The SMITH Chart Review: Reflection Coefficient Reflection Coefficients Transformation Reflection Coefficient (at the load) Reflection Coefficient (at the input) Complex Number: Rectangular form Γ=x+jy Polar form Γ=r∟θ Exponential form Γ = r ejθ Γ • The Smith chart is one of the most useful graphical tools for high frequency circuit applications. • Invented by Phillip H. Smith (1905–1987) • Designed for Electrical and Electronics Engineers specializing in radio frequency (RF) engineering to assist in solving problems with transmission lines and matching circuits. • From a mathematical point of view, the Smith chart is simply a representation of all possible complex impedances with respect to coordinates defined by the reflection coefficient. • The domain of definition of the reflection coefficient is a circle of radius one (1.0) in the complex plane. This is also the domain of the Smith chart. 𝑍𝐿 − 𝑍𝑂 Γ= = 𝑅𝑒(Γ) + 𝐼𝑚(Γ) 𝑍𝐿 + 𝑍𝑂 Possible values for |Γ| −𝟏 < 𝟎 < +𝟏 “Short” “Matched” “Open” SWR Circles From the origin, all impedance points along a fixed radius circle will have the same VSWR or |Γ| (magnitude of Γ) Typical use of a Smith Chart Find voltage reflection coefficient (Γ∠𝜃 or x+jy) Find value of the voltage standing wave ratio (VSWR) Find return Loss (dB) / Reflection Loss (dB) Find Zin at any given distance along the transmission line Find Γ at any given distance along the transmission line Find VSWR at any given distance along the transmission line Do impedance matching: ◦ Quarter Wave Transformer ◦ Lumped Network ◦ Stubs Etc. Special Consideration The Smith Chart is a very useful tool if we assume the transmission line to be lossless or distortionless. Since the reflection coefficient will have a uniform magnitude along the entire length of the line. If the transmission line is lossy, we can still use the smith chart, but the graphical analysis become more complicated. It would be better to use analytical methods for this case. The spiral pattern will appear since the magnitude of the reflection coefficient will decrease along the length of the line. Non-uniform curves forming spiral patterns becomes evident if the transmission line is lossy! Microwave Network Analyzer An HP8720A network analyzer showing a plot of the Smith Chart (also known as an RF Vector Analyzer) Additional Quantities Transmission Coefficient (T) Clarification: Voltage reflection coefficient ( ΓE = Γ ) Current reflection coefficient ( ΓI = Γ ) Power reflection coefficient ( ΓP = Γ2 ) RECALL: Normalized Impedance (z) 𝑍𝐿 𝑅𝐿 + 𝑗𝑋𝐿 𝑧= = = 𝑟 + 𝑗𝑥 𝑍𝑂 𝑍𝑜 𝑅𝐿 𝑟= 𝑍𝑂 Normalized Resistance Normalized Reactance 𝟎 < 𝒓 < +∞ −∞ < 𝒙 < +∞ 𝑋𝐿 𝑥= 𝑍𝑂 In order to use the Smith Chart, impedances must be normalized. Normalized Resistance Normalized Reactance Conversion: Γ(d) <-> Z(d) Γ(d) = reflection coefficient Z(d) = input impedance (d) = distance from the load (towards the generator) Recall: if ‘d’ is not specified: Γ(d) = ΓL Z(d) = ZL Find the ff: Case 1 (use analytical method) Case 1I (use graphical method) a.) ΓL (polar & rectangular) b.) TE (polar & rectangular) b.) VSWR c.) Return Loss (dB) d.) Reflection Loss(dB) Find ZL using smith chart: Given: ΓL = 0.665 ∟ 60° Zo = 100 Ω