Impedance and admittance circles on the Smith Chart

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Reflection Coefficient and Impedance

Reflection coefficient and impedance are related through Equation eq:SCDereqnreflectioncoefficient. We can find an impedance that corresponds to the reflection coefficient, Equation eq:SCDereqnimpedancerc. Every point on the Smith Chart represents one reflection coefficient $\Gamma$ and one impedance $Z_L$ .

$\begin{eqnarray} \Gamma = \frac{Z_L-Z_\circ}{Z_L+Z_\circ} \label{eq:SCDereqnreflectioncoefficient} \\ \nonumber \\ Z_L = Z_0 \frac{ 1+ \Gamma}{1- \Gamma } \label{eq:SCDereqnimpedancerc} \end{eqnarray}$

All impedances on the Smith Chart are normalized to the transmission line impedance $Z_0$ . The normalized impedance is denoted in Equation eq:SCDerNormImp with lowercase $z_L$ .

$\begin{eqnarray} z_L = \frac{Z_L}{Z_0} = \frac{ 1+ \Gamma}{1- \Gamma } \label{eq:SCDerNormImp} \end{eqnarray}$

Derivation of Impedance and Admittance Circles on the Smith Chart

Impedance and reflection coefficient are complex numbers. The normalized impedance has a real and imaginary part $z_L=r_L + j x_L$ , and the reflection coefficient can also be shown in Cartesian coordinates as $\Gamma = \Gamma _r + j \Gamma _i$ . We can now substitute these equations into Equation eq:SCDerRealImag.

$\begin{eqnarray} r_L + j x_L = \frac{ 1+\Gamma_r + j \Gamma_i}{1- (\Gamma_r + j \Gamma_i)} \label{eq:SCDerRealImag} \end{eqnarray}$

We can equate the real and imaginary parts on the left and right side of Equation eq:SCDerRealImag to get the equations of constant $r_L$ and $x_L$ .

$\begin{eqnarray} \left( \Gamma_r - \frac{r_L}{1+r_L} \right)^2 + \Gamma_i^2 =\left( \frac{1}{1+r_L} \right) ^2 \label{eq:SCDerRealCirc} \\ \left( \Gamma_r - 1 \right)^2 + \left(\Gamma_i - \frac{1}{x_L} \right)^2 =\left( \frac{1}{x_L} \right) ^2 \label{eq:SCDerImagCirc} \end{eqnarray}$

These are equations of a circle, with the constant resistance circle’s center at $(\frac {r_L}{1+r_L} ,0)$ and radius $\frac {1}{1+r_L}$ ; and the constant reactance imaginary circle center at $(1,\frac {1}{x_L})$ and radius of $\frac {1}{x_L}$ .

Figures fig:SCDerscresistance- fig:SCDerscreactance show circles on the Smith Chart that represent constant (normalized) reactances, and resistances.

Figure 1: All points on the circle have the constant real part of the impedance (resistance). Normalized resistance circles.

Figure 2: All points on the circle have the constant imaginary part of the impedance (reactance). Normalized reactance circles.

Admittance circles can be similarly derived using the fact that $Y_L=\frac {1}{Z_L}$ and the Equation eq:SCDerNormImp