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Smith Charts

Main source material from Rohde & Schwarz

https://www.youtube.com/watch?v=rUDMo7hwihs

Named after Philip Hagar Smith, first described in January 1939
http://smithchart.org/phsmith.shtml

  • When asked why he invented the chart, Smith explained, "From the time I could operate a slide rule, I've been interested in graphical representations of mathematical relationships."
  • In 1969 he published the book Electronic Applications of the Smith Chart: In Waveguide, Circuit, and Component Analysis, a comprehensive work on the subject. He retired from Bell Labs in 1970.

Smith Chart with Annotations

The Smith Chart:

  • Has many applications.
  • the most common are impedance matching and the design of matching networks
  • pre-computers this was an intensive process
  • the Smith Chart lets you solve the problem graphically with a compass, ruler and pencil
  • still useful to help visualize complex impedances, especially as a function of frequency.
  • widely used when tuning or verifying the performance of networks

A Smith Chart:

  • used when making 1 Port measurements (reflection coefficients e.g. $S_{xy})
  • Shows $Z_{L}$ relative to $Z_{0}$
  • can be used as
    • a single point
    • traces (impedance as a function of frequency)
  • essentially bends the right hand side of a cartesian coordinate plane
    • this side is all that would be used in measurement
    • postitive and negative reactance axes (top and bottom half circles) bend around to meet the resistance axis (horizontal/x axis)
    • top half is the Inductive region
    • bottom half is the Capacitative region
    • horizontal (x-axis) is the resistive axis
  • A complex impedance appears as a point on a Smith Chart

layout

Smith Chart Annotated

Prime Center

  • Point in the middle of chart
  • corresponds to the source impedance $Z_{0}$ and a VSWR of 1
  • in most RF systems the source is purely resistive 50 $\Omega$ load
  • using the Smith Chart this value is normalized to 1.0
  • 50 $\Omega$ / 50 == 1
  • if the point moves to the right along the resistive axis to 2.0, this corresponds to a resistive value of 2 * 50 or 100 $\Omega$.
  • moving the point to 4.0 would increase the resistive value to 4 * 50 or 200 $\Omega$.
  • All Values on the Smith Chart are normalized by the same Prime Center value.
    • This allows the Chart to be used regardless of impedance (50 $\Omega$ or 75 $\Omega$ etc)

Significance of the Prime Center:

  • ideally $Z_{L}$ = $Z_{0}$ ("matched")
  • $Z_{0}$ is always the Prime Center.
  • Measured $Z_{L}$ is plotted on the Smith Chart
  • The closer the values are to the center the better the impedance match
  • the farther away the values to the center, the higher the degree of mismatch
  • when viewing a trace (many points):
    • the load is resonant at the frequency where the trace moves through the center

Reistive Axis:

  • The horizontal centerline
  • left of prime center decreases until 0 on the outer edge - A Short Circuit
  • right of prime center increases until the outer edge where it reaches $\infty$ - an Open Circuit
  • therefore VSWR is $\infty$ on either end of the resistive axis. 100% reflected power

Resistance Circles:

Resistive Circles

  • Most loads have complex impedances
    • have conductive and inductive characteristics, so not flat on the resistive line

Circle R=1 (red):

  • all values within represent a normalized resistance of 1

Circle R=.2 (blue):

  • all values within represent a normalized resistance of .2

Circle R=4 (green):

  • all values within represent a normalized resistance of 4

and so on.

The Normalized Resistance of any point is found by following the resistance circle to the horizontal axis and reading the value

Reactance

Reactance Curve

  • the Reactive axis is the outer circle (circumference) since it's bent from a cartesian plane.
  • Normalized Values along the outside. Get larger from L to R.
  • Values increase rapidly as they get closer to the centerline at the right hand side of the chart (Open Circuit)
  • Normalized reactance are shown as Curves.
  • every point along the reactive curve has the same Imaginary part.
  • in the diagram every point along the line has a reactance value of 1
  • on the upper half of the chart all values are Inductive (positive)
  • on the upper half of the chart all values are Capacitative (negative)

Plotting the impedance on the Smith Chart

Complex impedance - Z = R $\pm$ jX

Our value is: 100 + j75 $\Omega$

  • normalize the impedance by dividing by $Z_{0}$
    • (100 + j75 $\Omega$ ) / 50 == 2 + j1.5
    • $R_{normalized}$ + $X_{normalized}$ (2 + 1.5)
    • Find and plot the resistance circle for this normalized resistance
      • $R_{normalized}$ == 2
    • Find and plot the reactance curve
      • $X_{normalized}$ == 1.5

plotting impedance

Reversing the process to read an impedance from a Smith Chart

  • determine which resistance circle the point lies on
  • determine which reactance curve the point lies on
  • multiply the normalized values by the source impedance ( $Z_{0}$ ) to obtain actual values

For R = .3 and X = 0.4, $Z_{normalized}$ = 0.3 - j0.4 (* 50) == 15 - 20

So in this case: $Z_{L}$ == 15 - j20 $\Omega$.