
### 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 $Z_L$.

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 $z_L$.

### 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.

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 $x_L$.

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.

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