Smith Chart analysis
Smith Chart: Derivation, Impedance and Admittance Circles Explained
The Smith Chart is a powerful graphical tool used in RF and microwave engineering to analyze transmission-line problems involving impedance, admittance, reflection coefficient, VSWR, and impedance matching. Instead of performing repeated complex-number calculations, the Smith Chart provides a graphical representation of the relationship between the normalized load impedance and the complex voltage reflection coefficient. Its construction is based directly on the mathematical transformation between the normalized impedance plane and the reflection-coefficient plane. The resulting chart contains families of constant resistance and constant reactance circles, which allow transmission-line quantities to be determined graphically.
What is an Impedance, Admittance, Immittance and Smith Chart?
The Smith Chart is commonly described as an impedance chart, but it can also be used for admittance analysis. The term immittance is a general term that includes both impedance \(Z\) and admittance \(Y\). Therefore, depending on the form of the chart and the analysis being performed, it may be referred to as an impedance chart, admittance chart, or immittance chart.
The fundamental idea behind the Smith Chart is to transform the complex impedance plane into the complex reflection-coefficient plane. This transformation makes it possible to represent impedance values as geometric circles and curves. Once the relationship has been established mathematically, quantities such as normalized resistance, normalized reactance, reflection coefficient, VSWR, and impedance transformation along a transmission line can be interpreted directly from the chart.
Fundamental Reflection Coefficient
The construction of the Smith Chart begins with the voltage reflection coefficient, \(\Gamma\), which describes the ratio of the reflected voltage wave to the incident voltage wave at a load. For a transmission line having characteristic impedance \(Z_0\) and load impedance \(Z_L\), the reflection coefficient at the load is:
\[ \Gamma = \frac{Z_L-Z_0}{Z_L+Z_0} \tag{1} \]
Here, \(Z_L\) is the complex load impedance and \(Z_0\) is the characteristic impedance of the transmission line. The load impedance can be written in rectangular form as:
\[ Z_L=R_L+jX_L \]
The magnitude of \(\Gamma\) indicates the amount of reflection produced by the load, while its phase indicates the phase relationship between the incident and reflected waves. A matched load produces zero reflection, whereas a mismatched load produces a nonzero reflection coefficient.
Normalization of Load Impedance
A major advantage of the Smith Chart is that the same chart can be used for transmission lines having different characteristic impedances. This is achieved by normalizing the load impedance with respect to the characteristic impedance \(Z_0\).
The normalized load impedance is defined as:
\[ z_L = \frac{Z_L}{Z_0} \]
Since:
\[ Z_L=R_L+jX_L \]
the normalized impedance becomes:
\[ z_L = \frac{R_L+jX_L}{Z_0} \]
Therefore:
\[ z_L=r_L+jx_L \]
where the normalized resistance and normalized reactance are:
\[ r_L=\frac{R_L}{Z_0} \]
\[ x_L=\frac{X_L}{Z_0} \]
Normalization removes the dependence on the particular value of \(Z_0\). For example, a load of \(100+j50\,\Omega\) produces different normalized values depending on whether the transmission line has a characteristic impedance of \(50\,\Omega\), \(75\,\Omega\), or another value. The Smith Chart itself represents the normalized quantities, while the actual impedance can be recovered by multiplying by \(Z_0\).
Relationship Between Normalized Impedance and Reflection Coefficient
Starting from the reflection-coefficient equation:
\[ \Gamma = \frac{Z_L-Z_0}{Z_L+Z_0} \]
Divide the numerator and denominator by \(Z_0\):
\[ \Gamma = \frac{ \dfrac{Z_L}{Z_0}-1 }{ \dfrac{Z_L}{Z_0}+1 } \]
Since:
\[ z_L=\frac{Z_L}{Z_0} \]
the reflection coefficient can be written as:
\[ \boxed{ \Gamma = \frac{z_L-1}{z_L+1} } \]
This equation forms the mathematical foundation of the Smith Chart. It establishes a direct transformation between the normalized impedance \(z_L\) and the reflection coefficient \(\Gamma\).
Inverting the Smith Chart Relationship
For Smith Chart construction, it is useful to express normalized impedance directly as a function of the reflection coefficient. Starting from:
\[ \Gamma = \frac{z_L-1}{z_L+1} \]
Multiplying both sides by \(z_L+1\):
\[ \Gamma(z_L+1) = z_L-1 \]
Expanding the left-hand side:
\[ \Gamma z_L+\Gamma = z_L-1 \]
Collecting the terms containing \(z_L\):
\[ z_L-\Gamma z_L = 1+\Gamma \]
Factoring \(z_L\):
\[ z_L(1-\Gamma) = 1+\Gamma \]
Therefore:
\[ \boxed{ z_L = \frac{1+\Gamma}{1-\Gamma} } \tag{2} \]
Equation (2) is the fundamental transformation used to derive the constant resistance and constant reactance circles of the Smith Chart.
Derivation of Constant Resistance and Reactance Circles
The Smith Chart is constructed in the complex \(\Gamma\)-plane. Since the reflection coefficient is complex, it can be expressed as:
\[ \Gamma = \Gamma_r+j\Gamma_i \]
where \(\Gamma_r\) is the real part and \(\Gamma_i\) is the imaginary part of the reflection coefficient.
Fig: Smith Chart Representation
Substituting \(\Gamma=\Gamma_r+j\Gamma_i\) into Equation (2):
\[ r_L+jx_L = \frac{ 1+(\Gamma_r+j\Gamma_i) }{ 1-(\Gamma_r+j\Gamma_i) } \]
Therefore:
\[ r_L+jx_L = \frac{ (1+\Gamma_r)+j\Gamma_i }{ (1-\Gamma_r)-j\Gamma_i } \]
To separate the real and imaginary parts, multiply the numerator and denominator by the complex conjugate of the denominator:
\[ r_L+jx_L = \frac{ \left[(1+\Gamma_r)+j\Gamma_i\right] \left[(1-\Gamma_r)+j\Gamma_i\right] }{ \left[(1-\Gamma_r)-j\Gamma_i\right] \left[(1-\Gamma_r)+j\Gamma_i\right] } \]
Expanding the numerator and denominator gives:
\[ r_L+jx_L = \frac{ 1-\Gamma_r^2-\Gamma_i^2+j2\Gamma_i }{ (1-\Gamma_r)^2+\Gamma_i^2 } \]
By equating the real and imaginary parts, the normalized resistance is:
\[ \boxed{ r_L = \frac{ 1-\Gamma_r^2-\Gamma_i^2 }{ (1-\Gamma_r)^2+\Gamma_i^2 } } \tag{3} \]
Similarly, the normalized reactance is:
\[ \boxed{ x_L = \frac{ 2\Gamma_i }{ (1-\Gamma_r)^2+\Gamma_i^2 } } \tag{4} \]
These two equations describe the relationship between the normalized impedance components and the real and imaginary components of the reflection coefficient. Rearranging them into standard circle equations produces the two families of curves that form the impedance portion of the Smith Chart.
Constant Resistance Circles
For a fixed value of normalized resistance \(r_L\), Equation (3) can be rearranged into the standard equation of a circle:
\[ \boxed{ \left( \Gamma_r-\frac{r_L}{1+r_L} \right)^2 + \Gamma_i^2 = \left( \frac{1}{1+r_L} \right)^2 } \tag{5} \]
This is the equation of a circle in the \(\Gamma\)-plane. Every point on a particular circle represents a normalized impedance having the same resistance \(r_L\), while the reactance changes as the point moves along the circle.
The center of the constant-resistance circle is:
\[ \boxed{ \left( \frac{r_L}{1+r_L},0 \right) } \]
and its radius is:
\[ \boxed{ \frac{1}{1+r_L} } \]
Therefore, each constant-resistance circle is centered on the horizontal axis of the Smith Chart. As the value of \(r_L\) increases, the center moves toward the right-hand edge of the chart and the radius becomes smaller.
Fig: Constant Resistance Circles
Constant Reactance Circles
For a fixed value of normalized reactance \(x_L\), Equation (4) can similarly be rearranged into a circle equation:
\[ \boxed{ (\Gamma_r-1)^2 + \left( \Gamma_i-\frac{1}{x_L} \right)^2 = \left( \frac{1}{x_L} \right)^2 } \tag{6} \]
This represents a family of constant-reactance circles. Every point on a particular circle has the same normalized reactance \(x_L\), while its normalized resistance changes.
The center of a constant-reactance circle is:
\[ \boxed{ \left( 1,\frac{1}{x_L} \right) } \]
and its radius is:
\[ \boxed{ \left|\frac{1}{x_L}\right| } \]
The sign of the reactance determines the location of the circle relative to the horizontal axis. Positive reactance corresponds to the inductive region of the Smith Chart, while negative reactance corresponds to the capacitive region.
Fig: Constant Reactance Circles
Smith Chart and VSWR
The Smith Chart also provides a direct graphical representation of the Voltage Standing Wave Ratio (VSWR). For a lossless transmission line, VSWR is related to the magnitude of the reflection coefficient by:
\[ \boxed{ \mathrm{VSWR} = \frac{1+|\Gamma|} {1-|\Gamma|} } \]
On the Smith Chart, points having the same magnitude of reflection coefficient lie on a circle centered at the origin. This circle is called a constant VSWR circle. Therefore, once a load is plotted on the Smith Chart, its reflection coefficient magnitude and corresponding VSWR can be determined graphically.
The radius of the constant VSWR circle represents \(|\Gamma|\). A point closer to the center of the chart has a smaller reflection coefficient and therefore a lower VSWR, while a point closer to the outer boundary has a larger reflection coefficient and therefore a higher VSWR.
Maximum Impedance on the Transmission Line
The maximum impedance occurs at a voltage maximum and current minimum of the standing-wave pattern. For a lossless transmission line, the maximum impedance is therefore:
\[ Z_{\max} = \frac{V_{\max}}{I_{\min}} \]
Using the incident and reflected wave magnitudes:
\[ V_{\max} = |V_0^+|(1+|\Gamma|) \]
and:
\[ I_{\min} = \frac{|V_0^+|}{Z_0}(1-|\Gamma|) \]
Therefore:
\[ Z_{\max} = Z_0 \frac{1+|\Gamma|} {1-|\Gamma|} \]
Since:
\[ \mathrm{VSWR} = \frac{1+|\Gamma|} {1-|\Gamma|} \]
the maximum impedance can be written as:
\[ \boxed{ Z_{\max} = Z_0\cdot\mathrm{VSWR} } \]
Dividing by \(Z_0\), the normalized maximum impedance is:
\[ \boxed{ z_{\max} = \frac{Z_{\max}}{Z_0} = \mathrm{VSWR} } \]
At the point of maximum impedance, the impedance is purely resistive because the voltage maximum and current minimum occur at the same location. Therefore, the imaginary component of the normalized impedance is zero:
\[ \Gamma_i=0 \]
At this point, the normalized resistance reaches its maximum value, giving:
\[ \boxed{ (r_L)_{\max} = \mathrm{VSWR} } \]
This provides an important graphical relationship between the constant-resistance circles and the constant-VSWR circle on the Smith Chart.
Fig: Smith Chart Representation of Maximum Impedance and VSWR