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 quan- tities 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 ca- bles) 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. 3 0.12 0.13 0.14 0.11 0.15 0.10 0.38 0.37 0.39 0.36 0.16 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 0.20 40 0.30 0.4 0.05 140 0.4 0.45 3 0.21 0.29 0.6 30 0.04 0.3 0.46 150 0.8 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 10 10 0.24 0.26 0.1 170 20 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 -20 0.24 0.26 -0.05 -10 -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 -0.3 -3 0.46 0.20 -40 0.4 0.30 0.05 -0.4 -140 0.45 0.19 0.31 -2 0.06 -50 -0.5 -1.8 0.2 0.18 0.44 -130 -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. 4: Example for a commonly used Smith chart. 3 Navigation in the Smith chart The representation of circuit elements in the Smith chart is discussed in this chapter starting with the important points inside the chart. Then several examples of circuit elements will be given and their representation in the chart will be illustrated. 3.1 Important points There are three important points in the chart: 1. Open circuit with Γ = 1,z → ∞ 2. Short circuit with Γ= 1,z = 0 − 3. Matched load with Γ = 0,z = 1 They all are located on the real axis at the beginning, the end and the center of the circle (Fig. 6). The upper half of the chart is inductive, since it corresponds to the positive imaginary part of the impedance. The lower half is capacitive as it is corresponding to the negative imaginary part of the impedance. 4 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 −| | 5 0.12 0.13 0.14 0.11 0.15 0.10 0.38 0.37 0.39 0.36 0.16 90 0.35 Γ = 1 0.09 0.40 100 80 0.34 70 0.17 0.41 1 0.08 110 0.9 1.2 | | 0.8 1.4 0.33 Γ = 0.75 0.7 60 0.18 0.42120 1.6 0.07 0.32 0.6 1.8 | | 0.43 50 0.19 Γ = 0.5 0.2 130 2 0.31 0.06 0.5 | | 0.44 0.20 Γ = 0.25 40 0.30 0.4 0.05 140 0.4 0.45 | | 3 0.21 Γ = 0 0.29 0.6 30 0.04 0.3 150 | | 0.46 0.8 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 10 10 0.24 0.26 0.1 170 20 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 -20 0.24 0.26 -0.05 -10 -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 -0.3 -3 0.46 0.20 -40 0.4 0.30 0.05 -0.4 -140 0.45 0.19 0.31 -2 0.06 -50 -0.5 -1.8 0.2 0.18 0.44 -130 -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.