Single-Stub Matching
Matches a complex load to the line with one shunt stub — a length of open- or short-circuited line tapped across the main line a distance d from the load. There are always two solutions per period: the stub is placed where the line admittance has the right conductance (Y₀), and its length is chosen so its susceptance cancels the remaining ±jB. The result is checked by transforming the load through the network and confirming the input VSWR is 1.
Equations & Parameters ▸
\(t=\tan\beta d=\dfrac{X_L\pm\sqrt{R_L\big[(Z_0-R_L)^2+X_L^2\big]/Z_0}}{R_L-Z_0}\quad(R_L\ne Z_0)\)
\(B=\dfrac{R_L^2\,t-(Z_0-X_L t)(X_L+Z_0 t)}{Z_0\big[R_L^2+(X_L+Z_0 t)^2\big]}\)
open stub: \(\ell/\lambda=\tfrac{1}{2\pi}\tan^{-1}(BZ_0)\); short stub: \(\ell/\lambda=-\tfrac{1}{2\pi}\tan^{-1}(Y_0/B)\)
\(B=\dfrac{R_L^2\,t-(Z_0-X_L t)(X_L+Z_0 t)}{Z_0\big[R_L^2+(X_L+Z_0 t)^2\big]}\)
open stub: \(\ell/\lambda=\tfrac{1}{2\pi}\tan^{-1}(BZ_0)\); short stub: \(\ell/\lambda=-\tfrac{1}{2\pi}\tan^{-1}(Y_0/B)\)
| RL, XL | Load resistance and reactance (Ω); ZL = RL + jXL. |
| Z0 | Characteristic impedance of the line and stub (Ω). |
| Stub | Open- or short-circuited shunt stub. |
| d, ℓ | Stub distance from load and stub length, in wavelengths. |
Reference: D. M. Pozar, Microwave Engineering, 4th ed., §5.2 (single-stub tuning).
Inputs
Ω
Real partΩ
Imag partΩ
ReferenceShunt stub
Results
Solution 1
Stub distance d—
Stub length ℓ—
Resulting VSWR—
Solution 2
Stub distance d—
Stub length ℓ—
Resulting VSWR—
Load
Reflection |ΓL|—
Load VSWR (unmatched)—
Diagram