One-dimensional modeling of TE devices using SPICE International Summerschool on Advanced Materials...

International Summerschool on Advanced Materials and Thermoelectricity 1

One-dimensional modeling of TE devices using SPICEOne-dimensional modeling of TE devices using SPICE

One-dimensional modeling of TE devices

Daniel Mitrani and Juan A. ChávezElectrical Engineering Department, Universitat Politècnica de Catalunya

Barcelona, Spain. Email: [email protected]



• Overview

• TEM description and formulae

• Steady-state electrical models and simulations

• Dynamic electrical model and simulations

• Conclusions

Contents

Presentation contents



TEM Description

Negative (-)

Positive (+)MoistureProtection

Ceramic Plates

Thermocouples

T

p J

x

x=0 x=L

Th

TcI np

Qh

Qc

TE module characteristics

• Solid-state devices.

• Couples connected electrically in series and thermally in parallel.

• Peltier mode: heat pump.

• Seebeck mode: electrical power generation.

Single couple unitFree standing pellet

Th

Tc



Interaction between thermaland electrical domains2T ds T

J JTx x dT x

1-D steady-state energy balance equation

Thomson Effect

Joule Effect

Fourier’s Law

TEM Formulae (I)

Tq sJT

x

Heat flow per unit area

Peltier Effect Fourier’s Law

TE J s

x

Electric field per unit length

Ohm’s Law Seebeck Effect

Electrical domain

Thermal domain

Peltier Effect

Thomson Effect

Seebeck Effect

ConductionConvection

Radiation

Joule Effect

Ohm’s Law



TEM Formulae (II)

( ) ( )h cL J L s T T

2 ( )

2h c

c c

J L T Tq s J T

L

2 2

2

( )T x J

x

Constant material properties

( 0) ( ) ( 0) 0Vc hT x T T x L T x

Dirichlet boundary conditions

2 22( )

2 2h c

c

J T T J LT x x x T

L

1-D temperature distribution

2 ( )

2h c

h h

J L T Tq s J T

L

222

2

( ) ( ) ( ) ( ) ( )( ) ( ) ( )

T x d T T x ds T T xT J T JT x

x dT x dT x

Heat flow per unit area at x=0 and x=L

Electrical potential at x=L

p-typeJ

xx=0 x=L

ThTc

qc qh



V h cI R S T T

212 ( )h h h cQ S I T I R K T T

2e h C sP Q Q I R I V

Heat absorbed at the cold side

212 ( )c c h cQ S I T I R K T T

Electrical power

Heat released at the hot side

Voltage across the terminalsp p n n

p n

A AK N

L L

p p n n

p n

L LR N

A A

p nS N s s

• Parallel thermal conductance of the N couples

• Serial electrical resistance of the N couples

• Seebeck coefficient of the N couples

TEM Formulae (III)

Qh

- +

np Inp np

Qc



Lumped Electrical Model (I)

Electrical steady-state three-port model

+

-

V

ThPe

+-

R

Vs

Px

TEM model

Rth

+

-

Tc

Qc Qh

I

+

-

Thermal and electrical analogies

Thermal Domain Electrical Domain

Heal Flow (W) Electrical Current (A)

Temperature Difference (K) Voltage (V)

Thermal Resistance (K/W) Electrical Resistance ()

Thermal Mass, (J/K) Electrical Capacitance (F)

Vs h cS T T 2e h CP Q Q I R I V

Model expressions21

2x cP SIT I R

• Thermal processes described in electrical terms.

• Flexible boundary conditions.

• Simulation of control electronics and thermal elements.

Material parameters are set prior to simulation !!!



Temperature profile for Qcmax case

25

30

35

40

45

50

55

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x (mm)

Tem

per

atu

re (

ºC)

Th=300K, Qc=Qcmax

Tm

Tc Th

Temperature profile for Tmax case

-40

-30

-20

-10

0

10

20

30

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x (mm)

Tem

per

atu

re (

ºC)

Th=300K, T=Tmax

Tm

Tavg

Tc

Th

2h c

avg

T TT

0

1( )

L

mT T x dxL

• Average temperature between hot and cold side • Mean module temperature

Lumped Electrical Model (II)



2

0

1 1( )

2 12

Lh c

m

T T I RT T x dx

L K

2h c

avg

T TT

• Mean module temperature:

• Hot side temperature

• Cold side temperature

• Average module temperature:

( ) ( )k

k T TS T S T ( ) ( )k

k T TR T R T ( ) ( )k

k T TK T K T

Material properties are calculated as:

+

-

V

Th

+

-

Tc

Qc Qh

I

+

-

Pe

+-Vs

Px

TEM model

Tcon+-

+ -

Vr

+-

S(Tk)+-

R(Tk)+-

K(Tk)+-

Tk

S R K Tk

Where Tk can be calculated as:

Tk

( )x c

conk

P QT

K T

( )r kV I R T

Additional VCVS’s are defined as:

Lumped Electrical Model (III)



+

-

V

Th

+

-Tc

Qc Qh

I

+

-

+

-T1Pe

Vs

Px

Tcon

+-

+ -

Vr

+ -

S1 R1 K1 Tm1

Pe

Vs

Px

Tcon

+-

+ -

Vr

+ -

S2 R2 K2 Tm2

+

-T2 Pe

Vs

Px

Tcon

+-

+ -Vr

+ -

S3 R3 K3 Tm3

+

-T3 Pe

Vs

Px

Tcon

+-

+ -

Vr

+ -

S4 R4 K4 Tm4

Steady-state equations are accurate as long as the thermoelectric properties do not vary over the region where they are applied.

• Divide the pellets of a TEM into many small segments• Each segment would be closer to meeting such criteria

Distributed Parameter Electrical Model

N

N-1

2

1

N

N-1

2

1

Tn

Tn-1

Qn-1

Qn

QN=Qh

Q0=Qc

TN=Th

T0=Tc

n nSn=S(Tn)Rn=R(Tn)Kn=K(Tn)



Steady-State Simulation Setup and Results (I)

s(T) = -0.0024 T2 + 1.7062 T - 73.929R2 = 0.9992

0

50

100

150

200

250

0 100 200 300 400T (K)

s (1

0-6V

·K-1

)

Experimental data

s(T)

s(T) = 0.0024 T2 - 1.6293 T + 326.5

R2 = 0.9972

0

50

100

150

200

250

300

0 100 200 300 400T (K)

s 103

1m

1

Experimental datasT

(T) = 3E-05 T2 - 0.0193 T + 4.1157

R2 = 0.9966

1.00

1.50

2.00

2.50

3.00

3.50

0 100 200 300 400T (K)

·m1K

1

Experimental data(T)

(Bi0.5Sb0.5)2Te3

L

Th

Th=300 K

L=1 mm

A=1 mm2



Temperature Difference vs. Electrical Current

ElecMod NumMod

NumMod

RelativeError 100T T

T

Model Tmax ITmaxRelative

Error

Lumped @ Th 59.83 31.21 4.06%

Lumped @ Tm 60.51 31.34 2.97%

Lumped @ Tavg 60.62 32.38 2.78%

Lumped @ Tc 59.65 35.68 4.34%

Distributed (10 FE) 62.34 32.97 0.03%

Numerical 62.36 32.99

0

10

20

30

40

50

60

70

0 10 20 30 40 50I (A)

T (

ºC)

0

2

4

6

8

10

12

14

Rel

ativ

e E

rro

r (%

)

Lumped @ThLumped @TmLumped @TavgLumped @TcDist. (10 FE)Numerical

Th=300K, Qc=0

58

59

60

61

62

63

30 31 32 33 34 35 36I (A)

T (

ºC)

Lumped @Th Lumped @Tc

Lumped @Tavg Lumped @Tm

Distributed (10 FE) Numerical

Th=300K, Qc=0

Steady-State Simulation Setup and Results (II)



0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0 10 20 30 40 50

I (A)

Qc

(W)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Rel

ativ

e E

rro

r (%

)

Lumped @Th=Tc=TavgLumped @TmDistributed (10 FE)Numerical

Th=300K, T=0

1.229

1.230

1.231

1.232

1.233

1.234

1.235

1.236

1.237

36.5 36.7 36.9 37.1 37.3 37.5 37.7 37.9

I (A)

Qc

(W)

Lumped @ Th=Tc=Tavg

Lumped @ Tm

Distributed (10 FE)

Numerical

Th=300K, T=0

Cooling Power vs. Electrical Current

ElecMod NumMod

NumMod

RelativeError 100Q Q

Q

Model Qcmax IQcmaxRelative

Error

Lump. @ Th =Tc=Tavg 1.237 W 37.74 A 0.37%

Lumped @ Tm 1.230 W 36.95 A 0.18%

Distributed (10 FE) 1.232 W 37.14 A 0.01%

Numerical 1.232 W 37.15 A

Steady-State Simulation Setup and Results (III)



218.50

212.56

208.88

195.61

190

195

200

205

210

215

220

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x (mm)

s (1

0-6V

·K-1

)

Lumped @ ThLumped @ TmLumped @ TavgLumped @ TcDistributed (10 FE)Numerical

17.37

16.16

15.44

13.20

12

13

14

15

16

17

18

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x (mm)

10

6 ·m


1.325

1.336

1.402

1.31

1.33

1.35

1.37

1.39

1.41

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x (mm)

·m1K

1

Lumped @ Th

Lumped @ Tm

Lumped @ Tavg

Lumped @ Tc

Distributed (10 FE)

Numerical

2.074

2.110

2.116

2.068

2.05

2.06

2.07

2.08

2.09

2.10

2.11

2.12

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x (mm)

z (1

0-3K

-1)


Spatial profiles for material parameters s(x), (x), (x), and z(x)

2sz

Steady-State Simulation Setup and Results (IV)



Proposed distributed parameter transient electrical model

Qpc Qph

+

-

Th

+

-

Tc

Qc Qh

Finite Elem. 1 Finite Elem. N-1 Finite Elem. N

+

-V

I

C

21 Rth 2

1 Rth

R Vs

+ -

QJ

C C

21 Rth 2

1 Rth

R Vs

+ -

QJ

C C

21 Rth 2

1 Rth

R Vs

+ -

QJ

CC

21 Rth 2

1 Rth

R Vs

+ -

QJ

C

Finite Elem. 2

2 2

2 2

T I T

x A t

No analytical

solution !!!

I → Electrical current → Electrical resistivity → Thermal conductivity → Thermal diffusivity

• Start-up and shut-down periods.• Operating conditions are varied with time.• Fast-response heat sources.• Similar TEC and Heat load thermal time constants• Pulse cooling analysis.

Dynamic Distributed Parameter Electrical Model

One-dimensional heat flow equationJustification

16

210

220

230

240

250

260

270

-5 0 5 10 15 20 25 30 35 40

Time (s)

Tc

(K)

SPICE Model

Parabolic Solution

Linear SolutionT pulse @ t min

T postpulse

T ss

Return to steady-state

Th=300 KP=2.5

Pulse Cooling

t ret

205

210

215

220

225

230

235

240

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

time (s)

Tc

(K)

t0

t1/2

t1

t2

Th=300 K

P=5

Pulse cooling simulation analysis examples

0

5

10

15

20

25

30

35

2 3 4 5 6 7 8 9 10

P = Ipulse/Imax

Tpu

lse

(K)

0.0

0.5

1.0

1.5

2.0

2.5

t min (

s)

SPICE ModelParabolic SolutionLinear Solution

Tpulse curves

tmin curves

Th =300 K

205

210

215

220

225

230

235

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

time (s)

Tc

(K)

L=1 mm

L=5 mm

L=10 mm

P=5

0

1

2

3

4

5

0 1 2 3 4 5 6 7 8 9 10P ellet L ength (mm)

t min

(s)

P =2.5P =5

P =10

Th=300 K

0.0

0.5

1.0

1.5

2.0

2.5

3.0

-5 0 5 10 15 20 25 30 35 40

Time (s)

Cur

rent

(A

)

I max (curent for maximum stedy-state T )

I pulse = P·I max Current Pulse

210

225

240

255

270

285

300

0 50 100 150 200 250 300Time (s)

Tc (

K)

Initial condition (Tc=Th)

Steady-state (Tc=Tcss)

Transient cooling (Tc=Tcmin)

Post pulse heating

Return to steady-state



Conclusions

Conclusions

• Based on 1-D steady-state analysis we propose

– Lumped parameter model

– Distributed parameter model

• Simulation of electrical and thermal domains with a single tool

– Control electronics

– Thermal elements

• Material parameters chosen according to different module temperatures

• Dynamic Distributed Parameter Electrical Model

– Start-up and shut down periods

– Similar TEM and heat load time constants

– Transient cooling operation

One-dimensional modeling of TE devices using SPICE International Summerschool on Advanced Materials...

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