Lecture 6-1 Transduction Based on Changes in the Energy Stored in an Electrical Field
Electric Field and Forces Suppose a charged fixed q 1 in a space, an exploring charge q is moving toward the fixed charge, the force between these two charges is given by Coulombs s law: The field established by q 1
Electric Field and Forces Recall, in mechanical system, the potential energy is given by l is the distance moved Voltage is defined as a potential energy per charge: V n θ E Voltage is also often referred as electrical potential. Consider a system of charges, if we have a volume in space bounded by a surface S, the field established by the charge q interior to the surface is directed radially outward. At any point on the surface, the normal component of the electric field q S
Electric Field and Forces To describe electric field, an additional field parameter, D, the electric displacement or charge density field, is often used. D [ ε ]E [ε ] is the matrix of dielectric permittivity of a dielectric materials
Electric Field and Forces For isotropic dielectric materials, [ε ] matrix reduced to scalar ε, Gauss s law may be rewritten as D da q S This explain why D is often called the charge density For dielectric materials, electric dipoles, i.e., closely coupled pairs of charges, will result in electric polarization, P, which is equal to the bound charge density. D ε o E + P Representation of total, free, and bound charge densities by field vector
Transducers Made with a Variable Gap Parallel Plate Capacitor Consider a parallel capacitor with a dielectric material separating the plates, the total charges related to the capacitance C By D ε E, CV ε EA DA D is charge density C is capacitance A is area of each plate E is electric field: EV/d Then ε AV d Therefore C ε A d This is the constitutive equation of parallel plate capacitance when edge effects may be neglected
Transducers Made with a Variable Gap Parallel Plate Capacitor We can rewrite the above equation to express the voltage in term of the charge and the capacitor material and the geometric dimension V d ε A The voltage can be change dynamically by using mechanical means to temporally vary the electrode gap and the electric permittivity ε A C d+x The electric energy stored by the capacitor is W 1 C ( d + x) εa x is the displacement of the electrode from equilibrium
Transducers Made with a Variable Gap Parallel Plate Capacitor The power stored by the variable gap parallel plate is given by the time rate of change of energy ( d + x) Recall: W Ρ dw dt W d dt W + x dx dt 1 C εa d dt dw d is the current flow associate with the capacitor must be the voltage Therefore ( d + x) V F ε ε A A dx dt dw dx Voltage Voltage Force is the velocity must be the force Displacement Force Charge
Relating Displacement with Voltage Let s first consider: V ( + x) d ε A The equation not only relates the voltage vs. displacement, but also vs. the charges on the plate Suppose we have DC bias voltage V o on the capacitor, and we either apply an AC voltage V ~ to generate a displacement (actuator) or sense an AC voltage V ~ in response to motion of the plate (sensor). V V o ~ + V The charge can be modeled as a static charge and a time varying charge o ~ +
Relating Displacement with Voltage Now relate the voltage to the displacement and charge ~ ~ ~ od o x d x V + Vo + + + ε A ε A ε A ε A Zero th -order (dc) term: The first order term: Displacement Voltage V o ~ V ~ V od ε A ~ o x d + ε A ε A o x ε A Voltage: Sensor Displacement: Actuator In transducers, the goal is to relate the gap change to the voltage change, thus we seek to minimize the effect of the changing charge. This can be done by electrically isolating the ac voltage and charge from dc voltage and charge, so that the charge on the plates is essentially the constant
Relating Displacement with Voltage Again: ~ V A As sensor, the sensitivity is: ε o x ~ Vo x o ε A Vo d The sensitivity can be improved as dc bias field increased or the plate separation is decreased Recall: C/V d V ε A As actuator, the sensitivity is: x ε A d ψ ~ V o V o The sensitivity can be enhanced by decreasing dc bias field or increasing the plate separation The second-order term and the neglected first-order term may be thought of producing a perturbation to the above expression. Effectively, these terms produce distortion of the signal, but it is generally small as long as the timevarying quantities are small relative to the static quantities
Relating Displacement with Voltage One way to eliminate the second-order distortion is to build transducer which uses three plates with center one moving. This push-pull configuration can effectively doubles the sensitivity. A differential capacitive sensor
Electrostatic Gap Closing Actuator Charge Control Consider: F ε A d Electrostatic force is Independent of the gap Proportional to the square of the charge
Electrostatic Gap Closing Actuator Voltage Control W 1 1 εav CV g Electrostatic Force with Voltage control is: Or: V ( g + x) ε A F ε A F εav g (when x0)
Voltage Controlled Actuator Graphical Solution g: new gap between two plates
Voltage Controlled Actuator Transfer Curve for Voltage Controlled actuator Bi-stability
Voltage Controlled Actuator
Voltage Controlled Actuator Pull-in Voltage