Free Online Smith Chart Calculator & Impedance Matching Tool

Step 1

Smith chart

Set the frequency and load, then add series and shunt components between the source and the load. Series elements move the point along constant-resistance circles, shunt elements along constant-conductance circles, and a transmission line rotates it around the chart toward the generator.

Admittance grid Q circles

Frequency & load

Design frequency

Reference Z₀

Load

Load resistance R_L

Load reactance X_L

Add component

Connection

Type

Value

Network source → load

Drag rows to reorder · edit values in place

Impedance chart Z₀-normalised · hover to read any point

Resistance circles rReactance arcs xAdmittance gridLoad Z_L (dashed: measured load across its file)Input Z_inConstant-VSWR circleConstant-Q arcs

Circuit diagram source → network → load

Free online Smith chart calculator

By the NextAntennas RF engineering team · Updated 19 October 2026

This free online Smith chart calculator shows how series and shunt inductors, capacitors, resistors, transmission lines and stubs transform an impedance, one element at a time. Enter a complex load and a frequency, or load a measured antenna as a Touchstone .s1p file, build a ladder network between the source and the load, and read the input impedance, reflection coefficient, S11, VSWR and mismatch loss as you go. A matching solver designs exact L, π and T networks to Z₀, a frequency sweep shows the matched bandwidth, and a set of geometry tools lets you work the chart by hand, as you would on paper. It runs entirely in your browser, needs no sign-up and uploads nothing.

What the calculator does

Impedance and admittance Smith chart with optional admittance grid and constant-Q arcs
Lumped elements: series and shunt L, C and R with live traces
Distributed elements: transmission lines, open and short stubs, quarter-wave transformers
Measured loads: Touchstone .s1p, S11 of a .s2p, or frequency, R, X columns
Impedance matching calculator: exact L-network, π-network and T-network synthesis
Bandwidth: −10 dB band, fractional bandwidth and Q of every solution
Frequency response: S11, VSWR, R_in and X_in against frequency, with markers
Graphical tools: VSWR circles, intersections, stub positions, V_max/V_min, line lengths in λ
Export: PNG charts, CSV data and Touchstone .s1p for ADS, AWR, Genesys or scikit-rf
Share: one link holds the whole design, measured data included

How to use it

Smith chart: set the design frequency, the reference impedance Z₀ and the load R_L + jX_L. Add series or shunt L, C and R, transmission-line sections, and open or short-circuited stubs (length in λ, degrees or mm with ε_eff); each element draws its own coloured arc from the load toward the source, and the constant-VSWR circle through Z_in shows how far you are from a match. Drag rows to reorder them, edit values in place, turn on the admittance grid to follow shunt elements, and add constant-Q arcs to see each node's loaded Q.
Matching networks: the solver matches the current input impedance to Z₀ with two-element L-networks and three-element π and T networks. Select a card to preview its path and circuit, set a design Q for the π and T networks, and press Apply to add the parts to the chart.
Chart geometry: a construction board for classic graphical Smith chart work. Place points by clicking or by entering Z, Y or Γ, draw circles (centre and point, fixed radius, or three points), lines that extend both ways, rays, segments, perpendiculars and parallels, and intersect any two curves. Smith tools draw the constant-VSWR, r, x, g and b curves through a point, reflect a point through the centre to swap impedance and admittance, draw constant-Q arcs, mark the first voltage maximum and minimum, rotate a point along a lossless line by a length in wavelengths, and measure the line length between two points on the wavelengths-toward-generator scale. Scroll the mouse wheel (or pinch on a touchscreen) to zoom in on any point up to 1000×; the grid adds finer r and x circles as you zoom so you can place points precisely.
Frequency response: sweep the network across any band to plot S11, VSWR or R_in and X_in against frequency, with the band below your match threshold shaded and its edges, bandwidth and fractional bandwidth reported. The same sweep is drawn on a Smith chart, markers read exact values at any frequency, and the sweep exports as CSV or as a Touchstone .s1p file for your simulator.

Worked example: matching 60 − j80 Ω to 50 Ω at 2.4 GHz

A load of 60 − j80 Ω on a 50 Ω system has |Γ| = 0.593 at −46.8°, a VSWR of 3.91 and S11 of −4.5 dB, so only about 65 % of the power is delivered. Entering it in step 1 and opening step 2 gives six exact solutions. The low-pass L-network, a 77.3 fF shunt capacitor at the load and a 5.07 nH series inductor toward the source, has a nodal Q of 0.45 and stays below −10 dB from 1.37 to 3.36 GHz. The three-element π and T networks run at a design Q of 2 to 2.3 and narrow the band to between about 1.9–2.8 GHz and 2.1–2.6 GHz, depending on the variant, in exchange for more control over harmonics.

The same load can be matched with a single open stub: 0.110 λ of 50 Ω line from the load reaches the g = 1 circle at y = 1 + j1.47, and a shunt open stub of 0.3456 λ cancels the +j1.47. Step 3 finds both lengths with a VSWR circle and an intersection; step 1 simulates them (|Γ| = 0.005), and step 4 shows the stub match covering 2.25 to 2.65 GHz, a 16 % fractional bandwidth, narrower than the lumped L-network because the line and stub store more energy.

Smith chart equations

Each impedance is plotted at its reflection coefficient Γ = (Z − Z₀)/(Z + Z₀). A series element adds its reactance, Z_in = Z + jX, and moves the point along a constant-resistance circle; a shunt element adds its susceptance, Y_in = Y + jB, and moves it along a constant-conductance circle. Return loss is −20 log₁₀|Γ| and VSWR = (1 + |Γ|)/(1 − |Γ|).

Frequently asked questions

What is a Smith chart?

A Smith chart is a polar plot of the reflection coefficient Γ with circles of constant normalised resistance and arcs of constant normalised reactance drawn on it. Every passive impedance maps to one point inside the unit circle, the centre is a perfect match to Z₀, and adding a series or shunt component moves the point along a resistance or conductance circle.

How do I match a load to 50 Ω with an L-network?

Enter the load impedance and frequency, open the Matching networks step and pick an L-network. The calculator solves the two reactances exactly for the complex load, converts them to inductor and capacitor values at your frequency and checks that the result lands on 50 Ω. Apply adds the parts to the chart so you can see the path.

Which way does a series inductor move on the Smith chart?

A series inductor adds positive reactance, moving the point clockwise along its constant-resistance circle toward the upper half. A series capacitor moves it anticlockwise. Shunt elements move along constant-conductance circles: a shunt capacitor clockwise and a shunt inductor anticlockwise.

What VSWR or return loss is good enough?

A common target is S11 below −10 dB, which is a VSWR of about 1.9:1 and means about 90% of the power reaches the load. Demanding systems aim for −15 to −20 dB (VSWR 1.4 to 1.2).

What does Q mean for a matching network?

Q sets the bandwidth: roughly f₀/Q. An L-network's Q is fixed by the resistance ratio, Q = √(R_high/R_low − 1). π and T networks pass through a virtual resistance and can be designed for a higher Q, giving a narrower but more selective match. The calculator reports the actual −10 dB band of each solution.

How do I find a line length or stub position on the Smith chart?

Plot the load and draw its constant-VSWR circle, a circle centred on the chart that every point along a lossless line stays on. Moving toward the generator rotates the point clockwise, one full turn per half wavelength. Intersect the VSWR circle with the r = 1 circle (or the g = 1 circle for a shunt stub) and measure from the load to the intersection on the wavelengths-toward-generator scale; that is the line length. Step 3 does each of these constructions and reads the length directly, and step 1 can then simulate the line and stub to confirm the match.

How do I find the bandwidth of a matching network?

Build or apply the network, then open step 4 and sweep a band around the design frequency. The calculator computes S11 at every point, using the measured load at each frequency if you loaded a file or a fixed load otherwise, and reports the band where S11 stays below your threshold (−10 dB by default, VSWR about 1.9), with its edges and fractional bandwidth. Fit to band zooms the sweep onto it. Higher-Q networks and stub matches give narrower bands than low-Q lumped L-networks.

Can I use measured antenna data or share my design?

Yes. In step 1, switch the load to From file and drop in a Touchstone .s1p (or the S11 of a .s2p) from a VNA or a simulator; S, Z or Y data in RI, MA or dB formats are read, as are plain columns of frequency, R and X. The load then follows the measurement at every frequency, so the matching network and the step 4 sweep show the real band. The Share button packs the whole design, including the measured load, into a link; nothing is uploaded, and the person who opens it can return to their own work with one click.

How do I calculate VSWR from an impedance?

Find the reflection coefficient Γ = (Z − Z₀)/(Z + Z₀), then VSWR = (1 + |Γ|)/(1 − |Γ|) and return loss = −20 log₁₀|Γ| dB. For example, 100 Ω on a 50 Ω line gives Γ = 1/3, a VSWR of 2.0 and 9.5 dB return loss. Enter the impedance in step 1 and the calculator shows all three, along with S11 and mismatch loss.

What is single-stub matching?

Single-stub matching uses a length of line from the load to reach a point where the normalised admittance is 1 + jb, then a short- or open-circuited stub in shunt that adds −jb to cancel it. On the Smith chart the first length is the rotation along the load's VSWR circle to the g = 1 circle. Step 3 draws that construction and step 1 simulates the line and stub to confirm the match.

What is a quarter-wave transformer?

A λ/4 line of impedance Z₁ = √(Z₀ R_L) transforms a resistive load R_L to Z₀ at one frequency. For a 100 Ω load on a 50 Ω system the line is 70.7 Ω. Add a transmission line of 0.25 λ with that impedance in step 1 to see the point rotate half a turn around the chart to the centre, and sweep it in step 4 to see its bandwidth.

Is the Smith chart calculator free?

Yes. It is free, needs no sign-up and runs entirely in your browser; nothing you enter is uploaded.

More RF tools: microstrip and stripline impedance calculator · microstrip patch antenna calculator · all NextAntennas calculators.

Next step — lay out the line Microstrip & Stripline Impedance Calculator Find the trace width for your 50 Ω feed, with FEM accuracy, loss and 3D fields — free, no sign-up.

Engineering Theory

Theory Behind the Equations

How the Smith chart maps impedance to reflection coefficient, how series and shunt elements move the point, and how the calculator synthesises exact L, π and T matching networks.

1. Complex Impedance

Impedance Z is a complex quantity describing opposition to alternating current. It combines resistance (dissipative) and reactance (energy-storing) in the complex plane.

Impedance

Z = R + j X [Ω]

Inductor

Z_{L} = j ω L = j 2 π f L

Capacitor

Z_{C} = \frac{1}{j ω C} = \frac{- j}{2 π f C}

Admittance Y = 1/Z = G + jB is often more convenient for shunt (parallel) elements, since parallel admittances simply add.

Admittance

Y = \frac{1}{Z} = G + j B [S]

2. Reflection Coefficient Γ

The reflection coefficient Γ relates the reflected wave to the incident wave at a reference impedance Z₀ (typically 50 Ω). It is a complex number with |Γ| ≤ 1.

Definition

Γ = \frac{Z - Z_{0}}{Z + Z_{0}} = Γ_{r} + j Γ_{i}

Inverse

Z = Z_{0} \frac{1 + Γ}{1 - Γ}

The Smith chart is a polar plot of Γ. Every point on the chart corresponds to a unique normalised impedance z = Z / Z₀. The centre (Γ = 0) is the perfect match point z = 1 + j0.

Normalised impedance — dividing by Z₀ makes the chart universal:

Normalised

z = \frac{Z}{Z_{0}} = r + j x Γ = \frac{z - 1}{z + 1}

★ Canvas mapping used in this tool: x_px = C_x + R·|Γ|·cos(∠Γ), y_px = C_y − R·|Γ|·sin(∠Γ), where C = (320, 320) and R = 290 px.

3. Constant-r and Constant-x Circles

The curved grid lines on the Smith chart are circles of constant normalised resistance (r = const) and arcs of constant normalised reactance (x = const), mapped through the Γ transformation.

r-circles

{(Γ_{r} - \frac{r}{1 + r})}^{2} + Γ_{i}^{2} = {(\frac{1}{1 + r})}^{2}

x-arcs

{(Γ_{r} - 1)}^{2} + {(Γ_{i} - \frac{1}{x})}^{2} = \frac{1}{x^{2}}

The r-circle centres lie on the real axis at Γ_r = r/(1+r) with radius 1/(1+r). The x-arcs are centred at (1, 1/x) with radius 1/|x|. In this tool, resistance circles are drawn in green and reactance arcs in orange, with the r = 1 circle and x = ±1 arcs emphasised; the optional admittance grid adds the mirrored constant-g circles and constant-b arcs in violet.

4. Series and Shunt Component Transformations

Adding a component to a network changes the impedance seen at the input. The direction of movement on the Smith chart depends entirely on topology and component type.

Series element — impedances add directly:

Series

Z_{in} = Z_{prev} + Z_{comp}

Movement: along the constant-r circle of Z_prev. +jX (inductor) → clockwise; −jX (capacitor) → counter-clockwise.

Shunt element — admittances add directly:

Shunt

Y_{in} = Y_{prev} + Y_{comp} ⟹ Z_{in} = \frac{Z_{prev} \cdot Z_{comp}}{Z_{prev} + Z_{comp}}

Movement: along the constant-g (conductance) circle. +jB shunt capacitor → clockwise; −jB shunt inductor → counter-clockwise on the admittance chart.

5. L-Network Impedance Matching

An L-network uses one series and one shunt reactance to move any impedance with positive resistance to Z₀. For a load Z_L = R_L + jX_L there are two arrangements, and the calculator solves both exactly, so the load reactance is absorbed without approximation (Pozar §5.1).

Shunt element next to the load, series element toward the source. It exists whenever R_L² + X_L² ≥ Z₀R_L, which always holds for R_L > Z₀:

Shunt at the load

B = \frac{X_{L} \pm \sqrt{R_{L} / Z_{0}} \sqrt{R_{L}^{2} + X_{L}^{2} - Z_{0} R_{L}}}{R_{L}^{2} + X_{L}^{2}} X = \frac{1}{B} + \frac{X_{L} Z_{0}}{R_{L}} - \frac{Z_{0}}{B R_{L}}

B = shunt susceptance (S) · X = series reactance (Ω)

Series element next to the load, shunt element toward the source. It exists when R_L < Z₀:

Series at the load

X = \pm \sqrt{R_{L} (Z_{0} - R_{L})} - X_{L} B = \pm \frac{1}{Z_{0}} \sqrt{\frac{Z_{0} - R_{L}}{R_{L}}}

take the upper or the lower sign in both equations

Each sign choice gives a different network, so up to four L-networks are listed. Positive X becomes a series inductor and negative X a series capacitor; positive B becomes a shunt capacitor and negative B a shunt inductor. The bandwidth follows from the network Q, fixed by the resistance ratio:

L-network Q

Q = \sqrt{\frac{R_{high}}{R_{low}} - 1}

6. π-Network and T-Network Matching

Three-element networks are two back-to-back L-sections that meet at a virtual resistance R_v. Choosing R_v sets the Q, so bandwidth can be designed independently of the impedance ratio. The design Q must exceed the L-network value; by default the calculator uses Q = max(1.5·Q_min, 2), or the value you enter.

π-network (shunt – series – shunt). Write the load as an admittance G_L + jB_L with R_p = 1/G_L. The virtual resistance lies below both ends:

π virtual resistance

R_{v} = \frac{max (Z_{0}, R_{p})}{1 + Q^{2}} Q_{1} = \sqrt{\frac{Z_{0}}{R_{v}} - 1} Q_{2} = \sqrt{\frac{R_{p}}{R_{v}} - 1}

π element values

B_{1} = \pm \frac{Q_{1}}{Z_{0}} X_{s} = \pm (Q_{1} + Q_{2}) R_{v} B_{2} = \pm \frac{Q_{2}}{R_{p}} - B_{L}

B₁ at the source, B₂ across the load; the load susceptance is absorbed into B₂

T-network (series – shunt – series). The virtual resistance lies above both ends:

T virtual resistance

R_{v} = min (Z_{0}, R_{L}) (1 + Q^{2}) Q_{1} = \sqrt{\frac{R_{v}}{Z_{0}} - 1} Q_{2} = \sqrt{\frac{R_{v}}{R_{L}} - 1}

T element values

X_{1} = \pm Q_{1} Z_{0} B_{sh} = \pm \frac{Q_{1} + Q_{2}}{R_{v}} X_{2} = \pm Q_{2} R_{L} - X_{L}

X₁ at the source, X₂ next to the load; the load reactance is absorbed into X₂

The upper signs give the low-pass form (series L, shunt C) and the lower signs the high-pass form (series C, shunt L), although absorbing the load can flip the element next to the load. Every solution is verified by recomputing Z_in and is listed only if |Γ| < 0.01; its −10 dB band is found by sweeping the network across frequency with the load held constant.

7. Transmission Lines and Stubs

A lossless line of characteristic impedance Z_c and electrical length βℓ transforms the impedance at its far end. On a Smith chart normalised to Z_c, the point simply rotates clockwise around the centre, one full turn every half wavelength, which is why the wavelengths-toward-generator scale runs around the rim. When Z_c differs from the chart's Z₀, the path is still a circle, but centred on Z_c rather than on the chart centre.

Line input impedance

Z_{in} = Z_{c} \frac{Z_{L} + j Z_{c} \tan β ℓ}{Z_{c} + j Z_{L} \tan β ℓ} β ℓ = \frac{2 π ℓ \sqrt{ε_{eff}}}{λ_{0}}

λ₀ = free-space wavelength · a length in λ or degrees is taken as electrical length at f₀

A stub is a line section terminated in a short or an open circuit, so it behaves as a pure reactance. Below a quarter wave, a short-circuited stub is inductive and an open-circuited stub is capacitive; at λ/4 they swap roles, which is why stub reactance changes much faster with frequency than a lumped part's.

Stub reactance

Z_{sc} = j Z_{c} \tan β ℓ Z_{oc} = - j Z_{c} \cot β ℓ

connected in shunt, a stub adds susceptance and moves the point along a constant-g circle

Classic single-stub matching: move along the line from the load until the admittance reaches the g = 1 circle (1 + jb), then cancel the jb with a shunt stub of susceptance −jb. The construction tools in step 3 find the line length, and step 1 lets you add the line and the stub to check the match and its bandwidth.

8. Voltage Maxima, Minima and VSWR

Along a mismatched line the incident and reflected waves add in phase every half wavelength and cancel a quarter wave later. The voltage maximum occurs where Γ is real and positive, on the right of the chart, and the minimum where it is real and negative. At those points the impedance is purely resistive, equal to Z₀·VSWR and Z₀/VSWR.

Standing wave

VSWR = \frac{V_{\max}}{V_{\min}} = \frac{1 + | Γ |}{1 - | Γ |} R_{\max} = Z_{0} \cdot VSWR R_{\min} = \frac{Z_{0}}{VSWR}

the first V_max and V_min lie a quarter wavelength apart

The distance from a load to its first voltage minimum, read on the wavelengths-toward-generator scale, is the quantity a slotted-line measurement gives. Together with the VSWR it fixes the load impedance, which is how loads were measured before vector network analysers. For 25 + j40 Ω on a 50 Ω line the VSWR is 3.49, the first maximum is 0.1305 λ toward the generator (174.7 Ω) and the first minimum 0.3805 λ (14.3 Ω).

9. S-Parameters & Return Loss

S₁₁ (input reflection coefficient) is identical to Γ when all other ports are terminated in Z₀. Return loss and VSWR are scalar metrics derived from |Γ|.

S₁₁

S_{11} = Γ = \frac{Z_{in} - Z_{0}}{Z_{in} + Z_{0}}

Return Loss

RL = - 20 \log_{10} (| Γ |) [dB]

VSWR

VSWR = \frac{1 + | Γ |}{1 - | Γ |}

|Γ| circles

| Γ | = \sqrt{Γ_{r}^{2} + Γ_{i}^{2}} \in [0, 1]

A perfect match gives |Γ| = 0, RL = ∞ dB, VSWR = 1:1. An open or short circuit gives |Γ| = 1, RL = 0 dB, VSWR = ∞.

10. Touchstone Files and Measured Loads

Vector network analysers and EM simulators export reflection data as Touchstone files. A one-port .s1p holds one line per frequency with S11 as real and imaginary parts (RI), magnitude and angle (MA) or dB and angle (DB). The option line, for example # GHz S MA R 50, sets the frequency unit, the parameter, the format and the reference resistance. The calculator converts each point to an impedance and interpolates between points, so a matching network can be designed and swept against the antenna as it was actually measured.

S11 to impedance

Z_{L} = R_{ref} \frac{1 + S_{11}}{1 - S_{11}} | S_{11} | = 10^{{dB}_{} / 20}

Z and Y data in a Touchstone file are normalised to R_ref

A match designed at one frequency on a measured antenna is only as good as the measurement: de-embed cables and fixtures to the antenna feed point first, and check the matched band in step 4 before building it.

11. Extracting L and C from Reactance

Once the required reactance X is known at the design frequency f, the physical component value is recovered by inverting the impedance formula.

Inductance

L = \frac{X}{ω} = \frac{X}{2 π f} [H]

Capacitance

C = \frac{1}{ω | X |} = \frac{1}{2 π f | X |} [F]

Values are automatically expressed in the most readable SI prefix unit (pH/nH/µH or pF/nF/µF) depending on the frequency range. At microwave frequencies (GHz), typical matching inductors are in the single-digit nH range and capacitors in the low-pF range.

12. Admittance, Conductance and Susceptance

For shunt (parallel) elements it is more natural to work in admittance space because parallel admittances simply add, just as series impedances do.

Admittance

Y = \frac{1}{Z} = \frac{Z_{r} - j Z_{i}}{Z_{r}^{2} + Z_{i}^{2}} = G + j B

Conductance G

G = \frac{Z_{r}}{Z_{r}^{2} + Z_{i}^{2}} [S]

Susceptance B

B = \frac{- Z_{i}}{Z_{r}^{2} + Z_{i}^{2}} [S]

Inductor susceptance is negative (B_L = −1/ωL) while capacitor susceptance is positive (B_C = ωC). On the Smith chart, shunt elements move the point along constant-conductance circles — a complementary grid to the resistance circles used for series elements.

Shunt add

Y_{in} = Y_{prev} + Y_{shunt} ⟹ Z_{in} = \frac{1}{Y_{in}}

13. Q Factor and Bandwidth

The quality factor Q of a matching network characterises how sharply it filters signals. Higher Q means narrower bandwidth — a critical trade-off in RF design.

Definition

Q = \frac{f_{0}}{B W} = \frac{| X |}{R}

For a two-element L-network, Q is fixed by the impedance transformation ratio — it cannot be chosen freely. The required Q is:

L-net Q

Q = \sqrt{\frac{R_{high}}{R_{low}} - 1}

For three-element networks (π and T), Q can be chosen freely as long as it exceeds this minimum. This tool uses Q = max(1.5 · Q_min, 2) by default, or the design Q you enter, to give a practical bandwidth while keeping element values sensible. The resulting 3 dB bandwidth is:

Bandwidth

B W = \frac{f_{0}}{Q} ⟹ f_{low} = f_{0} - \frac{B W}{2}, f_{high} = f_{0} + \frac{B W}{2}

★ Each matching solution card shows its Q and the −10 dB band found by a frequency sweep — lower Q gives a broader match.

14. Constant-Q Circles and Bandwidth

The nodal quality factor of a point on the chart is the ratio of its reactance to its resistance. Every point with the same Q lies on one circular arc through the short- and open-circuit points, so a family of Q arcs shows at a glance how much stored energy a matching path passes through. Keeping the whole path inside a low-Q arc is the classic way to design a broadband match.

Nodal Q

Q_{n} = \frac{| x |}{r} u^{2} + {(v \pm \frac{1}{Q_{n}})}^{2} = 1 + \frac{1}{Q_{n}^{2}}

Γ = u + jv · the upper (inductive) arc takes the + sign, the lower (capacitive) arc the − sign

For a two-element L-network the loaded Q is roughly half the highest nodal Q on its path, and the fractional bandwidth is about 1/Q. A 60 − j80 Ω load matched to 50 Ω at 2.4 GHz shows the effect: the low-Q L-network stays below −10 dB from about 1.37 to 3.36 GHz, while a single-stub match of the same load covers only about 2.25 to 2.65 GHz, because the line and stub store far more energy.

Turn on Q circles in step 1 to overlay arcs for any Q values, or use the Q arc tool in step 3 to draw the arc through a particular point.

15. Reference Impedance and Normalisation

The Smith chart is drawn for a specific reference impedance Z₀. In this tool Z₀ = 50 Ω, the universal RF/microwave standard. All impedances are normalised before plotting.

Normalise

z = \frac{Z}{Z_{0}} = r + j x

De-normalise

Z = Z_{0} \cdot z = Z_{0} (r + j x)

The chart centre (Γ = 0) corresponds to z = 1 + j0, i.e. Z = 50 + j0 Ω — a perfect impedance match. Points on the outer boundary (|Γ| = 1) represent either a short circuit (left edge, z = 0) or an open circuit (right edge, z = ∞).

Key points

\begin{array}{ll} Z = Z_{0} & ⟹ Γ = 0 (match) \\ Z = 0 & ⟹ Γ = - 1 (short) \\ Z \to \infty & ⟹ Γ = + 1 (open) \end{array}

16. Return Loss, Mismatch Loss and Power Transfer

Three key metrics characterise how well the network matches at a given frequency. All are derived from the reflection coefficient Γ at the network input.

Return Loss

RL = - 20 \log_{10} (| Γ |) [dB]

Mismatch Loss

ML = - 10 \log_{10} (1 - {| Γ |}^{2}) [dB]

Power delivered

P_{del} = P_{avail} \cdot (1 - {| Γ |}^{2})

VSWR

VSWR = \frac{1 + | Γ |}{1 - | Γ |}

A perfect match has |Γ| = 0, RL = ∞ dB, ML = 0 dB, and VSWR = 1:1. An open or short circuit has |Γ| = 1, RL = 0 dB, and VSWR = ∞. S11 in dB, as displayed by the tool, is the negative of the return loss: S11 = −RL.

17. Complex Arithmetic Used in the Engine

Every impedance in this tool is represented as a complex number Z = {r, i} where r is the real part and i is the imaginary part. The following primitive operations are used throughout the trace computation, matching solver, and coordinate transforms.

Addition

a + b = (a_{r} + b_{r}) + j (a_{i} + b_{i})

Multiplication

a \cdot b = (a_{r} b_{r} - a_{i} b_{i}) + j (a_{r} b_{i} + a_{i} b_{r})

Division

\frac{a}{b} = \frac{a_{r} b_{r} + a_{i} b_{i}}{b_{r}^{2} + b_{i}^{2}} + j \frac{a_{i} b_{r} - a_{r} b_{i}}{b_{r}^{2} + b_{i}^{2}}

Modulus

| Z | = \sqrt{Z_{r}^{2} + Z_{i}^{2}}

Parallel combination of two impedances (used for shunt elements) is computed as:

Parallel Z

Z_{par} = \frac{Z_{1} \cdot Z_{2}}{Z_{1} + Z_{2}}

★ Division is implemented by multiplying numerator and denominator by the complex conjugate of the denominator, keeping all arithmetic real-valued.

References

D. M. Pozar, Microwave Engineering, 4th ed., Wiley, 2012, ch. 2 and §5.1.
C. Bowick, RF Circuit Design, 2nd ed., Newnes, 2008, ch. 4.
P. H. Smith, "Transmission line calculator," Electronics, vol. 12, pp. 29–31, January 1939.