RF Electronic Devices and Systems

# Single-stub and double-stub matching

Stub tuning is an impedance matching technique, when an open-circuited or short-circuited transmission line is connected to the main transmission line. A stub is usually made as part of circuit which allows the avoidance of lumped elements. Coplanar waveguides or slot lines are usually connected to a stub in series; microstrips in parallel.

Parallel stub tuning is depicted in Figure 1. The parameter $d$ is chosen, so admittance is $Y={Y}_{0}+jB$ , and susceptance $–jB$ .

Figure 1.

Series stub tuning is depicted in Figure 2. The parameter $d$ is chosen so the impedance is  $Z={Z}_{0}+jX$, where reactance is $–jX$.

Figure 2.

Admittance and impedance are related with $Y=\frac{1}{X}$ .

By varying the parameter distance to the load, we can achieve the desired values of reactance and susceptance.

For this type of impedance, matching parameters and are important. Here is the analytical solution.

${Z}_{L}={R}_{L}+j{X}_{L}$ is a load impedance.

Then, the distance between stub and load can be found as:

The stub length equations are more complex (stub can be opened or short cutted):

When we have a series connection of stubs, the analytical solution will be the following:

${Y}_{L}={G}_{L}+j{B}_{L}$ is a load admittance.

The distance between load and stub is

The stubs lengths are (for short cut and open stub):

.

Double-stub matching is a type of matching where two stubs are shunted to main transmission line on a fixed distance.

This type of tuning is more favourable from a practical point of view. Double stub tuning is schematically depicted in Figure 3.

Figure 3.

Here two stubs are shunted to the main transmission line in a fixed position. Here is an analytical solution for double stub tuning.

Admittance of the first stub ${Y}_{1}={G}_{L}+j\left({B}_{L}+{B}_{1}\right)$, where load admittance ${Y}_{L}={G}_{L}+j{B}_{L}$ . Then for admittance of a second stub we have: ${Y}_{2}={Y}_{0}\frac{{G}_{L}+j\left({B}_{L}+{B}_{1}+{Y}_{0}\mathrm{tan}\beta d\right)}{{Y}_{0}+j\left(\mathrm{tan}\beta d\right)\left({G}_{L}+j{B}_{1}+j{B}_{L}\right)}$,

here ${G}_{L}={Y}_{0}\frac{1+\left(\mathrm{tan}\beta d{\right)}^{2}}{2\left(\mathrm{tan}\beta d{\right)}^{2}}\left[1±\sqrt{1–\frac{4\left(\mathrm{tan}\beta d{\right)}^{2}\left({Y}_{0}–{B}_{L}\mathrm{tan}\beta d+{B}_{1}\mathrm{tan}\beta d{\right)}^{2}}{{{Y}_{0}}^{2}\left(1+\mathrm{tan}\beta {d}^{2}{\right)}^{2}}}\right]$.

The lengths of stubs are:

.

Here ${B}_{1}=–{B}_{L}+\frac{{Y}_{0}±\sqrt{{Y}_{0}{G}_{L}\left(1+\left(\mathrm{tan}\beta d{\right)}^{2}\right)–{{G}_{L}}^{2}\left(\mathrm{tan}\beta d{\right)}^{2}}}{\mathrm{tan}\beta d}$, and ${B}_{2}=\frac{{G}_{L}{Y}_{0}±{Y}_{0}\sqrt{{Y}_{0}{G}_{L}\left(1+\left(\mathrm{tan}\beta d{\right)}^{2}\right)–{{G}_{L}}^{2}\left(\mathrm{tan}\beta d{\right)}^{2}}}{{G}_{L}\mathrm{tan}\beta d}$