1 = r ∞ = r 0 = r (short) (open) 0 1 0.5 im Γ re Γ Figure 1: The r -circles in the complex plane re im (, ) Γ Γ. All the r -circles pass through the point re im ( 1, 0) Γ = Γ =.The r -circles become progressively smaller as r increases from 0 to ∞, ending at the re im ( 1, 0) Γ = Γ = point for an open circuit.It is centred at the origin and has a radius of 1. The circle where there is no resistance ( r = 0) is the largest.The centres of all r -circles lie on the re Γ -axis.The following properties of the r -circles are noted: Different values of r yield circles of different radii with centres at different positions on the re Γ -axis. (8)Ģ This equation is a relationship in the form of a parametric equation 2 2 2 ) ( ) ( R b y a x = − + − in the complex plane re im (, ) Γ Γ of a circle centred at the coordinates ,0 1 r r ⎛ ⎞ ⎜ ⎟ + ⎝ ⎠ and having a radius of 1 1 + r. (5) Multiplying both the numerator and the denominator of (5) by the complex conjugate of the denominator and separating the real and imaginary parts, we obtain 2 2 re im 2 2 re im 1 (1 ) r − Γ − Γ = − Γ + Γ (6) and 2 im 2 2 re im 2 (1 ) x Γ = − Γ + Γ. (3) The inverse relation of (3) is 1 1 1 1 j L L L j L L e z e θ θ + Γ + Γ = − Γ − Γ (4) or re im re im (1 ) (1 ) j r jx j + Γ + Γ + = − Γ − Γ. (2) With this simplification, we can rewrite the reflection coefficient formula in (1) as 0 0 re im 0 0 ( ) / 1 ( ) / 1 L L L L L Z Z Z z j Z Z Z z − − Γ = Γ + Γ = + +. We can then define the normalised load impedance by 0 0 / ( ) / L L z Z Z R jX Z r jx = + = +. The characteristic impedance Z 0 is often a constant and a real industry normalized value, such as 50 Ω, 75 Ω, 100 Ω, and 600 Ω. To understand how the Smith chart for a lossless transmission line is constructed, examine the voltage reflection coefficient of the load impedance defined by refl 0 re im inc 0 L L L V Z Z j V Z Z − Γ = Γ + Γ +, (1) where re Γ and im Γ are the real and imaginary parts of the complex reflection coefficient L Γ. The Smith chart is a polar plot of the complex reflection coefficient, or equivalently, a graphical plot of normalized resistance and reactance functions in the reflection-coefficient plane. The only effort required is the reading and following of values along the circles. When used correctly, impedance matching can be performed without any computation. The Smith chart is a circular plot with a lot of interlaced circles on it. The best known and most widely used graphical chart is the Smith chart. This tedium can be alleviated using a graphical method of solution. 1 Transmission Lines – Smith Chart & Impedance Matching (Intensive Reading) 1 The Smith Chart Transmission line calculations − such as the determination of input impedance using equation (4.30) and the reflection coefficient or load impedance from equation (4.32) − often involves tedious manipulation of complex numbers.
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