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EEL5225: Principles of MEMS Transducers (Fall 2003) Instructor: Dr. Hui-Kai Xie Lumped Modeling in Thermal Domain ; Last lecture z z Lumped modeling Self-heating resistor z Today: Self-heating resistor z Other dissipation mechanisms z z z Friction, dielectric, magnetic, damping Coupled flows Reading: Senturia, Chapter 11, p.278-296 10/31/2003 EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie Lecture 28 by 1 H.K. Xie 10/31/2003 Resistor Self-Heating z Temperature dependent resistance z Voltage-Current relation, V=IR(T) (Ohm’s Law) z Now we consider resistor selfheating and temperature dependence of resistance, R(T) z z fI + e - V R Joule Heating, Pdissipation=VI=I2R >0 For small to moderate temperature variation, approximate R(T) using first order Taylor series expansion. R (T ) ≅ R (T0 )[1 + α R (TR − T0 )] where α R = temperature coefficient of resistance [1/K] 10/31/2003 EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie 2 Resistor Self-Heating z Circuit model for resistor self-heating-Current Source Thermal reservoir at temperature, T0 Governing equation is: dTR TR 2 =− + i R0 (1 + α RTR ) CT dt RT i 2 R0 1 dTR 2 =− 1 − α R R0 RT i ) TR + ( dt RT CT CT 10/31/2003 Ref. Senturia, Microsystem Design, p. 231. EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie 3 Resistor Self-Heating z Circuit model for resistor self-heating 2 i R0 dTR 1 2 R R i T =− 1 − α + ( R 0T )R C dt RT CT T i 2 R0 Linear first-order system with input, . CT The time constant is: RT CT τi = 1 − α R R0 RT i 2 Steady-state temperature rise is given by TSS ,i R0 RT i 2 = 1 − α R R0 RT i 2 Example: Metal fuse 10/31/2003 i < (α R R0 RT ) −1/ 2 Ref. Senturia, Microsystem Design, p. 231. EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie 4 Resistor Self-Heating z Circuit model for resistor self-heating-Voltage Source Governing equation is: dTR TR V2 =− + CT dt RT R0 (1 + α RTR ) Ref. Senturia, Microsystem Design, p. 232. 10/31/2003 EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie 5 Resistor Self-Heating z Circuit model for resistor self-heating-Voltage Source V2 V2 Since the T.C.R., α R , is small, ≅ (1 − α RTR ) R0 (1 + α RTR ) R0 α RTRV 2 V 2 1 + + R0 R0 This linear first-order system has a time constant, RT CT and steady-state temperature rise, τv = 2 α RV 1+ R T R0 dTR T =− R dt RT CT TSS ,V 10/31/2003 RT V 2 / R0 Note that this is stable with no singularities = 2 α RV for positive TCR. 1+ R T R0 EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie 6 Resistor Self-Heating z Implanted resistor embedded in a thermally conducting silicon substrate z Need to estimate lumped elements for this distributed system Ref. Senturia, Microsystem Design, p. 234. ρL R= Thermal capacitance: CT = ρ mVCˆ p = ρ m ( aWL ) Cˆ p A = L aW σ e Electrical resistance: Cˆ p : specific heat per unit mass (J/(kg-Kelvin)) 10/31/2003 EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie 7 Resistor Self-Heating z Heat conduction to substrate Begin with Fourier's law, J Q = −κ∇T . Consider a semi-cylindrical boundary. ∂T Q = J Q A(r ) = −κ (π rL ) neglecting end conduction effects. ∂r TS r0 Q dr ∫T dT = − ∫a κπ L r R Q r0 TR − TS = ln = RT Q κπ L a where RThermal = r ln 0 [Kelvin/Watts] κπ L a 1 T=TS Ref. Senturia, Microsystem Design, p. 234. 10/31/2003 EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie 8 Resistor Self-Heating z Heat conduction to substrate Geometry: L=300µ m, W=4µ m, a=2µ m, r0 = 10a Silicon: n-type, N DD =1017 cm −3 , σ e = 15 S / cm, α R = 2.5 ×10−3 K −1 , ρ m = 2330 kg / m3 , Cˆ p = 712 J / kg − K , κ = 148W / m − K Lumped Elements: ρL L = = 2.5 × 104 Ω R0 = A aW σ e CT = ρ mVCˆ p = ρ m ( aWL ) Cˆ p = 4 × 109 J / K RThermal = 10/31/2003 r ln 0 = 16.5 K / W κπ L a 1 EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie Ref. Senturia, Microsystem Design, p. 234. 9 Resistor Self-Heating z Heat conduction to substrate ρL L = 2.5 ×104 Ω A aW σ e CT = ρ mVCˆ p = ρ m ( aWL ) Cˆ p = 4 × 10−9 J / K R0 = = r0 RThermal = ln = 16.5 K / W κπ L a For current drive at low current, 1 τi = RT CT ≅ RT CT 2 1 − α R R0 RT i τ i = 16.5 K / W × 4 × 10−9 J / K = 66ns TSS ,i R0 RT i 2 = 0.04K if i = 0.3mA 2 1 − α R R0 RT i Ref. Senturia, Microsystem Design, p. 236. 10/31/2003 EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie 10 Resistor Self-Heating z Effect of Self-Heating in Resistive Transducers z Example: Piezoresistor z Two sources of resistance change resistance change due to self-heating resistance change due to piezoresistance z z Maximum ∆R ≅ 1%. If we measure this to 1% accuracy, R piezoresistance we need to discern 1% of 1% or 1 part per 10,000 change in resistance. ⇒ Need ∆R < 1 part per 10,000=10-4 (or 100 ppm). R self-heating From R(TR ) ≅ R0 [1 + α R (TR − T0 )], ∆R 10-4 10-4 -4 = α R (TR − T0 ) <10 . Therefore, TR − T0 < = = 0.04 K −6 −1 R self-heating α R 2500 ×10 K This can be satisfied if i < 0.3mA (which corresponds to 7.5V across R(=25kΩ)) 10/31/2003 EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie 11 Other Dissipation Mechanisms z Contact Friction z Dielectric Losses z Viscoelastic Losses z Magnetic Losses z Fluid Viscosity Losses 10/31/2003 EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie 12 Other Dissipation Mechanisms z Contact Friction Retarding force, Fr that opposes motion in response to opposite tangential driving force, F z Related to normal force pressing down z Approximate model only, evaluated via experiment z Ref. Senturia, Microsystem Design, p. 237. Linear model of contact friction → Fr ∝ Fn Fr = µ f Fn where 0 ≤ µ f ≤ 1 Irreversible process, power dissipated as heat energy The retarding force is present only when there is relative motion, x. Fr = bx. Modeled as resistor with value, b. 10/31/2003 EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie 13 Other Dissipation Mechanisms z Contact Friction Depends on surface roughness, velocity, and initial condition z Static friction > Sliding friction z Non-linear friction (Coulomb friction) z z z z Threshold tangential force before motion begins One key originating force is Coulomb attraction between charged surfaces Ref. Senturia, Microsystem Design, p. 238. Internal Friction z Viscoelastic effects z z 10/31/2003 Internal retarding forces that oppose deformation Internal friction produces heat, heat flow, entropy EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie 14 Other Dissipation Mechanisms z Dielectric Losses z z z 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. Relevant forces: applied electric field, retarding Coulomb attraction in dipole, internal friction G G D = (ε ′ + jε ′′) E 10/31/2003 EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie Ref. Jordan and Balmain, Electromagnetic Waves and Radiating Systems, p. 307. 15 Other Dissipation Mechanisms z Dielectric Losses Consider Ampere's Law for a dielectric with finite conductivity, σ e . K K ' " K K K ∂ ( ε r + jε r ) ε 0 E ∂D ∇× H = J + = σeE + ∂t ∂t Assuming sinusoidal steady-state, e − jωt : K K K K K ' σ e K " ' " ∇ × H = σ e E − jωε r E + ωε r E = − jω ε r ε 0 + j ε r ε 0 + E ω Lumped element equivalent circuit for a parallel plate capacitor: C= ε r' ε 0 A g R= (σ g " ωε + rε 0 ) A e ε = ε r ε 0 where ε 0 = vacuum permitivity=8.854 ×10-14 F / cm 10/31/2003 EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie 16 Other Dissipation Mechanisms z Magnetic Losses I2R Joule heating in finite coil resistance (inductor or transformer winding) z Eddy currents induced in conducting magnetic core (and also winding conductors) K K K ∂B Consider Faraday's Law, ∇ × E = − ∂t The time-varying magnetic flux gives rise to a voltage (opposing flux change) that induces electrical currents in the magnetic core. z Ref. Erickson, Fundamentals of Power Electronics, p. 472. These 'eddy currents' cause i 2 R heating of the core. 10/31/2003 EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie 17 Other Dissipation Mechanisms z Magnetic Hysteresis z Internal friction due to magnetic domain wall motion when magnetic flux density attempts to align with magnetic field, H. z z z z z Coercive force, HC, before alignment occurs similar to Coulomb friction. Remanent flux density, BR, exists at H = 0. Magnetically hard materials: large HC, such as Sm-Co alloy. Magnetically soft materials: small HC, such nickel-iron alloy. The internal area of a hysteresis loop is a measure of the energy lost due to magnetic hysteresis. 10/31/2003 Ref. Senturia, Microsystem Design, p. 241. 18 EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie Coupled Flows z Coupled flows: z z z Temperature gradient and concentration gradient Simultaneous heat flow and diffusion Seebeck effect and thermocouples G αs: Seeback coefficient E = α s ∇T αs,1 TH TV αs,2 TC TH TV TV TC TH V TC VTC = ∫ α s ,1 dT + ∫ α s ,2 dT + ∫ α s ,1 dT = TH ∫ (α s ,1 − α s ,2 ) dT = (α s ,1 − α s ,2 ) (TH − TC ) TC 10/31/2003 EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie 19 Coupled Flows z Peltier effect and Refrigerators Q = ∏ It cold hot Q = I ( ∏ n − ∏ p ) N Peltier effect is the converse of Seebeck effect. The two coefficients have the following relation, ∏ = Tα s 10/31/2003 EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie 20