# 5.1: Introduction

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This chapter describes what happens when two transmission lines are so close together that the fields produced by one line interfere with the other line. Then a portion of the signal energy on one line is transferred to the other resulting in coupling. For microstrip lines the coupling reduces as the lines separate and usually the coupling is small enough to be ignored if the separation is at least three times the height of the strips. Transmission line coupling may be undesirable in many situations, but the phenomenon is exploited to realize many novel RF and microwave elements such as filters. A coupled pair of transmissions lines also enables the forward- and backwardtraveling waves on a line to be separately measured.

The chapter begins with a discussion of the physics of coupling which leads to the preferred description of propagation on a pair of **parallel coupled lines** (**PCL**s) as supporting an even mode and an odd mode, and forward- and backward-traveling versions of each. Many microwave circuit elements are based on PCL elements such as PCL filters. The separation of the fields into even and odd modes comes naturally from a consideration of the fields. This is leads to a low-frequency circuit model described in Section 5.4. Section 5.4.1 presents a high frequency model of a PCL with the \(L\) and \(C\) model of a single transmission line replaced by multiple inductances, capacitances, and mutual inductances and capacitances. As with single microstrip lines, formulas have been developed for the propagation characteristics of a pair of coupled microstrip lines, and these are presented in Section 5.6. The rest of the chapter builds up design concepts and presents some particular applications on the way.

Section 5.7 presents the analysis used to determine reflection coefficients of terminated coupled lines. The next section, Sections 5.8, describe two types of couplers that use a pair of coupled lines to selectively separate and tap off a portion of the power in either the forward-traveling wave, the backward-traveling wave, or both. A design technique for synthesizing them is presented and this is key to nearly all design using a pair of coupled lines as a circuit element. Section 5.9 presents the circuit models for various coupled line configurations and these models are essential to the synthesis of PCL-based microwave components. There are other ways to partition the modes on a pair of coupled lines and with RFICs the preferred description uses differential and common modes as described in Section 5.10. The difference between these modes and odd and even modes comes down to bookkeeping with a preference to use differential and common modes with circuits, and a preference to use odd and even modes when relating to fields. The final section, Section 5.11, describes another type of coupling, common impedance coupling. This is not always related to PCLs but this is the appropriate place to consider this phenomenon.