Thermal Modeling© M. T. Thompson, Power Electronics Notes 29 Thermal Circuit Modeling and Introduction to Thermal System Design Marc T. Thompson, Ph.D. Thompson Consulting, Inc. 9 Jacob Gates Road Harvard, MA Phone: (978) Website: Jeff W. Roblee, Ph.D. VP of Engineering Precitech, Inc. Keene, NH

Thermal Modeling© M. T. Thompson, Summary Basics of heat flow, as applied to device sizing and heat sinking Use of thermal circuit analogies –Thermal resistance –Thermal capacitance Examples –Picture window examples –Magnetic brake –Plastic tube in sunlight

Thermal Modeling© M. T. Thompson, 2009 All components (capacitors, inductors and transformers, semiconductor devices) have maximum operating temperatures specified by manufacturer High operating temperatures have undesirable effects on components: Need for Component Temperature Control Capacitors Electrolyte evaporation rate increases significantly with temperature increases and thus shortens lifetime Magnetic Components Losses (at constant power input) increase above 100 °C Winding insulation (lacquer or varnish) degrades above 100 °C Semiconductors Unequal power sharing in parallel or series devices Reduction in breakdown voltage in some devices Increase in leakage currents Increase in switching times 3

Thermal Modeling© M. T. Thompson, 2009 Control voltages across and current through components via good design practices Snubbers may be required for semiconductor devices. Free-wheeling diodes may be needed with magnetic components Maximize heat transfer via convection and radiation from component to ambient Short heat flow paths from interior to component surface and large component surface area. Component user has responsibility to properly mount temperature-critical components on heat sinks. Apply recommended torque on mounting bolts and nuts and use thermal grease between component and heat sink. Properly design system layout and enclosure for adequate air flow Temperature Control Methods 4

Thermal Modeling© M. T. Thompson, 2009 Heat Transfer Heat transfer (or heat exchange) is the flow of thermal energy due to a temperature difference between two bodies Heat transfers from a hot body to a cold one, a result of the second law of thermodynamics Heat transfer is slowed when the difference in temperature between the two bodies reduces 5

Thermal Modeling© M. T. Thompson, Intuitive Thinking about Thermal Modeling Heat (Watts) flows from an area of higher temperature to an area of lower temperature Heat flow is by 3 mechanisms –Conduction - transferring heat through a solid body –Convection - heat is carried away by a moving fluid Free convection Forced convection - uses fan or pump –Radiation Power is radiated away by electromagnetic radiation You can think of high- thermal conductivity material such as copper and aluminum as an easy conduit for conductive power flow…. i.e. the power easily flows thru the material

Thermal Modeling© M. T. Thompson, Thermal Circuit Analogy Use Ohm’s law analogy to model thermal circuits Thermal resistance k = thermal conductivity (W/(mK)) Thermal capacitance: analogy isn’t as straightforward c p = heat capacity of material (Joules/(kg-K))

Thermal Modeling© M. T. Thompson, Thermal Circuit Analogy ELECTRICALTHERMAL Forcing variableVoltage (V)Temperature (K) Flow variableCurrent (A)Heat (W) ResistanceResistance (V/A)Thermal resistance (K/W) CapacitanceCapacitance (V/C)Thermal capacitance (J/K) Heat transfer can be modeled by thermal circuits Using Ohm’s law analogy: Reference: M. T. Thompson, Intuitive Analog Circuit Design, Newnes, 2006.

Thermal Modeling© M. T. Thompson, Thermal Circuit Analogy Elementary thermal network Reference: M. T. Thompson, Intuitive Analog Circuit Design, Newnes, 2006.

Thermal Modeling© M. T. Thompson, Thermal Resistance Thermal resistance quantifies the rate of heat transfer for a given temperature difference k = thermal coefficient (W/(mK)) A = cross section (m 2 ) l = length (m)

Thermal Modeling© M. T. Thompson, Thermal Capacitance Thermal capacitance is an indication of how well a material stores thermal energy It is used when transient phenomena are considered Analogy isn’t as straightforward M = mass (kg) c p = heat capacity of material (Joules/(kg-K))

Thermal Modeling© M. T. Thompson, Heat Flow Mechanisms Heat flows by 3 mechanisms; the driving force for heat transfer is the difference in temperature 1.Conduction 2.Convection Free convection Forced convection 3. Radiation Reference: R. E. Sonntag and C. Borgnakke, Introduction to Engineering Thermodinamics, John Wiley, 2007

Thermal Modeling© M. T. Thompson, Conduction Heat is transferred through a solid from an area of higher temperature to lower temperature To have good heat conduction, you need large area, short length and high thermal conductivity Example: aluminum plate, l = 10 cm, A=1 cm 2, T 2 = 25C (298K), T 1 = 75C (348K), k = 230 W/(m-K)

Thermal Modeling© M. T. Thompson, Thermal Conductivity of Selected Materials References: 1. B. V. Karlekar and R. M. Desmond, Engineering Heat Transfer, pp. 8, West Publishing, Burr Brown, Inc., “Thermal and Electrical Properties of Selected Packaging Materials”

Thermal Modeling© M. T. Thompson, 2009 Thermal Equivalent Circuits Heat flow through a structure composed of layers of different materials Thermal equivalent circuit simplifies calculation of temperatures in various parts of structure. P R  sa  cs R  jc R Junction Case SinkAmbient j T c T s T a T Chip T j Case T c Isolation pad Heat sink T s Ambient Temperature T a T i = P d (R  jc + R  cs + R  sa ) + T a If there parallel heat flow paths, then thermal resistances combine as do electrical resistors in parallel. 15

Thermal Modeling© M. T. Thompson, Thermal Conductivity of Selected Materials Reference: International Rectifier, Application note N-1057, “Heatsink Characteristics”

Thermal Modeling© M. T. Thompson, Heat Capacity of Selected Materials Reference: B. V. Karlekar and R. M. Desmond, Engineering Heat Transfer, West Publishing, 1977 Heat capacity is an indication of how well a material stores thermal energy

Thermal Modeling© M. T. Thompson, Heat Capacity of Alloys Reference:

Thermal Modeling© M. T. Thompson, Convection Reference: International Rectifier, Application note N-1057, “Heatsink Characteristics” Convection can be free (without a fan) or forced (with a fan)

Thermal Modeling© M. T. Thompson, Free Convection Reference:

Thermal Modeling© M. T. Thompson, Heat Transfer Coefficient for Convection Heat is transferred via a moving fluid Convection can be described by a heat transfer coefficient h and Newton’s Law of Cooling: Heat transfer coefficient depends on properties of the fluid, flow rate of the fluid, and the shape and size of the surfaces involved, and is nonlinear Equivalent thermal resistance: Reference: B. V. Karlekar and R. M. Desmond, Engineering Heat Transfer, pp. 14, West Publishing, 1977

Thermal Modeling© M. T. Thompson, Free Convection Heat is drawn away from a surface by a free gas or fluid Buoyancy of fluid creates movement For vertical fin: A in m 2, d vert in m Example: square aluminum plate, A=1 cm 2, T a = 25C (298K), T s = 75C (348K)

Thermal Modeling© M. T. Thompson, Free Convection Heat Transfer Coefficient (h) For vertical fin: Area A in m 2, fin vertical height d vert in m

Thermal Modeling© M. T. Thompson, Forced Convection With a fan Reference: International Rectifier, Application note N-1057, “Heatsink Characteristics”

Thermal Modeling© M. T. Thompson, Forced Convection In many cases, heat sinks can not dissipate sufficient power by natural convection and radiation In forced convection, heat is carried away by a forced fluid (moving air from a fan, or pumped water, etc.) Forced air cooling can provide typically 3-5  increase in heat transfer and 3-5  reduction in heat sink volume –In extreme cases you can do  10x better by using big fans, convoluted heat sink fin patterns, etc.

Thermal Modeling© M. T. Thompson, Thermal Performance Graphs for Heat Sinks Reference: Curve #1: natural convection (P vs.  T sa ) Curve #2: forced convection curve (R sa vs. airflow) 1 2

Thermal Modeling© M. T. Thompson, Radiation Energy is transferred through electromagnetic radiation Reference: International Rectifier, Application note N-1057, “Heatsink Characteristics”

Thermal Modeling© M. T. Thompson, Radiation Energy is lost to the universe through electromagnetic radiation  = emissivity (0 for ideal reflector, 1 for ideal radiator “blackbody”);  = Stefan-Boltzmann constant = 5.68  W/(m 2 K 4 ) Example: anodized aluminum plate,  = 0.8, A=1 cm 2, T a = 25C (298K), T s = 75C (348K)

Thermal Modeling© M. T. Thompson, Radiation Incident, reflected and emitted radiation; e.g. body in sunlight Reference:

Thermal Modeling© M. T. Thompson, Emissivity Reference: International Rectifier, Application note N-1057, “Heatsink Characteristics”

Thermal Modeling© M. T. Thompson, Emissivity Reference: International Rectifier, Application note N-1057, “Heatsink Characteristics”

Thermal Modeling© M. T. Thompson, Comments on Radiation In multiple-fin heat sinks with modest temperature rise, radiation usually isn’t an important effect –Ignoring radiation results in a more conservative design Effective heat transfer coefficient due to radiation for ideal blackbody (  = 1) at with surface temperature 350K radiating to ambient at 300K is h rad = 6.1 W/(m 2 K), which is comparable to free convection heat transfer coefficient –However, radiation between heat sink fins is usually negligible (generally they are very close in temperature)

Thermal Modeling© M. T. Thompson, IC Mounted to Heat Sink Interfaces –Heat sink-ambient: convection (free or forced) –Heat sink-case of IC: conduction –Case – junction: conduction

Thermal Modeling© M. T. Thompson, 2009 Multiple Fin Heat Sink Reference: 34

Thermal Modeling© M. T. Thompson, IC Mounted to Heat Sink Reference: International Rectifier, Application Note AN-997

Thermal Modeling© M. T. Thompson, IC Mounted to Heat Sink --- Close-up Reference: International Rectifier, Application Note AN-997 Thermal compound is often used to fill in the airgap voids

Thermal Modeling© M. T. Thompson, IC Mounted to Heat Sink --- Contact Resistance vs. Torque (TO-247) Reference: International Rectifier, Application Note AN-997

Thermal Modeling© M. T. Thompson, IC Mounted to Heat Sink --- Contact Resistance vs. Interface Material (TO-247) Reference: International Rectifier, Application Note AN-997

Thermal Modeling© M. T. Thompson, IC Mounted to Heat Sink --- Contact Resistance vs. Interface Material (TO-247) Reference: International Rectifier, Application Note AN-997 Dry vs. thermal compound vs. electrically-insulating pad

Thermal Modeling© M. T. Thompson, Thermal Grease

Thermal Modeling© M. T. Thompson, Heat Sink Pad

Thermal Modeling© M. T. Thompson, 2009 Transient Thermal Impedance Heat capacity per unit volume C v = dQ/dT [Joules / o C] prevents short duration high power dissipation surges from raising component temperature beyond operating limits. Transient thermal equivalent circuit. C s = C v V where V is the volume of the component. P(t)  R j T (t) a T C s Transient thermal impedance Z  (t) = [T j (t) - T a ]/P(t)   = π R  C s /4 = thermal time constant T j (t =   ) = P o R  42

Thermal Modeling© M. T. Thompson, 2009 Use of Transient Thermal Impedance Response for a rectangular power dissipation pulse P(t) = Po {u(t) - u(t - t 1 )}. T j (t) = Po { Z  (t) - Z  (t - t 1 ) } Symbolic solution for half sine power dissipation pulse. P(t) = P o {u(t - T/8) - u(t - 3T/8)} ; area under two curves identical. T j (t) = Po { Z  (t - T/8) - Z  (t - 3T/8) } 43

Thermal Modeling© M. T. Thompson, 2009 Multilayer Structures Multilayer geometry Transient thermal equivalent circuit Transient thermal impedance (asymptotic) of multilayer structure assuming widely separated thermal time constants. 44

Thermal Modeling© M. T. Thompson, 2009 Heat Sinks Aluminum heat sinks of various shapes and sizes widely available for cooling components. Often anodized with black oxide coating to reduce thermal resistance by up to 25%. Sinks cooled by natural convection have thermal time constants of minutes. Forced-air cooled sinks have substantially smaller thermal time constants, typically less than one minute. Choice of heat sink depends on required thermal resistance, R  sa, which is determined by several factors. Maximum power, P diss, dissipated in the component mounted on the heat sink. Component's maximum internal temperature, T j,max Component's junction-to-case thermal resistance, R  jc. Maximum ambient temperature, T a,max. R  sa = {T j,max - T a,max }P diss - R  jc P diss and T a,max determined by particular application. T j,max and R  jc set by component manufacturer. 45

Thermal Modeling© M. T. Thompson, 2009 Heat Conduction Thermal Resistance Generic geometry of heat flow via conduction Heat flow P cond [W/m 2 ] =kA (T 2 - T 1 ) / d = (T 2 - T 1 ) / R  cond Thermal resistance R  cond = d / [k A] Cross-sectional area A = hb k = Thermal conductivity has units of W-m -1 - o C -1 (k Al = 220 W-m -1 - o C -1 ).Units of thermal resistance are o C/W 46

Thermal Modeling© M. T. Thompson, 2009 Radiative Thermal Resistance Stefan-Boltzmann law describes radiative heat transfer. P rad = 5.7x10 -8 EA [( T s ) 4 -( T a ) 4 ] ; [P rad ] = [Watts] E = emissivity; black anodized aluminum E = 0.9 ; polished aluminum E = 0.05 A = surface area [m 2 ] through which heat radiation emerges. T s = surface temperature [  K] of component. T a = ambient temperature [  K]. ( T s - T a )/Prad = R ,rad = [T s - T a ][5.7x10 -8 EA {( T s /100) 4 -( T a /100) 4 }] -1 Example - black anodized cube of aluminum 10 cm on a side. T s = 120  C and T a = 20  C R ,rad = [ ][(5.7) (0.9)(6x10-2){(393/100) 4 - (293/100) 4 }] -1 R ,rad = 2.2  C/W 47

Thermal Modeling© M. T. Thompson, 2009 Convective Thermal Resistance P conv = convective heat loss to surrounding air from a vertical surface at sea level having height d vert [in meters] less than one meter. P conv = 1.34 A [Ts - Ta] 1.25 d vert A = total surface area in [m 2 ] T s = surface temperature [  K] of component. T a = ambient temperature [  K]. [T s - T a ]/P conv = R ,conv = [T s - T a ] [d vert ] 0.25 [1.34 A (T s - T a ) 1.25 ] -1 R ,conv = [d vert ] 0.25 {1.34 A [T s - T a ] 0.25 } -1 Example - black anodized cube of aluminum 10 cm on a side. T s = 120  C and T a = 20  C. R ,conv = [10 -1 ]0.25([1.34] [6x10 -2 ] [ ] 0.25 ) -1 R ,conv = 2.2  C/W 48

Thermal Modeling© M. T. Thompson, 2009 Combined Effects of Convection and Radiation Heat loss via convection and radiation occur in parallel. Steady-state thermal equivalent circuit R ,sink = R ,rad R ,conv / [ R ,rad + R ,conv ] Example - black anodized aluminum cube 10 cm per side R ,rad = 2.2  C/W and R ,conv = 2.2  C/W R ,sink = (2.2) (2.2) /( ) = 1.1  C/W 49

Thermal Modeling© M. T. Thompson, Cost for Various Heat Sink Systems Reference: Note: heat pipe and liquid systems require eventual heat sink

Thermal Modeling© M. T. Thompson, Comparison of Heat Sinks Reference: STAMPEDEXTRUDED“CONVOLUTED”FAN

Thermal Modeling© M. T. Thompson, N3904 Static Thermal Model

Thermal Modeling© M. T. Thompson, Liquid Cooling Advantages –Best performance per unit volume –Typical thermal resistance C/W Disadvantages –Need a pump –Heat exchanger –Possibility of leaks –Cost

Thermal Modeling© M. T. Thompson, Heat Pipe Heat pipe consists of a sealed container whose inner surfaces have a capillary wicking material Boiling heat transfer moves heat from the input to the output end of the heat pipe Heat pipes have an effective thermal conductivity much higher than that of copper

Thermal Modeling© M. T. Thompson, Thermoelectric (TE) Cooler “Cooler” is a misnomer; a TE cooler is a heat pump Peltier effect: uses current flow to pump heat from cold side to warm side Pumping is typically 25% efficient; to pump 2 Watts of waste heat takes 8 Watts or more of electrical power However, device cooled device can be at a lower temperature than ambient TE coolers can heat or cool, depending on current flow

Thermal Modeling© M. T. Thompson, Thermoelectric (TE) Cooler

Thermal Modeling© M. T. Thompson, Fan

Thermal Modeling© M. T. Thompson, Example 1: Picture Window Consider picture window with A = 1 m 2, 2.5 mm thick T i = 70F (25C); Approximate T o = 32F (0C) for 6 months (long winter !) What is total cost for heat loss at $0.10/kW-hr

Thermal Modeling© M. T. Thompson, Example 1: Picture Window Assumptions: –Window glass k = 0.78 W/(m-K) –Inside and outside window, heat transfer dominated by free convection; h = 10 W/(m 2 K) R iw = R ow = 1/(hA) = 0.1 C/Watt R w = w/(kA) =0.0025/(0.78)(1) = C/Watt R total = C/Watt P =  DT/R total = 25C/0.2032C/Watt = 123 Watts E = 3 kW-hr/day or 539 kW-hr for winter Cost = $53.9

Thermal Modeling© M. T. Thompson, Example 2: Picture Window with Double Pane Assumptions: –Still air in airgap k = W/(m-K) –Ignore radiation 1 cm airgap: R airgap = g/(kA) =0.01/(0.027)(1) = 0.37 C/Watt R total = 0.58 C/Watt P =  DT/R total = 25C/0.58C/Watt = 43 W E = 1 kW-hr/day or 188 kW-hr for winter Cost = $18.80 –Cost will be lower if gap has vacuum

Thermal Modeling© M. T. Thompson, Example 3: Temperature Rise in Magnetic Brake Train mass M = 12,300 kg Initial speed = 16 meters/second Brake aluminum fin length 10 meters Stopping time: a few seconds Cycle time: 1200 seconds What is temperature rise in aluminum fin and in steel ?

Thermal Modeling© M. T. Thompson, Example 3: Magnetic Brake Thermal Model Model for 1 meter long section of brake Guesstimated dominant time constant = 4,500 seconds (0.5  x 9000 F) based on thermal model above

Thermal Modeling© M. T. Thompson, Example 3: Magnetic Brake Temperature Profile PSPICE simulation

Thermal Modeling© M. T. Thompson, Example 4: White Pipe in the Hot Sun How hot does the surface of a white pipe get? Assume R = m, pipe length = 1m, sunlight = 1200 W/m 2, h = 8 W/m 2 -K,  = 0.9 and solar absorption coeff.  solar = 0.26 Assume no conductive heat transfer

Thermal Modeling© M. T. Thompson, Example 4: Pipe in the Hot Sun Q sun = 1356 W Q refl = (1-  solar )Q sun = 1003 W Therefore, 353 Watts is absorbed by the pipe, then dissipated via radiation and convection

Thermal Modeling© M. T. Thompson, Example 4: Pipe in the Hot Sun Given these assumptions, temperature rise above ambient (T s – T A )  7 degrees C with Q conv = 195 W and Q rad = 158 W For radiation: with  = 0.9,  = 5.68  W/(m 2 K 4 ) and surface area A = 1.0 m 2. For free convection: with free convection heat transfer coefficient estimated as h  8 W/(m 2 -K).

Thermal Modeling© M. T. Thompson, Example 5: What Happens if Pipe is Black? Q refl goes way down (solar energy absorption goes up, as  solar = 0.9)

Thermal Modeling© M. T. Thompson, Other Important Thermal Design Issues Contact resistance –How to estimate it –How to reduce it Thermal pads, thermal grease, etc. Proper torque for mounting screws Geometry effects –Vertical vs. horizontal fins –Fin efficiency (how close together can you put heat sink fins ?)

Thermal Modeling© M. T. Thompson, Some Heat Sinks TO-92 (small transistor package) Reference: Aavid-Thermalloy

Thermal Modeling© M. T. Thompson, Some Heat Sinks TO-220 Reference: Aavid-Thermalloy

Thermal Modeling© M. T. Thompson, Some Heat Sinks TO-247 Reference: Aavid-Thermalloy

Thermal Modeling© M. T. Thompson, Some Heat Sinks Vicor power brick Reference: Aavid-Thermalloy

Thermal Modeling© M. T. Thompson, Some Heat Sinks Liquid cooled plate Reference: Aavid-Thermalloy

Thermal Modeling© M. T. Thompson, Extrusions Reference: Aavid-Thermalloy

Thermal Modeling© M. T. Thompson, Cooling Fins References: J H. Lienhard IV and J H. Lienhard V, “A Heat Transfer Textbook,” 3 rd edition, Phlogiston Press, Cambridge, MA 2008

Thermal Modeling© M. T. Thompson, Improving Conductive Heat Transfer References: International Rectifier, Application note N-1057, “Heatsink Characteristics”

Thermal Modeling© M. T. Thompson, Improving Forced Convection Heat Transfer References: International Rectifier, Application note N-1057, “Heatsink Characteristics”

Thermal Modeling© M. T. Thompson, Improving Forced Convection Heat Transfer References: International Rectifier, Application note N-1057, “Heatsink Characteristics”

Thermal Modeling© M. T. Thompson, Improving Radiation Heat Transfer References: International Rectifier, Application note N-1057, “Heatsink Characteristics”

Thermal Modeling© M. T. Thompson, Conversion Factors References: J H. Lienhard IV and J H. Lienhard V, “A Heat Transfer Textbook,” 3 rd edition, Phlogiston Press, Cambridge, MA 2008