tailieunhanh - Mechanics of Microelectromechanical Systems - N.Lobontiu and E.Garcia Part 9

Tham khảo tài liệu 'mechanics of microelectromechanical systems - and part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 228 Chapter 4 where D is the dielectric displacement vector and ecơ is the electrical permittivity matrix the subscript Ơ indicates that the matrix is determined under constant-stress conditions . The vector D is defined as D - Dị D2 D Jr and the symmetric permittivity matrix is ell g12 e J -2 22 22 31 2 2 633. When premultiplying Eq. by the vector 1 2 ơ T the following equation is obtained which contains only specific energy energy per unit volume terms ơ T Em ưm UPZT where ưm is the mechanical energy and is formulated as ưm ơf cE ơ and is the piezoelectric energy defined as UPZT ơ T d E The following equation can be obtained from Eq. through leftmultiplication by 1 2 E r E T D UPZT Ue where the electric energy Je is Ee E T EJ 4. Microtransduction actuation and sensing 229 The energy formulation is useful as it allows introducing an amount the piezoelectric coupling factor which is defined as and which gives the measure of the degree of energy conversion efficiency. Example Determine the coupling factor kpe for the case defined in Example knowing that the electrical permittivity Ễ33 is 15 X 10 9 F m. Solution For the particular problem of the previous example the stress vector reduces to the Ơ3 component. Similarly the compliance matrix is singletermed as it only contains C33 the permittivity matrix reduces to its Ễ33 component and the electric field vector reduces to E3. The piezoelectric energy will be in this case UpzT - 2 33 3Ơ3 The mechanical energy is rr 2 and the electrical energy simplifies to By substituting Eqs. 4 117 and into the definition equation - Eq. the following equation is obtained for the coupling factor kpe - 33 ylcĩĩEĩĩ and it can be shown by analyzing the strain-stress relationship of Eq. - Example - that C3 3 1 E where E is the Young s modulus about direction 3 or z . As a consequence Eq. yields a coupling .