Impedance matching techniques

A mismatched load causes reflections, standing waves, power loss and possible damage to the source. Matching makes the line see a load equal to Z₀, so Γ = 0 and SWR = 1.

Matching techniques

1. Quarter-wave transformer (λ/4 section).
2. Single-stub matching (short- or open-circuited stub in shunt).
3. Double-stub (and triple-stub) matching: fixed stub positions, adjustable lengths.
4. Lumped L-C matching networks at low frequencies.
5. Tapered lines (exponential, linear) for broadband matching.
6. Changing the load itself (antenna design, loading coils).

Quarter-wave transformer

A λ/4 line of characteristic impedance Z₀′ inserted between the line and the load gives Z_in = Z₀′²/Z_L. For a match we need Z_in = Z₀:

Z₀′ = √(Z₀ Z_L) (for a real Z_L)

 Z0 line ----[ Z0' , λ/4 ]---- Z_L

Example: to match a 100 Ω load to a 50 Ω line, Z₀′ = √(50×100) = 70.7 Ω.

For a complex Z_L, move toward the generator to the nearest voltage maximum (or minimum). There the impedance is purely resistive (R_max = Z₀·S at a maximum, R_min = Z₀/S at a minimum), and the transformer is placed there with Z₀′ = √(Z₀ R_max) (or √(Z₀ R_min)). Limitation: it matches at one frequency only (narrow band).

Single-stub matching

A short-circuited stub is connected in parallel with the main line at a distance d from the load.

Generator ---------+--------- d ---------- Z_L
 (Z0)              |   (Z0)
                   +-- stub (length l, Z0, shorted end)

Working with admittances (y = Y/Y₀):

1. Find y_L = 1/z_L.
2. Move toward the generator a distance d until the normalised admittance becomes y = 1 ± jb (it meets the g = 1 circle).
3. The stub, of length l, supplies a susceptance ∓jb that cancels the imaginary part. A shorted stub has y = −j cot βl, so l comes from cot βl = ±b.
4. The total admittance is now Y₀, so the line is matched from the stub position back to the generator.

On the Smith chart: mark y_L, rotate it clockwise on its constant-|Γ| circle to the g = 1 circle (this gives d), read b, then move from the short-circuit point (y = ∞) toward the generator to the point −jb (this gives l). Only d and l have to be found, and no adjustable element is needed along the line, but the stub must be fixed at that particular point; this is why double-stub matching is used when the stub position cannot be changed.