## Principles and functions

A transmission line is composed of two conductors, one representing the signal, and the other the ground. If we add a third conductor in the vicinity of the first two, we obtain a coupled line; one of the conductors always being connected to ground. The other two conductors Cdt1 and Cdt2 have potentials V1 (t) and V2 (t). If these two conductors are close enough to one another, then they can exchange energy.

We have a coupling phenomenon => coupled line.

– A signal that propagates in an unshielded line generates an electromagnetic field around it. A signal induced by this electromagnetic field appears at the terminals of a second line placed in the

vicinity of the first.

– The coupling of lines is by its nature distributed and the closer the lines are, the stronger the coupling will be.

– To build a coupled line, we therefore need at least three conductors close to each other, but it is also possible to use a larger number of conductors.

**Excitations of a coupled line with a geometric symmetry in the transverse plane (Cdt1 and Cdt2 are identical and positioned similarly with respect to the ground plane):**

• We define two types of excitation for a coupled line:

– Excitation in even mode: the currents in the conductors are of equal magnitude and flow in the same direction.

– Excitation in odd mode: the currents in the conductors are of equal magnitude and flow in opposite directions.

• Any excitation of a coupled line can always be decomposed into a superposition of a pair mode and an odd mode with appropriate amplitudes.

• The even and odd modes correspond to the two TEM (quasi TEM) modes, which can propagate in the coupled line.

**Homogeneous coupled line:**

• This propagation structure formed by three distinct conductors embedded in a homogeneous dielectric medium (e.g. triplate) propagates two TEM modes (even and odd mode).

• These two modes have the same propagation constant and the same phase velocity.

** Inhomogeneous coupled line:**

• The energy propagated by the inhomogeneous coupled line is distributed in several dielectrics with different permittivities (e.g. microstrip).

• We still have two propagation modes (even and odd), but they are no longer TEM. The phase velocities of the two modes are different. Nevertheless, below a frequency threshold that depends on the technology used, it is possible to use quasi-static (quasi-TEM mode) analysis methods, with very satisfactory results. We can therefore still apply the formalism of the even and odd modes.

When the lines operate in TEM or quasi-TEM mode (quasi-static approximation), their electrical properties can be fully determined by the capacitances and propagation velocities.

**Modelling of a coupled line by capacitances**

- C12: Capacitance of the two conductors in the absence of the ground plane.

• C11; C22: Capacitance between a conductor and the ground plane, in the absence of the other conductor.

• In the symmetric case, i.e. if the two conductors are identical and positioned similarly with respect to the ground plane, then: C11 = C22. Here, we consider this case.

Note: We consider an asymmetric coupled line (no geometric symmetry in the cross-section), operating in TEM or quasi-TEM mode. This asymmetric line propagates two independent fundamental modes, also called normal modes:

. C mode

. π mode

The study of this asymmetrical structure and its two fundamental modes is much more complicated than for a symmetric coupled line. This asymmetrical structure is analysed in the

following article:

V.K. Tripathi; “Asymmetric Coupled Transmission Lines in an Inhomogeneous Medium” IEEE Transactions on Microwave Theory and Techniques, Sep 1975, Vol 23, pp:734 – 739

## Coupled lines: even mode

Excitation in even mode: the currents in the conductors are of equal magnitude and flow in the same direction. The potentials of the two conductors are thus equal.

Ligne couplée en mode pair

An electrical open-circuit plane occurs between the two conductors. There is no difference in potential between the C12 terminals (the two conductors are at the same potential), which is therefore excluded from the equivalent schema

In even mode, the capacitance resulting from each line is:

Ce=C11=C22

Impédance caractéristique de la ligne couplée en mode pair :

ve : velocity of propagation on the line in even mode

L : inductance per unit length

εreff e : effective relative permittivity in even mode

Note: we considered μ = μ0

## Coupled lines: odd mode

Excitation in odd mode: the currents in the conductors are of equal magnitude and flow in opposite directions. The potentials of the two conductors are thus opposite.

An electrical short-circuit plane occurs between the two conductors. Thus, there exists a plane between the two conductors of the line, where the voltage is zero. There is a difference in potential here between the two terminals of the capacitance C12 (the two conductors have potentials of opposite sign, but the same amplitude), which is therefore conserved on the equivalent schema. The presence of the electrical short-circuit plane is thus modelled by a connection of the ground plane to “the middle” of C12.

The effective capacitance between each conductor and the ground plane is:

Co = C11 + 2C12 = C22 + 2C12

We define the characteristic impedance of the coupled line in odd mode:

vo: velocity of propagation on the line in odd mode

L: inductance per unit length

εreff o: effective relative permittivity in odd mode

Note: we considered μ = μ0

## Coupled lines in microstrip technology

Microstrip technology => Inhomogeneous coupled lines

Configurations of the electromagnetic field lines

The distributions and proportions of the electromagnetic energy in the two propagation media (air and substrate) are specific to each of the two propagation modes:

=> εreff e ≠ εreff o

=> ve ≠ vo