1. Transducers Devices to transform signals between different physical domains. Taxonomy –Active: input signal modulates output energy; e.g. liquid crystal display. –Passive: input signal transformed to output energy; Bidirectional, e.g. motor/generator; Unidirectional, e.g. photodiode [<Lat. Trans, across + ducere, to lead]

2. Signals may occur in many physical forms Examples –Electrical signals on wires –Sound in free air –Light fields in free space Transducers connect signal domains –input signals of one physical form –produce signals in a different physical form. Signal domains

3. Signals, power and energy Physical signals are carried by energy –Energy = force x distance (mechanical) [Joules] May take many equivalent forms. –Within a domain: potential vs kinetic; –In different domains: electrical, nuclear, heat, etc. Is conserved: can transform but not create or destroy. –Power = time derivative of energy [Watts] Signal strength is a measure of signal power. Signal to noise ratio is a ratio of powers. A transducer receives signals in one domain and generates them in another.

4. Potential energy is stored in reactive tension: –Voltage on a capacitor –Force compressing a spring Kinetic energy is stored in reactive motion: –current in an inductor –velocity of a mass Reactance types –Potential: voltage, force, electric field, etc.; –Kinetic: current, velocity, magnetic field, etc. Potential and kinetic energy

5. Voltage, current, force, &cet. Power is expressible by pairs of variables: –Voltage and current, –Force and velocity, –Pressure and flow, etc. These are potential-kinetic variable pairs in particular physical domains. Each domain has an associated measure of impedance and its reciprocal, admittance, although nomenclature varies.

6. An electrostatic microphone We first explore the physics relating voltage to pressure and current to volume- velocity. The result is non-linear. We use small-signal assumptions to linearize the model. We then represent these results in terms of two-port matrices which linearly relate variables between domains. Passive bidirectional transducer example

7. Microphone construction

8. Parallel plate capacitor physics Consider a capacitor with fixed lower plate and movable upper plate, both of area A, separated by distance x. An amount Q of charge has been moved from one plate to the other, causing an electric field between them to develop a voltage V. The field of the lower plate acts on the charge of the upper plate to produce a force F on it. These effects provide the coupling between the electrical and mechanical domains.

9. Electric field of the lower plate Coulomb’s law in differential and integral forms: where D =  E, E is electric field,  is charge density, and  8.85 x 10 -12 for air dielectric. The first integral is of the field normal to a closed surface, and the second is of the charge in the volume enclosed by the surface. Thus the field near a flat plate of area A with charge Q is,

10. Capacitance between the plates The definition of capacitance is Q=CV. The total field between the plates is the superposition of fields from both plates, or E=Q/  A. By the definition of voltage we have, The latter because the field between the plates is nearly uniform. From this we can obtain the formula for a parallel plate capacitor:

11. Force on the upper plate The field E from the charge on the lower plate exerts force on the charge Q on the upper plate which is negative: the latter because the field is nearly uniform over the plate area. Substituting for the electric field, we get a nonlinear force as a function of Q, which is always attractive.

12. Force - voltage relationship Let the total voltage V T = V 0 + V, the sum of the bias and signal voltages respectively. Similarly, F T = F 0 + F, where F 0 is force due to the bias voltage alone and F is the signal force. Also, x T = x 0 + x, where x 0 is the plate separation at rest, and x is the signal displacement. The total force is, If the signal displacement x is held to zero, This is a non-linear relationship which is always attractive.

13. Linearizing the relationship This behavior can be “linearized” by assuming that Expanding F T and V T we make the approximation,

14. Velocity - current relationship

15. Two-port transducer matrices Relate two input variables to two output variables. –The product of the input variables must be power –The product of the output variables must be power –Input and output variables may be in any domains A microphone has inputs p and u, outputs V and I. The hybrid, or h-parameter matrix H is written, * (See Hanspeter Schmid, “Tables: Two-Port Matrices,” people.ee.ethz.ch/~hps/publications/twoport.pdf )

16. The microphone’s H matrix

17. The microphone’s T matrix It was convenient to use the H matrix because we could derive the behaviors of p and I when u and V were held fixed, respectively. It is also useful to have a transmission matrix T providing electrical outputs, given acoustic inputs. Conversion of the H matrix to the T matrix is a problem in linear algebra, the solution of which is,

18. The microphone’s Z matrix It is also sometimes useful to represent this as an impedance matrix Z facilitating impedance matching calculations at the electrical and acoustic ports. In this case, the solution is,