Stub impedance matching

What is the process of stub impedance matching and what are the elements included in the calculations relevant to it?

• This question is too broad. Please include context and your research so far. May 13 '15 at 14:13

"Stubs" are sections of transmission line which are usually less than a half-wavelength long and either shorted or open on one end.

The two connections on the other end look like two terminals on a lumped impedance which can be either an inductor or a capacitor, depending on the length of the stub.

For a short-circuited stub, the impedance is:

$$j Z_0 \tan\left({2\pi\over\lambda}l\right)$$

where:

• $j$ is the imaginary unit,
• $Z_0$ is the characteristic impedance of the transmission line,
• $\lambda$ is the wavelength, and
• $l$ is the length of the line (in the same units as the wavelength).

An open-circuited stub is the same but negative:

$$-j Z_0 \tan\left({2\pi\over\lambda}l\right)$$

It is thus possible to construct a stub which has the same effect as any reactive component at a particular frequency. A distinct advantage is that stubs are in some cases more easily fabricated than capacitors or inductors, for example at microwave frequencies where transmission lines are easily fabricated on PCBs.

• In your second formula, shouldn't $\tan$ be $\cot$? Jun 20 '16 at 20:54 Quarter wave impedance transformer and quarter wave stub

The input impedance Zin of a length of coax depends on four factors: 1) its characteristic impedance, Z0, which is independent of the RF frequency; 2) its load impedance, Zload and 3) its length in terms of wavelength, the latter both at the fourth factor: the frequency of the applied RF. Note in all this that the wavelengths are in terms of the reduced velocity of RF in the feeder. The formula relating these is complex, but one simple case is often used, however note that this formula applies only to quarter wavelength lines:

The second use is a quarter wave matching transformer. 