This is the second installment of my article about coaxial stubs for filters on transmitters. In the first installment I derived the basic attenuation features of a 1/4 wave length stub connected between atransmitter and an antenna. In this installment I will attempt to answer the question about using 2 stubs together to get better rejection. Before I start this time I must make one note about the previous article... I made an over simplification in the calculation of the loss function that added about 6db of attenuation, the max attenuation for the single stub should have been about 31db instead of 37db as plotted.

I am now using the new version of Mathcad for Windows 3.0, it is much nicer than the original and I highly recommend it for anyone seriously planning to work on anything like this. (for you legal eagles out there: Mathcad is a product of MathSoft, Windows 3.0 is from Microsoft)

In working on this document I changed it quite a bit from the last version, I added separate parameters for the cable for the stubs so I could look at using different cable for the stubs than for the transmission line. I also added the formulas to vary the distance between the two stubs as well as the length of each stub. This results in a much more complex problem and it becomes very difficult at times to visuallize what is happening. I will attempt to show the various relationships by the use of plots of attenuation vs frequency and the distance between the stubs. First I will define the parameters for the feedline.

Z.0 is the characteristic impedance of the line

V is velocity factor,

a is loss in db/100'.

Z.0 = 50W

V = .66

a = 1 db/100'

Z.stub is the characteristic impedance of the stubs,

V.stub is the velocity factor and alpha.stub is the loss for the stub.

d.1 and d.2 are the lengths of the stubs and Z.L is the termination impedance.(0 ohms is of course a shorted stub).

Z.stub = 50W

V = .66

a.stub = 1 db/100'

d.1 = 11.5' d.2 = 11.5' Z.L = 0 W

The formulas for Z.trans and Z.par are defined at the end of the document. What they do is simple, Z.trans just uses Smith Chart formulas to transform an impedance from one end of a line to the other. Then Z.par uses the basic formula for paralleling two impedances. The index values (n) and (m) added to some of the variables enable me to vary them over a range of frequencies (for (n)) or to change the gap between stubs (for (m)).

• 1. Calculate Z.1 which is the impedance at the end of the first stub
• 2. Parallel first stub with main line = Z.a
• 3. Transform Z.a to connection point of other stub = Z.gap
• 4. Calculate Z.2 for impedance of second stub
• 5. Parallel second stub with Z.gap to get Z.net
• 6. Use Z.net to get reflection coeficient and then 'loss' value

Z.1(n)=Z.trans(Z.stub,V.stub,a.stub,Z.L,f(n),d.1)

Z.a(n)=Z.par(Z.1(n),Z.0)

Z.gap(n,m)=Z.trans(Z.0,V,a,Z.a(n),f(n),gap(m))

Z.2(n)=Z.trans(Z.stub,V.stub,a.stub,Z.L,f(n),d.2)

Z.net(n,m)=Z.par(Z.2(n),Z.gap(n,m))

Now we can play with fancy graphs to see what we have. First lets plot attenuation vs frequency for a gap of 0' (i.e. the two stubs are connected to the same point).

This shows the attenuation as the frequency is swept across the 20m band. Note that the max attenuation is about 6db better than a single stub.

Now lets see what happens when the distance between the stubs changes. For this plot I will hold frequency constant and vary the spacing. This shows the change from a spacing of 0' to 46', or about 1 wave length at 14.1 Mhz. Note the dips at 0, 1/4, 1/2, and 3/4 wavelengths.

This is a very interesting result, nothing at all like I had expected. But after long contemplation it seems to make sense. Consider first the farthest stub from the transmitter. It presents a very low impedance to the harmonic that is trying to be rejected, this low impedance is then moved along the transmission line toward the second stub. As this impedance is transformed along the 50 ohm line it changes from low to high and back over each 1/2 wave of the line so that after each 1/2 wave it is back to the original low value. At a point 1/4 wave along the line it presents a very high impedance. When the second stub is added it also presents a low impedance, but the total impedance seen by the source is a parallel combination of the transformed impedance from the first stub and the impedance of the second one. If they are both low at the same time the net impedance is even lower and better rejection results. When the transformed impedance of the first stub is high at 1/4 wave from its connection then the 2 stubs are fighting each other and the rejection is lower.

Now lets take a look at the pass band characteristics, in particular the impedance presented to the radio.

This shows the frequency dependance of the impedance at a spacing of 0' between stubs. Note how it changes over the width of the 20m band. It shouldn't be enough to affect most transmitters, but it does add another factor to tuning.

Now we take and hold frequency constant and vary spacing. Note that at 1/4 wave and 3/4 wave spacing the impedance is almost exactly 50 ohms.

And now lets sweep the frequency across the band at a spacing of 1/4 wave to see how this arrangement behaves. This change shouldn't bother any of todays transmitters or amps.

Now the trade off... For best rejection of harmonic, as I showed above, put the two stubs at the same point. For least effect on tuning of radio/amp put the two stubs 1/4 wave apart. Personally I put both stubs at the same point, it is easier to build, and I don't feel the change in impedance is severe enough to spend the extra time and effort to space them out.

For parallel impedances x and y. This can be used to figure net impedance of stub connected to feed line

Z.par(x,y)=1/((1/x) + (1/y))

Reflection coefficient given impedance of transmission line Z.0 and impedance of load Z.load.

r=(z.load - z.0)/(z.load + z.0)

Complex wave number, given f=frequency, V=velocity factor, alpha=loss of line. Constant c=velocity of light

b(f,V,a)=a+2ipf/Vc

Formula for transformed impedance along length of coax cable.
Input values are:
Z.line = characteristic impedance of transmission line
V = velocity factor
a = loss of line
Z.start = impedance at starting point(far end of line)
freq = frequency
len = length of line (in feet)

Z.trans(Z.line,V,a,Z.start,freq,len) = 

       ((Z.start + Z.line tanh(b(freq,V,a) len )
Z.line (---------------------------------------)
       ((Z.line + Z.start tanh(b(freq,V,a) len )

This is formula for loss due to reflection from mismatched load. Used here to figure reflection from intersection of stub with main transmission line.

loss(Z.load,Z.line) = 20 log(1 + r(Z.load,Z.line))