**2. Dispersion properties of one-dimensional dielectric discontinuity**

Simplest structure suitable for electromechanical alteration of microwave characteristics is presented in Figure 1. In this structure two dielectrics are placed between infinite metal plates. The thickness *d* of the dielectric in the domain 2 may be variable.

**Figure 1.** One-dimensional dielectric discontinuity

Electromagnetic field of this structure can be described in terms of LM and LE modes. Transverse wavenumber of the LM mode can be found from dispersion equations:

$$\begin{aligned} \frac{\mathbb{B}\_{y1}^{e}}{\mathbb{E}\_{1}} \tan \left( \mathbb{B}\_{y1}^{e} h \right) + \frac{\mathbb{B}\_{y2}^{e}}{\mathbb{E}\_{2}} \tan \left( \mathbb{B}\_{y2}^{e} d \right) &= 0; \\ \mathbb{B} \left( \mathbb{E}\_{1} - \mathbb{E}\_{2} \right) k^{2} &= \mathbb{B}\_{y1}^{e} \overset{2}{\cdots} - \mathbb{B}\_{y2}^{e} \overset{2}{\cdots} \end{aligned} \tag{1}$$

where 1(2) *e <sup>y</sup>* is the transverse wavenumber in the region 1 or 2 respectively, 1(2) is the permittivity of the region 1 or 2 respectively, *h* and *d* are thicknesses of regions 1 and 2 respectively, *<sup>k</sup> <sup>c</sup>* is the wavenumber in free space, is the circular frequency, *c* is the light velocity in vacuum.

282 Dielectric Material

Application of piezoelectric or electrostrictive actuators opens an opportunity for electromechanical control over effective dielectric permittivity in microwave devices [1]. However for such applications a tuning system should be highly sensitive to rather small displacement of device components. The key idea how to achieve such a high sensitivity of system characteristics to small displacement of device's parts is to provide a strong perturbation of the electromagnetic field in the domain influenced directly by the mechanical control. For that, a tunable dielectric discontinuity (the air gap) should be created perpendicularly to the pathway of the electric field lines. This air gap is placed between the dielectric parts or the dielectric plate and an electrode. An alteration of the air gap dimension leads to essential transformation in the electromagnetic field, and revising of components' characteristics such as resonant frequency, propagated wave phase, and so on.

The goal of this chapter is to describe electromagnetic field phenomena in structures suitable

Simplest structure suitable for electromechanical alteration of microwave characteristics is presented in Figure 1. In this structure two dielectrics are placed between infinite metal

Electromagnetic field of this structure can be described in terms of LM and LE modes.

1 2

,

*<sup>y</sup>* is the transverse wavenumber in the region 1 or 2 respectively, 1(2) is the

(1)

tan tan 0;

*h d*

2 2 <sup>2</sup> 1 2 1 2

permittivity of the region 1 or 2 respectively, *h* and *d* are thicknesses of regions 1 and 2

Transverse wavenumber of the LM mode can be found from dispersion equations:

*k*

1 2

1 2

*e e y y e e y y e e y y*

**2. Dispersion properties of one-dimensional dielectric discontinuity** 

for electromechanical control of effective dielectric permittivity.

**Figure 1.** One-dimensional dielectric discontinuity

where 1(2) *e*

plates. The thickness *d* of the dielectric in the domain 2 may be variable.

Using equations (1) calculations of the transverse wavenumbers are carried out in a wide range of the permittivities and thicknesses of dielectrics in domains 1 and 2. It is found that the transverse wavenumbers as the solutions of equations (1) depend on frequency, permittivities of dielectrics and sizes of domains 1 and 2. Computed results are shown in Figure 2. This figure illustrates a dependence of normalized transverse wavenumber of domain 1 for fundamental LM mode versus the normalized air gap while permittivity of domains 2 is <sup>2</sup> 1 .

**Figure 2.** Normalized transverse wavenumber of fundamental LM mode versus normalized air gap size: (*a*) for certain normalized wavenumbers *kh* while 1 = 80; (*b*) for various permittivities of dielectric in domain 1 while normalized wavenumber is *kh* = 2.

As it is seen, transverse wavenumber of LM mode is very sensitive to variation of air gap between dielectric and metal plate. The change in only tenth or even hundredth part of percent from size of dielectric in domain 1 is sufficient for considerable alteration of transverse wavenumber. Required absolute change of air gap for significant alteration is not more than tens or hundreds micrometres depending on wavelength band and permittivity of dielectric in domain 1.

In contrast to LM mode distribution of electromagnetic field of LE mode is significantly less sensitive to variation of air gap. Transverse wavenumbers <sup>1</sup> *m <sup>y</sup>* and <sup>2</sup> *m <sup>y</sup>* of the LE mode are solutions of the dispersion equations:

$$\begin{aligned} \mathbb{B}\_{y1}^{m}\cot\left(\mathbb{B}\_{y1}^{m}h\right) + \mathbb{B}\_{y2}^{m}\cot\left(\mathbb{B}\_{y2}^{m}d\right) &= 0;\\ \mathbb{B}\_{y1}\left(\mathbb{B}\_{1} - \mathbb{B}\_{y1}\right)k^{2} &= \mathbb{B}\_{y1}^{m} - \mathbb{B}\_{y2}^{m}.\end{aligned} \tag{2}$$

Figure 3 illustrates a dependence of transverse wavenumber <sup>1</sup> *m <sup>y</sup>* of the basic LE mode on normalized air gap thickness. As it is seen, for the LE mode required change of air gap is comparable with size of dielectric in domain 1 and quantitatively the alteration is appreciably less than for the LM mode.

Electromechanical Control over Effective Permittivity Used for Microwave Devices 285

this structure can be described in terms of LM and LE modes represented by *y*-component of

**Figure 4.** Imaginary part of normalized transverse wavenumber of fundamental LM mode versus normalized air gap size: (*a*) for certain normalized wavenumbers *kh* while 1=80; (*b*) for various

(a) (b)

An incident wave in domain 1 is described by a sum of partial waves of LM and LE types:

*<sup>n</sup> j z e ee e i i*

1 0

1

*m mm m j z i i*

domain 1, *<sup>e</sup> X x* and *<sup>m</sup> X x* are solutions of the Helmholtz equation

*cY yX xe*

*cY y X x e*

*i n*

*e*

*<sup>i</sup> c* are amplitudes of partial waves, <sup>1</sup>

*i*

1

*m*

,

*zi*

*zi*

*<sup>e</sup> Y y <sup>i</sup>* and <sup>1</sup>

,

(3)

*<sup>m</sup> Y y <sup>i</sup>* are eigen functions of the

permittivities of dielectric in domain 1 while normalized wavenumber is *kh* = 2.

**Figure 5.** Structure that illustrated scattering problem

where *e m*( )

electrical *<sup>e</sup>* and magnetic *<sup>m</sup>* Hertz vectors.

**Figure 3.** Normalized transverse wavenumber of basic LE mode versus normalized air gap size: (*a*) for certain normalized wavenumbers *kh* while 1 = 80; (*b*) for various permittivities of dielectric in domain 1 while normalized wavenumber is *kh* = 2.

Peculiarity of the LM mode is existence of *<sup>y</sup> E* -component of electrical field which is directed normally to the border of dielectric discontinuity. For the LE mode the component *<sup>y</sup> E* is equal to zero. Therefore to achieve considerable alteration of electromagnetic field a border should be located between dielectric and air to perturb normal component of the electric field. This principle should be applied to all of electromechanically controlled microwave devices.

If the domain 1 contains lossy dielectric characterized by the loss tangent tan, then transverse wavenumber is a complex value and its imaginary part defines dielectric loss. Figure 4 demonstrates dependences of imaginary part of normalized transverse wavenumber of domain 1 for fundamental LM mode versus the normalized air gap.

Negative values of the imaginary part say, that dielectric losses in the structure would be reduced in comparison with homogeneous structure. Moreover, for certain frequency and air gap size the dielectric loss reaches a minimum. This effect is fundamental and is observed in more complicated tunable structures.

Rigorous simulation of electromechanically controllable microwave devices requires solving of scattering problem on dielectric wedge placed between metal plates, Figure 5. Solution of the problem by the boundary element method (BEM) is discussed below.
