Immittance chart and duality parameters

An immittance chart (ZY Smith chart) is a combined chart that shows both immittance forms, impedance and admittance, on the same reflection-coefficient (Γ) plane. It overlays the normal impedance Smith chart (constant r and x circles) and the admittance Smith chart (constant g and b circles). The admittance chart is the impedance chart rotated by 180°. Any point can therefore be read as z = r + jx or as y = g + jb directly, without the λ/4 rotation.

Sketch

               inductive (+jx, −jb)
              .----~~~~~~~~----.
           .'   r-circles  ↘    '.
          /   (touch at OC) ↘     \
   SC  o(----------( 1 )----------)o  OC
 y = ∞    \  g-circles ↗          /  z = ∞
 z = 0      '. (touch at SC)   .'    y = 0
              '----~~~~~~~~----'
               capacitive (−jx, +jb)

Properties

• Constant-r circles all touch at the open-circuit point (right). Constant-g circles all touch at the short-circuit point (left).
• Constant-x arcs leave the right point, and constant-b arcs leave the left point.
• The centre (z = y = 1) is the matched point.
• Upper half: inductive in both readings (+jx, −jb). Lower half: capacitive.
• A series element moves the point along a constant-r circle. A shunt element moves it along a constant-g circle.
• It is used for lumped L, T and π matching networks and for amplifier input and output matching.

Duality parameters used in microwave network design

In each pair, swapping one quantity for the other turns a valid circuit relation into another valid one.

Why duality matters: a design for a shunt (admittance) network can be reused for a series (impedance) network by swapping the dual quantities. On the immittance chart this swap is only a change of scale (Z or Y), not a recalculation.