Smith chart

The Smith chart is a circular nomogram used in radio frequency (RF) engineering to solve transmission line and impedance-matching problems. It plots a complex reflection coefficient ( \Gamma ) on a grid of normalized electrical impedance ( z ). Depending on the region, it is also known as the Smith diagram, Volpert–Smith chart, or Mizuhashi chart. Because normalized impedance is complex, the chart maps two families of curves: circles of constant resistance ( \operatorname{Re}(z) ) and arcs of constant reactance ( \operatorname{Im}(z) ). Standard charts focus on passive circuits where resistance is non-negative ( \operatorname{Re}(z)\ge0 ). Regions outside the unit circle ( \operatorname{Re}(z) ) correspond to negative resistance, which is used for oscillator design and stability analysis. The display allows engineers to evaluate multiple RF parameters at once, including admittance, S_{nn} scattering parameters, noise figures, and stability boundaries. Although paper charts have mostly been replaced by software for calculations, the display format remains standard across RF simulation software and vector network analyzers to visualize how parameters change with frequency.

The Smith chart is a circular nomogram used in radio frequency (RF) engineering to solve transmission line and impedance-matching problems. It plots a complex reflection coefficient ( \Gamma ) on a grid of normalized electrical impedance ( z ). Depending on the region, it is also known as the Smith diagram, Volpert–Smith chart, or Mizuhashi chart. Because normalized impedance is complex, the chart maps two families of curves: circles of constant resistance ( \operatorname{Re}(z) ) and arcs of constant reactance ( \operatorname{Im}(z) ). Standard charts focus on passive circuits where resistance is non-negative ( \operatorname{Re}(z)\ge0 ). Regions outside the unit circle ( \operatorname{Re}(z) ) correspond to negative resistance, which is used for oscillator design and stability analysis. The display allows engineers to evaluate multiple RF parameters at once, including admittance, S_{nn} scattering parameters, noise figures, and stability boundaries. Although paper charts have mostly been replaced by software for calculations, the display format remains standard across RF simulation software and vector network analyzers to visualize how parameters change with frequency. ==Overview== The Smith chart converts normalized impedance into the complex reflection coefficient using a Möbius transformation. Impedances with positive real parts (passive loads) plot inside the unit circle, while those with negative real parts fall outside it. set up to display measured data on a Smith chart. For an impedance chart, the transformation is: \Gamma = \frac{Z - Z_0}{Z + Z_0} = \frac{z - 1}{z + 1}, where z = Z / Z_0 is the complex impedance Z normalized by the reference impedance Z_0 . Charts can show normalized impedance (Z chart), normalized admittance (Y chart), or both (YZ chart). On combined YZ charts, different colors distinguish the two grids. The outer border has scales marked in wavelengths and degrees. The wavelength scale measures distance along a transmission line in distributed-element circuits, while the degree scale shows the phase angle of the reflection coefficient. Because the chart uses normalized values, it applies to any system impedance. Its center represents the reference impedance (typically 50 Ω). To find the actual impedance or admittance, multiply the chart value by Z_0 or Y_0 . Reflection coefficients are unitless and read directly off the chart. Engineers use the chart for both distributed and lumped-element circuit analysis. For manual calculations, plotting a single point per frequency works well for narrowband applications (typically 5–10% bandwidth). Across wider bandwidths, connecting points from multiple frequencies forms a locus. This path shows how capacitive or inductive a load is, how difficult it is to match, and how well the component performs across the frequency range. If a locus covers a wide impedance range, details can become hard to read, though specific regions can be enlarged for accuracy. A transmission line with characteristic impedance Z_0 has a characteristic admittance Y_0 = 1 / Z_0 . Any load impedance Z or admittance Y is normalized by dividing it by the corresponding characteristic value: z = \frac{Z}{Z_0}, \quad y = \frac{Y}{Y_0} = \frac{1}{z}. While impedance is measured in ohms (Ω) and admittance in siemens (S), dividing by Z_0 or Y_0 cancels these units. The resulting normalized values are dimensionless quantities, which are the only values plotted directly on a Smith chart. \begin{align} V_\text{F}(\ell) &= A e^{+\gamma \ell}, \\ V_\text{R}(\ell) &= B e^{-\gamma \ell}, \end{align} where A and B are the complex voltage amplitudes at the load ( \ell = 0 ). The common time factor e^{j\omega t} is suppressed assuming steady-state single-frequency operation. The complex propagation constant is: \gamma = \alpha + j \beta, where \alpha is the attenuation constant and \beta is the phase constant. \Gamma = \frac{V_\text{R}}{V_\text{F}} = \frac{B e^{-\gamma \ell}}{A e^{+\gamma \ell}} = C e^{-2 \gamma \ell}, where C is a constant. For a uniform line with constant propagation constant \gamma , \Gamma varies with position. On a lossy line ( \alpha > 0 ), this variation forms a spiral toward the center. On a lossless line ( \alpha = 0 ), attenuation is zero, simplifying the expression to: \Gamma = \Gamma_\text{L} e^{-2j\beta\ell} = \Gamma_\text{L} \exp\left(-j \frac{4\pi \ell}{\lambda}\right), where \Gamma_\text{L} is the reflection coefficient at the load, \ell is line length from the load, and \lambda = 2\pi / \beta is the line wavelength. Because the angle changes by 2\pi radians over a distance of \lambda / 2 , the reflection coefficient and impedance repeat every half-wavelength. The outer scale of the Smith chart measures line length from 0 to 0.5\,\lambda . \begin{align} V_\text{F} + V_\text{R} &= V, \\ V_\text{F} - V_\text{R} &= Z_0 I. \end{align} Dividing these equations gives the relationship between normalized load impedance z and reflection coefficient \Gamma (a Möbius transformation): z = \frac{1 + \Gamma}{1 - \Gamma} \quad \iff \quad \Gamma = \frac{z - 1}{z + 1}. Because both \Gamma and z are dimensionless, frequency-dependent complex quantities, measurement values are plotted on a polar reflection diagram. Points inside the unit circle represent passive loads ( |\Gamma| \le 1 ). Substituting \Gamma(\ell) = \Gamma_\text{L} e^{-2j\beta\ell} into the impedance expression yields the standard transmission-line input equation: Z_\text{in} = Z_0 \frac{Z_\text{L} + j Z_0 \tan (\beta \ell)}{Z_0 + j Z_\text{L} \tan (\beta \ell)}. On the Smith chart, this transformation corresponds to drawing a circle centered at the origin through z . Moving along the line corresponds to rotating along this circle using the outer wavelength scale. Angles on the chart are measured counterclockwise from the positive horizontal axis. The positive real axis extends from the center ( z = 1 ) to infinity ( z \to \infty ). The upper half-plane represents inductive impedances ( \operatorname{Im}(z) > 0 ), while the lower half-plane represents capacitive impedances ( \operatorname{Im}(z) ). A perfectly matched load ( \Gamma = 0 ) sits at the center. Pure reactances lie on the outer boundary ( |\Gamma| = 1 ), with open and short circuits at extreme ends. Writing normalized impedance in terms of resistance and reactance as z = r + jx gives: \Gamma = \frac{r^2 + x^2 - 1}{(1+r)^2 + x^2} + j \frac{2x}{(1+r)^2 + x^2}. Plotting lines where r or x is constant forms two orthogonal families of circles on the complex plane. Standard charts plot these grid lines for passive conditions ( r \ge 0 ). Converting a point between normalized impedance z and normalized admittance y = 1/z corresponds to a 180° rotation around the origin at the same radius. For example, point \text{P}_1 ( z = 0.80 + j1.40 ) rotates to diametrically opposite point \text{Q}_1 ( y = 0.30 - j0.54 ). Engineers choose between Z and Y charts based on circuit topology. Series components add directly as impedances ( Z_\text{series} = Z_1 + Z_2 + \dots ), while parallel components add directly as admittances ( Y_\text{parallel} = Y_1 + Y_2 + \dots ). The normalized impedance and admittance for basic passive elements are: \begin{aligned} \text{Resistor:} &\quad z = \frac{R}{Z_0}, &\quad y = \frac{G}{Y_0} = \frac{Z_0}{R} \\[0.5em] \text{Inductor:} &\quad z = j \frac{\omega L}{Z_0}, &\quad y = -j \frac{1}{\omega L Y_0} \\[0.5em] \text{Capacitor:} &\quad z = -j \frac{1}{\omega C Z_0}, &\quad y = j \frac{\omega C}{Y_0} \end{aligned} where Z_0 = 1 / Y_0 is the characteristic impedance. Suppose a lossless air-spaced transmission line of characteristic impedance Z_0 = 50\,\Omega , operating at 800 MHz, is terminated with a circuit comprising a 17.5 Ω resistor in series with a 6.5 nH inductor. The reactance of the inductor at 800 MHz is: j \omega L = j 2 \pi f L = j32.7\,\Omega, giving a total load impedance of: Z_\text{T} = 17.5 + j32.7\,\Omega, and a normalized impedance of: z_\text{T} = \frac{Z_\text{T}}{Z_0} = 0.35 + j0.65. This point is plotted on the Z Smith chart as \text{P}_{20} . Extending the line from the origin \text{O} through \text{P}_{20} to the outer wavelength scale intersects it at L_1 = 0.098\lambda . Because the line is lossless, a circle centered at the origin through \text{P}_{20} represents the path of constant-magnitude reflection coefficient. At point \text{P}_{21} , this circle intersects the r = 1 circle of constant normalized resistance at: z_{\text{P}_{21}} = 1.00 + j1.52. Extending the line \text{OP}_{21} to the wavelength scale gives an intersection at L_2 = 0.177\lambda , making the distance from the termination: L_2 - L_1 = 0.177\lambda - 0.098\lambda = 0.079\lambda. For an air-spaced transmission line, the wavelength at 800 MHz equals its free-space value: \lambda = \frac{c}{f}, where c is the speed of light and f is frequency. This yields \lambda = 375\,\text{mm} , placing the matching component 29.6 mm from the load. The conjugate match for the impedance at \text{P}_{21} is: z_\text{match} = -j1.52. Since the chart is in the normalized impedance plane, a series capacitor C_\text{m} is required: z_\text{match} = -j1.52 = -\frac{j}{\omega C_\text{m} Z_0} = -\frac{j}{2 \pi f C_\text{m} Z_0}. Rearranging gives: C_\text{m} = \frac{1}{1.52\,\omega Z_0} = \frac{1}{1.52 \times 2 \pi f Z_0}, which yields C_\text{m} = 2.6\,\text{pF} . To match the load at 800 MHz, this capacitor is placed in series with the line 29.6 mm from the termination. An alternative shunt match can be found by transforming the chart from normalized impedance to normalized admittance. Point \text{Q}_{20} is the admittance equivalent of \text{P}_{20} ; extending \text{OQ}_{20} to the wavelength scale gives L_3 = 0.152\lambda . The nearest position toward the generator for a shunt conjugate match is \text{Q}_{21} (at the same location as \text{P}_{21} , but representing admittance): y_{\text{Q}_{21}} = 1.00 + j1.52. The distance along the line for this shunt match is: L_2 + L_3 = 0.177\lambda + 0.152\lambda = 0.329\lambda, or 123 mm. The conjugate matching component requires a normalized admittance of: y_\text{match} = -j1.52. A negative susceptance requires an inductor connected in parallel with the line. For an inductance L_\text{m} : -j1.52 = -\frac{j}{\omega L_\text{m} Y_0} = -\frac{jZ_0}{2\pi f L_\text{m}}, yielding L_\text{m} = 6.5\,\text{nH} . A suitable shunt match is therefore a 6.5 nH inductor in parallel with the line, 123 mm from the load. The circuit shown below is analyzed at an operating frequency of 100 MHz (wavelength \lambda = 3\,\text{m} ). It consists of two series capacitors and two shunt elements between the input and a load resistor. Because the component dimensions are on the order of millimeters, the lumped-element assumption holds. A reference impedance is still required for normalization; Z_0 = 50\,\Omega is chosen because it equals the load resistance R_1 . The analysis begins at the load and proceeds toward the input. Looking into R_1 alone, the normalized impedance is 1, represented by point \text{O} at the chart center. Points in the Z plane are labeled \text{P} and points in the Y plane are labeled \text{Q} ; series elements are added in the Z plane and shunt elements in the Y plane. The table records the complete path around the chart, returning to the center and confirming a matched 50 Ω condition at the input. The admittance (Y) Smith chart expresses the voltage reflection coefficient in terms of normalized admittance y instead of impedance z . Because y = 1/z : y = \frac{1-\Gamma}{1+\Gamma}, \quad \Gamma = \frac{1-y}{1+y}. The Y chart is identical to the Z chart, but rotated 180°. Writing y = g + jb for conductance g and susceptance b gives: \Gamma = \frac{1-g^2-b^2}{(1+g)^2+b^2} - j\frac{2b}{(1+g)^2+b^2}. The region above the horizontal axis represents capacitive admittances ( b > 0 ), and the region below represents inductive admittances ( b ). A matched load ( \Gamma = 0 ) sits at the center, while pure reactances lie on the bounding unit circle ( |\Gamma| = 1 ). ==Footnotes== ==References== ==External links== 3D Smith chart tool – interactive 2D and 3D Smith chart for active and passive circuits QuickSmith – open-source, browser-based Smith chart and impedance matching tool Smith Chart Studio – interactive Smith Chart with dynamic high-DPI zoom, VSWR overlays, and real-time impedance matching trajectory simulation

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