recall Euler’s equations:
e + j β A = cos β A + j sin β A
e − j β A = cos β A − j sin β A
Jim Stiles
The Univ. of Kansas
Dept. of EECS
1/26/2005
Transmission Line Input Impedance.doc
3/9
Using Euler’s relationships, we can likewise write the input
impedance without the complex exponentials:
⎛ Z L cos β A + j Z 0 sin β A ⎞
⎟
A
A
+
β
β
Z
cos
j
Z
sin
L
0
⎝
⎠
Zin = Z 0 ⎜
⎛ Z + j Z 0 tan β A ⎞
= Z0 ⎜ L
⎟
+
A
β
tan
Z
j
Z
L
⎝ 0
⎠
Note that depending on the values of β , Z 0 and A , the input
impedance can be radically different from the load impedance
ZL !
Special Cases
Now let’s look at the Zin for some important load impedances
and line lengths.
Æ You should commit these results to memory!
1. A = λ
2
If the length of the transmission line is exactly one-half
wavelength ( A = λ 2 ), we find that:
βA =
meaning that:
cos β A = cos π = −1
Jim Stiles
2π λ
=π
λ 2
and
The Univ. of Kansas
sin β A = sin π = 0
Dept. of EECS
1/26/2005
Transmission Line Input Impedance.doc
and therefore:
4/9
⎛ Z L cos β A + j Z 0 sin β A ⎞
⎟
⎝ Z 0 cos β A + j Z L sin β A ⎠
Zin = Z 0 ⎜
⎛ Z ( − 1) + j Z L (0) ⎞
= Z0 ⎜ L
⎟
(
1)
(0)
−
+
Z
j
Z
L
⎝ 0
⎠
= ZL
In other words, if the transmission line is precisely one-half
wavelength long, the input impedance is equal to the load
impedance, regardless of Z0 or β.
Zin = Z L
Z0, β
A = λ
2. A = λ
ZL
2
4
If the length of the transmission line is exactly one-quarter
wavelength ( A = λ 4 ), we find that:
βA =
meaning that:
cos β A = cos π 2 = 0
Jim Stiles
2π λ π
=
λ 4 2
and
The Univ. of Kansas
sin β A = sin π 2 = 1
Dept. of EECS
1/26/2005
Transmission Line Input Impedance.doc
5/9
and therefore:
⎛ Z L cos β A + j Z 0 sin β A ⎞
Zin = Z 0 ⎜
⎟
cos
sin
β
β
+
Z
j
Z
A
A
L
⎝ 0
⎠
⎛ Z (0) + j Z 0 (1) ⎞
= Z0 ⎜ L
⎟
(0)
(1)
+
Z
j
Z
L
0
⎝
⎠
(Z )
=
2
0
ZL
In other words, if the transmission line is precisely one-quarter
wavelength long, the input impedance is inversely proportional to
the load impedance.
Think about what this means! Say the load impedance is a short
circuit, such that Z L = 0 . The input impedance at beginning of
the λ 4 transmission line is therefore:
Z )
(
=
2
Zin
0
ZL
Z )
(
=
2
0
0
=∞
Zin = ∞ ! This is an open circuit! The quarter-wave transmission
line transforms a short-circuit into an open-circuit—and vice
versa!
Zin = ∞
Z0 , β
A = λ
Jim Stiles
ZL=0
4
The Univ. of Kansas
Dept. of EECS
1/26/2005
Transmission Line Input Impedance.doc
6/9
3. Z L = Z 0
If the load is numerically equal to the characteristic impedance
of the transmission line (a real value), we find that the input
impedance becomes:
⎛ Z L cos β A + j Z 0 sin β A ⎞
⎟
Z
j
Z
A
+
A
β
β
cos
sin
L
⎝ 0
⎠
Zin = Z 0 ⎜
⎛ Z cos β A + j Z 0 sin β A ⎞
= Z0 ⎜ 0
⎟
+
Z
j
Z
A
A
β
β
cos
sin
0
⎝ 0
⎠
= Z0
In other words, if the load impedance is equal to the
transmission line characteristic impedance, the input impedance
will be likewise be equal to Z0 regardless of the transmission
line length A .
Zin = Z 0
Z0 , β
ZL=Z0
A
4. Z L = j X L
If the load is purely reactive (i.e., the resistive component is
zero), the input impedance is:
Jim Stiles
The Univ. of Kansas
Dept. of EECS
1/26/2005
Transmission Line Input Impedance.doc
7/9
⎛ Z L cos β A + j Z 0 sin β A ⎞
⎟
A
A
cos
sin
+
Z
β
j
Z
β
L
⎝ 0
⎠
Zin = Z 0 ⎜
⎛ j X L cos β A + j Z 0 sin β A ⎞
= Z0 ⎜
⎟
2
⎝ Z 0 cos β A + j X L sin β A ⎠
⎛ X cos β A + Z 0 sin β A ⎞
= j Z0 ⎜ L
⎟
A
A
−
cos
sin
β
β
Z
X
L
⎝ 0
⎠
In other words, if the load is purely reactive, then the input
impedance will likewise be purely reactive, regardless of the
line length A .
Z in = j X in
Z0 , β
ZL=jXL
A
Note that the opposite is not true: even if the load is purely
resistive (ZL = R), the input impedance will be complex (both
resistive and reactive components).
Q: Why is this?
A:
Jim Stiles
The Univ. of Kansas
Dept. of EECS
1/26/2005
Transmission Line Input Impedance.doc
8/9
5. A λ
If the transmission line is electrically small—its length A is
small with respect to signal wavelength λ --we find that:
βA =
and thus:
cos β A = cos 0 = 1
and
2π
λ
A = 2π
A
λ
≈0
sin β A = sin 0 = 0
so that the input impedance is:
⎛ Z L cos β A + j Z 0 sin β A ⎞
⎟
A
A
+
Z
cos
j
Z
sin
β
β
L
⎝ 0
⎠
Zin = Z 0 ⎜
⎛ Z (1) + j Z L (0) ⎞
= Z0 ⎜ L
⎟
⎝ Z 0 (1) + j Z L (0) ⎠
= ZL
In other words, if the transmission line length is much smaller
than a wavelength, the input impedance Zin will always be equal
to the load impedance Z L .
This is the assumption we used in all previous circuits courses
(e.g., EECS 211, 212, 312, 412)! In those courses, we assumed
that the signal frequency ω is relatively low, such that the
signal wavelength λ is very large ( λ A ).
Jim Stiles
The Univ. of Kansas
Dept. of EECS
1/26/2005
Transmission Line Input Impedance.doc
9/9
Note also for this case ( the electrically short transmission
line), the voltage and current at each end of the transmission
line are approximately the same!
V (z = −A) ≈ V (z = 0)
and
I(z = −A) ≈ I (z = 0) if A λ
If A λ , our “wire” behaves exactly as it did in EECS 211 !
Jim Stiles
The Univ. of Kansas
Dept. of EECS