Why does a series inductor move the point along a circle?

In the **z** plane, **Z_new = Z_old + jωL** changes only the **imaginary part** while **R** stays the same. That constraint traces one of the **constant resistance** curves when mapped through Γ(z); on the Smith chart it is one of the magenta family of circles (not a straight line in Γ coordinates).

Why are shunt steps computed through admittance?

A parallel element adds **susceptances** in **admittance** (**Y = 1/Z**), not in **Z** directly. The code updates **Y ← Y + jb** (or **−jb** for shunt **L** in our convention), then **Z = 1/Y**, which reproduces the usual “flip to the admittance chart / rotate the Smith chart 180°” procedure taught in textbooks.

Can I read absolute henries and farads from the step size?

Not directly. The slider sets a **normalized** reactance or susceptance step **Δx** or **Δb** at one implicit **ω**. Real values satisfy **ωL = (Δx)·Z₀** or **ωC = (Δb)/Z₀** only after you fix **f** and **Z₀**.

What is missing compared to a professional RF simulator?

**Electrical length** of lines, **loss**, **dispersion**, and **parasitics** are omitted. The chart here is a **lumped-element, single-frequency** teaching aid. For filters and wideband designs you would cascade full ABCD/S-parameter models or use field solvers.

Smith Chart & Matching

Click to set normalized Z; series L/C along constant-r arcs and shunt L/C via Y = 1/Z; Γ, SWR, matching trail.

About this model

The Smith chart is a polar graph of the complex reflection coefficient Γ in the unit disk |Γ| ≤ 1, with a curvilinear grid of constant normalized resistance r = R/Z₀ and constant normalized reactance x = X/Z₀. It encodes the bilinear map Γ = (z − 1)/(z + 1) and its inverse z = (1 + Γ)/(1 − Γ) where z = r + jx is the load normalized to the reference impedance Z₀ (usually the line’s characteristic impedance). This simulator draws the classic r- and x-circles (clipped to the unit circle), lets you click to pick a point (Γ, hence Z), and exposes four lumped-element ladder steps at an implicit single frequency: series L and series C change only x while keeping r fixed in the z plane—paths that appear as motion along a constant-r circular arc on the chart; shunt C and shunt L add susceptance in parallel, computed by converting to admittance Y = 1/Z, adding ±jb in normalized form, then converting back—corresponding to motion along constant-conductance arcs. The pink trail connects successive Γ points so students see how matching networks “walk” the load toward the chart center (Γ ≈ 0, Z ≈ Z₀). Displayed quantities include Z in ohms, Γ, |Γ|, its angle, and SWR = (1+|Γ|)/(1−|Γ|) for passive |Γ|<1. The model is quasi-static and lossless: it does not include transmission-line electrical length, distributed matching, or component parasitics—those belong to fuller CAD or classroom extensions.

Who it's for: Undergraduates in RF/microwave and circuits labs, amateur-radio operators learning antenna tuners, and anyone bridging transmission-line reflection with lumped matching.

Key terms

Smith chart
Reflection coefficient Γ
Normalized impedance
Characteristic impedance Z₀
Standing wave ratio (SWR)
Series and shunt matching
Admittance Y = 1/Z
Constant r and constant g circles

How it works

Interactive Smith chart: click to set a normalized impedance, then step series or shunt L/C and watch the reflection coefficient move along the classic constant-r and constant-g circles.

Frequently asked questions

Why does a series inductor move the point along a circle?: In the z plane, Z_new = Z_old + jωL changes only the imaginary part while R stays the same. That constraint traces one of the constant resistance curves when mapped through Γ(z); on the Smith chart it is one of the magenta family of circles (not a straight line in Γ coordinates).
Why are shunt steps computed through admittance?: A parallel element adds susceptances in admittance (Y = 1/Z), not in Z directly. The code updates Y ← Y + jb (or −jb for shunt L in our convention), then Z = 1/Y, which reproduces the usual “flip to the admittance chart / rotate the Smith chart 180°” procedure taught in textbooks.
Can I read absolute henries and farads from the step size?: Not directly. The slider sets a normalized reactance or susceptance step Δx or Δb at one implicit ω. Real values satisfy ωL = (Δx)·Z₀ or ωC = (Δb)/Z₀ only after you fix f and Z₀.
What is missing compared to a professional RF simulator?: Electrical length of lines, loss, dispersion, and parasitics are omitted. The chart here is a lumped-element, single-frequency teaching aid. For filters and wideband designs you would cascade full ABCD/S-parameter models or use field solvers.