Magnitude and phase of I2/V1 for a 90° transmission line section

Magnitude and phase of V2/V1 for a transmission line section ended with:

The quantities I2/V1 and V2/I1 can be more interesting in some applications.

This article discusses the quantity I2/V1 for a special case, a 90° transmission line section.

90° transmission line sections are often used as a splitter / combiner / impedance transformer for two nearly identical antennas. I say “nearly” because no two antennas are likely to be exactly identical. They are sometimes called phasing harnesses.

Above, an example.

90° transmission line sections are often used as a splitter / combiner / impedance transformer for two nearly identical antennas. I say “nearly” because no two antennas are likely to be exactly identical.

So, a practical application is to feed two nominally 50Ω Yagis with a quarter wave of 75Ω line, and to parallel the inputs of those line sections. In broad terms, the 50Ω of each Yagi is transformed to \(Z_{in}=\frac{75^2}{50}=112.5\Omega\), the two in parallel is 56.25Ω… close enough to 50Ω.

Such a feed arrangement is intended to deliver almost identical drive to the almost identical Yagis.

What is identical drive?

Well for many antennas, one might prefer to deliver identical current (in magnitude and phase) irrespective of small differences in feed point impedance.

One of the interesting properties of the quarter wave section is that the quantity I2/V1 is approximately 1/Zo and fairly insensitive to Z2 or small changes in the electrical length.

Let’s look at an example, a 4 element Yagi for 144MHz.

Above is a Simsmith model of the feed components for one branch. Note the wide variation in Yagi feed point impedance (the magenta curve).

The magnitude and phase of I2/V1 are calculated and plotted.

Above, a plot of magnitude and phase of I2/V1. Note that the magnitude of I2/V1 is almost independent of frequency at 0.0133S, and phase is very close to 90° over the 4MHz sweep, even though feed point impedance varies quite widely. Note that the observed 0.0133S is approximately Yo=1/Zo.

So, when the two branches are feed in parallel (ie, V1 is identical), the feed point currents are almost equal even though there might be significant variation in feed point impedance from one Yagi to the other.

This property is sometimes spoken of as the current forcing property of a quarter wave.

Similar behavior is delivered by 3/4 wave branches, just the VSWR bandwidth is a little less. I have used 1/4 and 3/4 wave branches together for convenience (one Yagi inverted relative to the other), the difference in magnitude and phase of the drive current to each Yagi is tiny.

Note that connecting the inputs in series is not the same.

The maths

We can develop an approximate solution for the value of I2/V1.

Considering the transmission line section to be a two port network, we can say that \(\frac{I_2}{V_1}=\frac1{Z_2 cosh(\gamma l)+Z_0 sinh(\gamma l)}\). The result is dominated by \(Z_0 sinh(\gamma l)\) for the expected Z2. Since in this case \(\gamma l \approx \frac{j \pi}2\), \(sinh(\gamma l) \approx j\) in the region of interest so we can simplify to \(\frac{I_2}{V_1} \approx \frac1{jZ_0}=\frac1{Z_0} \angle -90 ^{\circ}
\).

Here is an example calc in Python for RG11A/U at 144MHz. gamma and length are specified in units of wavelength.

>>> import math
>>> import cmath
>>> z2=60+8j #a not very exact nominally 50 ohm antenna
>>> gamma=1.244e-2+6.283e+0j #from TLLC /wavelength
>>> l=0.25
>>> zo=75
>>> gl=gamma*l
>>> gl #approx j pi/2
(0.00311+1.57075j)
>>> y=1/(z2*cmath.cosh(gl)+zo*cmath.sinh(gl))
>>> y #approx -j/Zo
(-3.907497487344187e-06-0.01330011142300598j)
>>> cmath.polar(y)
(0.013300111997006295, -1.5710901211286914)
>>> cmath.phase(y)/cmath.pi*180
-90.0168331753713
>>> 1/abs(y) #note this is approximately Zo
75.18733678521569

Conclusions

The explanation applies only to the type of two way divider / combiner / transformer that has a quarter wavelength of line in each of its branches.

The common quarter wave divider / combiner / transformer has benefits beyond good VSWR.