Magnitude and phase of V2/V1 for a 180° transmission line section

The discussion at Magnitude and phase of I2/V1 for a 90° transmission line raises the question whether something special happens for a 180° line section.

This article discusses the quantity V2/V1 for a special case, a 180° transmission line section.

180° transmission line sections are often used as part of a balun for VHF/UHF antennas.

Above is an application of a 180° line, a ‘half wave balun’, the U shaped section is 180° in electrical length.

In this role, the intention is to deliver at the end of the line section a voltage equal in magnitude and opposite in phase to that at the input, fairly independently of the impedance at the end of the section (within reason). Readers will recognise that this is the behavior of a good Voltage Balun.

Let’s look at an example, a 4 element Yagi for 144MHz with a folded dipole feed. The feed point data is derived from a single ended feed, and multiplied by two in the Simsmith model to simulate the impedance of one side of the folded dipole connected to the end of the 180° line section.

Above is a Simsmith model of the feed components for the 180° feed branch. Note the wide variation in Yagi feed point impedance (the magenta curve). Note that 50Ω coax is used (similar to RG/58AU), so there is a significant standing wave on the balun section, this is common practice and it illustrates an important aspect of the line section… that impedance looking in (the blue curve) is fairly similar to the load impedance (the magenta curve). Recall that the line section is only exactly 180° at one frequency, 144.2 MHz in this case. If lower loss coax was used, the two curves would be closer.

The magnitude and phase of V2/V1 are calculated and plotted.

Above, a plot of magnitude and phase of V2/V1. The scales are expanded, the magnitude is almost 1 with little variation and phase approximately 180° with only small variation… especially in the few hundred kHz of typical operating range for SSB.

So, when the two branches are feed in parallel (ie, V1 is identical), the feed point voltages are almost equal in magnitude and opposite in phase even though there might be significant variation in feed point impedance.

This property could be thought of  as the voltage forcing property of a half wave.

The maths

We can develop an approximate solution for the value of V2/V1.

Considering the transmission line section to be a two port network, we can say that \(\frac{V_2}{V_1}=\frac1{cosh(\gamma l)+\frac{Z_0}{Z_2} sinh(\gamma l)}\). The result is dominated by \(cosh(\gamma l)\) when Z2 is of the order of Zo. Since in this case \(\gamma l \approx j \pi\), \(cosh(\gamma l) \approx -1\) in the region of interest so we can simplify to \(\frac{V_2}{V_1} \approx -1=1 \angle -180 ^{\circ}
\).

Here is an example calc in Python for RG59A/U at 144MHz. gamma and length are specified in units of wavelength.

>>> import math
>>> import cmath
>>> z2=113+20j #
>>> gamma=3.447e-2+6.283e+0j #from TLLC /wavelength
>>> l=0.5
>>> zo=50
>>> gl=gamma*l
>>> gl #approx j pi
(0.017235+3.1415j)
>>> v2v1=1/(cmath.cosh(gl)+zo/z2*cmath.sinh(gl))
>>> v2v1 #approx -1
(-0.992518269569061-0.0013300496437132702j)
>>> cmath.polar(v2v1)
(0.9925191607522839, -3.1402525786628877)
>>> cmath.phase(v2v1)/cmath.pi*180
-179.92321936245702
>>> abs(v2v1) #note this is approximately 1
0.9925191607522839

Conclusions

A half wave of transmission line can be used as a phase inverter, and gives good results even in the presence of moderate standing waves on the line section.