How do series and shunt elements move on the combined Smith chart?

Series elements follow impedance chart circles: a series inductor moves clockwise along the constant-r (resistance) circle toward increasing positive reactance; a series capacitor moves counterclockwise toward increasing negative reactance. Shunt elements follow admittance chart circles: a shunt capacitor moves clockwise along the constant-g (conductance) circle toward increasing positive susceptance; a shunt inductor moves counterclockwise toward increasing negative susceptance. The arc length traveled depends on the component value: for a series inductor, the reactance added is X_L = 2 pi f L ohms (normalized by Z_0); for a shunt capacitor, the susceptance added is B_C = 2 pi f C siemens (normalized by Y_0). Transmission line sections rotate clockwise along constant-VSWR circles on either chart.

RF Design

Combined Smith Chart

Q: How does the combined Smith chart simplify matching network design?

A matching network typically uses both series elements (which move along constant-resistance circles on the impedance chart) and shunt elements (which move along constant-conductance circles on the admittance chart). Without the combined chart, the designer must manually convert between Z and Y representations at each transition from series to shunt topology. The combined chart eliminates this step: after adding a series inductor (moving clockwise along a constant-r circle on the Z overlay), the designer can immediately read the admittance and add a shunt capacitor (moving clockwise along a constant-g circle on the Y overlay) without any calculation. This visual workflow makes it straightforward to design L, pi, and T matching networks by alternating between the two overlaid coordinate systems.

Q: What is the relationship between the impedance and admittance charts?

The admittance Smith chart is a 180-degree rotation of the impedance Smith chart about the center point. Every point on the unit-circle boundary maps to itself under this rotation. A point at normalized impedance z = r + jx on the impedance chart maps to normalized admittance y = g + jb at the diametrically opposite point. The center of the impedance chart (z = 1 + j0, matched) is also the center of the admittance chart (y = 1 + j0). Short circuit (z = 0) on the left becomes open circuit (y = 0) on the right of the admittance chart, and vice versa. The reflection coefficient magnitude is the same at any point on both charts; only the angle shifts by 180 degrees.

/kom-bynd smith chart/

An overlay of the standard impedance Smith chart (normalized resistance r and reactance x circles) with the admittance Smith chart (normalized conductance g and susceptance b circles) on a single diagram. Since the admittance chart is a 180° rotation of the impedance chart, any point simultaneously shows both its normalized impedance z = r + jx and its normalized admittance y = g + jb. This enables direct visual design of matching networks that use both series elements (which move along constant-r circles) and shunt elements (which move along constant-g circles) without manual Z-to-Y conversion, streamlining the design of L-section, π-network, and T-network topologies.

Category: RF Design

Also Called: ZY Smith Chart

Relationship: Y = 180° rotation of Z

Understanding the Combined Smith Chart

Phillip Smith introduced his famous chart in 1939 as a graphical tool for solving transmission line problems. The standard (impedance) chart maps all passive impedances inside the unit circle of the complex reflection coefficient plane, with constant-resistance circles and constant-reactance arcs. The admittance chart is its complement: it maps all passive admittances with constant-conductance circles and constant-susceptance arcs. Because Y = 1/Z, and the reflection coefficient for admittance is the negative of the reflection coefficient for impedance, the admittance chart is obtained by rotating the impedance chart by 180 degrees.

The combined chart places both coordinate grids on the same diagram, typically drawing impedance circles in one color (red or solid) and admittance circles in another (blue or dashed). Any point on the chart can be read in both coordinate systems simultaneously. This is particularly powerful for matching network design: when adding a series element (inductor or capacitor), the designer traces along the impedance chart's constant-resistance circle. When switching to a shunt element, the designer immediately reads the current admittance value and traces along the admittance chart's constant-conductance circle. The target is always the center of the chart (z = 1, y = 1, VSWR = 1:1, perfect match).

Impedance-Admittance Conversion

Normalized Conversion:
y = 1/z  →  g + jb = 1/(r + jx)

Explicit Formulas:
g = r / (r² + x²)    b = −x / (r² + x²)

Reflection Coefficient:
Γ_Z = (z − 1)/(z + 1)    Γ_Y = −Γ_Z

Where z = Z/Z₀ (normalized impedance), y = Y×Z₀ (normalized admittance), Z₀ = reference impedance (typically 50 Ω). Graphically: the admittance of any point is found at its diametrically opposite position on the chart.

Element Movement on Combined Chart

Element	Topology	Chart Used	Movement Direction	Circle Followed	Value Added
Series inductor	Series	Impedance (Z)	Clockwise	Constant r	+jX = +j2πfL/Z₀
Series capacitor	Series	Impedance (Z)	Counterclockwise	Constant r	−jX = −j/(2πfCZ₀)
Shunt capacitor	Shunt	Admittance (Y)	Clockwise	Constant g	+jB = +j2πfCZ₀
Shunt inductor	Shunt	Admittance (Y)	Counterclockwise	Constant g	−jB = −jZ₀/(2πfL)
Transmission line	Either	Both	Clockwise	Constant VSWR	θ = βl (electrical length)

Common Questions

Frequently Asked Questions

How does the combined Smith chart simplify matching network design?

Matching networks alternate between series and shunt elements. Without the combined chart, manual Z-to-Y conversion is needed at each transition. The overlay lets designers add a series element (tracing on the Z grid), then immediately read the admittance and add a shunt element (tracing on the Y grid) with no calculation. This makes L, π, and T networks straightforward to design visually.

What is the relationship between the impedance and admittance charts?

The admittance chart is a 180° rotation of the impedance chart. Every point at z = r + jx maps to y = g + jb at the diametrically opposite position. The center (matched) is the same on both. Short circuit (z=0, left) maps to open circuit (y=0, right) and vice versa. Reflection coefficient magnitude is identical; only the angle shifts by 180°.

How do series and shunt elements move on the combined chart?

Series elements follow Z-chart constant-r circles: inductors clockwise (adding +jX), capacitors counterclockwise (adding −jX). Shunt elements follow Y-chart constant-g circles: capacitors clockwise (adding +jB), inductors counterclockwise (adding −jB). Transmission lines rotate clockwise on constant-VSWR circles on either chart.

Related Terms

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