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0 . Test 8. Measurement World, July 2001 How does a Smith chart work? A venerable calculation aid retains its allure in a world of lightning—ﬁrst computers and graphical user
interfaces. By Rick Nelson, Senior Technical Editor The Smith chart appeared in 1939 (Ref. 1) as a graph—based method of simplifying the complex math (that is,
calculations involving variables of the form x + jy) needed to describe the characteristics of microwave
components. Although Iculators and computers can now make short work of the problems the Smith chart
was designed to solve, the Smith chart, like other graphical calculation aids (Ref. 2). remains a valuable tool. Smith chart inventor Philip H. Smith
explained in Ref. 1. “From the time I
could operate a slide rule, I’ve been
interested in graphical representations
of mathematical relationships." It's the
insights you can derive from the Smith
chart’s graphical representations that
keep the chart relevant for today's
instrumentation and design . automation applications. On
instruments (Ref. 3). Smith chart
displays can provide an easyto
decipher picture of the effect of
tweaking the settings in a microwave
network; in an EDA program (Figure
1). a Smith chart display can
graphically show the effect of altering
component values. Although the Smith chart can look
imposing. it's nothing more than a
special type of 2D graph. much as
polar and semilog and loglog scales  l ' _ constitute special types of 2D graphs. ,1 ,1" in essence, the Smith chart is a “a   __A...: ._.r"' special plot of the complex 5— parameter 511 (R91 4) Which is Figure 1. RF eledronic—designautomation programs use Smith
equivalent to the complex reﬂection charts to display the results of operations such as Sparameter
coefﬁcient G for single—port microwave simulation. Courtesy of Agilent Technologies. components. Note that in general,
 'ii
r= z: + l 1:=l/le’ . and that I G el‘9 is often expressed as Glu. Note that this latter format omits the absolute—value bars around
magnitude G; in complexnotation formats that include the angle sign (I), the preceding variable or constant is http://www.wcb—ce.com/primers/ﬁles/SmithCharts/smith_charts.htrn 2/21/2005 —_——————_— ————————_
Page 2 of10 assumed to represent magnitude. Figure 2 shows the speciﬁc se of a complex G value 0.6 + j0.3 piotted in
rectangular as well as polar coordinates (0.67/26.6°). Why the circles? That’s all well and good, you may say, but _, 1 l __ where do the Smith chart’s familiar circles , “ “ a.
(shown in gold in Figure 1) come from? The f “
outer circle (corresponding to the dashed circle in Figure 2) is easy—it corresponds to a /
reﬂection coefﬁcient of magnitude 1. Beuse
reﬂectioncoefﬁcient magnitudes must be 1 or
less (you can‘t get more reﬂected energy than
the incident energy you apply), regions outside
this circle have no signiﬁcance for the physical
systems the Smith chart is designed to
represent. It's the other circles (the gold nonccncentric
circles and circle segments in Figure 1) that give \. the Smith chart its particular value in solving  ,
problems and displaying results. As noted  1 , . "“
above, a graph such as Figure 2's provides for 7 convenient plotting of complex reﬂection coefﬁcients, but such plots aren't particularly Figure 2. The Smith chart resides in the complex plane useful by themselves. Typically, you’ll want to f ﬂ   = + I: a: G, At
relate reﬂection coefﬁcients to complex source, 0 re emu" coefﬁqentG Gr 6' IGIeI I I”. line. and load impedances. To that end, the mm A' G ' 0'6 +103 ' 067/265 '
Smith chart transforms the rectangular grid of
the complex impedance plane into a pattern of
circles that can directly overlay the complex
reﬂection caeﬂicr'ent plane of Figure 2. Ref. 5 provides a Quicktime movie of a
rectangular graph of the compleximpedance
plane morphing into the polar plot of the typical
Smith chart. The following section shows the
mathematical derivation that underlies the Smith
chart. In effect. the Smith chart performs the
algebra embodied in equations 2 through 16. The algebra Recall that nonzero reﬂection coeﬂicients arise
when a propagating wave encounters an
impedance mismatch—for example, when a
transmission line having a characteristic
impedance 20 = R0 + jXO is terminated with a
load impedance ZL = RL + j)(,_ D 20. In that case,
the reflection coefﬁcient is Fiure 3. Points of constant resistance form circles on
the complex reﬂectioncoefﬁcient plane. Shown here are
Z the circles for various values of load resistance.
L _ 1
ZL+ZO _Z_L..+]_ Zo "ﬂ Page 3 of 10 In Smith charts, load impedance is olten expressed in the dimensionless normalized form z,_ = rL + xL =ZL/ZO.
so Equation 2 becomes f=z_L.—_1
3L+1 Equation 3 is amenable to additional manipulation to obtain zL in terms of G: sz+ f=zL—l sz—zL=—F—1 zL(l—f)=l+f
_1+F _1—r 3L Explic'rtly stating the real and imaginary parts of the complex variables in Equation 4 yields this equation: . _1+r:+jr: which can be rearranged to clearty illustrate its real and imaginary components. The ﬁrst step is to multiply the numerator and denominator of the righthand side of Equation 5 by the complex conjugate of its
denominator, [(lIDj/ﬂ[(1—/;)+j1;]
a—arur =11f+j2!;'—If =(1131?)+i(21§)
(balm (bah/f I ’L +JxL = thereby enabling a form in which real and imaginary parts are readin identiﬁable and separate: ‘ ....___ ______h Page 4 of 10 l—ﬁ—ﬂ . 2/; (l—nz+zf+’(1—1:)2+If r TL +13% = The real part is then rzl—ﬁ—E
L (Ham? and the imaginary part is _ 2/?
(1—02“? xL You n further manipulate equations 8 and 9 in the hope of getting them into a form that might suggest a
meaningful graphical interpretation. Equation 8, for example, n be altered as follows: _ Eff1?
1—2/?+/f+1f
rLZrL!:+rL/f+rL/?=1Jf1?
rLCZ+jf_2rLj:+TL+r£fii+fii=l
1:2(rL+1)—2rL_/;+ [£201 +1)=1—rL
1:2_EE_£+1;Q=1—rL rL+l rL+l I"L The last line of Equation 10 might look familiar. It’s suggestive of this equation you might remember from high
school analytic geometry: (ac—Hf +(y55')2 =r2 Equation 11 represents a circle plotted in an xy plane with radius rand centered at x = a. y = b. In Equation
10. you can add rL2l(r,_ + 1)2 to each side to convert the G ,terms into a polynomial that you can factor: Page 5 of 10 . 2 2 r +
rL+l rL+1 (l—rLXrLH + if (TL + 1)2 (TL + 1y 2 2 (’1 +1)2 (TL +1): You can then arrange Equation 12 into the form of a circle centered at [rLl(rL + 1). 0] and having a radius of 1!
(rL + 1): 2 2 1
I: — ’1 +(1: —0i =
rL+1 rL+1 . Figure 3 shows the circles for several values of rL. Note that the Q = 0 circle corresponds to the  G  = 1
circle of Figure 2. You can similarly rearrange Equation 9: ._ 2!:
1—2/:+/f+!? xL *ZxL1:+xL/;Q+xL/?=2£ ijf_2ij:+xLIf_2j: =‘xL 2
If—21:+/f——1:=»1
3‘1. xi. adding a constant to make the G iterrns part of a factorable polynomial: 2 2 2
jfu2ﬂ+1+jf——j;+ i =—1+1+ i xL xL 33L . Equation 15 can then be wn'tten as follows: f Page 6 of 10 2
1‘2—21:+1+1;2—1;+ — =——1+1+ —————
xi; 35L 35L representing a circle of radius “XL centered at [1, 1IxL]. Figure 4 shows several of these circles or circle
segments for various values of XL. The segments lying in the top half of the compleximpedance plane
represent inductive reactances; those lying in the bottom half represent pacitive readances. Note that the circle centers all lie on
the blue G r: 1 vertical line. Only the circles segments that lie within the
green  G  =1 circle are relevant for
the Smith chart. Note that XL = 0 along the horizontal axis, which represents a
circle of inﬁnite radius centered at [1,
+y} or [1, —y] in the complex G plane. You can superimpose the circles of
Figure 3 and the segments lying within
the  G  = 1 circle of Figure4 to get
the familiar Smith chart (Figure 5).
Note that the Smith chart circles aren’t
a replacement for the complex
reﬂectioncoefﬁcient plane—in fact,
they exist on the plane, which is
represented in rectangular form by the
gray grid in Figure 5. Now what? The coexistence of complex
impedance and complex—reﬂection
coeﬁicient information on a single
graph allows you to easily determine
how values of one affect the other.
Typically, you might want to know
what complex reﬂection coefﬁcient
would result from connecting a
particular load impedance to a system
having a given characteristic
impedance. Consider. for example, the normalized
load impedance 1 + j2. You n locate
the point representing that value on
the Smith chart at the intersection of
the rL = 1 constantresistance circle
and the xL = 2 constant—reactance
circle segment; the intersection is point
A in Figure B. erh point A plotted,
you can directly read the resulting reﬂection coefﬁcient: G = 0.5 + j0.5, or
G = 0.707!45°. To graphically determine the polar
form, simply divide the length of line
segment CA by the radius of the rt = 0 {30.25 xlgz Figure 4. Values of constant imaginary load impedances x1 make
up circles centered at points along the blue vertil line. The
segments lying in the top half of the compleximpedance plane
represent inductive reactances; those lying in the bottom half
represent capacitive reactances. Only the circle segments within
the green circle have meaning for the Smith chart. circle. You can use a protractor to
measure the angle; many Smith
charts, such as one included in Adobe
PDF format on a CDROM supplied
with Ref. 6, include a protractor scale
around the circumference of the rL = 0 circle. Such a scale is suggested in
yellow in Figure 6. As another example. the complex
impedance value 1 —j1 is located at
point B in Figure 6; at point B, you can
read off the corresponding reﬂection
coefﬁcient 6 = 0.2 —j0.4, or G = 0.45!—
63". (Keep in mind here that this
example describes dimensionless
normalized impedances. For a system
characteristic impedance of 50 V, the
respective values of load impedances
at points A and B would be 50 + j100 V
and 10 —j20 V.) Standing wave ratio Smith charts can help you determine
input impedances as well as relate
load impedances to the reﬂection
coefﬁcient. To understand how that
works, ﬁrst review the operation of
standing waves in a transmission line
with a mismatched load. Such waves
take on a sinusoidal form such as that
shown in Figure Ta. In Figure 7,
standing waves result when a voltage generator of output voltage VG = (1 V)
sin(vt) and source impedance ZG drive
a load impedance ZL through a transmission line having characteristic
impedance 20, where 25 = 20 b ZL
and where angular frequency v
corresponds to wavelength t (Figure
7b). The values shown in Figure 7a result from a reﬂection coefﬁcient of
0.5. i won’t derive the equation that
describes the standing wave that
appears along the transmission line in
Figure 7b; for a derivation, see Ref. 5
or another text covering transmission—
line theory. I'm asking you to accept
that if you could connect an
oscilloscope to various points along
the transmission line, you would obtain
readings illustrated in Figure 7c. Here, probe A is located at a point at
which peak voltage magnitude is
greatest—the peak equals the 1V
peak of the generator output, or
incident voltage, plus the in—phase
peak reﬂected voltage of 0.5 V, so on Page 7 of 10 ﬂat ﬁzl Figure 5. The circles (green) of Figure 3 and the segments (red)
of Figure 4 lying within the  G  = 1 circle combine to form the
Smith chart, which lies within the complex reﬂectioncoefﬁcient
(G) plane, sh0wn in rectangular form by the gray grid. ISE‘U Figure 6. With a Smith chart. you can plot impedance values
using the red and green circles and circle segments and then
read reﬂectioncoefﬁcient values from the gray gn'd. Many Smith
charts include a scale (yellow) around their circumference that
lets you read angle of reﬂection coefﬁcient. Page 8 of 10 your oscilloscope you would see a
timevarying sine wave of 1.5V peak
amplitude (the gray trace in Figure 7c).
At point C, however, which is lecated
onequarter of a wavelength (II4)
closer to the load. the reﬂected voltage
is 180° out of phase with the incident
voltage and subtracts from the incident
voltage. so peak magnitude is the 1—V
incident voltage minus the 0.5V
reﬂected voltage, or 0.5 V, and you bl 20 = 26 =
would see the red trace. At .
intermediate points, you'll see peak
values between 0.5 and 1.5 V; at B
(offset Ila from the first peak) in Figure
To, for example, you'il ﬁnd a peak
magnitude of 1 V. Note that the
standing wave repeats every half
wavelength (U2) along the
transmission line. E.
Total voltage {V}
O .'
‘3‘ c:
a?
a.
i"
u
w
r
x
N U 1.5 sintmt) ail 1.5 v
The ratio of the maximum to minimum values of peak voltage amplitude ‘ .1
measured along a standing wave is ﬁt
the standing wave ratio SWR. For the K
Figure 7 system. SWR = 15105 = 3.
Note that 1+f
1—f it 1, rm not proving Equation 17 here, but Figure 7. (a) Standing waves, which repeat for every half SWR = O
.__.‘_m_____w.mm___mm substituting the 0.5 reﬂection wavelength of the source voltage, arise when (b) a matched coefﬁcient used in the Figure 7 generator and transmission line drive an unmatched load. (c) example into Equation 17 does Timevarying sine waves of different peak magnitudes appear at provide the desired result of 3, different distances along the transmission line as a function of
wavelength. The relationship between G and SWR suggests that SWR might have a place within the Smith chart. and indeed it does. In fact. lculations
involving SWR ﬁrst prompted Smith to invent his chart. "By taking advantage of the repetitive nature of the
impedance variation along a transmission line and its relation to the standingwave amplitude ratio and wave
position, I devised a rectangular impedance chart in which standing—wave ratios were represented by circles."
he explained in Ref. 1. Figure 8 helps to explain how those
circles arise. In Figure 8, point L
represents a normalized load
impedance 2L = 2.5 —j1 = 0.5118” (I
chose that particular angle primarily to
avoid the need for you to interpolate
between resistance and reactance
circles to verify the results). The
relationship of reﬂection coefficient
and SWR depends only on the
reﬂection coefficient magnitude and
not on its phase. If point L corresponds
to  G t = 0.5 and SWR = 3. then any
point in the complex reﬂection
coefﬁcient plane equidistant from the
origin must also correspond to l G l = Page 9 of 10 0.5 and SWR = 3, and a circle
centered at the origin and whose
radius is the length of line segment 0L
represents a locus of constantSWR
points. (Note that the SWR = 3 circle in
Figure 8 shares a tangent line with the rL = 3 circle at the real axis; this relationship between SWR and :1
circles holds for all values of SWR.) Using the standingwave circle, you
can determine input impedanoes
looking into any portion of a
transmission line such as Figure 7’s if
you know the load impedance. Figure
7, for instance, shows an input impedance Z,” to be measured at a distance l0 from the load (toward the generator). Assume that the load
impedance is as given by point L in
Figure 3. Then, assume that to is 0.139 wavelengths. (Again, I chose this
value to avoid interpolation.) One trip  
around the Smith chart is equivalent to f, u _5. .‘" 5W," o‘g‘ " 037 = W39
traversing onehalf wavelength along a \\ Eve "'9‘ ‘ mm" genera”!
standing wave, and Smith charts often
include 0 to 0.5wavelength scales
around their circumferences (usually
tying outside the reﬂectioncoefﬁcient
angle scale previously discussed). Figure 8. Constant SWR circles are centered at the origin of the
complex reﬂectioncoefﬁcient plane. The yellow scale represents
per—unit wavelength movements away from the load toward a
generator along a transmission line. Such a scale is show in yellow in Figure 8, where clockwise movement corresponds to movement away from
the load and toward the generator (some charts also include a c0unterclockwise scale for movement toward
the load). Using that Scale, you can rotate the red vector intersecting point L clockwise for 0.139 wavelengths, ending
up at the blue vector. That vector intersects the SWR = 3 circle at point I, at which you can read Figure 7’s input impedance 2,". Point  lies at the intersection of the 0.45 resistance circle and —0.5 reactanoe circle, so
2m = 0.45 j0.5. Still going strong The Smith chart remains an invaluable aid for a variety of applications, from the design of impedance
matching networks to the determination of the feedpoint impedance of an antenna based on a measurement
taken at the input of a random length of transmission line (Ref. 7). Whether you are using it as a
computational tool—as its inventor intended—or as the graphical interface to instrumentation or EDA
software, it provides insights to circuit operation not available from the raw data that number crunching
machines can produce from microwave component measurements and simulations. T&llllw References 1. “Philip Smith, Electrical Engineer," an oral history conducted in 1973 by Frank A. Polkinghorn, IEEE
History Center, Rutgers University, New Brunswick, NJ. www.ieee.orglorganizationslhistory_centerl
oral_histoﬁesltranscriptslsmith3.htrnl. 2. See, for example, "Parallel Resistance Nomograph," www.trrmorld.comlreferencelnom_
oddsendshtm. 3. Nelson, Rick, “Vector network analyzers grade microwave components,” Test & Measurement Wortd, April
2001. p. 16. www.tmwor1d.coml Page 10 oflO articles/2001m_VNAs.htm. 4. Nelson. Rick, “What are Sparameters, anyway?” Test & Measurement World. February 2001. p. 23.
www.trnwond.comlartidesl2001l02_sparameters.htm. 5. Agilent Technologies provides an interactive demonstration you can download. www.tm.agilent.
comldatalstatidengttmolNotesﬁnteradivel an951Iclassesﬁmatch.html. 6. Ulaby, Fawwa T., Fundamentals of Applied Electromagnetics, 1999 ed.. PrenticeHall. Upper Saddle
River, NJ. 1999. 7. Straw, R. Dean. ed., "Smith Chart Calculations," Chapter 28. The ARRL Antenna Book, 18th ed., The
American Radio Relay League. Newington. CT, 1997—98. Rick Nelson received a BSEE degree from Penn State University. He has six years experience designing
electronic industrialcontrol systems. A member of the IEEE, he has served as the managing editor of EDN,
and he became a senior technical editor at T&MW in 1‘ 998. Email: melson@Lrnwoirtd.com. Copyright 2001, Tesl 3. Measurement World. Published by Cahners Business Information. Newton, MA. Download a Smith chart from WebEE by clicking here 3D: I +3» LY The Complete Smith Chart
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