RF Engineering Basic Concepts: the Smith Chart

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RF engineering basic concepts: the Smith chart

F. Caspers CERN, Geneva, Switzerland

Abstract The Smith chart is a very valuable and important tool that facilitates interpre- tation of S-parameter measurements. This paper will give a brief overview on why and more importantly on how to use the chart. Its deﬁnition as well as an introduction on how to navigate inside the chart are illustrated. Use- ful examples show the broad possibilities for use of the chart in a variety of applications.

1 Motivation With the equipment at hand today, it has become rather easy to measure the reﬂection factor Γ even for complicated networks. In the “good old days” though, this was done measuring the electric ﬁeld strength1 at a coaxial measurement line with a slit at different positions in the axial direction (Fig. 1). A

movable electric ﬁeld probe

from generator DUT

Umax

Umin

Fig. 1: Schematic view of a measurement set–up used to determine the reflection coefficient as well as the voltage standing wave ratio of a device under test (DUT) [1] small electric field probe, protruding into the field region of the coaxial line near the outer conductor, was moved along the line. Its signal was picked up and displayed on a microvoltmeter after rectification via a microwave diode. While moving the probe, field maxima and minima as well as their position and spacing could be found. From this the reflection factor Γ and the Voltage Standing Wave Ratio (VSWR or SWR) could be determined using the following definitions:

– Γ is defined as the ratio of the electrical field strength E of the reflected wave over the forward travelling wave: E of reflected wave Γ = . (1) E of forward traveling wave

1The electrical ﬁeld strength was used since it can be measured considerably more easily than the magnetic ﬁeld strength.

95 F. CASPERS

– The VSWR is deﬁned as the ratio of maximum to minimum measured voltage:

Umax 1 + Γ VSWR = = | | . (2) Umin 1 Γ − | | Although today these measurements are far easier to conduct, the definitions of the aforementioned quantities are still valid. Also their importance has not diminished in the field of microwave engineering and so the reflection coefficient as well as the VSWR are still a vital part of the everyday life of a microwave engineer be it for simulations or measurements. A special diagram is widely used to visualize and to facilitate the determination of these quantities. Since it was invented in 1939 by the engineer Phillip Smith, it is simply known as the Smith chart [2].

2 Deﬁnition of the Smith chart The Smith chart provides a graphical representation of Γ that permits the determination of quantities such as the VSWR or the terminating impedance of a device under test (DUT). It uses a bilinear Moebius transformation, projecting the complex impedance plane onto the complex Γ plane:

Z Z0 Γ = − with Z = R + j X. (3) Z + Z0 As can be seen in Fig. 2 the half-plane with positive real part of impedance Z is mapped onto the interior of the unit circle of the Γ plane. For a detailed calculation see Appendix A.

X = Im (Z) Im (Γ)

R = Re (Z) Re (Γ)

Fig. 2: Illustration of the Moebius transform from the complex impedance plane to the Γ plane commonly known as Smith chart

2.1 Properties of the transformation In general, this transformation has two main properties: – generalized circles are transformed into generalized circles (note that a straight line is nothing else than a circle with inﬁnite radius and is therefore mapped as a circle in the Smith chart); – angles are preserved locally.

Figure 3 illustrates how certain basic shapes transform from the impedance to the Γ plane.

96 RF ENGINEERINGBASICCONCEPTS: THE SMITH CHART

X = Im (Z) Im (Γ)

R = Re (Z) Re (Γ)

Fig. 3: Illustration of the transformation of basic shapes from the Z to the Γ plane

2.2 Normalization

The Smith chart is usually normalized to a terminating impedance Z0 (= real): Z z = . (4) Z0 This leads to a simpliﬁcation of the transform:

z 1 1 + Γ Γ = − z = . (5) z + 1 ⇔ 1 Γ − Although Z = 50 Ω is the most common reference impedance (characteristic impedance of coaxial cables) and many applications use this normalization, any other real and positive value is possible. There- fore it is crucial to check the normalization before using any chart. Commonly used charts that map the impedance plane onto the Γ plane always look confusing at ﬁrst, as many circles are depicted (Fig. 4). Keep in mind that all of them can be calculated as shown in Appendix A and that this representation is the same as shown in all previous ﬁgures — it just contains more circles.

2.3 Admittance plane The Moebius transform that generates the Smith chart provides also a mapping of the complex admittance 1 1 plane (Y = Z or normalized y = z ) into the same chart:

y 1 Y Y0 1/Z 1/Z0 Z Z0 z 1 Γ = − = − = − = − = − . (6) −y + 1 −Y + Y0 −1Z/Z + 1/ 0 Z + Z0 z + 1 Using this transformation, the result is the same chart, but mirrored at the centre of the Smith chart (Fig. 5). Often both mappings, the admittance and the impedance plane, are combined into one chart, which looks even more confusing (see last page). For reasons of simplicity all illustrations in this paper will use only the mapping from the impedance to the Γ plane.

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0 = Γ -0.4 0.07 0.08 -130 0.2 0.5 0.42 − i.4: Fig. 0.42

z , 0.08 -0.5 120

1 0.09

z , 0.3

z , 0.41 -120 0.6 ∞ → 0.41 1 =

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0.11 0.6 0.8 0.39

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0.24 0.26 ASPERS 0.25 RF ENGINEERINGBASICCONCEPTS: THE SMITH CHART

Y Y0 Γ = − with Y = G + j B B = Im (Y ) − Y +Y0 Im (Γ)

G = Re (Y ) Re (Γ)

Fig. 5: Mapping of the admittance plane into the Γ plane

Im (Γ)

short circuit open circuit

Re (Γ)

matched load

Fig. 6: Important points in the Smith chart

Concentric circles around the diagram centre represent constant reflection factors (Fig. 7). Their radius is directly proportional to the magnitude of Γ, therefore a radius of 0.5 corresponds to reflection of 3 dB (half of the signal is reflected) whereas the outermost circle (radius = 1) represents full reflection. Therefore matching problems are easily visualized in the Smith chart since a mismatch will lead to a reflection coefficient larger than 0 (Eq. (7)).

1 a 2 Power into the load = forward power - reﬂected power: P = a 2 b 2 = | | 1 Γ 2 . (7) 2 | | − | | 2 − | |

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0.41 0.41 0.09 -0.6 0.09 -0.6

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0.35 0.35

2.0 a 2.0 0.15 0.15

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1.8 1.8

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0.30

20 0.30 20 -4 -4

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30 30

5 5

0.20 0.20 -5

-5 = 0.20 0.20

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0.29 0.29

10 10

0.30 0.30

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20 20

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0.21 0.21 -50 -50

-20 -20 0.21 0.21 0.28 0.28

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0.22 0.22

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0.29 − 0 0

0.27 0.27

0.22 0.22

0.23 0.23

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0.25 0.25 0.28 0.28

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3.2 Adding impedances in series and parallel (shunt) A lumped element with variable impedance connected in series is an example of a simple circuit. The corresponding signature of such a circuit for a variable inductance and a variable capacitor is a circle. De- pending on the type of impedance, this circle is passed through clockwise (inductance) or anticlockwise (Fig. 9). If a lumped element is added in parallel, the situation is the same as for an element connected in

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0.03 1 5 0.03 1 5 20 20 0.47 0.47 0.2 0.2

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10 10

0.24 0.24 0.26 0.26 0.1 0.1

170 170

20 20

0.01 0.01 0.49 0.49

50 50 0.05 0.05 0.25 0.25 0.25 0.25 0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 10 20 50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 10 20 50

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0.00 0.00 0.00 0.00

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0.24 0.24

0.26 0.26

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0.01 0.01

0.49 0.49

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0.27 0.27

0.02 0.02

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-0.15 -0.15

-5 -5

0.22 0.22

0.28 0.28

-160

-160 1 1

-0.2 -0.2

0.03 0.03 -4 -4

0.47

0.47 0.8 0.8

0.21 0.21

-30 -30

0.29 0.29

-150 -150

0.04

0.04 0.6 0.6

-0.3 -0.3 -3 -3

0.46 0.46

0.20 0.20

-40 -40 0.30

0.4 0.30 0.4 0.05 0.05

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-140 -140

0.45 0.45 0.19 0.19

0.31 0.31

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0.06 0.06 -50 -50

-0.5 -0.5

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0.18

0.2 0.18 0.2

0.44 0.44 -130 -130

-1.6 -1.6 0.32 0.32 0.07 0.07

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-60 -60

0.17 0.17

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0.43 0.43

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0.08 0.08

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0.33 0.33

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-70 -70 0.16 0.16

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-110 -110

0.42 0.42

0.09 0.09

-80 -80

0.15 0.15

0.34 0.34

-100 -100

-90 -90

0.10 0.10

0.14 0.14

0.41 0.41

0.11 0.11

0.13 0.13 0.35 0.35 0.12 0.12

0.40 0.40

0.36 0.36

0.39 0.39

0.37 0.37 0.38 0.38

Series L Series C Z Z

Fig. 9: Traces of circuits with variable impedances connected in series

series mirrored by 180◦ (Fig. 10). This corresponds to taking the same points in the admittance mapping. Summarizing both cases, one ends up with a simple rule for navigation in the Smith chart:

For elements connected in series use the circles in the impedance plane. Go clockwise for an added inductance and anticlockwise for an added capacitor. For elements in parallel use the circles in the admittance plane. Go clockwise for an added capacitor and anticlockwise for an added inductance.

This rule can be illustrated as shown in Fig. 11

3.3 Impedance transformation by transmission line The S matrix of an ideal, lossless transmission line of length l is given by

0 e jβl S = − (9) e jβl 0 − 2π where β = λ is the propagation coefficient with the wavelength λ (λ = λ0 for ǫr = 1). When adding a piece of coaxial line, we turn clockwise on the corresponding circle leading to a j2βl transformation of the reflection factor Γload (without line) to the new reflection factor Γin = Γloade− . Graphically speaking, this means that the vector corresponding to Γin is rotated clockwise by an angle of 2βl (Fig. 12).

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hunt 0.08 -0.5 120 0.09

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rcso iciswt aibeipdne once nparallel in connected impedances variable with circuits of Traces 0.9

0.38 0.12 0.00 90

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0.00 0.03 0.4 0.4 0.01 0.49 1.2

0.47 0.48 0.6 0.6 0.13

0.02 180 0.04 0.37 170 0.47 0.8

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0.03 0.14 0.46 160 1

0 1 0.05 0.46 -80

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0.05 -1.4

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4.0

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0.17

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0.09 0.31 0.32

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110 0.29

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0.7 -50

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0.5 10 0.22

0.40 -10 -110 0.29

0.10 -0.7 0.27 0.22

0.23

0.26

0.24 0.25

0.11 0.28

0.6 0.8 0.39 0.23

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C 0.24 0.26 100 0.25 L

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102 0.11

-100 0.9

0.8 0.12 0.38

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0.38 0.12 90

1.0 1

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1.2

0.6 0.6

0.13

0.37 0.8

1.4 0.8 1.2

0.36

0.14

1 1 -80

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0.14 0.48

0.36 1.8 0.01 0.00 0.49 0.03 1.4 0.47 0.48

2.0 0.35 0.02 180 0.04

0.15 -170 170 0.47

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0.3 140 0.44 1.8 0.44

60 0.16 -0.3 0.1

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4.0 0.33 0.4 0.43

2 0.43 0.17 130

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5.0 C

0.42 L

0.33 0.5 0.32 0.42

hunt

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0.18 0.3 0.09

0.18 0.41

-120 0.6

0.31 0.32

10 3 0.41 40

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0.40

0.7 4

20 0.30 -4

0.31 0.5

0.40

30 -110

5 -0.7 0.20 0.10 -5

0.20 -30 50 C

0.29 0.11

0.6 0.8 0.39

0.30

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20 100 -20 50 0.21 -50

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0.21

0.28 0.39 -0.8 0.7

0.11

0.22 -10 -100

0.29 0.9

0.27 0.8 0.12 0.38 0.22

0.23

0.26

0.24 0.25

0.28 -0.9 0.9

0.23 0.38

0.12 90

0.27 1.0 1 0.24

0.26

0.25 -1 0.2 0.2 0.37

-90 0.13 0.4 0.4

1.2

0.6 0.6

0.13 0.37 0.8

1.4 0.8 1.2

0.36

0.14

1 1

-80

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0.36 1.8 1.4

2.0 0.35 0.15

-1.4

0.15

-70

0.35 1.6

0.34

0.16

Z -1.6 3.0

1.8

60 0.16

0.34

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4.0 0.33 2

0.17

-2

0.17 5.0

50 0.33

0.32

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0.18

0.31 0.32

10 3 40

-3

0.19 -40 0.19

20 0.30 -4

0.31

0.20

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0.29

0.30

-10

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0.22

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0.27

0.22

0.23

0.26

0.24

0.25 0.28

0.23

0.27

0.24

0.26 0.25 ASPERS RF h ucini q 1)hsapl tatasiso ielnt of length line transmission a at pole a has (10) Eq. in function The oeeto h meac etrb 180 impedance by original vector impedance the the changing of to movement equivalent is this Again of change a in results length this with line transmission by given is lossless!) (if line a such of impedance The length. its on depending resistor mt hr) digatasiso ieo length of line chart). transmission Smith a adding chart), Smith NIERN AI CONCEPTS BASIC ENGINEERING h euirt fatasiso iei hti eae ihra nidcac,acpctr ra or capacitor, a inductance, an as either behaves it that is line transmission a of peculiarity The Fi .12: g. F Γ ig in 13: . lutaino digatasiso ieo length of line transmission a adding of Illustration Γ =

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( m -0.2 -150 0.06

− 0.05 0.3 0.44 140 0.44 -0.3 0.1 j 0.06 -140 0.07 2 capacitive : 0.4 0.43 Z βl 0.43 130 -0.4 TH 0.07 0.08 -130 0.2 ) 0.5 0.42

Γ = 0.42 0.08 -0.5 120

0.3 0.09 E 0.41 -120 0.6 Z 0.41

0.09 -0.6 S

◦ 0.4 0.10 in 110

load 0.40 0.7 seilywe trigwt hr ici (at circuit short a with starting when Especially . IHCHART MITH 0.5 0.40 -110 0.10 -0.7 =

0.11 0.6 0.8 0.39

e 100

0.39 -0.8 0.7

− 0.11 j

-100 0.9 Γ Z 0.8 0.12 0.38 j

2( -0.9 0.9 l 0

0.38 0.12 oa 90 103

1.0 1

λ/

-1 0.2 0.2 0.37

-90 0.13

2 tan( 0.4

λ 0.4 d λ 4

π 1.2 0.6

i 0.6

0.13 0.37

4 ) 0.8 nductive

1.4 0.8 1.2

0.36

l 0.14 1 1

-80 -1.2 1.6

rnfrsi noa pncrut(at circuit open an into it transforms l 0.14

βl 0.36 1.8

= 1.4 0.35

yafactor a by 2.0

2 0.15

70 -1.4 λ

) 0.15

β -70

0.35 1.6

0.34

l 0.16

Γ -1.6 3.0

. 1.8

60 0.16

z 0.34

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4.0 0.33 2

0.17

ad -2

0.17 5.0 oisadmittance its to

0.33

0.32

e -50 0.18 0.18

−

0.31 0.32

10 3 40

-3

0.19 -40 j 0.19

e( Re

π 4

Γ 20 0.30 − -4

λ/ 0.31

0.20 -5 0.20

-30 50

i 0.29

= 10

l 0.30

1: -10 n

-20 50 0.21

λ -50 2 -20 0.21

4 0.28

oa impedance an to 10

0.22 -10 0.29

Z 0.27

0.22

0.23

− 0.26 0.24

0.25 0.28

Fg 3.Teeoeadn a adding Therefore 13). (Fig. 0.23

0.27

) 0.24 0.26 0.25 Γ load . 1 l /z rteclockwise the or + − nthe in 1 nthe in 1 (10) (11) F. CASPERS

A line that is shorter than λ/4 behaves as an inductance, while a line that is longer acts as a capacitor. Since these properties of transmission lines are used very often, the Smith chart usually has a ruler around its border, where one can read l/λ — it is the parametrization of the outermost circle.

3.4 Examples of different 2-ports

In general, the reflection coefficient when looking through a 2-port Γin is given via the S-matrix of the 2-port and the reflection coefficient of the load Γload:

S12S21Γload Γin = S11 + . (12) 1 S Γload − 22 In general, the outer circle of the Smith chart as well as its real axis are mapped to other circles and lines. In the following three examples different 2-ports are given along with their S-matrix, and their representation in the Smith chart is discussed. For illustration, a simpliﬁed Smith chart consisting of the outermost circle and the real axis only is used for reasons of simplicity.

3.4.1 Transmission line λ/16 The S-matrix of a λ/16 transmission line is

j π 0 e− 8 S = π (13) e j 8 0 − with the resulting reﬂection coefﬁcient

j π Γin = Γloade− 4 . (14)

This corresponds to a rotation of the real axis of the Smith chart by an angle of 45◦ (Fig. 14) and hence a change of the reference plane of the chart (Fig. 14). Consider, for example, a transmission line terminated j π by a short and hence Γload = 1. The resulting reﬂection coefﬁcient is then equal to Γin = e− 4 . −

z = 0 increasing terminating

z = 1 resistor

z = ∞

Fig. 14: Rotation of the reference plane of the Smith chart when adding a transmission line

3.4.2 Attenuator 3 dB The S-matrix of an attenuator is given by

0 √2 S 2 (15) = √2 . " 2 0 #

104 RF ENGINEERINGBASICCONCEPTS: THE SMITH CHART

The resulting reﬂection coefﬁcient is Γ Γ = load . (16) in 2 In the Smith chart, the connection of such an attenuator causes the outermost circle to shrink to a radius of 0.53 (Fig. 15).

z = 0 z = 1 z = ∞

Fig. 15: Illustration of the appearance of an attenuator in the Smith chart

3.4.3 Variable load resistor Adding a variable load resistor (0 < z < ) is the simplest case that can be depicted in the Smith chart. ∞ It means moving through the chart along its real axis (Fig. 16).

z = 0 z = z = 1 ∞

Fig. 16: A variable load resistor in the simpliﬁed Smith chart. Since the impedance has a real part only, the signal remains on the real axis of the Γ plane

4 Advantages of the Smith chart — a summary – The diagram offers a compact and handy representation of all passive impedances4 from 0 to . Impedances with negative real part such as reflection amplifier or any other active device would∞ show up outside the Smith chart. – Impedance mismatch is easily spotted in the chart. 1 – Since the mapping converts impedances or admittances (y = z ) into reflection factors and vice versa, it is particularly interesting for studies in the radio frequency and microwave domain. For

3An attenuation of 3 dB corresponds to a reduction by a factor 2 in power. 4Passive impedances are impedances with positive real part.

105 F. CASPERS

reasons of convenience, electrical quantities are usually expressed in terms of direct or forward waves and reﬂected or backwards waves in these frequency ranges instead of voltages and currents used at lower frequencies. 1 – The transition between impedance and admittance in the chart is particularly easy: Γ(y = z ) = Γ(z). − – Furthermore the reference plane in the Smith chart can be moved very easily by adding a transmission line of proper length (Section 3.4.1). – Many Smith charts have rulers below the complex Γ plane from which a variety of quantities such as the return loss can be determined. For a more detailed discussion see Appendix B.

5 Examples for applications of the Smith chart In this section two practical examples of common problems are given, where the use of the Smith chart greatly facilitates their solution.

5.1 A step in characteristic impedance Consider a junction between two inﬁnitely short cables, one with a characteristic impedance of 50 Ω and the other with 75 Ω (Fig. 17). The waves running into each port are denoted with ai (i = 1, 2) whereas

a1 a2 Junction between a 50 Ω and a 75 Ω cable (inﬁnitely short cables) b1 b2

Fig. 17: Illustration of the junction between a coaxial cable with 50 Ω characteristic impedance and another with 75 Ω characteristic impedance respectively. Infinitely short cables are assumed – only the junction is considered the waves coming out of every point are denoted with bi. The reflection coefficient for port 1 is then calculated as Z Z1 75 50 Γ1 = − = − = 0.2 . (17) Z + Z1 75 + 50 Thus the voltage of the reflected wave at port 1 is 20% of the incident wave (a = a 0.2) and the 2 1 · reflected power at port 1 is 4%5. From conservation of energy, the transmitted power has to be 96% and 2 it follows that b2 = 0.96. A peculiarity here is that the transmitted energy is higher than the energy of the incident wave, since Eincident = 1, Ereflected = 0.2 and therefore Etransmitted = Eincident + Ereflected = 1.2. The transmission coefficient t is thus t = 1 + Γ. Also note that this structure is not symmetric (S = S ), but only 11 6 22 reciprocal (S21 = S12). The visualization of this structure in the Smith chart is easy, since all impedances are real and thus all vectors are located on the real axis (Fig. 18). As stated before, the reflection coefficient is defined with respect to voltages. For currents its sign inverts and thus a positive reflection coefficient in terms of voltage definition becomes negative when defined with respect to current.

5Power is proportional to Γ2 and thus 0.22 = 0.04.

106 RF nFg 19. Fig. in n ftems omncssweeteSihcati sdi h eemnto fteqaiyfactor quality the of task. when: determination given this of the illustration is the used to is dedicated chart is Smith section the This cavity. where a cases of common most the of One the of Determination 5.2 NIERN AI CONCEPTS BASIC ENGINEERING Fi aiycnb ecie yaparallel a by described be can cavity A Fo Z .18: g. oegnrlcs,e.g., case, general more a r G z = 1 = iulzto ftetopr omdb h w also ifrn hrceitcimpedance characteristic different of cables two the by formed two-port the of Visualization 50Ω i.19: Fig. V a b 1 I iulzto ftetopr eitdo h eti h mt chart Smith the in left the on depicted two-port the of Visualization la impedance) (load 1

0.49 0.00 0.01 Z 0.02 Q 0.48 0.00 0.49 0.03 0.47 0.01 0.48 0.02 180 I -170 170 0.47 0.04 = 0.46 0.03

z 160 atr facavity a of factors 0 0.05 0.46 -160 1 -0.05 0.1 0.05 0.04 -0.1 V 0.15 0.45 -0.15 0.2 150

Z 0.45 1+ = -150 -0.2 50+ 0.06

0.05 1 0.3 0.44 140 0.44 -0.3 0.1 0.06 -140 0.07 = : = Z 0.4 0.43 0.43 130 -0.4

0.07 I TH j 0.08 1 -130 0.2 80Ω 0.42

cdn wave ncident 0.5 j

0.42 a 1 a 50 = 0.08 -0.5 120 0.3 0.09 E . 0.41 -120 0.6 6 0.41 + − 0.09 -0.6 0.4 S 0.10 110 0.40 0.7 0.5 CHART MITH Ω 0.40 -110 0.10 -0.7 b b ωL I

0.11 0.6 0.8 0.39

100 1 and = RLC 0.39 -0.8 0.7 0.11 − Z -100 0.9

0.8 0.12 0.38 =

-0.9 0.9

0.38 0.12 1 b 90 1

107 1.0 Z

-1 0.2 0.2 0.37

-90 0.13 0.4 . 0.4 ω 1.2 2

0.00 0.01 0.6 2 0.6 0.13

0.49 0.37 b 1

ici Fg 0 hr h eoac odto is condition resonance the where 20) (Fig. circuit 0.02 C 0.8

0.48 1.4 0.8 1.2 80

0.00 0.03 0.36

0.49 j80 + 50 = 0.14 0.01 1

0.47a 0.48 1 -80

0.02 180 -1.2 1.6

-170 170 0.47 0.04 = 0.14

0.46 0.03 160 0.36 1.8

0 0.05 0.46 a

-160 -0.05 1.4 0.1 0.05 0.35

− -0.1 0.15 2.0 . 0.04 0.15

0.45 -0.15 0.2 150 70 0.45 -1.4

-0.2

-150 0.15 0.06 -70

+0 0.35

0.05 1.6

1 = 0.3 0.34

0.44 140 0.44

0.16

-0.3 b 0.1 -1.6 3.0

0.06 -140 0.07 1.8

60 0.16

0.4 0.43 0.34

-60 -1.8 0.43 0.33

130 4.0

-0.4 2

0.07 a 0.17 . 0.08 -2

0.2 V -130 0.17

0.42 5.0 2

0.5 50

0.33

0.42 0.32

Ω -50 1

0.08 -0.5 120 0.18

= 0.3 0.09 0.18

0.41

0.31 0.32

-120 0.6 10 3

h etr ntecataedepicted are chart the in vectors the , 40

0.41 -3

= 0.19

-40

0.09 -0.6 0.4 0.19

0.10 20 0.30 -4 110 0.31

1 0.40

0.7 5

0.20 -5

0.20 -30 50

0.5 0.29

0.40 -110

0.30

0.10 -0.7 -10

20 -20 50 0.21

a -50

-20 0.21

0.11 0.28

0.6 0.8 0.39

0.22

-10 0.29

0 100 0.27

0.22

0.23

0.39 -0.8 0.26 0.7 0.24

0.25 0.11 0.28 +

− 0.23

-100

0.9

0.27

0.8 0.12 0.24 0.38 0.26 0.25

-0.9 0.9

0.38 0.12 90 b

1.0 1

-1 0.2 0.2 0.37

-90 0.13 0.4 0.4

1.2

0.6 0.6

0.13

0.37 0.8

1.4 0.8 1.2

0.36

0.14

1 1

-80 -1.2 1.6

b 0.14

0.36 1.8 1.4

2.0 0.35 0.15

-1.4

0.15

-70

0.35 1.6

0.34

0.16

-1.6 3.0

1.8

60 0.16

0.34

-60 -1.8

4.0 0.33 2

0.17

-2

0.17 5.0

50 0.33

0.32

-50

0.18

0.31 0.32

10 3 40

-3

0.19 -40 0.19

20 0.30 -4

0.31

0.20 -5

0.20

-30 50

0.29

0.30

-10

-20 50 0.21 -50

-20 0.21 0.28

0.22

-10 0.29

0.27

0.22

0.23

0.26

0.24

0.25 0.28

0.23

(1 0.27

0.24 0.26 0.25 8 ) F. CASPERS

Zinput Zshunt

R L C Vbeam

Fig. 20: The equivalent circuit that can be used to describe a cavity. The transformer is hidden in the coupling of the cavity (Z 1 MΩ, seen by the beam) to the generator (usually Z = 50 Ω) ≈

This leads to the resonance frequency of 1 1 1 ωres = or fres = . (19) √LC 2π √LC

The Impedance Z of such an equivalent circuit is given by 1 Z(ω) = 1 1 . (20) R + jωC + jωL

The 3 dB bandwidth ∆f refers to the points where Re(Z) = Im(Z) which corresponds to two vectors with an argument of 45◦ (Fig. 21) and an impedance of Z( 3dB) = 0.707R = R/√2. | − |

Im (Z) f = f(−3dB) −

f = 0

45◦ f = f(res)

Re (Z) f → ∞

+ f = f( 3dB) − Fig. 21: Schematic drawing of the 3dB bandwidth in the impedance plane

In general, the quality factor Q of a resonant circuit is deﬁned as the ratio of the stored energy W over the energy dissipated in one cycle P : ωW Q = . (21) P

108 RF ENGINEERINGBASICCONCEPTS: THE SMITH CHART

The Q factor for a resonance can be calculated via the 3 dB bandwidth and the resonance frequency: f Q = res . (22) ∆f For a cavity, three different quality factors are deﬁned:

– Q0 (unloaded Q): Q factor of the unperturbed system, i. e., the stand alone cavity; – QL (loaded Q): Q factor of the cavity when connected to generator and measurement circuits; – Qext (external Q): Q factor that describes the degeneration of Q0 due to the generator and diag- nostic impedances.

All these Q factors are connected via a simple relation: 1 1 1 = + . (23) QL Q0 Qext The coupling coefficient β is then defined as Q β = 0 . (24) Qext This coupling coefficient is not to be confused with the propagation coefficient of transmission lines which is also denoted as β. In the Smith chart, a resonant circuit shows up as a circle (Fig. 22, circle shown in the detuned short position). The larger the circle, the stronger the coupling. Three types of coupling are defined depending on the range of beta (= size of the circle, assuming the circle is in the detuned short position):

0.12 0.13 0.14 0.11 0.15 0.10 0.38 Locus0.37 0.36 of Im ( ) = Re ( ) 0.39 0.16 Z Z 90 0.35 0.09 0.40 100 80 0.34 0.17 1 70 0.08 0.41 110 0.9 1.2 0.8 1.4 0.33 0.7 60 0.18 0.42120 1.6 0.07 0.32 0.6 1.8 0.43 50 0.19 0.2 130 2 0.31 0.06 0.5 0.44 f 1 0.20 40 0.30 0.4 0.05 140 0.4 0.45 f3 3 0.21 0.29 0.6 30 0.04 0.3 0.46 150 0.8f5 4 0.22 0.28

0.03 1 5 20 0.47 0.2

160 0.23 0.27

0.02 0.15 0.48

0.24 0.26 0.1

170

0.01 0.49

50 0.05 0.25 0.25 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 10 20 50 -50 0 180

0.00 0.00 f

0 -20

0.24

0.26

-0.05

-10

0.01

0.49

-170

-0.1 0.23

0.27

0.02

-20 0.48

-0.15

-5

0.22

0.28

-160 1

-0.2

0.03

-4

0.47 0.8

0.21

-30

0.29

-150

0.04 0.6 f -0.3

6 -3

0.46

0.20 -40

0.4 0.30

0.05 f

-0.4

-140 4 0.45 0.19

0.31

-2

0.06 -50

-0.5

-1.8

0.2 0.18

0.44 f

-130 2

-1.6 0.32 0.07

-0.6 -60

0.17

-1.4

0.43

-120 -0.7

0.08

-1.2

0.33

-0.8

-70 0.16 -0.9

-1

-110

0.42

0.09

-80

0.15

0.34

-100

-90

0.10

0.14

0.41

0.11

0.13 0.35 0.12

0.40

0.36

0.39 0.37 0.38

Fig. 22: Illustration of how to determine the different Q factors of a cavity in the Smith chart

– Undercritical coupling (0 < β < 1): The radius of resonance circle is smaller than 0.25. Hence the centre of the chart lies outside the circle. – Critical coupling (β = 1): The radius of the resonance circle is exactly 0.25. Hence the circle touches the centre of the chart.

109 F. CASPERS

– Overcritical coupling (1 < β < ): The radius of the resonance circle is larger than 0.25. Hence ∞ the centre of the chart lies inside the circle.

In practice, the circle may be rotated around the origin due to the transmission lines between the resonant circuit and the measurement device. From the different marked frequency points in Fig. 22 the 3 dB bandwidth and thus the quality factors Q0, QL and Qext can be determined as follows:

– The unloaded Q can be determined from f5 and f6. The condition to ﬁnd these points is Re(Z) = Im(Z) with the resonance circle in the detuned short position. – The loaded Q can be determined from f and f . The condition to ﬁnd these points is Im(S ) 1 2 | 11 | → max.

– The external Q can be calculated from f3 and f4. The condition to determine these points is Z = j. ± To determine the points f1 to f6 with a network analyzer, the following steps are applicable:

– f1 and f2: Set the marker format to Re(S11) + j Im(S11) and determine the two points, where Im(S11) = max. – f and f : Set the marker format to Z and ﬁnd the two points where Z = j. 3 4 ± – f5 and f6: Set the marker format to Z and locate the two points where Re(Z) = Im(Z).

Appendices A Transformation of lines with constant real or imaginary part from the impedance plane to the Γ plane This section is dedicated to a detailed calculation of the transformation of coordinate lines form the impedance to the Γ plane. The interested reader is referred to Ref. [3] for a more detailed study. Consider a coordinate system in the complex impedance plane. The real part R of each impedance is assigned to the horizontal axis and the imaginary part X of each impedance to the vertical axis (Fig. A.1). For reasons of simplicity, all impedances used in this calculation are normalized to an

Im(z) 5

4 z+= 3.5 j3 3

Re(z) 1 2 3 4 5 Fig. A.1: The complex impedance plane impedance Z0. This leads to the simpliﬁed transformation between impedance and Γ plane: z 1 Γ = − . (A.1) z + 1

110 RF ENGINEERINGBASICCONCEPTS: THE SMITH CHART

Γ is a complex number itself: Γ = a+jc. Using this identity and substituting z = R+ jX in equation (A.1) one obtains z 1 R + jX 1 Γ = − = − = a + jc . (A.2) z + 1 R + jX + 1 From this the real and the imaginary part of Γ can be calculated in terms of a, c, R and X:

Re: a(R + 1) cX = R 1; (A.3) − − Im: c(R + 1) + aX = X. (A.4)

A.1 Lines with constant real part To consider lines with constant real part, one can extract an expression for X from Eq. (A.4): 1 + R X = c (A.5) 1 a − and substitute this into Eq. (A.3):

R R 1 a2 + c2 2a + − = 0 . (A.6) − 1 + R R + 1 Completing the square, one obtains the equation of a circle:

R 2 1 a + c2 = . (A.7) − 1 + R (1 + R)2 From this equation the following properties can be deduced:

– The centre of each circle lies on the real a axis. – Since R 0, the centre of each circle lies on the positive real a axis. 1+R ≥ 1 – The radius ρ of each circle follows the equation ρ = 2 1. (1+R) ≤ – The maximal radius is 1 for R = 0.

A.1.1 Examples Here the circles for different R values are calculated and depicted graphically to illustrate the transformation from the z to the Γ plane.

0 1 1. R = 0: This leads to the centre coordinates (ca/cc) = 1+0 /0 = (0/0), ρ = 1+0 = 1 0.5 1 1 2 2. R = 0.5: (ca/cc) = 1+0.5 /0 = ( 3 /0), ρ = 1+0.5 = 3 1 1 1 1 3. R = 1: (ca/cc) = 1+1 /0 = ( 2 /0), ρ = 1+1 = 2 2 2 1 1 4. R = 2: (ca/cc) = 1+2 /0 = ( 3 /0), ρ = 1+2 = 3 1 5. R = :(ca/cc) = 1+∞ /0 = (1/0), ρ = 1+ = 0 ∞ ∞ ∞ This leads to the circles depicted in Fig. A.2.

111 F. CASPERS

Im (z) Im (Γ) 2 1

R = 0 R = 1 R = 2 1

1 R = 2 1 R = 0 R = 2 R = 1 R = 2 R = ∞ 1 2Re (z) 1 1 Re (Γ) −

1 −

1 2 − −

Fig. A.2: Lines of constant real part transformed into the Γ plane

A.2 Lines with constant imaginary part To calculate the circles in the Smith chart that correspond to the lines of constant imaginary part in the impedance plane, the formulas (A.3) and (A.4) are used again. Only this time an expression for R and R + 1 is calculated from Eq. (A.3) a + 1 cX 2 cX R = − and 1 + R = − (A.8) 1 a 1 a − − and substituted into Eq. (A.4): c a2 2a + 1 + c2 2 = 0 . (A.9) − − X Completing the square again leads to the equation of a circle:

1 2 1 (a 1)2 + c = . (A.10) − − X X2 Examining this equation, the following properties can be deduced:

– The centre of each circle lies on an axis parallel to the imaginary axis at a distance of 1. – The ﬁrst coordinate of each circle centre is 1. 1 – The second coordinate of each circle centre is X . It can be smaller or bigger than 0 depending on the value of X. – No circle intersects the real a axis. 1 – The radius ρ of each circle is ρ = X . – All circles contain the point (1/0). | |

A.2.1 Examples In the following, examples for different X values are calculated and depicted graphically to illustrate the transformation of the lines with constant imaginary part in the impedance plane to the corresponding circles in the Γ plane.

112 RF ENGINEERINGBASICCONCEPTS: THE SMITH CHART

113 F. CASPERS

Im (z) Im (Γ) 1 1 X = 2 X = 2 X = 1 X = 2 X = 1 X = 1 X = 2 ∞ X = 0 R = Re (z) 1 X = 0 1Re (Γ) 1 − X = 2 X = 2 − − X = 1 − 1 X = 1 X = − − 2 X = 2 1 − −

Fig. A.3: Lines of constant imaginary part transformed into the Γ plane

114 RF ENGINEERINGBASICCONCEPTS: THE SMITH CHART

B Rulers around the Smith chart Some Smith charts provide rulers at the bottom to determine other quantities besides the reflection coefficient such as the return loss, the attenuation, the reflection loss etc. A short instruction on how to use these rulers as well as a specific example for such a set of rulers is given here.

B.1 How to use the rulers First, one has to take the modulus (= distance between the centre of the Smith chart and the point in the chart referring to the impedance in question) of the reﬂection coefﬁcient of an impedance either with a conventional ruler or, better, using a compass. Then refer to the coordinate denoted as CENTRE and go to the left or for the other part of the rulers to the right (except for the lowest line which is marked ORIGIN at the left which is the reference point of this ruler). The value in question can then be read from the corresponding scale.

B.2 Example of a set of rulers A commonly used set of rulers that can be found below the Smith chart is depicted in Fig. B.1. For

Fig. B.1: Example for a set of rulers that can be found underneath the Smith chart

further discussion, this ruler is split along the line marked centre, to a left (Fig. B.2) and a right part (Fig. B.3) since they will be discussed separately for reasons of simplicity. The upper part of the ﬁrst

Fig. B.2: Left part of the rulers depicted in Fig.B.1

ruler in Fig. B.2 is marked SWR which refers to the Voltage Standing Wave Ratio. The range of values is between one and infinity. One is for the matched case (centre of the Smith chart), infinity is for total reflection (boundary of the SC). The upper part is in linear scale, the lower part of this ruler is in dB, noted as dBS (dB referred to Standing Wave Ratio). Example: SWR = 10 corresponds to 20 dBS, SWR = 100 corresponds to 40 dBS (voltage ratios, not power ratios). The second ruler upper part, marked as RTN.LOSS = return loss in dB. This indicates the amount of reflected wave expressed in dB. Thus, in the centre of the Smith chart nothing is reflected and the return loss is infinite. At the boundary we have full reflection, thus a return loss of 0 dB. The lower part

115 F. CASPERS

Fig. B.3: Right part of the rulers depicted in Fig.B.1 of the scale denoted as RFL.COEFF. P = reflection coefficient in terms of POWER (proportional Γ 2). | | There is no reflected power for the matched case (centre of the Smith chart), and a (normalized) reflected power = 1 at the boundary. The third ruler is marked as RFL.COEFF,E or I. With this, the modulus (= absolute value) of the reflection coefficient can be determined in linear scale. Note that since we have the modulus we can refer it both to voltage or current as we have omitted the sign, we just use the modulus. Obviously in the centre the reflection coefficient is zero, while at the boundary it is one. The fourth ruler is the voltage transmission coefficient. Note that the modulus of the voltage (and current) transmission coefficient has a range from zero, i.e., short circuit, to +2 (open = 1+ Γ with Γ =1). | | | | This ruler is only valid for Zload = real, i.e., the case of a step in characteristic impedance of the coaxial line. The upper part of the first ruler in Fig. B.3, denoted as ATTEN. in dB assumes that an attenuator (that may be a lossy line) is measured which itself is terminated by an open or short circuit (full reflection). Thus the wave travels twice through the attenuator (forward and backward). The value of this attenuator can be between zero and some very high number corresponding to the matched case. The lower scale of this ruler displays the same situation just in terms of VSWR. Example: a 10 dB attenuator attenuates the reflected wave by 20 dB going forth and back and we get a reflection coefficient of Γ = 0.1 (= 10% in voltage). The upper part of the second ruler, denoted as RFL.LOSS in dB refers to the reflection loss. This is the loss in the transmitted wave, not to be confused with the return loss referring to the reflected wave. 2 It displays the relation Pt = 1 Γ in dB. Example: If Γ = 1/√2 = 0.707, the transmitted power is − | | | | 50% and thus the loss is 50% = 3 dB. The third ruler (right), marked as TRANSM.COEFF.P refers to the transmitted power as a function 2 of mismatch and displays essentially the relation Pt = 1 Γ . Thus in the centre of the Smith chart − | | there is a full match and all the power is transmitted. At the boundary there is total reflection and for a Γ value of 0.5, for example, 75% of the incident power is transmitted.

References [1] H. Meinke, F.–W. Gundlach, Taschenbuch der Hochfrequenztechnik, Springer Verlag, Berlin, 1992 [2] P. Smith, Electronic Applications of the Smith Chart, Noble Publishing Corporation, 2000 [3] M. Paul, Kreisdiagramme in der Hochfrequenztechnik, R. Oldenburg Verlag, Muenchen, 1969 [4] O. Zinke, H. Brunswig, Lehrbuch der Hochfrequenztechnik, Springer Verlag, Berlin – Heidelberg, 1973

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