Smith Chart Basics

The Smith chart is a graphical tool used universally in RF, microwave, and antenna engineering to analyze and design transmission line circuits. It transforms the complex mathematics of impedance transformation on transmission lines into a visual, intuitive graphical operation. For GATE and university examinations, understanding how impedances are plotted and how they move on the chart is essential.

Core Concept Explanation

The Smith chart is a polar plot of the complex reflection coefficient Γ. Since |Γ| ≤ 1 for passive loads, all physically realizable impedances map to points within the unit circle on this chart. The chart is constructed so that the standard Cartesian coordinates (real and imaginary parts of Γ) are replaced by curves of constant normalized resistance r and constant normalized reactance x, where the normalized impedance is z = Z/Z₀ = r + jx.

The constant-r circles are complete circles centered on the real axis of the Γ plane. The center of each r-circle is at (r/(r+1), 0) and has radius 1/(r+1). For r = 0 (pure reactance, no resistance), the circle is the outer boundary of the chart. For r = 1 (normalized resistance equals Z₀), the circle passes through the center of the chart. For r → ∞ (open circuit), the circle shrinks to the rightmost point.

The constant-x arcs are portions of circles centered on the vertical line passing through the rightmost point of the chart (Γ = +1). For x = 0 (purely real impedance), the arc degenerates to the horizontal diameter. Positive x (inductive) arcs appear in the upper half of the chart, and negative x (capacitive) arcs appear in the lower half.

Three critical reference points on the chart are: the leftmost point (Γ = -1) representing a short circuit (Z = 0, z = 0), the rightmost point (Γ = +1) representing an open circuit (Z = ∞, z = ∞), and the center of the chart (Γ = 0) representing the matched condition (Z = Z₀, z = 1 + j0).

Mathematical Expression

The relationship between normalized impedance z = r + jx and reflection coefficient Γ = Γᵣ + jΓᵢ is:

Γ = (z - 1)/(z + 1), or equivalently z = (1 + Γ)/(1 - Γ)

Substituting and separating real and imaginary parts gives the circle equations. The constant-r circle equation is:

(Γᵣ - r/(r+1))² + Γᵢ² = (1/(r+1))²

The constant-x arc equation is:

(Γᵣ - 1)² + (Γᵢ - 1/x)² = (1/x)²

Moving along a lossless transmission line corresponds to rotating the Γ point along a circle of constant radius (constant |Γ|) centered at the chart center. Moving toward the generator (away from load) means rotating clockwise, while moving toward the load means rotating counter-clockwise. One full revolution corresponds to a distance of λ/2 on the line.

Practical Understanding

When a transmission line is terminated in a known load impedance, the first step is to normalize it: z_L = Z_L/Z₀. For example, if Z₀ = 50Ω and Z_L = 100 + j50Ω, then z_L = 2 + j1. This point is plotted at the intersection of the r = 2 circle and the x = +1 arc. The magnitude |Γ| is the distance from the chart center to this point, and the angle of Γ is read from the outer scale.

Moving along the transmission line rotates the impedance point along the Wavelengths Toward Generator (WTG) or Wavelengths Toward Load (WTL) scales printed on the outer rim of the chart. The radius of rotation remains constant (constant VSWR circle) since a lossless line does not change |Γ|.

Given:
Z₀ = 50Ω
Z_L = 100 + j75Ω

Why this formula applies:
Normalize load impedance to find the Smith chart point, then read |Γ|.

Formula:
z_L = Z_L / Z₀
Γ = (z_L - 1)/(z_L + 1)
|Γ| = distance from chart center to plotted point

Substitution:
z_L = (100 + j75)/50 = 2 + j1.5

Calculation:
Γ = (2 + j1.5 - 1)/(2 + j1.5 + 1)
  = (1 + j1.5)/(3 + j1.5)
Numerator magnitude: √(1² + 1.5²) = √3.25 = 1.803
Denominator magnitude: √(3² + 1.5²) = √11.25 = 3.354
|Γ| = 1.803/3.354 = 0.537

VSWR = (1 + |Γ|)/(1 - |Γ|) = 1.537/0.463 = 3.32

Final Answer:
The point z = 2 + j1.5 lies on Smith chart at intersection of r=2 circle and x=1.5 arc.
|Γ| = 0.537, VSWR = 3.32

Exam Tip: On the Smith chart, moving toward the generator (source) = clockwise rotation; toward load = counter-clockwise. One full rotation = λ/2 movement on the line. The VSWR equals the normalized resistance value where the constant-|Γ| circle crosses the positive real axis (right side of horizontal diameter).

Mechanism Summary

• Normalize impedance: z = Z/Z₀. Plot at intersection of r-circle and x-arc. The distance from chart center to this point equals |Γ|.
• Constant VSWR circle is a circle centered at chart center with radius |Γ|. VSWR value is read where this circle crosses the positive real axis.
• Moving along a lossless line rotates the impedance point on the constant |Γ| circle. Clockwise rotation = moving toward generator; counter-clockwise = toward load.
• One full revolution on the Smith chart corresponds to λ/2 of line length, since the reflection coefficient has period 2βl = 2π at l = λ/2.
• Admittance (Y = 1/Z) of a point is found by rotating exactly λ/4 (half revolution) on the Smith chart, which places the point diametrically opposite to the original impedance point.

Quick Revision

• Smith chart plots normalized impedance z = Z/Z₀ = r + jx as a point within the unit |Γ| circle.
• r-circles: centered at (r/(r+1), 0) with radius 1/(r+1). x-arcs: portions of circles tangent to right edge of chart.
• SC = left edge (Γ = -1), OC = right edge (Γ = +1), Match = center (Γ = 0).
• Clockwise rotation = toward generator, counter-clockwise = toward load. λ/2 = one full revolution.
• VSWR = value of r where constant-|Γ| circle intersects positive real axis (right half of horizontal diameter).
• Admittance found by 180° rotation (λ/4 movement) of impedance point on same VSWR circle.
• Common trap: forgetting to normalize impedance before plotting. Always compute z = Z/Z₀ first.