A double stub tuner is an impedance matching technique commonly employed in microwave circuitry. It consists of two lengths of transmission line (the stubs) connected to the main transmission line at fixed distances from the load impedance. The stubs are can be connected either in parallel or series, but parallel is much more common. The stubs may be shorted or open at the ends, but open is more common (the only way to go for microstrip).

The double stub tuner, like other common impedance matching techniques such as the single stub tuner and the quarter wavelength transformer, has narrow bandwidth. A circuit is generally tuned to a given frequency, and can be expected to work well only near (10% - 20%) that frequency.

The double stub tuner, like the single stub tuner has two degrees of freedom. These are the lengths of the two stubs. The distance of the first stub from the load is fixed, as is the distance of the second stub from the first. I'll call the distance from the load to the first stub d1, and the distance from the first stub to the second d2. The lengths of the lines will be denoted as l1 and l2.

       |      d2         |    d1
-----------------------------------| load (Zl) |
       |                 |
       | l2              | l1
       |                 |
       |

The lengths of the stubs may be found either by calculation or by using a Smith chart. Since the calculations involve complex numbers and can be somewhat tedious, I'll describe the method for using the Smith Chart.

The perform this procedure, you'll need to know the load impedance you are trying to match, the characteristic impedance of the transmission line, the wavelength of the signal you are matching for, and the distances of the stubs from the load.

  1. Normalize the load impedance by dividing it by the characteristic impedance of the transmission line. Plot this on the Smith chart.
  2. If your stub is in parallel with the transmission line, convert the impedance to an admittance by rotating it a quarter of a wavelength (half way) around the Smith chart.
  3. Convert the distances d1 and d2 to electrical distance by dividing them by the wavelength.
  4. Rotate your load point an electrical length of d1 toward generator. This gives you the load at the point on the transmission line where the first stub is connected. Call this point A.
  5. Starting at 0.25 λ, go toward the load a distance of d2. Draw a line between the origin of the Smith chart and the point on the outside edge you have just found. Find the center of this line and draw a circle around this point, so that the circle touches the origin and the edge of the chart. This is the unit resistance circle transformed from the second stub toward the load to the point where the first stub joins the transmission line. This circle is referred to as the L circle. We do this because we want to have the load impedance/admittance on the unit resistance circle when we are at the second stub, so that we can add a pure reactance to match the line.
  6. Move the load point along a line of constant resistance until it lies on the L circle. There will either be two possible points or none at all. Not all loads can be matched with the double stub technique. Assuming you find a point (call it B) that intersects the L circle, read the inductance / capacitance value off of the Smith chart.
  7. We are now at the point where the transmission line and the first stub from the load meet. At this point, we want our impedance / admittance point to be in our L circle. We know the value A, and the value of B. B = A + C, so find the complex part of C. This is the value you have to cancel out with the first stub.
  8. If the first stub is open, start at the point of infinite impedance (zero admittance) and rotate toward generator until you reach 0 - jA. If the first stub is shorted, start at zero impedance instead. The distance around the chart to get to this point is the electrical length of the stub; multiply by the wavelength to get l1.
  9. Now we want to examine the impedance / admittance at the point where the second stub from the load connects to the transmission line. To do this, we have to move point B along a circle of constant VSWR (a circle centered on the origin of the Smith chart) until it is on the circle of unit resistance. Call this point on the circle of unit resistance D.
  10. Now read off the complex part of D. This is the value we have to cancel out with the second stub. Find the negative of the complex part of D on the edge of the Smith chart. If the second stub is open, start at zero admittance (infinite impedance) and rotate toward the generator until you come to the aforementioned point. The distance traveled around the Smith chart, when multiplied by the wavelength for which we are tuning, gives l2.

The double stub tuner is slightly more complicated to design than the single stub tuner, the quarter wavelength transformer, or reactive matching. The main difference is that when finding the length of the first stub, you have to add the transformed load impedance or admittance to the stub impedance or admittance such that the result is located on the L circle.

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