Smith chart domains

pysmithchart can plot several different domains on the same Smith chart projection. The domain determines what you pass to ax.plot() / ax.scatter() / ax.text() and how the library converts that data internally for display.

This notebook explains each domain and shows small, self-contained examples.

%config InlineBackend.figure_format = 'retina'

import sys
import numpy as np
import matplotlib.pyplot as plt

if sys.platform == "emscripten":
    import piplite

    await piplite.install("pysmithchart")

import pysmithchart
from pysmithchart import Z_DOMAIN, Y_DOMAIN, R_DOMAIN, NORM_Z_DOMAIN, NORM_Y_DOMAIN
from pysmithchart import utils

text_box = dict(facecolor="lightyellow")

Domain quick reference

The Smith chart itself is drawn in terms of normalized impedance

\[z = \frac{Z}{Z_0} = r + jx\]

but users naturally work with different quantities (ohms, siemens, reflection coefficient). pysmithchart supports these through a domain= choice.

Domain constant	What you pass to plotting calls	Internal conversion	Typical use
Z_DOMAIN (default)	\(Z\) in ohms	\(z = Z/Z_0\)	Most RF matching, measured impedances
NORM_Z_DOMAIN	\(z\) normalized (dimensionless)	use as-is	data matches impedance grid
R_DOMAIN	\(\Gamma\) (a.k.a. S11), complex	\(z = (1+\Gamma)/(1-\Gamma)\)	VNAs / S-parameter workflows
Y_DOMAIN	\(Y\) in siemens	\(z = \frac{1}{Y Z_0}\)	Admittance-based matching, dual networks
NORM_Y_DOMAIN	\(y\) normalized (dimensionless)	use as-is	data matches admittance grid

Domains let you supply the quantity you naturally have: \(Z\), \(z\), \(Y\), or \(\Gamma\).

How to set the domain

You can reset the default domain for all subsequent plots on an axis:

ax = plt.subplot(111, projection="smith", domain=NORM_Z_DOMAIN)

or per plotting call (overrides the default):

ax.plot(Γ, domain=R_DOMAIN)
ax.text(z.real, z.imag, "label", domain=Z_DOMAIN)

A single physical load shown four ways

We will use the same physical load, and plot it using each domain.

• Reference impedance: Z0 = 50 Ω

• Load impedance: ZL = 25 + j25 Ω

From these we can compute:

• normalized impedance: z = ZL / Z0

• admittance: Y = 1 / ZL

• reflection coefficient: Γ = (ZL - Z0) / (ZL + Z0)

Z0 = 50
ZL = 30 + 1j * 30

z_norm = ZL / Z0
Y = 1 / ZL
Gamma = utils.calc_gamma(Z0, ZL)

print("ZL =", ZL)
print("z  =", z_norm)
print("Y  =", utils.cs(Y, 3))
print("Γ  =", utils.cs(Gamma, 3))

ZL = (30+30j)
z  = (0.6+0.6j)
Y  =  0.017 -  0.017j
Γ  = -0.096 +  0.411j

1) Z_DOMAIN: pass ohms

This is the default domain. You pass Z in ohms; pysmithchart divides by Z0 internally.

plt.figure(figsize=(6, 6))
ax = plt.subplot(111, projection="smith", Z0=Z0)

ax.plot(ZL, marker="o", linestyle="")
text_offset = 5 + 3j
ax.text(ZL + text_offset, "$Z_L =$" + utils.cs(ZL, 0), bbox=text_box)

ax.set_title("Z_DOMAIN")
plt.show()

2) NORM_Z_DOMAIN: pass normalized z (dimensionless)

In this domain, you pass z = Z/Z0 directly. These values should appear just where you expect them on the chart

Z0 = 50
ZL = 30 + 1j * 30
z_norm = ZL / Z0

plt.figure(figsize=(6, 6))
ax = plt.subplot(111, projection="smith", domain=NORM_Z_DOMAIN)

ax.plot(z_norm, "bo")
text_offset = 0.05 + 0.05j
ax.text(z_norm + text_offset, "$Z_L =$" + utils.cs(z_norm, 2), bbox=text_box)

ax.set_title("NORM_Z_DOMAIN")
plt.show()

3) R_DOMAIN: pass Γ (S11)

If you have S-parameters from a VNA or simulator, you might start with Γ directly. In the R_DOMAIN pysmithchart internally converts the datapoint to normalized impedance using the inverse Möbius transform:

\[z = \frac{1+\Gamma}{1-\Gamma}\]

Only points inside the unit circle |Γ| ≤ 1 are visible on the chart.

Z0 = 50
ZL = 30 + 1j * 30
Gamma = utils.calc_gamma(Z0, ZL)

plt.figure(figsize=(6, 6))
ax = plt.subplot(111, projection="smith", domain=R_DOMAIN)

ax.plot(Gamma, marker="o", linestyle="")
text_offset = 0.05 + 0.05j
ax.text(Gamma + text_offset, "$\Gamma =$" + utils.cs(Gamma, 3), bbox=text_box)

ax.set_title("R_DOMAIN")
plt.show()

4) Y_DOMAIN: pass Y in siemens

Admittance plotting is useful for “dual” matching workflows (e.g., shunt elements).

Internally, pysmithchart converts Y to normalized impedance using: \(z = \frac{1}{Y Z_0}\)

ZL = 30 + 1j * 30
YL = 1 / ZL

plt.figure(figsize=(6, 6))
ax = plt.subplot(111, projection="smith", grid="admittance", domain=Y_DOMAIN)

ax.plot(YL, marker="o", linestyle="")
ax.text(YL, "  " + utils.cs(YL, 4), bbox=text_box)

ax.set_title("Y_DOMAIN")
plt.show()

5) NORM_Y_DOMAIN: pass normalized admittance (dimensionless)**

Admittance plotting is useful for “dual” matching workflows (e.g., shunt elements).

y = 0.6 + 1j

plt.figure(figsize=(6, 6))
ax = plt.subplot(111, projection="smith", grid="admittance", domain=NORM_Y_DOMAIN)

ax.plot(y, marker="o", linestyle="")
ax.text(y, "  " + utils.cs(y, 4), bbox=text_box)

ax.set_title("NORM_Y_DOMAIN")
plt.show()

What about ax.text() and ax.annotate()?

pysmithchart applies the same domain logic to text() and annotate() as it does to plot().

All forms below work fine in all four domains

• ax.text(z, label, ...)

• ax.text(x, y, label, ...)

• ax.annotate(x, y, label, ...)

Note because annotate has two positions xy and xytext, you cannot use complex values with annotate

A simple example of ax.text()

Z0 = 50
ZL = 30 + 1j * 30
z_norm = ZL / Z0

plt.figure(figsize=(6, 6))
ax = plt.subplot(111, projection="smith", domain=NORM_Z_DOMAIN)

ax.plot(ZL, marker="o", linestyle="")
ax.text(z_norm, " label in normalized z", bbox=text_box)

ax.set_title("Text placement uses the same domain rules as plotting")
plt.show()

A simple example of ax.annotate()

plt.figure(figsize=(6, 6))
ax = plt.subplot(111, projection="smith", domain=NORM_Z_DOMAIN)

z = 1 + 0.5j
ax.plot(z, "ro")

# Annotate the point
ax.annotate(
    "Z = 1 + j0.5",
    xy=(z.real, z.imag),  # point being annotated (Smith coords)
    xytext=(0.6, 1.0),  # location of text
    arrowprops=dict(arrowstyle="->", color="blue", lw=3),
    bbox=text_box,
)

ax.set_title("Annotating a Single Impedance Point")
plt.show()