Canonical Forms of Multi-Port Dynamic Thermal Networks

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1 Canonical Forms o Multi-Port Dynamic Thermal Networs orenzo Codecasa Dario D Amore and Paolo Maezzoni Politecnico di Milano Dipartimento di Elettronica e Inormazione Piazza eonardo da Vinci Milan Italy {codecasa damore pmaezz}@elet.polimi.it Abstract In this paper it is shown that multi-port dynamic thermal networs admit our canonical representations which generalize the our canonical representations o passive lumped RC networs: Foster I and II canonical orms Cauer I and Cauer II canonical orms. In particular the generalized Foster I canonical orm is equivalent to the time-constant representation and the generalized Cauer I canonical orm is a passive multi-conductor RC transmission line. Keywords: Multi-Port Thermal Networs Time-Constant Representation Structure Function. Introduction Thermal networs are widely used or modeling heat diusion in components and pacages. At irst thermal networs have been proposed or modeling static heat diusion. More recently thermal networs have been proposed also or modeling dynamic heat diusion. The question o determining the canonical orms o one-port dynamic thermal networs has been considered in 6. It has been shown that passive one-port dynamic thermal networs admit our canonical orms which are the generalizations o Foster I Foster II Cauer I and Cauer II canonical orms o passive one-port RC lumped networs. In particular the generalized Foster I canonical orm is equivalent to the time-constant representation while the Cauer I canonical orm is equivalent to the structure unction representation 6 The question o determining canonical orms o multi-port dynamic thermal networs has not been tacled in literature yet. In this paper it is shown that all the results proved or passive one-port dynamic thermal networs can be extended to passive multi-port dynamic thermal networs. As a result the Foster I Cauer I Foster II and Cauer II representations o passive multiport RC lumped networs are extended one-port to multi-port passive dynamic thermal networs. In particular the generalized Foster I canonical orm is equivalent to the time-constant representation. Besides the generalized Cauer I canonical orm is a passive multi-conductor RC transmission line. It is also shown how all multi-port dynamic thermal networs can be represented by passive multi-port dynamic thermal networs. Thus the our canonical orms o passive multi-port dynamic thermal networs can be applied to all multi-port dynamic thermal networs. The rest o this paper is organized as ollows. In Section multi-port dynamic thermal networs are introduced. In Sections preliminary results on passive multi-port dynamic thermal networs are presented. The our canonical orms are shown in Sections 6 and 8 9. An application example is presented in Sections 7 and. Multi-Port Dynamic Thermal Networs In a bounded spatial region the relation between the power density Fr t and the temperature rise ur t with respect to ambient temperature unctions o the position vector r and o the time instant t is ruled by the heat conduction equation r ur t cr u r t = Fr t t cr is the volumetric heat capacity and r is the thermal conductivity. Eq. is completed by conditions on the boundary o and by initial condition or the temperature rise ur t. The boundary conditions assumed o Robin s type are r u r t = hrur t ν hr is the heat transer coeicient and νr is the outward unit vector normal to. Here hr is not assumed to be identically zero over that is pure Neumann s boundary conditions are excluded. The initial condition is assumed to be zero ur =. This is by no means a limitation. In act any heat diusion problem with non-zero initial condition ur = Ur and power density Fr t can be represented by an equivalent heat diusion problem with zero initial condition and power density Fr t crurδt. The heat diusion problem deined by Eqs. satisies the ollowing main physical properties: Theorem Passivity A non-negative unction Wt exists such that or each time t t t Wt Wt dt Fr tur tdr. t Theorem Reciprocity et u r s u r s be the aplace transorms o the temperature rises due to the power densities whose aplace transorms are F r s F r s respectively. It results in F r su r sdr = F r su r sdr. A multi-port dynamic thermal networ can be deined rom the heat diusion problem by introducing the powers and the

2 temperature rises measured at its ports. The powers P i t with i =...n elements o column vector Pt determine Fr t as Fr t = T rpt r is a column vector o shape unctions i r with i =... n. The temperature rise T i t with i =...n elements o column vector Tt are deined by Tt = grur t dr. gr is a column vector o shape unctions g i r with i =...n. The multi-port dynamic thermal networ deined by Eqs. - in general does not preserve the main physical properties o the heat diusion problem: reciprocity and passivity. However these physical properties are preserved i In act in this case it results in r = gr. 6 Theorem Passivity A non-negative unction Wt exists such that or each time t t t Wt Wt P T tttdt. t Theorem Reciprocity et T s T s be the aplace transorms o the temperature rise vectors due to the power vectors whose aplace transorms are P s P s respectively. It results in P T st s = P T st s. As shown in limitedly to the case o one-port dynamic thermal networs only i passivity and reciprocity hold canonical orms o dynamic thermal networs can be given. Thus hereater it will be assumed that Eq. 6 holds or equivalently that the multi-port dynamic thermal networ is passive. It can be observed that this is not a limitation. In act a multi-port dynamic thermal networ whose powers are deined by r and whose temperature rises are deined by gr has port responses equal to a subset o the port responses o the passive multi-port dynamic thermal networ whose shape unctions or both powers and temperature rises are all the distinct elements o r and gr. As a result a generic n-port dynamic thermal networ can always be substituted by a passive N-port dynamic thermal networ with n N n. Hereater it is also assumed that the shape unctions in r = gr are linearly independent. Again this is not a limitation. In act i r = gr are linearly dependent it results in gr = Rĝr ĝr is a vector o ˆn < n linearly independent shape unctions and R is an n ˆn rectangular matrix. Thus it results in Tt = RˆTt 7 ˆPt = R T Pt 8 and Fr t = ĝ T rˆpt 9 ˆTt = ĝrur t dr The passive n-port dynamic thermal networ is then the connection o a multi-port transormer 7 deined by Eqs. 7 8 to a passive ˆn-port dynamic thermal networ deined by Eqs. 9 with ˆn < n linearly independent shape unctions. Solutions o Passive Multi-Ports Dynamic Thermal Networs The solution o Eqs. can be expressed by the series expansion Tr t = a tz r z r are the eigenunctions o the eigenvalue problem associated to the thermal problem r z r = λ crz r r with boundary conditions r z ν r = hrz r r τ. The eigenvalues λ are real positive and constitute a divergent monotonically increasing sequence. The eigenunctions z r are real unctions o position satisying the orthonormality relations crz rz rdr = δ δ is Kronecher s delta. Coeicients a t in Eq. are solutions o equations d dt a t λ a t = z rgr tdr with zero initial conditions. The solutions to these initial value problems are a t = e λt z rgr tdr 6 is the convolution operator in the time domain. The solution o the passive multi-port dynamic thermal networ can then be expressed as ollows. From Eqs. 6 it results in a t = e λt Γ T Pt. 7 Γ = From Eqs. it results in Tt = z rgrdr. 8 Γ a t. 9

3 Thus substituting Eq. 7 into Eq. 9 it ollows Zt = Tt = Zt Pt Γ Γ T e λt is the power impulse thermal response matrix o the passive multi-port dynamic thermal networ. Taing the aplace transorm o Eq. it also ollows Zs = Ts = ZsPs Γ Γ T s λ is the thermal impedance matrix o the multi-port dynamic thermal networ. Preliminary Results on Passive Multi-Port Dynamic Thermal Networs Passive multi-port dynamic thermal networs are a generalization o passive multi-port lumped RC networs. In act their impedance matrices satisy properties common to passive multiport lumped RC networs. Theorem. Impedance matrix Zs with s = σ iω is symmetric and positive real 7. That is or σ > Zs is analytic Z s = Zs ReZs is positive deinite the bar indicates the complex conugate operator.. Poles o Zs are simple real negative and orm a divergent monotonically decreasing sequence λ λ..... The residues at the poles o Zs are real symmetric positive semi-deinite.. On the positive real axis Z σ is symmetric positive deinite. Since the shape unctions deining the passive multi-port dynamic thermal networ are linearly independent an admittance matrix Ys inverse o Zs exists. Such admittance matrix satisies the ollowing properties common to that o passive multi-port lumped RC networs. Theorem 6. Matrix Ys is symmetric and positive real 7. That is or σ > Ys is analytic Y s = Ȳs ReYs is positive deinite.. Poles o Ys are simple real negative and orm a divergent monotonically decreasing sequence µ µ..... The residues at the poles o Ys/s are real symmetric positive semi-deinite.. On the positive real axis Y σ is symmetric positive deinite. As a consequence o Theorems and 6 a multi-port passive distributed thermal networ can be approximated at s by the parallel connection o a passive resistive multi-port and a passive capacitive multi-port and at s by a passive capacitive multi-port as stated in the ollowing Theorem 7 At s the Zs impedance matrix converges to the impedance o the parallel connection o a passive resistive multi-port o resistance matrix R and o a passive capacitive multi-port o capacitance matrix C being R = Z C = Y. Theorem 8 For s with σ > the Ys admittance converges to the admittance o a passive capacitive multi-port o capacitance matrix Ys C = lim = lim s s s Y s. The R matrix the C matrix and the inverse o C matrix are hereater reerred to respectively as total resistance matrix total capacitance matrix and total elastance matrix o the passive multi-port dynamic thermal networ. Generalized Foster I Canonical Form As a consequence o Theorem Theorem 9 Zs = r s/λ = e s λ r = e /λ are real symmetric positive semi-deinite matrices. Eq. deines an ininite networ composed o ideal transormers passive resistors and passive capacitors which generalizes the Foster I canonical orm o a passive multi-port lumped RC networ 7. The resistance matrix o the series connections o the passive resistive multi-ports having r resistance matrices is the total resistance matrix R. Thus the generalized Foster I canonical orm deines partial resistance matrices o R. Similarly the elastance matrix o the series connections o the passive capacitive multi-ports having e elastance matrices is the total elastance matrix inverse o C. Thus the generalized Foster I canonical orm deines partial elastance matrices o the inverse o C. The Foster I canonical orm can be deined by the cumulative resistance matrix Rλ = r Hλ λ

4 H being Heaviside s step unction equivalent to the timeconstant representation or by the cumulative elastance matrix Eλ = e Hλ λ. 6 Generalized Foster II Canonical Form As a consequence o Theorem 6 the admittance matrix Ys can be represented as ollows Theorem Ys = sc R c are symmetric positive semi-deinite. sc s/µ 6 Eq. 6 deines an ininite networ composed o ideal transormers passive resistors and passive capacitors which generalizes the Foster II canonical orm o a passive multi-port lumped RC networ. The resistance matrix o the passive resistive multi-port is R. The capacitance matrix o the passive capacitive multi-port is C. The capacitance matrix o the parallel connections o the capacitive multi-ports having C and c capacitance matrices is the total capacitance matrix C. Thus the generalized Foster II canonical orm deines partial capacitances o the total capacitance matrix C. The Foster II canonical orm can be deined by the cumulative capacitance matrix and by R. Cλ = C c Hλ µ. 7 Application Example: Part I A cylinder o length area A thermal conductivity and heat capacity c is considered. The powers P t P t are uniormly generated within the lower and upper halves o the cylinder respectively. On the lower and upper ace o the boundary the temperature is set equal to the ambient temperature. On the rest o the boundary the thermal lux is set to zero. According to Eq. 6 the mean temperature rises in the lower and upper halves o the cylinder are the T t and T t temperature rise o a passive -port dynamic thermal networ. The thermal impedance matrix is Kp = p Zs = K c s p p p 7 Thus rom Eq. 7 and Eqs. it results in r = e λ = R = C = Ac C = Ac The generalized Foster I and II canonical orms o this thermal networ can be determined analytically in closed orm. Determining the Mittag-eler s partial ractions expansion o Kp the Foster I canonical orm ollows c π = λ = c π = c π = 6 π = 6 π 6 π. = = being any natural number. The Rλ cumulative resistance matrix deining the Foster I canonical orm is shown in Fig R = R R = R c λ Figure : Rλ cumulative resistance matrix deining Foster I canonical orm. and q = tanhq. q The admittance matrix Ys is Ys = H c s 8

5 Ac C = Ac C Ac C = Ac C c λ Figure : Cλ cumulative capacitance matrix deining Foster II canonical orm. Hp = p Determining the Mittag-eler s partial ractions expansion o Hp the Foster II canonical orm ollows c ξ = µ = c ξ = c ξ = Ac ξ = c = Ac ξ = Ac = ξ being any natural number and ξ being the -th positive root o the equation tanξ = ξ. The Cλ cumulative capacitance matrix deining the Foster II canonical orm is shown in Fig.. 8 Generalized Cauer I Canonical Form et us consider a passive multi-conductor RC transmission line o length x in the x dimension described at a complex requency s by equations V x s = rxix s x 9 ex I x s = svx s x Vx s Ix s are the aplace transorms o the voltage and current n vectors at x and rx ex are symmetric positive semi-deinite n n matrices representing the resistance matrix density and the elastance matrix density o the line at x. By introducing the impedance matrix Zx s at each x along the line Eqs. 9 can be reduced to the single Riccati-type matrix equation Z x x s rx = szx se xzx s is the pseudo-inverse operator. Thus i the output port o the line is short-circuited the impedance matrix Zx s and in particular the input impedance matrix Zs = Z s can be determined by solving Eq. with boundary condition Z x s =. The inverse problem can also be considered. By assigning the input impedance matrix Zs when the output port o the line is short-circuited a passive multi-conductor RC transmission line can be determined by solving Eqs. or rx and ex. In this way as a consequence o Theorem the ollowing result can be proved Theorem The Zs impedance matrix o a passive multiport dynamic thermal networ is the short-circuit input impedance matrix o a passive multi-conductor RC transmission line ruled by Eqs. 9. Thus passive multi-port dynamic thermal networs can be represented by passive multi-conductor RC transmission lines with short-circuited output ports. This is the generalization o Cauer I canonical orm o passive multi-port lumped RC networs to passive multi-port dynamic thermal networs. As shown in with passive one-port dynamic thermal networs the RC transmission line is not uniquely determined. Similarly with passive multi-port dynamic thermal networs the multi-conductor RC transmission line is not uniquely determined. Such passive RC multi-conductor transmission line deines partial resistance matrices o the R total resistance matrix and can be decomposed into the shunt connection o a passive capacitive multi-port o capacitance matrix C and a second passive multi-conductor RC transmission line. 9 Generalized Cauer II Canonical Form As a consequence o Theorem 6 it can be proved that a continued raction expansion can be perormed or the impedance matrix Zs o a passive multi-port dynamic thermal networ exactly as when determining the Cauer II canonical orm o a passive multi-port lumped RC networ 7.

6 Theorem Given the impedance matrix Zs o a passive multi-port dynamic thermal networ it results in e Zs = r s r e s... r r... and e e... are symmetric positive semi-deinite. This expansion deines an ininite networ composed o passive resistive multi-ports o resistance matrices r r... and o passive capacitive multi-ports o elastance matrices e e... which generalizes the Cauer II canonical orm o a passive multi-port dynamic thermal networ. The irst resistance matrix r is the total resistance matrix R the irst elastance matrix e is the inverse o the total capacitance matrix C. The elastance matrix o the series connections o the passive capacitive multi-ports o elastance matrices e e... is the inverse o matrix C. Thus the Cauer II canonical orm o a passive multi-port dynamic thermal networs deines partial elastance matrices o the inverse o matrix C. Application Example: Part II From Eqs. a generalized Cauer I canonical orm o the passive multi-port dynamic thermal networ o section 7 is rx= 6 { ex = Ac x < x x x as shown in Fig.. The Cauer II canonical orm can be determined by perorming a continued raction expansion o Zs. The irst resistance matrices are r = r = r = r being R. The irst elastance matrices are e = Ac e = Ac e = Ac e being the inverse o C r = r r = r.... x Figure : Resistance matrix o the Cauer I canonical orm. Conclusions In this paper our canonical orms have been introduced or multi-port dynamic thermal networs which generalize the our canonical orms o passive multi-port lumped RC networs. In particular it has been shown that the generalized Foster I canonical orm is equivalent to the time-constant representation and that the generalized Cauer I canonical orm is a passive multiconductor RC transmission line. Reerences C. asance H. Vine H. Rosten K.. Weiner A novel approach or the thermal characterization o electronic parts SEMI-THERM IX Vol. pp V. Széely T. Van Bien Fine Structure o Heat Flow Path in Semiconductor Devices: a Measurement and Identiication Method Solid-State Electronics Vol. pp M. Rencz V. Széely Dynamic thermal multi-port modeling o IC pacages IEEE Trans. Components and Pacaging Technologies Vol. No. pp V. Széely On the Representation o Ininite-ength Distributed RC One-Ports IEEE Trans. Circuits and Systems I Vol. 8 No. 7 pp Codecasa Canonical Forms o One-Port Passive Distributed Thermal Networs IEEE Trans. Components and Pacaging Technologies Vol. 8 No. pp Codecasa Structure Function Representation o Multi- Directional Heat Flows accepted in IEEE Trans. Components and Pacaging Technologies. 7 R. W. Newcomb inear Multi-port Synthesis McGraw- Hill 966.