The Amplitude Response of a Coupled Transmission Line, All-Pass Network Having Loss

The voltage transfer function of a first- and second-order coupled transmission line, all-pass network having loss is derived. The amplitude response is calculated from the transfer function. It is shown that the amplitude response is completely determined when α, the real part of propagation function, is known. A method for calculating α is presented and an example is given. The results show that the losses (dielectric and conductor) cause periodic dips in the amplitude response of the all-pass networks. However, for practical materials and configurations, the peak amplitude loss is less than 0.4 dB for the first two periodicities. The results show that, for most applications, it is possible to cascade many first-order or second-order lines before the amplitude response must be equalized.