Smith Chart Impedance Matching Calculator: L, Pi and T

All networks use a real source resistance $R_S$ and a load $R_L+jX_L$ at $f_0$. The Smith chart plots $\Gamma=(z-1)/(z+1)$ with $z=Z/R_S$. Own algebra implements the series/parallel transformations described by Long (UCSB Notes 5, pp. 3–8 and 26–29), Bowick and Pozar. Open worked numerical checks come from Analog Devices and Steer.

For a load-side L shunt, write the normalized load admittance as $g+jb$. Solve $b_t=\pm\sqrt{g-g^2}$, add shunt susceptance $b_t-b$, then add series reactance $x_s=b_t/g$. This requires $R_{L,\parallel}\ge R_S$. For a source-side shunt, solve $x_t=\pm\sqrt{r(1-r)}$, add series reactance $x_t-x$, then shunt susceptance $b_s=x_t/r$. This requires $R_L\le R_S$. Both signs are retained. Degenerate series elements become wires and degenerate shunts become opens.

Pi synthesis uses $R_{L,\parallel}=(R_L^2+X_L^2)/R_L$ and virtual resistance $R_v=\max(R_S,R_{L,\parallel})/(1+Q^2)$. T synthesis uses $R_v=\min(R_S,R_L)(1+Q^2)$. Each side is an L section with $Q_k=\sqrt{R_\mathrm{high}/R_\mathrm{low}-1}$. Combine the central series reactances in Pi or central shunt susceptances in T. Subtract the load susceptance from the Pi output shunt, or subtract its reactance from the T output series element. This is the maximum-section-Q approximation, not an exact stored-energy or 3 dB Q.

Convert with $X_L=\omega L$, $X_C=-1/(\omega C)$, $B_C=\omega C$, and $B_L=-1/(\omega L)$. E12/E24 mantissas follow the attached Vishay manufacturer sheet; nearest means minimum absolute difference, including the next decade.

For every sweep frequency, start with load current 1 and load voltage $Z_L$. Add each series impedance to voltage, or shunt admittance to current. A finite-Q part has series resistance $|X|/Q_\mathrm{part}$; shunt branches use the reciprocal of that same lossy series model. Independently accumulate $|I|^2R$ and $|V|^2G$ losses. Input reflection gives return loss and VSWR. Delivered load power is divided by available source power; dissipation loss is relative to accepted input power.

The sweep has 801 logarithmic points and includes $f_0$ exactly. Search outward from that point using uncapped reflection magnitude to bracket the band edges. Bisect each bracket against $|\Gamma(f)|=10^{-RL_\mathrm{goal}/20}$; the plotted 120 dB cap never enters this solve. Report clipped edges, or zero width if the center fails. Display return loss is capped at 120 dB and chart VSWR at 20; raw VSWR remains available in the calculation. The animation adds element reactance or susceptance at fixed $f_0$, not time-domain circuit dynamics. Every static and animated trace uses the same 129 points, uniform in circle angle between its exact endpoints. The normalized sagitta is at most $1-\cos(\pi/128)<0.000302$, below 0.077 pixel at the 760-pixel chart width.

• Lumped, linear, passive components; positive real source and load resistance. Source reactance is zero.
• The synthesis is lossless. Rounding and component Q are applied afterward without retuning.
• Each component Q is constant throughout the sweep. The selected value represents a model input, not a guaranteed component specification.
• Load resistance is constant. The series-L/C model uses $X(f)=X_0 f/f_0$ for $X_0\ge0$ and $X(f)=X_0 f_0/f$ for $X_0<0$. The alternative holds complex impedance fixed.
• P1 is nearest the load and P3 nearest the source. A zero in an L/C table column means no component of that type, not a zero-valued physical part.
• The two Pi/T families have equal-sign constituent L sections before absorbing load reactance. Mixed-sign Pi/T realizations and exhaustive topology optimization are outside this tool.
• Algebra, circuit solver and drawings are original implementations. Sources establish the method and independent worked values.

Use only while component dimensions and PCB interconnects are electrically small and all parts operate below self-resonance. The input range of 0.01–3000 MHz is a numerical range, not a guarantee that lumped models remain valid. Check manufacturer RF models, tolerances, parasitics and power ratings for an actual build.

The maximum-section Q convention is approximate, especially for comparable end resistances or appreciable load reactance. The calculated return-loss band is conditional on the selected load and constant-component-Q models. It is not an antenna bandwidth measurement, a 3 dB power bandwidth, or f₀/Q.

Band edges are refined by bisection of the uncapped reflection threshold after bracketing on the sweep. The plotted curve may visually undersample an exceptionally narrow peak, but the bandwidth solve is independent of its peak height. A clipped band is a lower bound within the shown window. No stubs, transmission-line matching sections, active or negative-resistance loads, nonlinear devices, layout optimization or component tolerance analysis are included. Preferred values do not imply stocked RF component values or appropriate ratings.