**5. Effective permittivity of microstrip and coplanar lines**

Microstrip and coplanar lines are the most widely used waveguide types in modern microwave systems. They interconnect oscillators, amplifiers, antennas and so on. Sections of transmission lines also used as coupling element for resonators. Usually characteristics of the transmission line are defined at design time and remain constant in fabricated device. However, transmission lines can get some agility. For example, movement of dielectric body above microstrip or coplanar line surface results in propagation constant change [2]. We have shown that the more efficient control can be achieved, if one of the conductors is detached from substrate's surface, Figure 10. Because controllable discontinuity crosses electric field strength lines, it results in higher sensitivity.

**Figure 10.** Mechanically controllable microstrip (*a*, *b*) and coplanar (*c*) lines

294 Dielectric Material

parameters set is observed.

Scattering matrix for effective permittivity approach can be found from equations:

11 21

constant in domain 2 filed by uniform dielectric with permittivity *eff* .

 1 2 1 2

where *<sup>z</sup>*1 is the propagation constant in domain 1 (Figure 5) but *<sup>z</sup>*2 is the propagation

Comparison of two techniques is demonstrated in Figure 9. Good agreement for reasonable

**Figure 9.** Comparison of *S*-parameters computation using boundary element method (BEM) and effective permittivity approach for the structure with parameters: 1 = 10, 2 = 1, 3 = 1, *d/h* = 10-3

**5. Effective permittivity of microstrip and coplanar lines** 

Hereby effective permittivity approach is efficient method for investigation of controllable microwave structures. Below this technique is extended for microstrip and coplanar lines.

Microstrip and coplanar lines are the most widely used waveguide types in modern microwave systems. They interconnect oscillators, amplifiers, antennas and so on. Sections of transmission lines also used as coupling element for resonators. Usually characteristics of the transmission line are defined at design time and remain constant in fabricated device. However, transmission lines can get some agility. For example, movement of dielectric body above microstrip or coplanar line surface results in propagation constant change [2]. We have shown that the more efficient control can be achieved, if one of the conductors is

12 12 <sup>2</sup> , *z z z z zz zz*

*S S* , (27)

Conventional methods of microstrip lines analysis, such as Whiller equations [3], Hammerstad equations [4], and their extensions to coplanar lines [5] exploit symmetry of the line with aid of conformal mappings. These methods also introduce effective permittivity to relate quasi-TEM wave propagation characteristics to those of equivalent TEM wave. Transmission lines in Figure 10 still possess symmetry, but their rigorous analysis becomes cumbersome. Thus, numerical techniques could be applied to accurately calculate electromagnetic field distribution.

Electromagnetic problem can be solved using electric and magnetic scalar *<sup>e</sup>* , *<sup>m</sup>* and vector *<sup>e</sup>* **A** , *<sup>m</sup>* **A** potentials:

$$\mathbf{E} = -i\alpha \mathbf{A}^{\varepsilon} - \nabla \boldsymbol{\phi}^{\varepsilon}; \quad \mathbf{H} = \frac{\nabla \times \mathbf{A}^{\varepsilon}}{\mu \mu\_0};\tag{28}$$

$$\mathbf{E} = \frac{\nabla \times \mathbf{A}^m}{\varepsilon \varepsilon\_0}; \quad \mathbf{H} = -i\alpha \mathbf{A}^m - \nabla \boldsymbol{\mathfrak{q}}^m. \tag{29}$$

Using these potentials one can introduce electromagnetic filed distribution types with one of components being zero. If electrical vector potential oriented along *z* axis ( *e e* **A e** *<sup>A</sup> <sup>z</sup>* , where *<sup>z</sup>* **<sup>e</sup>** is *<sup>z</sup>* -axis unit vector), then *<sup>e</sup> <sup>A</sup>* and *<sup>e</sup>* functions define E-type field, or TM-mode, for which <sup>0</sup> *Hz* . Similarly, if magnetic vector potential oriented along *z* axis ( *m m* **A e** *<sup>A</sup> <sup>z</sup>* ), then *<sup>m</sup> <sup>A</sup>* and *<sup>m</sup>* functions define Н-type filed, or ТЕ-mode, for which <sup>0</sup> *<sup>z</sup> <sup>E</sup>* . Equation (28) is more convenient in the systems with dielectric only discontinuities, but with uniform permeability.

Equations (28), (29) allow ambiguity in relation of vector and scalar potentials with electromagnetic field components. For example, if *<sup>e</sup>* **A** and *<sup>e</sup>* define certain electromagnetic filed distribution, then *<sup>e</sup>* **A** and *<sup>e</sup>* , where is differentiable function, define the same distribution. This ambiguity is removed applying Lorentz's calibration:

$$
\nabla \left( \varepsilon \nabla \boldsymbol{\phi}^{\varepsilon} \right) + \varepsilon^2 \mu \frac{\boldsymbol{\phi}^2}{c^2} \boldsymbol{\phi}^{\varepsilon} = \boldsymbol{0} \ . \tag{30}
$$

In case of axial symmetry and absence of external currents solution of (30) may be presented in the form:

$$
\boldsymbol{\Phi}^e = \boldsymbol{\Psi}(\boldsymbol{x}, \boldsymbol{y}) \boldsymbol{Z}(\boldsymbol{z}) \boldsymbol{\hspace{0.5cm}} \boldsymbol{\Phi}^e
$$

where *x y*, is distribution of scalar potential in О*ху* plane, *Z z* is distribution along propagation direction O*z*. Then (30) splits in two equations with two mentioned distribution functions. In most practical cases electric field component along direction of propagation is much smaller and could be neglected. This is so called quasi-TEM mode. Thus 3D electromagnetic problem reduces to 2D plane problem:

$$\nabla \cdot \left( \varepsilon \nabla \Psi \right) + \mathfrak{B}^2 \Psi = 0 \,, \tag{31}$$

Electromechanical Control over Effective Permittivity Used for Microwave Devices 297

**Figure 11.** Comparison of effective permittivity calculation in microstrip line using 3D FEM, FDTD and

In conventional microstrip line most of electromagnetic field is confined in substrate between the strip and ground plane. When conductor is lifted above substrate as in Figure 10 *a*, *b*, certain part of electromagnetic filed redistributes from substrate to the air filled domains close to the strip. Because of lower permittivity energy stored in air filled domains is lower comparing to that one in substrate. This leads to decrease of the system's effective permittivity, as it is shown in Figure 12. Effective permittivity of the line defines wavelength in the system or, equivalently, propagation constant. Thus, mutual displacement of transmission line parts results in change of propagation constant. Described method of effective permittivity control has strong sensitivity. As seen in Figure 12 displacement by 10% of substrate's thickness may change effective permittivity

Redistribution of electromagnetic energy to air filled domains also changes loss in the system. Because air is almost lossless medium, the portion of energy confined in air filled domains experience practically no dielectric loss. Consequently more energy reaches output

Presented quasi-static approximation can be applied for analysis of coplanar line as well. Dependencies of effective permittivity and loss on coplanar line with lifted signal strip

Derived values of effective permittivity and loss then can be used to design device similarly to strict TEM-mode devices. Controllability of effective permittivity for more complicated

quasi-static approximation (1 = 12, *w* = 0.5 mm, *h* = 1.5 mm)

terminal, resulting in lower effective loss, Figure 12.

qualitatively similar to those of coplanar line, Figure 13.

microwave devices was presented in [1].

more then by half.

where 2 2 2 <sup>2</sup> *<sup>z</sup> c* . Applying appropriate boundary conditions the problem is solved numerically using two dimensional finite element method (2D FEM). Then one may calculate electromagnetic field distribution as:

$$\begin{aligned} E\_x &= -\frac{\partial \Psi}{\partial x}; \qquad &E\_y = -\frac{\partial \Psi}{\partial y}; \qquad &E\_z = -i\frac{\mathbb{B}^2}{\sqrt{\varepsilon \mu \frac{\text{co}^2}{c^2} - \mathbb{B}^2}} \mathbb{W};\\ H\_x &= Z\_0^{-1} \sqrt{\frac{\varepsilon}{\mu}} \frac{\sqrt{\varepsilon \mu} \frac{\text{co}}{c}}{\sqrt{\varepsilon \mu \frac{\text{co}^2}{c^2} - \mathbb{B}^2}} \frac{\partial \Psi}{\partial y}; \qquad &H\_y = -Z\_0^{-1} \sqrt{\frac{\varepsilon}{\mu}} \frac{\sqrt{\varepsilon \mu} \frac{\text{co}}{c}}{\sqrt{\varepsilon \mu \frac{\text{co}^2}{c^2} - \mathbb{B}^2}} \frac{\partial \Psi}{\partial x}. \end{aligned}$$

where 0 0 0 *Z* 120 is free space characteristic impedance. In most practical cases 2 2 *<sup>z</sup>* <sup>2</sup> *c* , thus *Ez* 0 and wave is close to TEM.

Having solution of (31) we introduce effective permittivity *eff* relating total power in the system under consideration to power in the system with uniform filling:

$$\varepsilon\_{eff} = \frac{\sum\_{i=1}^{N} \left( \varepsilon\_i \iint\_{S\_i} \left( \frac{\partial \boldsymbol{\psi}}{\partial \boldsymbol{\chi}} \right)^2 + \left( \frac{\partial \boldsymbol{\psi}}{\partial \boldsymbol{\chi}} \right)^2 \right) d\mathbf{x} dy}{\iint\_S \left( \left( \frac{\partial \boldsymbol{\psi}\_1}{\partial \boldsymbol{\chi}} \right)^2 + \left( \frac{\partial \boldsymbol{\psi}\_1}{\partial \boldsymbol{\chi}} \right)^2 \right) d\mathbf{x} dy} \,\boldsymbol{\psi}\_1$$

where *Si* is *i*-th domain area with permittivity *<sup>i</sup>* , *S* is line's cross section total area, 1 is distribution of scalar potential in regular line with 1 = 1. Quasi-static approximation gives results, which coincide well with rigorous solution by 3D FEM and finite difference in time domain (FDTD) method (Figure 11). However, at small displacements of conductor above substrate rigorous solutions faced convergence difficulties, especially FDTD method.

296 Dielectric Material

where 2

where 0

0

0

 2 2

*<sup>z</sup>* <sup>2</sup> *c*

2 2 <sup>2</sup> *<sup>z</sup> c*

calculate electromagnetic field distribution as:

(,)() *<sup>e</sup> xyZz* ,

where *x y*, is distribution of scalar potential in О*ху* plane, *Z z* is distribution along propagation direction O*z*. Then (30) splits in two equations with two mentioned distribution functions. In most practical cases electric field component along direction of propagation is much smaller and could be neglected. This is so called quasi-TEM mode. Thus 3D

numerically using two dimensional finite element method (2D FEM). Then one may

*x yz*

*E E E i x y*

*x y*

, thus *Ez* 0 and wave is close to TEM.

system under consideration to power in the system with uniform filling:

*eff*

*N*

1

*i i S*

*i*

*S*

1 1

Having solution of (31) we introduce effective permittivity *eff* relating total power in the

where *Si* is *i*-th domain area with permittivity *<sup>i</sup>* , *S* is line's cross section total area, 1 is distribution of scalar potential in regular line with 1 = 1. Quasi-static approximation gives results, which coincide well with rigorous solution by 3D FEM and finite difference in time domain (FDTD) method (Figure 11). However, at small displacements of conductor above

substrate rigorous solutions faced convergence difficulties, especially FDTD method.

*c c H Z H Z*

<sup>2</sup> 0 , (31)

2 2 2

2

*c*

; ,

,

*y x*

. Applying appropriate boundary conditions the problem is solved

2 2

2 2

*c c*

;; ;

0 0 2 2

*Z* 120 is free space characteristic impedance. In most practical cases

2 2

*dxdy x y*

2 2 1 1

*dxdy x y*

electromagnetic problem reduces to 2D plane problem:

**Figure 11.** Comparison of effective permittivity calculation in microstrip line using 3D FEM, FDTD and quasi-static approximation (1 = 12, *w* = 0.5 mm, *h* = 1.5 mm)

In conventional microstrip line most of electromagnetic field is confined in substrate between the strip and ground plane. When conductor is lifted above substrate as in Figure 10 *a*, *b*, certain part of electromagnetic filed redistributes from substrate to the air filled domains close to the strip. Because of lower permittivity energy stored in air filled domains is lower comparing to that one in substrate. This leads to decrease of the system's effective permittivity, as it is shown in Figure 12. Effective permittivity of the line defines wavelength in the system or, equivalently, propagation constant. Thus, mutual displacement of transmission line parts results in change of propagation constant. Described method of effective permittivity control has strong sensitivity. As seen in Figure 12 displacement by 10% of substrate's thickness may change effective permittivity more then by half.

Redistribution of electromagnetic energy to air filled domains also changes loss in the system. Because air is almost lossless medium, the portion of energy confined in air filled domains experience practically no dielectric loss. Consequently more energy reaches output terminal, resulting in lower effective loss, Figure 12.

Presented quasi-static approximation can be applied for analysis of coplanar line as well. Dependencies of effective permittivity and loss on coplanar line with lifted signal strip qualitatively similar to those of coplanar line, Figure 13.

Derived values of effective permittivity and loss then can be used to design device similarly to strict TEM-mode devices. Controllability of effective permittivity for more complicated microwave devices was presented in [1].

Electromechanical Control over Effective Permittivity Used for Microwave Devices 299

Demonstrated high sensitivity of effective permittivity to microwave device parts displacement opens an opportunity to employ piezoelectric or electrostrictive actuators to control characteristics of the microwave devices by the electromechanical manner. Properties of materials for piezoelectric and electrostrictive actuators are discussed in the next section.

Application of usual piezoelectric ceramics for the microwave device tuning was described previously [1, 2]. However, in a strong controlling field piezoelectric ceramics show electromechanical hysteresis that produces some inconveniences. Much more prospective are relaxor ferroelectrics that have better transforming properties and practically no hysteresis.

Ferroelectrics with partially disordered structure exhibit diffused phase transition properties. Relaxor ferroelectrics near this transition show an extraordinary softening in their dielectric and elastic properties over a wide range of temperatures. Correspondingly, dielectric permittivity of the relaxor shows large and broad temperature maximum where giant electrostriction is observed (because the strain *x* is strongly dependent on the dielectric

Relaxors are characterized by the large ~ (2 – 6)104 and, consequently, by very big induced polarization *Pi* . A comparison of *Pi* in the relaxor ferroelectric Pb(Mg1/3Nb2/3)O3 = PMN and *Pi* of paraelectric material Ba(Ti0.6,Sr0.4)O3 = BST (that also has rather big ~ 4000) is shown in

**Figure 14.** *a* – electrically induced polarization *Pi* in the relaxor of PMN and in the paraelectric BST; *b* – dielectric permittivity of PMN without () and under bias field *Eb*=10 kV/cm (*b*); *Pi* is the induced

Induced polarization in PMN many times exceeds one of BST. Moreover, in relaxor, the *Pi* depends on the temperature (like *PS* of ferroelectrics), as it can be seen in Figure 14, *b*. An

example is electrically induced piezoelectric effect that is explained in Figure 15.

polarization in the relaxor PMN, obtained by pyroelectric measurements

**6. Piezoelectric and electrostrictive materials for actuators** 

permittivity: *x* ~ 2).

Figure 14, *a*.

**Figure 12.** Effective permittivity in near 50 microstrip line with micromechanical control (*w*/*h* = 2). <sup>0</sup> *eff* and <sup>0</sup> tan *eff* are effective permittivity and loss tangent at zero *d/h* = 0

**Figure 13.** Effective loss in near 50 coplanar line with micromechanical control (*b*/*a* = 0.72). <sup>0</sup> *eff* and 0 tan *eff* are effective permittivity and loss tangent at *d/h* = 0

Demonstrated high sensitivity of effective permittivity to microwave device parts displacement opens an opportunity to employ piezoelectric or electrostrictive actuators to control characteristics of the microwave devices by the electromechanical manner. Properties of materials for piezoelectric and electrostrictive actuators are discussed in the next section.
