This is the text accompanying a letter I sent to QST in January 1996, suggesting this for a possible technical correspondence column. It was rejected by the editors.

**Kevin Schmidt, W9CF**

The general magic bullet would produce any desired relative phase. In the following I show that such a network cannot be realized with a reciprocal lossless network except for phase differences of 0 or 180 degrees. A nonreciprocal lossless network can produce other phase shifts. I show how to make a nonreciprocal transmission line by using a circulator. This produces the magic bullet, but the requirement that nonreciprocal elements are needed will probably limit the usefulness to UHF and microwave frequencies.

(1) |

(1 - y_{22}Z_{2})I_{2} - y_{23}Z_{3}I_{3} = y_{21},
| |||

- y_{32}Z_{2}I_{2} + (1 - y_{33}Z_{3})I_{3} = y_{31}.
| (2) |

Solving for the ratio of the currents gives,

= . | (3) |

= exp(j).
| (4) |

y_{22} = y_{23} = y_{32} = y_{33} = 0.
| (5) |

For the cases of 0 and 180 degrees, Lewallen's current forcing
solution using
1/4 wavelength lines, gives
*y*_{22} = *y*_{32} = *y*_{23} = *y*_{33} = 0 ,
and
*y*_{21}/*y*_{31} = 1 , as required.

A lossless network cannot absorb power. This requirement is expressed by the equation

V^{ *}_{m}(y_{mn} + y^{ *}_{nm})V_{n} = 0,
| (6) |

If I now assume that the network is reciprocal,
*y*_{mn} = *y*_{nm} . Therefore the off diagonal
elements must be purely imaginary as well. The ratio of two of
these admittances can have only a phase of 0 or 180 degrees.

This says that for a reciprocal lossless network, only 0 or 180 degree phase differences can be ``current forced.'' Since capacitors, inductors, transformers, and normal transmission lines are reciprocal elements, no combination of them can produce the magic bullet.

If I introduce ferrites, or other nonreciprocal elements, it is possible to current force other phase differences.

The form of the admittance matrix required for current forcing is suggestive,

(7) |

(8) |

= | (9) |

(10) |

The admittance matrix can be derived from the ABCD matrix to give:

= exp(- ja)/Z_{0}
| (11) |

A standard circulator can be used to produce current forcing lines with any phase shift. The general line can be produced using a circulator where power into port 1 goes to port 2, power into port 2 goes to port 3, and power into port 3 goes to port 1. If I put an adjustable shorted line on port 3, the forward wave travels ``directly'' to port 2, the reverse wave travels from 2 to 3 where it travels up the adjustable line, reflects, and then travels back to port 1. By adjusting the line, I can make the electrical length in the reverse direction anything I want. I now simply add extra line to port 2 so the electrical length from 1 to 2 is what I need, and adjust the shorted stub on 3 to give a total electrical length that is an odd multiple of a half wavelength.

The problem with this nonreciprocal technology is that it works well only at UHF or higher frequencies.

The special case of the nonreciprocal element with *a* = 0 and
*b* = was postulated by Tellegen[2] in 1948. He
called it a gyrator using
a mechanical analogy where nonreciprocal elements correspond to
spinning flywheels (i.e. gyroscopes).
Within a few years, a ferrite gyrator and other nonreciprocal elements
for microwave frequencies had been described[3].
If a medium or high frequency gyrator
could be constructed, it would provide the necessary element
to produce a quadrature feed without measuring element impedances.

For a quadrature feed, one antenna is connected to a quarter wavelength transmission line, and the other to a gyrator of the same characteristic impedance. The ends of the gyrator and the quarter wavelength line are connected together and driven by the same source.

**1**-
R.W. Lewallen, page 8-14 in
*The ARRL Antenna Book*, 17th Edition, edited by R.D. Straw, (ARRL,Newington,1994). **2**-
B.D.H. Tellegen,
*The Gyrator, a New Electric Network Element*, Philips Research Reports**3**, 81 (1948). **3**-
C.L. Hogan,
*The Ferromagnetic Faraday Effect at Microwave Frequencies and Its Applications*, Reviews of Modern Physics**25**, 253 (1953).