The Smith Chart

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 cart are illustrated. Useful examples show the broad possibilities for usage 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 of a coaxial measurement line with a slit at different positions in axial direction (Fig. 1). A movable electric field probe

from generator DUT

Umax

Umin

Fig. 1: Schematic view of a measurement setup 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 micro–voltmeter 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 following definitions:

– Γ is defined as the ratio of the electrical field strength E of the reflected wave over the forward traveling wave: E of reflected wave Γ= (1) E of forward traveling wave – The VSWR is defined as the ratio of maximum to minimum measured voltage:

Umax 1+ Γ VSWR = = | | (2) Umin 1 Γ −| | 1The electrical field strength was used since it can be measured considerably easier than the magnetic field strength. Although today this 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 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

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

2 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, there is any other real and positive value possible. Therefore 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 ﬁgures before — 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 −1/Z + 1/Z0 Z + Z0 z + 1 Using this transformation, the result is the same chart, only mirrored at the center 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 = xml o omnyue mt chart. Smith used commonly a for Example 0 = 0.09 -0.6 0.4 0.10 110 0.40 0.7 0.5 0.40 -110 0.10 -0.7

0.11 0.6 0.8 0.39

100

0.39 -0.8 0.7 0.11

-100 0.9

0.8 0.12 0.38

-0.9 0.9

0.38 0.12 90 4

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 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

mgnr ato h impedance. the of part imaginary

1.8

60 0.16

0.34

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4.0 0.33 2

0.17 -2

ato h impedance. the of part

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50 0.33

0.32

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0.31

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10 3 trsatn ihthe with starting pter 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

0.27

0.24 0.26 0.25 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 center 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 (equation (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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0.3 1 + − 0.3 0.44 140 0.44 0.44 140 0.44 -0.3 0.1 -0.3 0.1 0.06 -140 0.07 0.06 -140 0.07 0.4 0.43 0.4 0.43 0.43 130 1 0.43 130 -0.4 2 -0.4 0.07 0.08 0.07 0.08 -130 0.2 -130 0.2 0.5 0.42 0.5 0.42 sue,weepwr= power where used, is =

0.42 - Reﬂection 0.42 0.08 -0.5 120 0.08 -0.5 120 0.3 0.09 0.3 0.09 0.41 0.41 -120 0.6 -120 0.6 1 0.41 + 1 0.41 0.09 -0.6 0.4 0.09 -0.6 0.4 0.10 0.10 110 110 0.40 0.40 0.7 − 0.7 0.5 0.5 0.40 -110 0.40 -110 0.10 -0.7 0.10 -0.7 Reﬂection Impedance

| 0.11 0.11 z 0.6 0.8 0.39 z 0.6 0.8 0.39 a

100 100

| 0.39 -0.8 0.7 0.39 -0.8 0.7 0.11 0.11 2 -100 -100 = 0.9 0.9

0.12 0.12 y 0.8 0.8 0.38 0.38 hs ovnin aen mato –aaeesbtte are they but S–parameters on impact no have conventions These .

-0.9 0.9 -0.9 0.9

0.38 0.38 0.12 0.12 90 90 1 1 6

1.0 1.0

0.2 0.2 -1

-1 0.2 0.2

0.37 0.37

-90 -90 0.13 0.13

0.4 0.4 0.4 0.4

1.2 1.2

0.6 0.6 z

0.6 z 0.6

0.13 0.13

0.37 0.37

0.8 0.8 1.2

1.4 0.8 1.2 1.4 0.8

80 80

0.36 0.36

0.14 0.14

1 1 − 1 1 + 1

z 1

-80 -80

-1.2

-1.2 1.6 1.6

0.14 0.14 0.36

0.36 1.8 1.8

1.4 1.4

0.35 0.35

2.0 1 2.0 0.15 0.15

70 70

-1.4 -1.4

0.15 0.15

-70 -70

0.35 0.35 1.6 1.6

0.34 0.34

0.16 0.16

-1.6

-1.6 3.0 3.0

1.8 1.8

60 60 0.16 0.16

or 0.34 0.34

-60 -60 -1.8 -1.8 0.33 0.33

4.0 | 4.0

2 2

0.17 0.17

a -2 -2

0.17

0.17 5.0 2 5.0

| 50 50 0.33

0.33 Γ

0.32 0.32 Γ(

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-50 -50

0.18 0.18

0.18 0.18 Furthermore .

ln sue,oecnesl s tto it use easily can one used, is plane

0.31 0.31

0.32 0.32 3

10 3 10 40 40

-3

-3 z

0.19 0.19 -40 -40 0.19 0.19

4 4

0.30

20 0.30 20

-4 -4

0.31

0.31 = )

30 30

5 5

0.20 0.20 -5 -5

0.20 0.20

-30 -30 50 50

0.29 0.29

10 10

0.30 0.30

-10 -10

20 20

-20 -20 50 50

0.21 0.21 -50 -50

-20 -20 0.21 0.21 0.28 0.28

10 10

0.22 0.22

-10 -10 0.29 0.29

0 0

0.27 0.27

0.22 0.22

0.23 0.23

0.26 0.26

0.24 0.24

0.25 0.25 − 0.28 0.28

0.23 0.23

0.27 0.27

0.24 0.24

0.26 0.26 0.25 h mt chart. Smith the n 0.25 Γ( | | | | | Γ Γ Γ Γ Γ y180 by e factor. z | | | | | ) 0 = 0.25 = 0.5 = 0.75 = 1 = (1 | − ◦ Γ Fg 8). (Fig. | 2 ) corresponds (8) 3.2 Adding impedances in series and parallel (shunt) A lumped element with variable impedance connected in series as an example of a simple circuit. The corresponding signature of such a circuit for a variable inductance and a variable capacitor is a circle. Depending on the type of the impedance this circle is passed through clockwise (inductance) or coun- terclockwise (Fig. 9). If a lumped element is added in parallel, the situation is the same as it is for an

0.12 0.13 0.14 0.12 0.13 0.14 0.11 0.15 0.11 0.15 0.10 0.38 0.37 0.10 0.38 0.37 0.39 0.36 0.16 0.39 0.36 0.16 90 0.35 90 0.35 0.09 0.40 100 80 0.09 0.40 100 80 0.34 0.17 0.34 0.17 1 70 1 70 0.08 0.41 110 0.9 1.2 0.08 0.41 110 0.9 1.2 0.8 1.4 0.33 0.8 1.4 0.33 0.7 60 0.18 0.7 60 0.18 0.42120 1.6 0.42120 1.6 0.07 0.32 0.07 0.32 0.6 1.8 0.6 1.8 0.43 50 0.19 0.43 50 0.19 0.2 0.2 130 2 0.31 130 2 0.31 0.06 0.5 0.06 0.5

0.44 0.20 0.44 0.20 40 40 0.30 0.30 0.4 0.4 0.05 140 0.4 0.05 140 0.4 0.45 0.45 3 0.21 3 0.21 0.29 0.29 0.6 30 0.6 30 0.04 0.3 0.04 0.3 0.46 150 0.46 150 0.8 4 0.22 0.8 4 0.22 0.28 0.28

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

160 0.23 160 0.23 0.27 0.27

0.02 0.15 0.02 0.15 0.48 0.48

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

-50 -50 0 0 180 180

0.00 0.00 0.00 0.00

-20 -20

0.24 0.24

0.26 0.26

-0.05 -0.05

-10 -10

0.01 0.01

0.49 0.49

-170 -170

-0.1 -0.1 0.23 0.23

0.27 0.27

0.02 0.02

-20 -20 0.48 0.48

-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

-0.4 -0.4

-140 -140

0.45 0.45 0.19 0.19

0.31 0.31

-2 -2

0.06 0.06 -50 -50

-0.5 -0.5

-1.8 -1.8

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

-0.6 -0.6

-60 -60

0.17 0.17

-1.4 -1.4

0.43 0.43

-120 -120

-0.7 -0.7

0.08 0.08

-1.2 -1.2

0.33 0.33

-0.8 -0.8

-70 -70 0.16 0.16

-0.9 -0.9

-1 -1

-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. element connected in 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:

1 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 spoken, this means that the vector corresponding to Γin is rotated clockwise by an angle of 2βl (Fig. 12).

7 0.00 0.01 0.49 0.02 0.48 0.00 0.49 0.03 0.47 0.01 0.48 0.02 180 -170 170 0.47 0.04 0.46 0.03 160 0 0.05 0.46 -160 -0.05 0.1 0.05 0.04 -0.1 0.15 0.45 -0.15 0.2 150 0.45 -150 -0.2 0.05 0.06 0.3 0.44 140 0.44 -0.3

i.11: Fig. 0.1 0.06 -140 0.07 0.4 0.43 0.43 130 -0.4 0.07 0.2 0.08 Shunt -130 0.5 0.42 0.42 0.08 -0.5 120 i.10: Fig. 0.3 0.09 0.41 -120 0.6

0.41 0.09 -0.6 0.4 0.10 110 0.40 0.7

lutaino aiaini h mt hr hnadn lu adding when chart Smith the in navigation of Illustration 0.5 0.40 -110 0.10 -0.7 L 0.11 0.6 0.8 0.39

100

0.39 -0.8 0.7 0.11

-100 0.9

0.8 0.12 0.38 rcso iciswt aibeipdne once npa in connected impedances variable with circuits of Traces -0.9 0.9

0.38 0.12 0.00 90

0.01 1

0.49 0.02 1.0

-1 0.2 0.2 0.37 -90 0.48 0.13

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

-170 1.4 0.8 1.2 80 0.36

0.03 0.14 0.46 160 1

0 1 0.05 0.46 -80

-160 -0.05 0.1 0.05 -1.2 1.6

-0.1 0.15 0.14 0.04 0.36

-0.15 150 1.8

0.45 0.2 1.4 0.45 0.35 2.0

-150 -0.2 0.15

0.06 70

0.05 -1.4

0.15

0.3 -70

0.35 0.44 140 0.44 1.6

-0.3 0.1 0.34

0.16

0.06 -140 0.07 Z -1.6 3.0

1.8 60

0.4 0.43 0.16

0.34

-60 0.43 -1.8 130 0.33

4.0

-0.4 2

0.17

0.07 0.08 -2

0.2 0.17

-130 5.0

50 0.42 0.33

0.5 0.32

0.42 -50

0.18

0.08 -0.5 120 0.18

Series

0.09 0.31

0.32

0.3 10 3

Shunt 40

0.41 -3

0.19

-40 -120 0.6 0.19

0.41 20 0.30 -4 0.31

0.09 -0.6 5

0.20 0.4 -5 0.20 -30 50

0.10

110 0.29

0.40 10 0.30

-10

-20 50 0.21

0.7 -50

-20 0.21 0.28

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

0.27

C 0.24 0.26 100 0.25 L

0.39 -0.8 0.7 0.11

-100 0.9

0.8 0.12 0.38

-0.9 0.9

8 0.38 0.12 90

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 0.00 0.01 0.49 0.02

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

Shunt Series 70 0.46 0.03 160

-1.4 0 0.05 0.46

-160 -0.05 0.1 0.05

0.15

-70 0.04 -0.1 0.15 0.35

1.6 0.45 -0.15 0.2 150 0.45 0.34 -0.2

-150 0.06

0.16 0.05 -1.6 3.0

0.3 140 0.44 1.8 0.44

60 0.16 -0.3 0.1

-140 0.07 0.34 0.06

-60 -1.8

4.0 0.33 0.4 0.43

2 0.43 0.17 130

-0.4

-2 0.07 Shunt 0.17 0.08

5.0 -130 0.2

C 0.42 L

0.33 0.5

0.32 0.42

-50 0.08 -0.5 120

0.18 0.3 0.09

0.18 0.41

-120 0.6

0.31 0.32

10 3 0.41 40

-3 0.09 -0.6 0.4 0.19 -40 0.19 0.10 110

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 C -30 50

0.29 0.11

0.6 0.8 0.39

0.30

-10

20 100 -20 50 0.21 -50

-20

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 pdelements. mped

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 -1.2 1.6

rallel. 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

Z -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

0.27

0.24 0.26 0.25 h ucini qain(0 a oea rnmsinln eghof length line transmission a at pole a has (10) equation in function The gi hsi qiaett hnigteoiia impedance original the changing to equivalent is this Again oeeto h meac etrb 180 by vector impedance the of movement eitrdpnigo t egh h meac fsc ie(flossless!) (if line a such of impedance The length. its on depending resistor mt hr) digatasiso ieo length of line transmission a adding chart), Smith rnmsinln ihti eghrslsi hneof change a in results length this with line transmission chart). h euirt fatasiso iei,ta tbhvsete sa inductanc an as either behaves it that is, line transmission a of peculiarity The i.12: Fig. i.13: Fig. Γ in lutaino digatasiso ieo length of line transmission a adding of Illustration Γ =

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j 0.4 0.43 Z 2 0.43 130 -0.4

βl 0.07 0.08 -130 0.2 ) 0.5 0.42 0.42 0.08 -0.5 120 Γ = 0.3 0.09 0.41 -120 0.6

0.41

Z 0.09 -0.6 0.4 ◦ 0.10 110 0.40 0.7 seilywe trigwt hr ici (at circuit short a with starting when Especially . in

load 0.5 0.40 -110 0.10 -0.7

0.11 0.6 0.8 0.39 =

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− 0.8 0.12 0.38 j λ 4 Z -0.9 0.9 load j

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yafactor a by 2.0 2 0.15

Γ = 70

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) 0.34

0.16

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1.8

60 0.16

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load 0.17

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0.18 0.18

0.31 0.32

10 3 40

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e( Re 4

Γ 20 0.30

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j 0.31

π 5

0.20 -5 0.20

-30 50

in 0.29

0.30

1: -10

-20 50 0.21

λ -50 2 -20 0.21

= 0.28

oa impedance. an to

0.22 -10

0.29

λ 0.27

Z 0.22

0.23

0.26

0.24

0.25 0.28

0.23

− 0.27

Fg 3.Teeoeadn a adding Therefore 13). (Fig. 0.24 ) 0.26 0.25 sgvnby given is Γ load l . z 1 ,acpctro a or capacitor a e, rteclockwise the or + nteSmith the in 1 − nthe in 1 (10) (11) λ A line that is shorter than 4 behaves as an inductance, while a line that is longer acts a capacitor. Since these properties of transmission lines are used very often, the Smith chart usually has a ruler around l its border, where one can read 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 of 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 simplicity reasons.

λ 3.4.1 Transmission line 16 λ The S–matrix of a 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 resultinig reﬂection coefﬁcient is then equal to Γin = e− 4 . −

z = 0 increasing terminating

resistor z = 1

z = ∞

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

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

0 √2 S 2 (15) = √2 " 2 0 # The resulting reﬂection coefﬁcient is Γ Γ = load (16) in 2

10 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 = 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 convenience reasons 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.

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

11 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 (3.4.1). – Many Smith charts have rulers below the complex Γ plane from which a variety of quantities such as e. g. 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 facilitates their solving very much.

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

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

Fig. 17: Illustration of the junction between a coaxial cable with 50Ω characteristic impedance and another with 75Ω characteristic impedance respectively. Infinetly 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 (a2 = a1 0.2) and the 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 thus is 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 considerably 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 gets negative when defined with respect to current.

For a more general case, e.g. Z1 = 50 Ω and Z2 = 50 + j80 Ω, the vectors in the chart are depicted in Fig. 19.

5Power is proportional to Γ2 and thus 0.22 = 0.04

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0.49 0.00 0.01 Z

ω 0.02 Q 0.48 0.00 0.49 0.03 0.47 0.01 0.48 res 0.02 180 I -170 170 0.47 0.04 fcoso cavity a of –factors 50+ = 0.46 0.03 160 z 0 0.05 0.46

-160 1 -0.05 0.1 0.05 0.04 -0.1 V 0.15 -0.15 150 = 0.45 0.2 0.45 Z

1+ = -0.2 -150 0.06

0.05 1 0.3 0.44 140 0.44 -0.3 0.1 -140 0.07 √ 0.06 =

= 0.4 0.43 0.43 130 -0.4 LC 0.07 wave Incident j

1 0.08 -130 0.2 80Ω 0.5 0.42 j

0.42 a a 1 0.08 -0.5 120 0.3 0.09

. 0.41 -120 0.6 6 0.41 + − 0.09 -0.6 0.4 0.10 110 0.40 0.7 0.5 0.40 -110

0.10 -0.7 b b I

0.11 0.6 0.8 0.39 ωL

100 1 1 = RLC 0.39 -0.8 0.7 0.11 − Z -100 0.9 or 0.8 0.12 0.38

-0.9 0.9

0.38 = 0.12 13 b 90

1.0 1

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0.00 0.01 0.6 2 0.6 0.13

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

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80 0.36

1 0.00 0.03 0.49 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 = 0.14

0.46 0.03 160 0.36 1.8

0 0.05 0.46 a 1.4 -160 -0.05 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

-150 -0.2 0.15

0.06 -70

0.35 0.05 1.6 1 =

res 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

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0.43 130 0.33

= 4.0

-0.4 2 a 0.17

0.07 . 0.08 -2

0.2 V -130 0.17

0.42 5.0 2

0.5 50 0.33

0.42 0.32

-50 -0.5 1 120 0.18

2 0.08

1 = 0.3 0.09 0.18

1 0.41

0.31 0.32

π -120 0.6 10 3 40

0.41 -3 = 0.19

-40

0.09 -0.6 0.4 0.19

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-4

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√ 0.40

0.7 5

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0.20 rn hrceitcimpedance. characteristic erent -30 50

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LC -50 a

1 -20 0.21

0.11 0.28

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0.29

0 100 0.27

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0.23

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− 0.23

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0.8 0.12 0.24 0.38 0.26 0.25 .

-0.9 0.9

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1.0 1

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chart. h 1.4

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h ult factor quality the 1.8

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0.17

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10 3 40

-3

0.19 -40 0.19

20 0.30 -4

0.31

0.20 -5

0.20

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0.29

0.30

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-20 50 0.21 -50

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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

(18) 0.27 (19)

0.24 0.26 0.25 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 = 50Ω). ≈

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

The 3dB 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 3 dB bandwidth in the Impedance plane.

In general, the quality factor Q of a resonant circuit is deﬁned as the ration of the stored energy W over the energy dissipated in one cycle P : ωW Q = (21) P The Q factor for a resonance can be calculated via the 3dB bandwidth and the resonance frequency: f Q = res (22) ∆f For a cavity three different quality factors are deﬁned:

14 – 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 is, 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 center 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 center of the chart. – Overcritical coupling (1 <β< ): The radius of the resonance circle is larger than 0.25. Hence the center 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 3dB bandwidth and thus the quality factors Q0, QL and Qext can be determined as follows:

15 – 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 f1 and f2. The condition to ﬁnd these points is Im(S11) 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)

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] O. Zinke, H. Brunswig, Lehrbuch der Hochfrequenztechnik, Springer Verlag, Berlin – Heidelberg, 1973 [4] M. Paul, Kreisdiagramme in der Hochfrequenztechnik, R. Oldenburg Verlag, Muenchen, 1969

16 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 t the Γ plane. The interested reader is furthermore referred to [4] 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 vertical axis and the imaginary part X of each impedance to the horizontal axis (Fig. A.1) For reasons of simplicity, all impedances used in this calculation are normalized to an impedance Im(z) 5

4 z =3.5+j3 3

Re(z) 1 2 3 4 5 Fig. A.1: The complex impedance plane.

Z0. This leads to the simpliﬁed transformation between impedance and Γ plane: z 1 Γ= − (A.1) z + 1 Γ is a complex number itself: Γ = a+jc. Using this identity and substitutingz = 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 equation (A.4) 1+ R X = c (A.5) 1 a − and substitute this into equation (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 following properties can be deduced:

17 – The center of each circle lies on the real a – axis. – Since R 0, the center 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 center 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.

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 formula (A.3) a + 1 cX 2 cX R = − and 1+ R = − (A.8) 1 a 1 a − − and substituted into equation (A.4): c a2 2a +1+ c2 2 = 0 (A.9) − − X

18 Completing the square again leads to an 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 center of each circle lies on an axis parallel to the imaginary axis in a distance of 1. – The ﬁrst coordinate of each circle center is 1. 1 – The second coordinate of each circle center 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.

19 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.

20 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 of how to use these rulers as well as a specific example for such a set of rulers is given here.

B.1 Instruction how to use the rulers First, one has to take the modulus (= distance between the center 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 to CENTER 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 center, 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 (center 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 center 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

21 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 (center of the Smith chart), and a (normalized) reflected| | power = 1 at the boundary. The third ruler is marked as RFL.COEFF,E or I. Whit 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 center 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 is travelling 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 10dB attenuator attenuates the reflected wave by 20dB 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 is referring to the reflection loss. This is the loss in the transmitted wave, not to be confounded with the return loss referring to the reflected 2 wave. 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%−| = 3dB.| | | Third ruler / right, marked as TRANSM.COEFF.P refers to the transmitted power as a function of 2 mismatch and displays essentially the relation Pt = 1 Γ . Thus, in the center of the Smith chart there −| | is a full match and all the power is transmitted. At the boundary there is total reflection and e.g. for a Γ value of 0.5 75% of the incident power is transmitted.

22 NAME TITLE DWG. NO.

DATE SMITH CHART FORM ZY-01-N Microwave Circuit Design - EE523 - Fall 2000

NORMALIZED IMPEDANCE AND ADMITTANCE COORDINATES

0.12 0.13 0.11 0.14 0.38 0.37 0.15 0.1 0.39 0.36 90 0.4 100 80 0.35 0.16 0.09 45

50 1.0

0.9 70 0.34

110 1.2 40 0.41 1.0 0.8

0.9 0.17 55 1.2

0.08 1.4 0.8

0.7 35 0.33 1.4 60

0.42 0.7 120 1.6

60 Yo) / 0.6 0.18

(+jB 1.6 0.07 0.6 E 30 1.8 NC 0.32 A 0.2 0.43 PT 0.2 1.8 CE 50

65 2.0 S 130 SU 0.5

E 2.0 0.19 0.5 0.06 ITIV 0.31 C 25 A 0.44 P A C

70 R 0.4 O 0.4 0.4 40

0.2 140 0.4 Zo), 0.05 / 0.3 jX 0.45 20 (+ 3.0 T N 3.0 75 0.6 E 0.6 N 0.21 O P

30 0.29 0.04 0.3 M 0.46 150 O C > 0.8 E 0.8 15 — 4.0 C 80 4.0 R N O A 0.22 T T 1.0 0.28 A C 1.0

R A

E 5.0 E 1.0

0.47 5.0 0.2 R 1.0 N 0.2 20 E E

160 85 V G 0.8 I

T 0.8 10 D C R 0.23 U A 0.27

D A W 0.6

N N A I 0.6 O 0.48 90 N T G

G S L

E L

H 10

10 E T 0.1 O

0.1

0.4 O F G 0.4

170 F N T 0.24 0.26 E R R

A L E

E F N 0.49

V 20

20 S E

A 0.2 M 0.2 C

I W T

S > 50 50 O

— 0.25 0.25

C 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 20

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

180

E 50

50 F F

RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) F C

—

C < I

0.2 E

0.2 I

20 N

D 20 E

0.24

A T

0.26

0.49

0.4

R 0.1 0.4 G

10 E

-170

E R

E E

0.23 E

0.6

0.27

S S

) 0.6

-90 o

0.48

j 0.8

-10 N -

(

E 0.8

-20

E L

C 0.2

-85 1.0

5.0

V -160 N

0.2 0.22

5.0

A A 1.0 0.28

1.0 T

P 1.0

0.47

—

4.0

S 0.8

-80 -15

E 0.8

0.21 V

I -30

0.3 0.29 C

0.04

-150 0.3 U

0.46

0.6

3.0

O -75

)

0.2

-20

0.3 0.05 /

0.4 -40

0.45

(-jX

0.4

0.4 T

-140 0.4

-70 0.19 O

0.06 0.31

-25

0.5 E

2.0

0.44 C

-50

0.5 N

2.0

-130 C

0.18

0.2

A 1.8

0.2 -65

0.07

1.8

0.6 0.32

-30

1.6

0.43 0.6

1.6 C -60

0.7 0.17

-60

-120

1.4

0.08 0.7

0.33

0.8 1.4 -35

1.2

0.42 0.9

0.8 1.0 0.16

-55

-70

1.2

0.09

-110 0.9

1.0 -40

0.34

-50

-45

0.15 0.41

-80 0.1

-100

-90

0.14

0.35

0.11

0.4

0.13 0.12

0.36

0.39

0.37 0.38

RADIALLY SCALED PARAMETERS RTN. LOSSSWR [dB] TOWARD LOAD —> <— TOWARD GENERATOR ∞10040 20 10 5 4 3 2.5 2 1.8 1.6 1.4 1.2 1.1 1 15 10 7 5 4 3 2 1 ATTEN. [dB] RFL. COEFF,dBS P RFL. COEFF, E or I ∞40 30 20 15 10 8 6 5 4 3 2 1 1 1 1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 ∞ S.W. LOSS COEFF

0 1 2 3 4 5 6 7 8 9 10 12 14 20 30 ∞ 0 0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 5 6 10 15∞ RFL. LOSS [dB] 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.05 0.01 0 0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 10 ∞ S.W. PEAK (CONST. P)

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 TRANSM. COEFF, P

CENTER 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 TRANSM. COEFF, E or I

ORIGIN