ECE4122 Chap.4 Smith Chart and Impedance Matching
Smith Chart and Impedance Matching
The Smith chart is a coordinate system for normalised impedance, admittance and reflection coefficient. Its centre is the matched point. Motion along a lossless line follows a constant-|Γ| circle, while the rim wavelength scales record electrical distance.
Matching is a route chosen by topology. Series reactance adds naturally in impedance; shunt susceptance adds naturally in admittance.
A complete solution normalises, plots, moves in the stated direction, cancels the reactive part, denormalises and verifies the final reflection coefficient.
What this chapter covers
- 01
Normalised impedance coordinates
- 02
Admittance as the opposite chart point
- 03
Constant-VSWR motion
- 04
Wavelength scales and direction
- 05
Series and shunt matching
- 06
Quarter-wave transformer design
Worked example · free
Design a quarter-wave transformer
- +1Use transformer impedance Z_t = √(Z₀Z_L).
- +1Evaluate √(50 × 200) = 100 Ω.
- +1Choose electrical length λ_t/4 in the transformer medium.
- +1Verify Z_in = Z_t²/Z_L = 10000/200 = 50 Ω.
- +1Confirm Γ_in = 0 at the design frequency.
Key terms
- Normalised Impedance
- Load or input impedance divided by the system characteristic impedance.
- Normalised Admittance
- Admittance divided by characteristic admittance, equal to the reciprocal of normalised impedance.
- VSWR Circle
- A constant-reflection-magnitude circle followed by a lossless line transformation.
- Stub Matching
- Impedance matching using an open- or short-circuited line section that supplies a required susceptance or reactance.
- Quarter-Wave Transformer
- A λ/4 line section whose characteristic impedance transforms one real resistance into another at a design frequency.
Smith Chart and Impedance Matching FAQ
Why must impedance be normalised?
Normalisation lets one chart represent any characteristic impedance. Plot z = Z/Z₀, perform the transformation on the dimensionless chart, then multiply by Z₀ to recover ohms.
How do I switch from impedance to admittance?
Move to the diametrically opposite point through the chart centre, which represents taking the reciprocal. Keep the same constant-|Γ| circle and relabel the coordinate as y.
Why are there often two stub solutions?
The constant-VSWR circle can intersect the unity-conductance circle at two locations. Each location needs an opposite susceptance from a stub, producing two valid distance-and-length combinations with different practical properties.
Why is a quarter-wave match narrowband?
Its impedance inversion requires an exact electrical length of λ/4. A fixed physical section changes electrical length with frequency, so the upstream impedance and reflection move away from the match.
Exam move
Narrate every chart move in verbs. Normalise, plot, draw the circle, choose direction, rotate, convert to admittance if needed, cancel, denormalise and verify. Practise reading both z and y without confusing their signs. Use an algebraic Γ calculation as the final check, and write the wavelength distance separately from the stub electrical length. Build chart fluency with a written route beside every construction.
Record the real reference impedance, divide the load by it, and identify the resistance and reactance curves that locate the point. Draw the constant-reflection-magnitude circle and predict whether the load is inside or outside the unity-resistance region. Use the open, short and matched anchors to confirm orientation. When moving along a line, circle the requested direction and read the wavelength scale before rotating.
Choose coordinates from circuit topology. Remain in impedance when adding a series reactance. Rotate through the centre to admittance when adding a shunt susceptance, and label the reciprocal point explicitly. For a shunt-stub match, find every unity-conductance intersection, record the main-line distance for each, and then find the stub susceptance of the opposite sign.
Keep stub length separate from stub location because they answer different physical questions and can use different structures. Verify the graphical result algebraically. Convert the final normalised coordinate into reflection coefficient and check that its magnitude is zero or within plotting tolerance.
Denormalise impedance by multiplying by the reference value; denormalise admittance with the corresponding reference admittance. Convert electrical distance into physical length using the phase velocity of the actual line or stub. If the design uses different dielectric media, do not apply one wavelength to both sections. Practise bandwidth reasoning by perturbing frequency around the design point.
A quarter-wave transformer performs exact inversion only at its chosen electrical length, and a reactive stub supplies the required susceptance only at the chosen frequency. Track how the chart point leaves the centre rather than saying merely that bandwidth is limited. Compare two valid stub solutions for physical length, loss, component accessibility and sensitivity.
In a final check, confirm that passive points remain inside the unit reflection circle and that a half-wavelength move repeats the load. These anchors make a hand-drawn route auditable even when reading precision is modest.
Working through Smith Chart and Impedance Matching in ECE4122? Sia is AskSia’s AI Electrical Engineering tutor — ask any ECE4122 Smith Chart and Impedance Matching question and get a clear, step-by-step explanation grounded in how ECE4122 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
