Understanding the Smith Chart

Introduction

• Overview of the Smith chart and its use in representing complex impedance.
• Pre-requisites: Background in network analysis, S-parameters, and return loss.

History

• Invented by Philip Hager Smith.
• Described in an article in Electronics Magazine in January 1939.
• Smith wrote a 200-page book on the applications of the Smith chart.

Applications

• Primarily used for impedance matching and designing matching networks.
• Important before modern computational resources as a graphical method to solve problems.
• Still useful for visualizing complex impedances, especially as a function of frequency.
• Widely used in tuning or verifying the performance of matching networks.
• Used mainly in one-port measurements (reflection coefficients).

Complex Impedance on the Smith Chart

• Shows load impedance $Z_L$ relative to source impedance $Z_0$.
• Easier to visualize complex impedance values (single points or frequency-dependent lines).
• Complex impedance: Resistive part $R$ + Reactive part $X$ (Inductive $X_L$ or Capacitive $X_C$).

Representation Issues on Cartesian Coordinates

• Resistance (always positive) leads to using only the right-hand plane.
• Impedance can range from 0 to infinity.
• Smith chart bends the right half of the Cartesian plane.
• Top half: Inductive region; Bottom half: Capacitive region; Resistance axis separates them.

Middle Point of the Smith Chart

• Prime center: Source impedance $Z_0$, typically 50 ohms (normalized to 1.0).
• Normalization allows using the same Smith chart regardless of system impedance (e.g., 75 ohms).

Impedance Matching

• Goal: Match $Z_L$ to $Z_0$ for minimal reflected power and maximal power transfer.
• Closer $Z_L$ to center of the chart, the better the impedance match.
• Resonance: $Z_L$ at/near the center of the chart.

Resistance Axis

• The only straight line on the Smith chart.
• Left edge: Resistance = 0 (short-circuit); Right edge: Resistance = ∞ (open circuit).
• Points on resistance axis are pure resistance with no reactive part.
• Most loads have both resistive and reactive parts.

Resistance Circles

• Prime center = normalized resistance of 1.
• Circle through the prime center represents a constant normalized resistance circle of 1.
• Similar circles for other normalized resistances (e.g., 0.2, 0.4, etc.).
• Determine resistive part by following the resistance circle to the horizontal resistance axis.

Reactance Curves

• Reactance axis along the circumference of the Smith chart.
• Upper half: Positive reactance; Lower half: Negative reactance.
• Reactance curves show normalized reactance values (e.g., 0.7, 1.0, 1.6, 3.0, 10, etc.).
• Points along a curve have the same reactive part.

Example Calculation

• Given impedance: 100 + 75j ohms.
• Normalize: (100/50) + (75j/50) = 2 + 1.5j.
• Plot the resistance circle for 2.0 and the reactance curve for 1.5.
• Intersection gives the impedance on the Smith chart.

Summary

• Smith chart displays complex impedances as points or frequency-dependent lines.
• Enables graphical instead of algebraic solutions for impedance matching.
• Charts consist of resistance and reactance axes, resistance circles, and reactance curves.
• Values are normalized to system impedance.

Conclusion: Smith chart remains a vital tool for understanding and visualizing impedance matching.