Lumped Modeling in Thermal Domain

Transcription

1 EEL55: Principles of MEMS ransducers (Fall 003) Instructor: Dr. Hui-Kai Xie Lumped Modeling in hermal Domain Last lecture oday: Lumped modeling Self-heating resistor Self-heating resistor Other dissipation mechanisms Friction, dielectric, magnetic, damping Coupled flows eading: Senturia, Chapter 11, p /31/003 1 Lecture 8 by H.K. Xie 10/31/003

2 esistor Self-Heating emperature dependent resistance Voltage-Current relation, V=I() (Ohm s Law) Now we consider resistor selfheating and temperature dependence of resistance, () Joule Heating, P dissipation =VI=I >0 + e V - f I For small to moderate temperature variation, approximate () using first order aylor series expansion. ( ) ( )[1 + α ( )] 0 0 where α = temperature coefficient of resistance [1/K] 10/31/003

3 esistor Self-Heating Circuit model for resistor self-heating-- Current Source hermal reservoir at temperature, 0 Governing equation is: d C = + i (1 + α ) 0 dt d 1 i = ( 1 α i 0 ) + dt C C 0 ef. Senturia, Microsystem Design, p /31/003 3

4 esistor Self-Heating Circuit model for resistor self-heating d 1 i = + dt C C i 1 α i 0 0 ( 1 α i 0 ) 0 SS, i 1 α i 0 i0 Linear first-order system with input,. C he time constant is: τ = C Steady-state temperature rise is given by i = ( ) 1/ i < α 0 Example: Metal fuse ef. Senturia, Microsystem Design, p /31/003 4

5 esistor Self-Heating Circuit model for resistor self-heating-- Voltage Source Governing equation is: d V C = + dt (1 + α ) 0 ef. Senturia, Microsystem Design, p /31/003 5

6 esistor Self-Heating Circuit model for resistor self-heating-- Voltage Source V V Since the.c.., α, is small, (1 α) (1 + α ) v α V SS, V d α V V = 1+ + dt C 0 0 his linear first-order system has a time constant, τ C = 1+ V / = α V and steady-state temperature rise, Note that this is stable with no singularities + for positive C. 10/31/003 6

7 esistor Self-Heating Implanted resistor embedded in a thermally conducting silicon substrate Need to estimate lumped elements for this distributed system ef. Senturia, Microsystem Design, p. 34. Electrical resistance: hermal capacitance: C = ρvcˆ = ρ ( ) awl Cˆ Cˆ : specific heat per unit mass (J/(kg-Kelvin)) p ρl = = A L awσ 10/31/003 7 e m p m p

8 Heat conduction to substrate Begin with Fourier's law, J = κ. Consider a semi-cylindrical boundary. esistor Self-Heating Q = JQ A( r) = κ( πrl) neglecting end conduction effects. r S r0 Q dr d = κπ L r a Q r0 S = ln = Q L a κπ 1 r0 where hermal = ln [Kelvin/Watts] κπ L a Q = S ef. Senturia, Microsystem Design, p /31/003 8

9 Heat conduction to substrate esistor Self-Heating Geometry: L=300µ m, W=4µ m, a=µ m, r = 10a 17 3 Silicon: n-type, N DD =10 cm, σ e = 15 S / cm, α K, m 330 kg/ m, Cˆ = 71 J / kg K, κ = 148 W / m K p = ρ = 0 Lumped Elements: 0 4 = = = Ω hermal ρl A L awσ.5 10 e ˆ ( ) ˆ m p ρm p / C = ρ VC = awl C = J K r = κπ L = a 1 ln / K W ef. Senturia, Microsystem Design, p /31/003 9

10 Heat conduction to substrate esistor Self-Heating 0 4 = = =.5 10 Ω e ˆ ( ) ˆ m p ρm p / 1 r ln 0 hermal = 16.5 K / W κπ L a = For current drive at low current, i 1 α i 0 i ρl A L awσ C = ρ VC = awl C = J K C τ = C 9 τ = 16.5 K / W 4 10 J / K = 66ns i = 0.04K if i = 0.3mA 0 SS, i 1 α i 0 ef. Senturia, Microsystem Design, p /31/003 10

11 esistor Self-Heating Effect of Self-Heating in esistive ransducers Maximum Example: Piezoresistor wo sources of resistance change resistance change due to self-heating resistance change due to piezoresistance piezoresistance -4 Need 1 part per 10,000=10 (or 100 ppm). self-heating 1%. If we measure this to 1% accuracy, we need to discern 1% of 1% or 1 part per 10,000 change in resistance. From < self-heating ( ) ( ) [1 + α ], = α <10. herefore, < = = 0.04K ( 0) α K his can be satisfied if i < 0.3mA (which corresponds to 7.5V across (=5k Ω)) 10/31/003 11

12 Other Dissipation Mechanisms Contact Friction Dielectric Losses Viscoelastic Losses Magnetic Losses Fluid Viscosity Losses 10/31/003 1

13 Other Dissipation Mechanisms Contact Friction etarding force, F r that opposes motion in response to opposite tangential driving force, F elated to normal force pressing down Approximate model only, evaluated via experiment ef. Senturia, Microsystem Design, p. 37. Linear model of contact friction F = µ F where 0 µ 1 r f n f he retarding force is present only when there is relative motion, x. F r = bx. Modeled as resistor with value, b. F 10/31/ r F n Irreversible process, power dissipated as heat energy

14 Contact Friction Other Dissipation Mechanisms Depends on surface roughness, velocity, and initial condition Static friction > Sliding friction Non-linear friction (Coulomb friction) hreshold tangential force before motion begins One key originating force is Coulomb attraction between charged surfaces ef. Senturia, Microsystem Design, p. 38. Internal Friction Viscoelastic effects Internal retarding forces that oppose deformation Internal friction produces heat, heat flow, entropy 10/31/003 14

15 Dielectric Losses Other Dissipation Mechanisms Joule heating due to small conductivity Internal friction that retards orientation of dipoles Since polarization requires displacement of charged particles with an inertial mass, the polarization is frequency dependent. So, a phase delay occurs. elevant forces: applied electric field, retarding Coulomb attraction in dipole, internal friction D = ( ε + jε ) E ef. Jordan and Balmain, Electromagnetic Waves and adiating Systems, p /31/003 15

16 Dielectric Losses Other Dissipation Mechanisms Consider Ampere's Law for a dielectric with finite conductivity, σ e. ' " D ( εr + jεr) ε0e H = J + = σ ee+ t t jωt Assuming sinusoidal steady-state, e : ' " ' " σ e H = σee jωεre+ ωεre = jω εε r 0 + j εε r 0 + ω E Lumped element equivalent circuit for a parallel plate capacitor: εε A ' C= r 0 = g g " ( σe + ωεrε0) A ε = εε ε = r where 0 vacuum permitivity= / F cm 10/31/003 16

17 Other Dissipation Mechanisms Magnetic Losses I Joule heating in finite coil resistance (inductor or transformer winding) Eddy currents induced in conducting magnetic core (and also winding conductors) B Consider Faraday's Law, E = t he time-varying magnetic flux gives rise to a voltage (opposing flux change) that induces electrical currents in the magnetic core. hese 'eddy currents' cause i heating of the core. ef. Erickson, Fundamentals of Power Electronics, p /31/003 17

18 Magnetic Hysteresis Other Dissipation Mechanisms Internal friction due to magnetic domain wall motion when magnetic flux density attempts to align with magnetic field, H. Coercive force, H C, before alignment occurs similar to Coulomb friction. emanent flux density, B, exists at H = 0. Magnetically hard materials: large H C, such as Sm-Co alloy. Magnetically soft materials: small H C, such nickel-iron alloy. he internal area of a hysteresis loop is a measure of the energy lost due to magnetic hysteresis. ef. Senturia, Microsystem Design, p /31/003 18

19 Coupled flows: Coupled Flows emperature gradient and concentration gradient Simultaneous heat flow and diffusion Seebeck effect and thermocouples E = α α s : Seeback coefficient s H α s,1 αs, V V C C C s,1 s, s,1 H V = α d + α d + α d V C H H ( α α ) d ( α α )( ) = = V s,1 s, s,1 s, H C C 10/31/003 19

20 Coupled Flows Peltier effect and efrigerators Q = It cold hot ( n p) Q = I N Peltier effect is the converse of Seebeck effect. he two coefficients have the following relation, =α s 10/31/003 0