Using the Smith Chart for Matching — Practical Impedance Matching with the Smith Chart
This lesson applies the Smith chart concepts from the previous lesson to real matching problems. The Smith chart excels at graphical matching because every matching operation — adding a series component, adding a shunt component, moving along a transmission line — corresponds to a specific type of geometric motion on the chart. Once you internalize the four basic moves, you can design and visualize matching networks without any calculation beyond plotting and reading.
This lesson assumes you have read the Introduction to the Smith Chart lesson and understand normalized impedance, resistance circles, reactance arcs, and constant-SWR circles. If you have not yet read that lesson, do so before continuing here.
The Four Basic Chart Moves
Every matching operation corresponds to movement on the Smith chart. The type of movement depends on what you are adding to the circuit:
| Operation | Chart move | Direction |
|---|---|---|
| Add series inductor (+jX) | Move along constant-r circle | Clockwise (upward, increasing x) |
| Add series capacitor (−jX) | Move along constant-r circle | Counter-clockwise (downward, decreasing x) |
| Add shunt inductor (+jB unwanted; actually −jB) | Move along constant-g circle on admittance chart | Counter-clockwise (toward short) |
| Add shunt capacitor (+jB) | Move along constant-g circle on admittance chart | Clockwise (toward open) |
| Move toward generator along line | Rotate around constant-SWR circle | Clockwise |
| Move toward load along line | Rotate around constant-SWR circle | Counter-clockwise |
The key distinction is whether you are adding a series element or a shunt (parallel) element. Series elements: work in the impedance (Z) domain and move along constant-r circles. Shunt elements: work in the admittance (Y) domain and move along constant-g circles. The trick to manual Smith chart work is remembering to switch between Z and Y representations as you add elements of each type.
The four basic chart moves. Series elements move along constant-resistance circles (impedance chart); shunt elements move along constant-conductance circles (admittance chart). A two-step L-network path is shown navigating from an arbitrary load impedance to the center (matched point).
View LargerAdding Series Elements
Adding a series element changes the total impedance by adding its reactance to the existing impedance. On the Smith chart, this moves the operating point along the constant-resistance circle that passes through the current point.
Starting at z = 2 − j1 (a resistance of 100 Ω with −50 Ω capacitive reactance on a 50 Ω system):
- Adding a series inductor with XL = +50 Ω moves along the r = 2 circle toward increasing x, arriving at z = 2 + j0 = 100 Ω purely resistive.
- From there, adding a series inductor with XL = +100 Ω moves further along the r = 2 circle to z = 2 + j2 = 100 + j100 Ω.
The key insight: series elements cannot change the resistive part of the impedance (cannot move you between r-circles). Only a lossless reactive element in series can be added; only a line section or a resistive element can change the real part.
Adding Shunt Elements
Adding a shunt (parallel) element is most conveniently handled in the admittance domain, since admittances add in parallel: Ytotal = Yload + Yshunt.
To add a shunt element on the Smith chart:
- Convert the current impedance point to its admittance equivalent by rotating 180° around the chart center. (On a ZY chart, the admittance point is already shown.)
- Move along the constant-conductance circle through the admittance point: clockwise for adding shunt capacitance (+jB), counter-clockwise for shunt inductance (−jB in susceptance terms).
- Rotate back 180° to return to the impedance representation.
Shunt elements cannot change the conductance (cannot move between g-circles in the admittance chart). They can only change the susceptance, which corresponds to moving along a constant-conductance arc.
L-Network Matching on the Chart
An L-network uses exactly two elements: one series and one shunt (or one shunt and one series). On the Smith chart, L-network matching is a two-step path from the load point to the center:
L-network example: Match ZL = 25 + j20 Ω to 50 Ω at 7.2 MHz.
1. Normalize: zL = (25+j20)/50 = 0.5 + j0.4. Plot this point.
2. We need to reach the center (z = 1, perfect match) in two steps. Choose a path through the r = 1 circle (which passes through the center).
3. First step — add a series element to move along the r = 0.5 circle until we reach the intersection with the r = 1 circle. The two intersections are at x ≈ +0.5 and x ≈ −0.5 on the r = 0.5 circle relative to r = 1 circle crossing. Starting at z = 0.5 + j0.4, moving along r = 0.5 by adding a series capacitor (decreasing x) brings us to z = 0.5 − j0.5 (intersection with x = −0.5 reactance arc on the r = 0.5 circle, at the r = 1 crossing: actually z = 0.5 + j0 when x = 0; we need to reach the r = 1 circle, not just the x = 0 axis). Using the Smith chart, the r = 0.5 circle intersects the r = 1 circle at approximately z = 0.5 ± j0.7. Moving from 0.5+j0.4 to 0.5−j0.7 requires adding ΔX = −1.1 × 50 = −55 Ω of series reactance → series capacitor: C = 1/(2π × 7.2×10⁶ × 55) ≈ 402 pF.
4. Now at z = 0.5 − j0.7 → Y = Z₀/Z = 50/(25−j35) = 50(25+j35)/(25²+35²) ≈ 0.72 + j1.01 normalized (inverted: y = 1/(0.5−j0.7) = (0.5+j0.7)/(0.25+0.49) = 0.68 + j0.95). The conductance g = 0.68 ≈ not yet 1. Re-read: at the intersection of r = 0.5 circle and r = 1 circle, the admittance has g = 1. That is the rule for L networks: get to the r = 1 circle, then add a shunt element to reach the center.
5. Second step — from z = 0.5 − j0.7 (which is on the r = 1 circle in admittance space = y = 1 + jB), add a shunt element to cancel the susceptance +jB and reach the center. Shunt inductor: B = ω/Z₀ of the element that cancels the admittance imaginary part.
The resulting L network: series capacitor ~400 pF + shunt inductor sized to cancel the remaining susceptance at 7.2 MHz, converting 25+j20 Ω to 50 Ω.
Single-Stub Matching on the Chart
The Smith chart provides the most intuitive approach to single-stub matching from the previous lesson. The graphical procedure replaces the formulas with two circle-following steps:
- Plot the normalized load admittance yL = 1/zL on the chart (rotate the load impedance point 180° to get the admittance point).
- Rotate clockwise around the constant-SWR circle (moving toward generator) until you reach the g = 1 circle. Mark this point — this is the stub connection position. Read the distance traveled in wavelengths from the outer ring scale.
- Read the susceptance b at the intersection. You need a stub with susceptance −b (to cancel the existing susceptance and land on y = 1 + j0 = the center).
- Design the stub length: for a short-circuit stub, the susceptance = cot(θ) / Z0; solve for θ. For an open-circuit stub, use tan(θ) / Z0. The stub length in wavelengths is read from the outer scale of the chart by moving from the short-circuit or open-circuit point to the required susceptance point.
The Smith chart intersection with the g = 1 circle has two solutions per revolution — a shorter stub placement (smaller d) with one stub length, and a larger d with a shorter stub. The first solution is usually preferred for minimum insertion loss and maximum bandwidth.
Quarter-Wave Transformer on the Chart
The quarter-wave transformer is particularly clean on the Smith chart. Starting at any purely resistive load z = r on the horizontal axis, rotating exactly 180° (a half-revolution of the chart, representing λ/4 of line) takes you to the point z = 1/r on the other side of the horizontal axis. The matching condition is that the characteristic impedance of the transformer section equals Z0,new = √(Zsource × Zload).
In Smith chart terms: the transformation maps the SWR circle through z = r to the same SWR circle but to the z = 1/r point. This does not reach the center unless the two points straddle the center (unless r × 1/r = 1, which means r = 1 — already matched). The quarter-wave transformer works because you choose a new characteristic impedance (new Z₀) that puts both the source and load on the r = 1 circle of the transformer section's normalized chart.
Reading Component Values from Chart Position
Once you have identified the starting impedance point and the target point after adding a component, the component value is determined by how far you moved along the circle:
ΔX = (xfinal − xinitial) × Z₀
If ΔX > 0 (moving up): series inductor L = ΔX / (2πf)
If ΔX < 0 (moving down): series capacitor C = −1 / (2πf × ΔX)
Shunt element value from chart move (admittance domain):
ΔB = (bfinal − binitial) / Z₀
If ΔB > 0 (shunt C): C = ΔB / (2πf)
If ΔB < 0 (shunt L): L = −1 / (2πf × ΔB)
Stub length from chart:
Read stub length in wavelengths from the outer ring of the Smith chart
Physical length = (wavelengths) × λ × velocity factor = (wavelengths) × vp × c / f
Practical Matching Workflow
A systematic workflow for Smith chart matching at a real workbench:
- Measure the load impedance with a vector network analyzer or antenna analyzer. Get R and X at the frequency of interest.
- Normalize: divide by Z₀ (50 Ω) to get r and x.
- Plot the load on the Smith chart at the intersection of the r-circle and x-arc.
- Choose a matching strategy:
- Load inside the r = 1 circle: L network with shunt element first (toward source side)
- Load outside the r = 1 circle: L network with series element first, or stub matching
- Pure reactance (on outer circle): short-circuit or open-circuit stub to resonate; then match the resistive component
- Execute the matching moves step by step on the chart, noting the magnitude of each move.
- Convert chart moves to component values using the formulas above.
- Verify: check that the final path arrives at or very close to the center of the chart (z = 1 + j0).
- Build and measure: measure the result with the analyzer and iterate if the actual impedance differs from the calculated value (due to component parasitics, component tolerances, or measurement errors).
In professional RF design, software tools such as Advanced Design System (ADS), NI AWR, and Keysight's PathWave perform all of these operations automatically. But the graphical procedure above produces identical results and builds physical intuition that no software can replace. Every experienced RF engineer can execute these steps mentally for simple matching problems.
Frequently Asked Questions
What if the load point is near the edge of the chart (high SWR)?
A load point near the edge of the chart represents a large mismatch (high SWR, |Γ| approaching 1). The matching path will require larger component values, and the design will be more sensitive to component tolerances and parasitic effects because small errors in impedance produce large errors in SWR when the impedance is far from the center. For very large mismatches (SWR greater than 10:1), a two-stage matching network is more practical than trying to match in a single step. Stage the matching: first bring the load somewhere on the r = 1 circle (into a reasonable mismatch range), then do the final match to center. This is especially important at VHF and above where distributed effects make precise component values difficult to achieve.
How do I match a complex load to a complex source impedance (not 50 Ω)?
The procedure is the same but you normalize to the source impedance rather than to 50 Ω. If the source impedance is ZS = 75 + j20 Ω, normalize everything to Z₀ = 75 Ω and use 75 Ω as the system reference for all chart operations. Alternatively, first conjugate-match the load to 50 Ω (using the chart as described in this lesson), then add a separate 50 Ω to ZS matching network. For maximum power transfer, the matching network must transform the load to the conjugate of the source impedance ZS* — so if ZS = 75 + j20 Ω, the target at the load end of the matching network is ZS* = 75 − j20 Ω, not 75 + j20 Ω.
Does the Smith chart handle resistive (lossy) elements?
Yes. Adding a series resistor moves the operating point along the constant-reactance arc from the current point toward higher r (to the right on the horizontal axis if no reactance, or along the x = constant arc). Adding a shunt conductance moves along the constant-susceptance arc toward higher conductance. Lossy transmission lines also move inward on the constant-SWR spiral rather than rotating on a constant-SWR circle. However, adding resistive elements is almost never the goal in a matching network since resistance dissipates the power you are trying to deliver. The lossy-line case is handled by reading the attenuation from the line's specification (dB/100 ft) and using the "reflection coefficient toward center" scale on some printed Smith charts to track the inward spiral.
Test Your Knowledge
Answer the questions below to check your understanding. Every answer can be found in the lesson above.