How to Use a Smith Chart
Samuel Ng
Department of Electrical and Computer Engineering
The University of British Columbia
samueln@ece.ubc.ca
October 17, 2018
1
Introduction
The Smith chart provides a simple, graphical method for finding various properties of a
transmission line, including Zin , Γ, VSWR and many more. This document provides a basic
overview for Smith chart operations that have been taught in-class so far.
2
Normalization and Denormalization
Operations on the Smith chart are performed with impedances normalized to the transmission line’s characteristic impedance. Starting with an arbitrary wave impedance Zw ∈ C, it
must be normalized to Z0 :
Zw
= r + jx
(1)
zw =
Z0
with zw the normalized load impedance, r the normalized resistance, and x the normalized
reactance. Denormalization takes the following form:
Zw = zw Z0 = R + jX
(2)
with R the resistance and X the reactance. In practice, Zw may be the load impedance ZL ,
input impedance Zin , or the wave impedance anywhere on a transmission line. Note that
the admittance Yw and yw may be acquired by taking the inverse of Zw and zw .
3
Understanding the Smith Chart
As covered in previous notes and videos (check Part 1: Transmission Lines → Applications
→ Smith Chart – Part 1 and 2 on edX), the Smith chart is made up of two families of
circles: one family for various r values as shown in Fig. 1a, and one for various x values as
shown in Fig. 1b. A Smith chart is acquired by superimposing those two families of circles
as shown in Fig. 2. Each point within the Smith chart represent a normalized impedance or
admittance, zw or yw .
1
(b) Circles for various x values.
(a) Circles for various r values.
Figure 1: Two families of circles which make up the Smith chart.
Figure 2: A crude Smith chart with the two families of circles superimposed.
2
4
Indicating Normalized Impedance and Admittance
Note that the included images are vector graphics, you are expected to zoom in to the Smith
chart and inspect fine details due to the space constraint.
Assume that a transmission line with characteristic impedance Z0 = 50 Ω is terminated
=
with a load impedance ZL = (50 + 25j) Ω. The normalized load impedance is zL = 50+25j
50
1 + 0.5j. On the Smith chart, zL is located at the intersection between the r = 1 circle and
the x = 0.5 circle as shown in Fig. 3.
x=0.5
r=1
1+j0.5
Figure 3: Locating zL on the Smith chart.
5
Using the Bottom Bars
The bars beneath the Smith chart provide convenient ways to find the values of various coefficients and ratios. Graphical operations to find the reflection coefficient Γ and transmission
coefficient τ for zL = 1 + 0.5j are shown in Fig. 4.
The following step-by-step procedure describes how to find Γ on the Smith chart:
3
0.5
06
0.
44
70
0.
0
14
(+
jX
/Z
5
5
0.0
4
0.
45
1.4
1.2
1.0
50
55
0.8
1.6
0.8
4.0
15
5.0
0.2
IND
UCT
IVE
20
10
0.25
0.26
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
REFLECTION COEFFICIENT IN DEG
REES
LE OF
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
EES
ANG
8
0.
0.6
10
0.1
0.4
20
0.2
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
1.0
50
0.1
50
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2
20
0.4
)
/Yo
(-jB
CE
AN
PT
CE
-75
US
0
0.
S
4
E
-1
06
IV
0.
CT
DU
IN
R
-70
O
),
o
Z
X/
1.0
1.0
4.0
0.8
9
0.6
5
3.0
0.0
-20
0.1
0.4
1
-110
0.0
9
0
.4
2
0.0
-12
8
0
CAP
0.4
A
C
ITI
3
VE
0.0
RE
7
AC
-1
TA
30
NC
E
C
OM
PO
N
EN
T
(-j
-4
0
0.4
2.0
1.8
0.2
0.6
1.4
1.2
1.0
-70
5
0.14
-80
-4
0
0.15
0.35
-4
6
4
TR S. RF S. A
A W. L. W. TT
N P L L EN
SM EA O O
.
. C K SS [ SS C [dB
O
O (C dB O ]
EF
EF O ]
EF
F,
F, NS
F
E
P T.
or
P)
I
0.12
0.37
0.4
0.39
0.38
RADIALLY SCALED PARAMETERS
∞40 30
0
10
20
1
0.9
5
0.8
3
15
2
0.7
4
0.6
2.5
2
1.8
1.6
1.4
8
6
5
4
3
10
3
4
0.5
0.4
5
6
0.3
7
8
0.2
9
10
0.1
12
14
0.05
1.2 1.1 1
2
20
0.01
1
15
TOWARD LOAD —>
10
7
5
1 1
1.1
30 ∞ 0
0.1
0 0
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 1
0.99
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
CENTER
1
1.1
0.2
1.2
1.3
0.95
4
1.2
1.3 1.4
0.4
0.6
1.4
0.8
1.5
0.9
<— TOWARD GENERATOR
2
1
3
1.6
1
1.8
2
1.5
2
1.6 1.7 1.8 1.9 2
0.8
0.7
3
4
3
4
5
0.6
0.5
10
5
2.5
3
0.4
4
0.3
20
∞
10 15 ∞
6
0.2
10 ∞
5
0.1
SM
R BS B] , P r I
SW d S [d EFF , E o
S
O CO EFF
.L .
N FL CO
RT R FL.
R
20
.C
0.11
-100
-90
0.13
0.36
0
2
TR
A
N
0.9
0.1
0.3
0
3
-5
-35
0.3
0.8
-60
0.7
7
0.1
1.6
-30
2
-55
0.3
0.5
8
0.1
0
-5
-25
31
0.
-65
19
0.
-60
0.3
4
0.
44
0.2
5
0.4
-15
4
0.0
0
-15 -80
5.0
1
0.2
0.2
-30
0.3
0.28
0.22
0.47
-20
-85
8
0.
-10
0.48
10
0.6
0.2
∞100 40
1
0.22
0.28
1.0
1.0
80
0.3
30
9
0.2
RE
AC
TA
75
NC
EC
OM
PO
N
EN
T
20
3.0
0.6
1
0.2
0.4
6
15
0
0.2
0.4
25
0.4
0.3
85
8
2
40
6
0.1
0.3
50
31
0.0
4
0.2
0.1
0.4
3
30
0.
0.0 —> WAVELE
0.49
NGTH
S TOW
ARD
0.0
0.49
AD <—
GEN
ARD LO
ERA
0.48
S TOW
± 180
H
TO
T
170
NG
R—
-170
ELE
V
0.47
>
WA
160
0
<—
6
-90
90
-1
0.1
7
0.3
60
19
R
,O
o)
4
0.
VE
TI
CI
PA
CA
6
0.3
35
2.0
65
3
0.4
0
13
)
/Yo
(+jB
CE
AN
PT
CE
S
SU
0.1
70
40
1.8
1
7
0.0
0.35
80
0.7
20
0.6 60
2
0.4
0.15
0.36
90
0.9
110
1
0.4
8
0.0
0.14
0.37
0.38
0.39
100
0.4
9
0.0
0.13
0.12
0.11
0.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
ORIGIN
Figure 4: Finding Γ and τ .
1. Indicate the location of zL on the Smith chart.
2. Draw a straight line from the center of the Smith chart, intersecting the zL point, and
extending all the way outside of the Smith chart.
3. Read the angle from the ring “ANGLE OF REFLECTION COEFFICIENT IN DEGREES”. In this example it is roughly 77◦ .
4. Measure the distance between the center and the zL point, and compare that distance
with the bar “RFL. COEFF, E or I” beneath the Smith chart to get |Γ|. In this
example it is 0.21.
The procedure for τ is similar, except the straight line in step 2 should originate from
the left side (origin) of the Smith chart, the angle in step 3 should be read off the “ANGLE
4
OF TRANSMISSION COEFFICIENT IN DEGREES” ring, and the magnitude in step 4
should be found with the bar “TRANSM. COEFF, E or I”.
To find the VSWR, measure the distance between the center point and the zL point just
like step 4 in finding Γ, and compare that distance with the bar “SWR”. In this example,
the VSWR is around 1.55 reading from the bar.
6
Rotation to Find zw at Various Points on the Line
Given an arbitrary zw1 in the middle of a transmission line, you may acquire the normalized wave impedance at various points of the transmission line by rotating about the center
starting from the zw1 location. As indicated on the outer two rings of the Smith chart, clockwise rotation represents probing the zw at locations closer to the generator, counterclockwise
rotation represents probing closer to the load.
4
70
0.4
0
(+
jX
/Z
5
14
45
1.2
1.0
50
0.9
55
0.8
1.6
1.8
2.0
65
RE
AC
TA
75
NC
EC
OM
PO
N
EN
T
0.4
5
0.0
0.4
0.8
4.0
15
20
0.2
IND
UCT
IVE
0.28
5.0
0.22
1.0
1.0
80
0.4
0.3
9
0.2
1
0.2
4
20
3.0
0.6
30
10
0.8
0.25
0.26
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
REFLECTION COEFFICIENT IN DEG
REES
LE OF
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
EES
ANG
0.6
10
0.1
0.4
20
0.2
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
1.0
50
0.1
50
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2
20
0.4
1.0
0.8
4
0.0
0
E
IV
CT
DU
IN
-80
1.0
4.0
-15
5.0
-15
-
R
O
),
Zo
X/
2.0
1.8
6
0.0
0.6
1.6
0.7
0.36
1.0
0
0.14
-80
-4
0
0.9
1.2
1.4
0.15
0.35
-5
-70
5
-4
4
0.8
-35
6
0.1
0.3
-90
0.13
0.12
0.37
0.11
-100
0.38
-55
-60
0.3
3
-75
-70
0.2
-30
7
0.1
-60
2
0.3
0.1
0.4
1
-110
0.0
9
0
.4
2
0.0
-12
8
0
CAP
0.4
AC
ITI
3
VE
0.0
RE
7
AC
-1
TA
30
NC
E
C
OM
PO
N
EN
T
(-j
0
14
0.4
0
-65 .5
1
0.3
4
9
0.1
0.4
0
8
0.1
0
-5
-25
5
0.6
3.0
-20
-4
0.3
0.0
0.2
9
0.2
5
0.4
0.28
0.2
1
-30
0.3
0.4
0.22
o)
jB/Y
E (NC
TA
EP
SC
SU
0.8
0.2
-20
-85
10
0.6
-10
0.0
0.2
6
15
0
25
0.4
0.3
85
8
40
6
0.4
0.1
0.3
2
50
1
0.0 —> WAVELE
0.49
NGTH
S TOW
ARD
0.48
0.0
—
0.49
GEN
D LOAD <
ERA
OWAR
0.48
± 180
HS T
TO
170
NGT
R—
-170
ELE
V
0.47
>
WA
160
<—
60
-90
90
-1
0.3
3
30
0.2
0.1
0.47
0.1
7
60
0.3
R
,O
o)
Yo)
jB/
E (+
NC
TA
EP
SC
U
S
VE
TI
CI
PA
CA
6
0.3
4
35
9
0.1
0.0
6
3
0.4
0
13
0
12
0.1
70
40
0.5
7
0.0
0.15
0.35
80
1.4
2
0.4
110
0.36
90
0.7
8
0.0
0.6 60
1
0.4
0.14
0.37
0.38
0.39
100
0.4
0.13
0.12
0.11
0.1
9
0.0
0.4
0.39
RADIALLY SCALED PARAMETERS
Figure 5: Finding zin for a line 0.2λ long terminated with zin .
An example is given in Fig. 5, where the normalized input impedance zin of a 0.2λ
long transmission line (where λ is the wavelength of the driven wave) terminated with a
normalized load zL = 1 + 0.5j is found using the following procedure:
1. Indicate the zL location on the Smith chart.
2. Rotate 0.2λ clockwise (toward the generator) about the center.
5
3. Read-out the final point, that is the normalized input impedance zin ≈ 1.1 − 0.5j.
In the case where zin is given and zL is to be found, the zin location should be indicated on
the Smith chart then rotated counterclockwise (toward the load) about the center in order
to find zL .
When rotating about the center, a few points carry particular properties as indicated in
Fig. 6. Please consult P.339-342 of the textbook if you would like to learn more about the
Vmax and Vmin points.
(+
jX
/Z
45
50
1.2
0.9
0.8
55
1.0
1.6
1.8
2.0
65
0.5
0.0
4
0.4
70
0.4
5
0.0
5
14
0
9
0.3
0.8
4.0
15
20
0.2
IND
UCT
IVE
0.28
5.0
0.22
1.0
1.0
4
0.4
RE
AC
TA
75
NC
EC
OM
PO
N
EN
T
1
10
0.8
0.25
0.26
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
REFLECTION COEFFICIENT IN DEG
REES
LE OF
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
EES
ANG
0.6
0.1
50
20
10
5.0
4.0
3.0
2.0
1.0
4.0
-80
1.0
-15
0.8
0.6
3.0
0
-4
0.4
0.4
2.0
1.8
0.2
0.6
1.4
1.2
0.36
1.0
0.14
-80
-4
0
5
-4
0.15
0.35
0.9
-70
0
6
4
-5
-35
0.1
0.3
0.8
3
0.3
-60
0.7
7
0.1
1.6
-30
2
-90
0.13
0.12
0.37
0.11
-100
0.38
-55
0.3
0.1
0.4
1
-110
0.0
9
0.4
2
0.0
-12
8
0
CAP
0.4
AC
ITI
3
VE
0.0
RE
7
AC
-1
TA
30
NC
E
CO
M
PO
N
EN
T
(-j
0.5
8
0.1
0
-5
-25
1
0.3
-65
9
0.1
-60
0.3
-20
4
0.0
0
5.0
9
0.2
-15
0.28
0.22
0.3
0.2
5
0.4
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.5
0.4
0.3
0.2
0.6
0.8
0.2
0.2
1
-30
o)
jB/Y
5
E (0.0
NC
TA
EP
4
SC
-75
U
0
0.4
S
4
E
6
-1
IV
0.0
CT
DU
IN
R
-70
,O
o)
Z
X/
0.6
-20
-85
0.4
-10
0.2
50
10
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
Vmax
point
20
0.2
0.1
0.4
Vmin
point
20
50
0.0
20
3.0
0.6
0.2
6
15
0
25
0.4
0.2
30
0.4
8
0.2
80
0.1
0.3
2
50
0.3
85
0.2
40
6
0.3
3
30
1
0.4
0.1
7
60
0.3
0.0 —> WAVELE
0.49
NGTH
S TOW
ARD
0.48
0.0
D <—
0.49
GEN
RD LOA
ERA
TOWA
0.48
± 180
TO
THS
170
R—
-170
ENG
L
E
V
0.47
>
WA
160
<—
60
-90
90
-1
4
10
0.1
0.47
6
0.3
35
9
R
,O
o)
VE
TI
CI
PA
CA
)
/Yo
(+jB
CE
AN
PT
CE
S
SU
0.1
70
40
0.1
6
3
0.4
0
13
0.15
0.35
80
1.4
0
12
0
7
0.0
110
0.36
90
0.7
.42
0.6 60
0
8
0.0
.41
0.14
0.37
0.38
0.39
100
0.4
9
0.13
0.12
0.11
0.1
0.0
0.4
0.39
RADIALLY SCALED PARAMETERS
Figure 6: Vmin and Vmax locations on the Smith chart when rotated from zL as the starting
point.
6