tailieunhanh - RF MEMS and Their Applicatio P2
Electromechanical mobility analogies [42] Mechanical parameter Electrical parameter Voltage Current Conductance Inductance Capacitance Inductance per unit length Capacitance per unit length Characteristic impedance Impedance Admittance Short circuit Open circuit Current Voltage Variable Lumped network elements Transmission lines Immitances Source immitances Velocity, angular velocity Force, torque Damping Compliance Mass, mass moment of inertia Compliance per unit length Mass per unit length Characteristic mobility Mobility Impedance Clamped point Free point Force Velocity ABCD matrix form as: x1 ˙ F1 where Z0 = β= vp = 1 √ A ρE ω vp E 1 =√ ρ Cl Ml Cl Ml () () cos βx = j sin βx Z0 j. | ELECTROMECHANICAL TRANSDUCERS 17 Table Electromechanical mobility analogies 42 Mechanical parameter Electrical parameter Variable Velocity angular velocity Force torque Voltage Current Lumped network elements Damping Compliance Mass mass moment of inertia Conductance Inductance Capacitance Transmission lines Compliance per unit length Mass per unit length Characteristic mobility Inductance per unit length Capacitance per unit length Characteristic impedance Immitances Mobility Impedance Clamped point Free point Impedance Admittance Short circuit Open circuit Source immitances Force Velocity Current Voltage ABCD matrix form as where JC1 Fi cos ßx jZosin ßx H r . x2 sin ßx cos ßx _ F2 Zo Zo M VP v - E-- P p CWi 1-1 1 C A E Mt ß In these equations j V 1 x1 and x2 are velocities F1 and F2 forces at two ends of a transmission line Z0 ft and vp are the characteristics impedance propagation constant and phase velocity of the transmission line A is the cross-sectional area of the mechanical transmission line E its Young s modulus and p the density. Quantities Cl and Mi are compliance and mass per unit length of the line respectively. Now looking at the electromechanical analogies in Johnson 1983 the expressions for an equivalent electrical circuit can be obtained in the same form as Equation V1 cos ßx jZ0 sin ßx V2 1. j a sin ßx Lzo p cos ßx _ 2 _ 1-5 18 MEMS AND RF MEMS Table Direct analogy of electrical and mechanical domains Mechanical quantity Electrical quantity Force Voltage Velocity Current Displacement Charge Momentum Magnetic flux linkage Mass Inductance Compliance Capacitance Viscous damping Resistance Source Tilmans 1996. In Equation V and I are the voltage and current on the transmission line with subscripts representing its ports . The other quantities in the matrix are also represented by equivalent electrical parameters as Z 1 6 vp LQ In Equations and L and C. represent the inductance and capacitance per unit length of
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