ELECTRICAL-THERMAL-MECHANICS MODELING OF CIRCUIT …€¦ · circuit breakers use two types of tripping, thermal tripping for slight increase of load demand, and magnetic tripping

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International Journal of Mechanical Engineering and Technology (IJMET)

Volume 9, Issue 2, February 2018, pp. 461–481, Article ID: IJMET_09_02_047

Available online at http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=9&IType=2

ISSN Print: 0976-6340 and ISSN Online: 0976-6359

© IAEME Publication Scopus Indexed

ELECTRICAL-THERMAL-MECHANICS

MODELING OF CIRCUIT BREAKER THERMAL

TRIPPING UNIT USING MULTIPHYSICS

PARTIAL ELEMENT EQUIVALENT CIRCUIT

METHOD COMBINED WITH LUMPED

BEHAVIORAL TRIPPING MECHANISM

MODEL

Vjosa Shatri, Lavdim Kurtaj* and Ilir Limani

Faculty of Electrical and Computer Engineering,

University of Prishtina “Hasan Prishtina”,

10000 Prishtina, Kosovo.

*Corresponding Author, Email: [email protected]

ABSTRACT

Main components of any electrical circuit are a power source that provides energy

and a load that will use that energy for fulfilling the designed purpose. Wires are the

most common means used to enable this energy transfer. To enable controlled energy

transfer switch is used. Unexpected working situation can happen, that can cause rise

in power demand from power source fare beyond the normal working conditions. To

protect actors of the circuit from permanent damage fuses and circuit breakers are

used to interrupt current path in case of over-current conditions. Most of the common

circuit breakers use two types of tripping, thermal tripping for slight increase of load

demand, and magnetic tripping if load increase is rapid and intense. Parametric

electrical-thermal-mechanics partial element equivalent circuit model of the thermal

tripping unit was developed, with possibility of exploring structural and material

properties on final functional behavior. Multiphysics partial element equivalent circuit

method (muphyPEEC) was used for deriving bimetallic strip model. It was combined

with lumped and behavioral model for other parts of trip mechanism to emulate

circuit breaker functional behavior. LTspice circuit simulation software is used for

simulations on combined circuit breaker thermal tripping model.

Key words: Multiphysics PEEC, electrical-thermal-mechanics model, bimetallic strip,

thermal tripping, circuit breaker.

Vjosa Shatri, Lavdim Kurtaj and Ilir Limani


Cite this Article: Vjosa Shatri, Lavdim Kurtaj and Ilir Limani, Electrical-Thermal-

Mechanics Modeling of Circuit Breaker Thermal Tripping Unit Using Multiphysics

Partial Element Equivalent Circuit Method Combined with Lumped Behavioral

Tripping Mechanism Model, International Journal of Mechanical Engineering and

Technology 9(2), 2018, pp. 461–481.

http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=9&IType=2

1. INTRODUCTION

Simulation is very powerful tool for analyzing system behavior [1]. It is based on models that

are created for elements that compose the system. What types of models will be used, and at

what level of detail, is highly dependent on purpose of simulation, also conditioned by

computational power available at time. With selected models simulations can be run to obtain

solution for selected problem. For simple problems models can be in form of linear ordinary

differential equation (ODE) with constant coefficients, and solution can be found analytically.

Problems in different physical domains can be described with linear ODEs that are of similar

form. This inspired abstraction and generalized use of solutions found to some physical

domain to other, less analyzed and harder to understand, physical domains by using analogies

[2] between quantities of corresponding differential equations. They were used initially to

gain better understanding for electrical problems from results in mechanics domain [3], being

more intuitive at that time. With developments in field of electronics, the opposite is more

common [4], especially as a tool for obtaining solution for differential equations by

simulation. This started by using analog computers [5], and got highly developed with

transition to digital computers [6], or hybrid solutions [7].

Single domain model buildup will use many idealizations and approximations, resulting in

simple models. During this process interaction of specific domain with other physical

domains is lost, and they will be represented with some steady state conditions, present

indirectly at parameters of resulting equations [8]. With more complex problems this

separation to single domain problems cannot be done, or it is not a desirable form, as

approximate solution may give results from not so accurate, caused from high deviations in

parameters that were assumed as constant, to totally wrong results, caused by reduced degrees

of freedom of resulting model and its inability to capture important phenomena present in real

system. Tests for possible inconsistencies between real physical system and a model are done

with prototypes. This can be costly and time consuming process. Some solutions that

contribute to better match can use real physical systems (full or scaled versions) for critical

parts of the model, while others are simulated, leading to hybrid solutions of development,

like rapid development [9] with hardware-in-the loop simulations [10]. Ideal solutions would

be to have to have reliable prototype, but with a simulated implementations, i.e. a virtual

prototype [11].

Implementation of the virtual prototype will require inclusion of multiple physical

domains in the model and their interactions during system simulations. This will offer

possibilities to perform parametric and structural influence of design to final performance.

Method used in this paper for multiphysics simulations, having its roots to analogies [2] and

equivalent electrical circuits [12], is partial element equivalent circuit method (PEEC) [13]. It

is a numerical method that was used to solve many problems in electromagnetics [14].

Resulting finite element model is in form of electrical circuit model. This model can be used

in simulations with circuit solvers of SPICE family, where other lumped parts can be easily

used to build full model, ex. electromagnetic-circuit model [15], [16]. Now PEEC method is

considered reliable and feasible approach for wide range of electromagnetic and combined

Electrical-Thermal-Mechanics Modeling of Circuit Breaker Thermal Tripping Unit Using Multiphysics

Partial Element Equivalent Circuit Method Combined with Lumped Behavioral Tripping Mechanism

Model


electromagnetic-circuit problems. Almost nothing was done for extending method to other

physical domains. Building thermal models for use in SPICE environment based on finite-

difference method, followed by model reduction, was done in [17]. Following similar steps of

the PEEC method used for electromagnetic domain [18], [16], extension of the PEEC method

to other physical domains following general finite element methodology (FEM) [19] was

done in [20]. PEEC method [21] and FEM [19] define basic building units, and then use

composition of these units to describe general problem of specific domain. Both methods use

“stamps” [21] of basic building unit, as element’s set of ordinary differential (or sometimes

algebraic) equations in matrix form, to obtain full set of ODEs expressed as matrix differential

equation for problem in question. Approach of [20] was in using vast knowledge of FEM for

other domains to enrich PEEC method, while keeping its initial SPICE orientation and using

SPICE solvers, mostly abandoned or used in co-simulation with dedicated PEEC solvers [22].

Keeping SPICE as a solver for PEEC problems was done in [23], [16], in form of a PEEC

Toolbox for MATLAB that generates LTspice [24] compatible netlist, with LTspice being

used as environment for integrating PEEC model with other electrical and electronic circuit

elements [16] and finally solving the problem. Since SPICE internally builds matrix

differential equations from element stamps [21], extension to other domains of [20] was in

building electrical equivalent circuits for element “stamps” that FEMs [19] of other domains

are using. This approach resulted in straightforward extensions of PEEC method to thermal

and mechanics domains, named multiphysics PEEC (muphyPEEC) [20]. Later extension to

thermal domain with matrix differential equation as result was done in [25].

In this paper we follow approach of [20] in using SPICE (specifically LTspice [24]) as a

solver for multiphysics partial element equivalent circuit method. Building blocks for

electrical, thermal and mechanics domains are presented, including electrical-to-thermal and

thermal-to-mechanics couplings. Multiphysics PEEC was used to build electrical-thermal-

mechanics circuit model for bimetallic strip of circuit breaker thermal tripping unit, while

combinations with lumped model for other parts of trip mechanism and simulations were

done with free circuit simulator LTspice [24] from Linear Technology, without limitation in

number of nodes or elements. Paper covers short overview of internal functional behavior for

circuit breakers with thermal and electromagnetic tripping units.

2. METHODS

Main components of any electrical circuit are a power source that provides energy, and a load

that will use that energy for fulfilling the designed purpose. Wires are the most common

means used to enable this energy transfer. To enable controlled energy transfer switch is used.

Unexpected working situation can happen, that can cause rise in power demand from power

source fare beyond the normal working conditions. To protect actors of the circuit from

permanent damage fuses and circuit breakers are used to interrupt current path in case of over-

current conditions.

Brief functional description of circuit breaker with thermal and magnetic tripping is given.

It is followed with explanation of Partial Element Equivalent Circuit method (PEEC) and

rationale of extending it to other physical domains.

2.1. Circuit Breakers and Tripping System

Most of the common circuit breakers [25] use two types of tripping, thermal tripping for slight

increase of load demand, and magnetic tripping if load increase is rapid and intense.

Functional diagram of circuit breaker with thermal and magnetic tripping is shown in Figure



1. Diagram shows normal working condition with circuit breaker contacts closed (Moving

Contact touching Fixed Contact), allowing load current to flow (when Switch is closed).

Inside circuit breaker load current passes through trip condition detection units, Coil (part of

Magnetic Tripping) and Bimetallic Strip (part of Thermal Tripping). Any tripping action will

move Slider to the right, and unlatch contact mechanism by moving Latch Bar aside, beyond

holding position of Moving Contact Extension Bar, allowing Spring to disconnect contact.

Small over-currents are detected by Bimetallic Strip that gets deflected in proportion to load

current, and potentially reaches tripping point. High over-currents will cause fast Plunger

movement toward Slider, causing fast unlatching of contact mechanism. Head of a plunger is

visible at the right position of coil image in Figure 1. Plunger is held in non-active position by

spring (not visible in Figure 1) inside the Coil. Unlatched position of tripping mechanism is

shown as shaded position of corresponding parts: Slider, Latch Bar, and Moving Contact with

Extension Bar. Bimetallic Strip is in deflected condition while under normal working load

currents.

Figure 1 Functional diagram of circuit breaker with thermal and magnetic tripping. Diagram shows

normal working condition with circuit breaker contacts closed (Moving Contact touching Fixed

Contact), allowing load current to flow (when Switch is closed). Inside circuit breaker load current

passes through trip condition detection units, Coil (part of Magnetic Tripping) and Bimetal Strip (part

of Thermal Tripping). Any tripping action will move Slider to the right, and unlatches contact

mechanism by moving Latch Bar aside, beyond holding position of Moving Contact Extension Bar,

allowing Spring to disconnect contact. Small over-currents are detected by Bimetallic Strip that gets

deflected in proportion to load current, and potentially reaches tripping point. High over-currents will

cause fast Plunger movement toward Slider, causing fast unlatching of contact mechanism. Plunger is

held in non-active position by spring inside the Coil. Unlatched position of tripping mechanism is

shown as shaded position of corresponding parts: Slider, Latch Bar, and Moving Contact with

Extension Bar. Bimetallic Strip is in deflected condition while under normal working load currents.

Figure does not show parts of mechanism for setting it manually to latching position. For

this action it should move Extension Bar up with tip beyond holding position of Latch Bar,

and move Latch Bar to the left to the holding position, and finally letting Extension Bar tip to

lie over Latch Bar lock under tension of the Spring.

When there is no current flowing through circuit breaker between terminals CB1 and CB2

Bimetallic Strip will be straight (marked as shaded position in Fig. 1). Between Bimetallic



Model


Strip and freely moving Slider there is a gap for allowed Bimetallic Strip deflection during

normal load working currents. Some circuit breaker may have some mechanism for adjusting

this gap, for calibration purpose if they are fixed current circuit breakers, or for setting

tripping current in case of adjustable current circuit breakers. Force for moving Slider is

neglected. At some deflection point Bimetallic Strip, Slider and Latch Bar will come in

contact. Further increase in currents will not change positions of these parts, but actuating

force of Bimetallic Strip will increase. When this force overcomes friction force between tip

of Moving Contact Extension Bar and Latch Bar, Slider will rotate Latch Bar to the right

(shaded position of Slider and Latch Bar), and then Spring will attract Extension Bar causing

disconnection of circuit breaker contacts (shaded position of Moving Contact Extension Bar).

2.2. Multiphysics Circuit Model

One of numerical methods for general solution of engineering problems is finite element

method. It will convert problem of solving partial differential equations that describe problem

to a system of algebraic equations. According to the method, geometry of the problem will be

partitioned (meshed) to a number of non-overlapping cells (finite elements) of specific shape,

each having given number of nodes, edges and faces. If cells are small enough original partial

differential equations can be approximated with a set of simple equations (algebraic or

ordinary differential with respect to time), defined on cell nodes, that describe behavior over

cells. Simplest approximation is obtained by treating specific quantities inside cell as

constant. Each physical domain of interest will result with corresponding set of ODEs. Each

ODE, irrespective of physical domain it belongs to, can be interpreted as electrical circuit, by

relaying on analogies [2] between quantities of ODEs of different domains with that of

electrical domain. This flow is pictured in Figure 2. All these cell circuits assembled together

and its solution will represent a circuit model and approximate solution for the problem in

question. Couplings between different physical domains are present in the dependence of

coefficients of ODE of one domain from quantities of other domains, leading to dependence

of circuit element values at elements forming equivalent circuit.

Figure 2 Obtaining multiphysics circuit model. Geometry of the problem is partitioned into a number

of cells. Behavior inside cell is approximated with ODEs of corresponding physical domain, for each

physical domain of interest. ODEs are interpreted as electrical circuit by using analogies. Multiphysics

circuit model is composed from all resulting cell “circuits” properly connected on cell nodes. Cells

may contain couplings between domains, or between cells of the same domain.



Couplings are modeled in form of dependent voltage and current sources, dependent

resistances, dependent capacitances, and dependent inductances. Couplings may exist also

between cells of the same physical domain. At the circuit level (similar to ODEs level)

identity of the domain is lost, and there is no distinction in implementation of intra- or inter-

domain couplings. General circuit solvers, e.g. a member of SPICE (Simulation Program with

Integrated Circuit Emphasis) family, can be used to solve the circuit model by enforcing

boundary conditions and connecting the model to other lumped elements.

2.3. Electromagnetic Partial Element Equivalent Circuit Method

Following similar route to previous subsection, general form of a new method for modeling

arbitrary shaped three dimensional electromagnetic problems was developed by Dr. Ruehli

[13], named Partial Element Equivalent Circuit (PEEC) method, based on circuit

interpretation of electric field integral equation [13], [21], [27]. Derivation can be found in

many sources, including first paper of the author [13], but book from Nitsch et.al. [27] and

recent book from Ruehli et.al. [21] cover many aspects of the method. Method refers to cells

as partial elements, and associates an equivalent circuit that approximates its behavior to each

of them, as two attributes used to devise the name of the method. Specific to PEEC method is

that it defines two types of meshes, surface and volume ones, with surface and volume cells as

building units. Quantities inside these cells, being surface charge and volume current

distributions, are assumed uniform. Volume cells are created between neighboring nodes in

axis direction, while surface cells are surfaces surrounding nodes. Surface cells seem as

shifted for half-cell length of volume cells in the corresponding dimension. Each cell is

associated with circuit model composed of electrical components: resistances, inductances,

capacitances, and controlled voltage and/or current sources [17], [28], [27]. Figure 3(a) shows

circuit for one volume (current) cell, branch between nodes φi and φj, and two surface

(charge) cells, between mentioned nodes and reference (assumed infinity), for quasi-static

case. Charge cells are not present for nodes inside conductor. Since derivation is based on

integral equation, couplings with all other cells of electromagnetic domain are included by

dependent voltage and current sources, representing inductive and capacitive couplings

respectively. Both types of sources can be used to represent either type of coupling by

transfiguring the circuit [27]. Depending on the type of the problem to be solved, different

derived models by neglecting specific interactions or circuit elements can be obtained and

used [27], [16], [29].

Collection of all these partial circuits will compose circuit model for selected problem.

Circuit model can be used in simulations with circuit solvers, where combination with other

lumped elements can be done, for solving combined electromagnetic-circuit problems [15],

[16].

Bimetallic thermal tripping used in circuit breakers has Manhattan geometry, having all

cells of rectangular parallelepiped-like form. Current passing through circuit breaker is mainly

dictated from other parts forming the electrical circuit, while for bimetallic thermal tripping

only power losses that generate heat will be of importance. This enables us to neglect all

couplings, current cell inductances, surface cell capacitances and influence of external

electromagnetic field, and simplest PEEC model of type [R] [27], [16], [29] can be used.

Since electromagnetic PEEC model assumes electric currents are flowing parallel to cell

lengths and are uniform over cell cross-sections, resistances of these cells can be calculated as

S

lR

, (1)



Model


where Rα, ρα, lα and Sα are resistance, specific electrical resistance, length and cross-

sectional area of the cell α, correspondingly.

2.4. Thermal Partial Element Equivalent Circuit

Circuit representation for thermal circuit is common mean of treating thermal problems in

electrical engineering. Even terminology used for thermal quantities that constitute thermal

circuit, thermal conductivity and thermal capacitance, are directly analogous to corresponding

electrical ones, by replacing term thermal with term electrical. Typically lumped elements are

used to represent thermal problem in gross. One of numerical methods used when geometrical

details are to be taken into account for thermal problem in hand is finite element method. Heat

equation is a starting point for general unsteady state heat transfer problems. Typical finite

element derivation process [19], usually using Galerkin’s residual method, will result in a

matrix differential equation [30], [31], [32], where only space is discretized. Circuit for each

cell is topologically similar to one of electrical PEEC in Figure 3(a), but much simpler (when

radiation transfer is not accounted) given in Figure 3(b) [20], [25]. As first, there are no

couplings between cells of the same domain, resulting in absence of all dependent sources.

Second, there is no equivalent of inductivity to thermal domain.

Figure 3 Building units of the electromagnetic and thermal PEEC model. (a) Electromagnetic PEEC

cell: current (volume) cell equivalent circuit and charge (surface) cell equivalent circuit. Dependent

voltage and current sources represent couplings with other cells of the electromagnetic domain. Rα and

Lαα are resistance and self-inductance of volume cell α. Node self-capacitances are represented with

1/pii, with pii being coefficients of potential. Uαinc

models effects of incident electric field. (b) Thermal

PEEC cell: thermal conductivity (volume) cell equivalent circuit, and equivalent circuits for thermal

capacity (volume) cells with and without heat sources. Current source Qith represent heat sources

inside thermal capacity cell i, Kαth is thermal conductivity between corresponding nodes, and Ci

th is

thermal capacity of the node i. Additional segments at nodes of both domains represent other possible

branches to other neighboring nodes in axes directions. Indices in (a) and (b) are for same meshing at

both domains.

Third, voltage generator accounting for incident electric field is missing. Fourth, if there

are heat sources present inside the cell they will be represented with current sources

(dependent or independent) parallel to capacitances. Quantities on nodes are temperatures,

while those at branches are heat flows. Reference node for capacitances is infinity, as in

electromagnetic PEEC case.

Same meshing (partitions) used for generating electromagnetic PEEC model can be used

for generating thermal PEEC model, but also different meshing can be used for both domains

[20]. If same meshing is used then current cells of electromagnetic domain will be the same to

thermal conductivity cells of thermal domain. Thermal capacity cell is volume surrounding



corresponding node, being composed of parts (usually halves) of thermal conductivity cells

connected to that node [20]. For uniform cell sizes, internal thermal capacity cells will be

equal to thermal conductivity cells but boundary cells will be different, whereas for non-

uniform cell sizes and possibly of different materials we should treat them as composition of

corresponding parts of thermal conductivity cells. These cells are present at all nodes,

including nodes inside conductors, contrasted to electromagnetic PEEC case. Values for

components of the thermal PEEC cells (of rectangular parallelepiped form) are calculated as

follows

th

thth

th

th

th

ththth

l

S

l

S

l

SKR

1/1/1

, (2)

thiVi

thi VcC , and (3)

ncc

k

thikik

thi QQ

1 . (4)

Thermal resistance Rαth

is calculated with the similar expression to one of electrical

resistance Rα. Specific thermal resistance ραth

is reciprocal of specific thermal conductance κα.

Thermal conductivity cell is defined by its length lαth

and cross-sectional area Sαth

. Thermal

capacity at node i, Cαth

, is proportional to thermal capacity volume Vith

of that node and to

volumetric specific heat capacity cVi. Each electromagnetic PEEC current cell connected to

node i will generate heat Qikth

, where k is one of ncc current cells connected to node i, but only

part of it ηik (ranging from 0 to 1) will be located inside i-th thermal capacity cell. If meshing

for electrical and thermal domains are different, there may be additional current cells not

connected to node i but located inside corresponding thermal capacity cell. In later case ncc

includes these cells too.

2.5. Mechanics Partial Element Equivalent Circuit

For building mechanics domain PEEC model for bimetallic strip, quasi-static [K] model

(equivalent of electrical [R] model) was used

FdK , (5)

where K is global stiffness matrix, d is global displacement vector, and F is global force

vector. This global equation is assembled from stamps of basic building elements, shown in

Figure 4(a), similar to modified nodal analysis equation assembly process for electrical

circuits used in SPICE. For two-dimensional meshing, basic building elements are beam-

columns in length and height direction, and have similar equation to (5) where quantities have

subscript e for element. Specific to mechanics domain is that each node has three degrees of

freedom with die={uie, vie, θie}, nodal displacement in two directions and rotation, in contrast

to electrical and thermal models that have only one degree of freedom per node. Basis for

building cell equivalent circuit model is element stiffness matrix Ke [33], shown in Figure

4(a), where L is element length, A is element cross section, I is the second moment of inertia,

and E is Young’s modulus for linear elastic material the element is made of. Figure 4(b)

shows mechanics cell equivalent circuit for beam-column basic element, built by

decomposing stiffness matrix to equivalent electrical stamps. Three groups of colored matrix

members correspond to three electrical conductance stamps, when displacement vector

members are analogous to voltages and force vector members are analogous to currents.

Members of the matrix that represent intra-cell couplings are implemented with controlled

current sources, bI1 to bI3 current sources. When two nodes have same currents but of



Model


opposite direction, they can be merged to one, as is the case for bI1 current source. Forces

acting at nodes in the corresponding direction are modeled also by current sources, with bI4

and bI5 given as examples when forces are acting in u directions of both nodes of a given

element.

Figure 4 Building unit of mechanics PEEC model. (a) Governing equilibrium equation for beam-

column element. Ke is beam-column element stiffness matrix, de is element displacement vector, and

Fe is element force vector. Three groups of colored matrix elements mark three equivalent electrical

conductance stamps. Other nonzero matrix members will result with intra-cell couplings. (b)

Mechanics PEEC cell. Colored resistances correspond to three colored groups of equivalent

conductance stamps in stiffness matrix. Current sources bI1 to bI3 represent intra-cell couplings. bI4

and bI5 represent forces acting at u direction of nodes 1 and 2 of a given element.

2.6. Couplings between Domains at Partial Element Equivalent Circuit Method

Thought in principle couplings between each domain can be present, for many problems only

some of them will be more pronounced and have to be accounted for. Couplings that directly

create functionality of bimetallic strip, part of circuit breaker thermal tripping unit, are from

electrical to thermal domain and from thermal to mechanics domain. First one is in form of

power losses at bimetallic strip conductors carrying load electric current, which will be

converted to heat flow, i.e. Joule effect. This coupling is modeled as controlled current source

acting at corresponding node of thermal PEEC cell, current source Qikth

shown in Figure 3(b),

and representing all heat generated inside volume of thermal capacity cell [19], [20], as in (4).

Each contributor of (4) represents power losses Pα in single current cell of the electromagnetic

(electrical) PEEC model

2 IRQP th

ik . (6)

Current Iα(t) will generally be time-dependent, making power losses also time-dependent

Pα(t), and also its dependent quantities at thermal domain cell heating and cell temperature,

and further to quantities of mechanics domain cell stress and cell strain.

Second coupling in bimetallic strip, between thermal and mechanics domains, is in form

of thermal stress induced in corresponding mechanics model building elements. Stress will act

as actuating force causing displacements and bimetallic deflection. This coupling is modeled

as controlled current source acting at corresponding node of mechanics PEEC cell, only

current sources bI4 and bI5 in Figure 4(b) by assuming negligible thermal effects in other

directions. Thermally induced forces in element x-direction will be

2

2121

TTAEFF thexex

, (7)



where αth is the coefficient of linear thermal expansion, ΔT1 and ΔT2 are temperature

changes at element nodes relative to temperature without thermal stresses, usually ambient

temperature.

3. RESULTS AND DISCUSSIONS

Combined multiphysics partial element equivalent circuit (muphyPEEC) model and lumped

behavioral circuit model were used to model circuit breaker thermal tripping unit. Bimetallic

strip, a part of circuit breaker thermal tripping unit, was modeled in three domains,

electromagnetic (electrical in this case), thermal and mechanics, including functionally

dominant interactions between domains, using muphyPEEC. Interaction of bimetallic strip

with other parts of tripping mechanism is modeled behaviorally with lumped circuit elements.

Implementation of muphyPEEC is in form of Toolbox for MATLAB [20], [16], [23], and

covers steps from geometry description to LTspice compatible netlist generation. LT spice

[24] is used as circuit solver, where PEECs of all domains, including interactions between

domains, are combined with other lumped elements and solved simultaneously.

3.1. Bimetallic Strip Electrical PEEC Model

muphyPEEC Toolbox for MATLAB was used to build PEEC model for thermal tripping unit

of the circuit breaker. Geometry of the Bimetallic Strip (Figure 1) is given as inset in Figure 5

(for clearer view dimensions are not in proportion). It is composed from two conductors of

different material, Conductor 1 (FeNi20Mn6) and Conductor 2 (FeNi36) in Figure 5.

Difference in thermal expansion coefficient is of main importance, but it has no influence on

electrical model. Dimensions of the strip are 27 mm in length, 5.25 mm in width, and 0.8 mm

thick (height). 2D meshing was applied for this problem, in length and height dimension, with

10 volumetric cells in length direction and 2 volumetric cells in height dimension for each

conductor. Width was not meshed, and there is only one cell in this direction. For general

electromagnetic problems FastCap2 [34] and FastHenry2 [35] are used to calculate

parameters of the electrical circuit elements, including capacitive and inductive interactions.

Bimetal strip electrical PEEC model generated from PEEC Toolbox is Quasi-Static of [R]

type [27], [16], [29]. It is in form of LTspice netlist. Part of this netlist with elements around

first six nodes of first conductor is shown in Figure 6. Since cells are orthogonal with edges

aligned with global coordinate system axis (3Dxyz type of meshing [16]) circuit parameters

are calculated analytically [17] inside toolbox, without calling external applications. Inset in

Figure 5 shows surface potential (capacitive) cells, while main picture shows normal upper

view to length-height face. Overlaid over this picture are node positions, and by one

representative volumetric current (resistive-inductive) cells for two directions with arrow

denoting referent direction.

Current cells on the edges are of half-width, while those inside the same body are of full-

width. Same figure shows node numbering, serving as reference when adding lumped

elements, for composing electromagnetic-circuit models (electrical-circuit for this problem).

This is shown in Figure 7, where nodes of the PEEC model are connected with other lumped

parts of the circuit, to build full circuit for performing



Model


Figure 5 Bimetallic strip meshed geometry. Height dimension is zoomed better view. Bimetallic strip is

composed from two conductors with different thermal expansion coefficient: Conductor 1 and Conductor 2.

Each conductor is 25 mm long, 2.5 mm wide and 0.2 mm thick (height dimension). Meshing was done in length

and height directions (2D meshing). There are 10 cells in length direction, and 2 in height direction. Width

direction has only one cell (normal to the viewing plane). Nodes are numbered with Nbbeesswwhh designation,

where bb is for body, ee for element, and sswwhh for node coordinates in length, width and height direction. bb

is equal to 01 for nodes in Conductor 1 and 02 for nodes in Conductor 2. Since each conductor (body) is

composed from single element (LR element) ee is alway 01. For shown discretization, range for ss is from 01 to

11, for ww is always 01, and for hh it is from 01 to 03. Squares around nodes represent capacitive cells. Two

representative current cells in length direction (cells with arrows in Conductor 2) and height directions (cells

with arrows in Conductor 1) are marked with different color. Current cells at the edge and at the interface

between two conductors, in both directions are of half width, while those inside the same conductor are of full

width.

electrical simulations. During PEEC model generation, it was selected option for

composing parameterized model. Final parameter values for simulation are set in LTspice

with dot command param, or they can be set by directly by editing netlist file. This is a

convenient form of the model, allowing different simulation runs without repeating all steps

of PEEC model building. Specific electrical resistances for two conductors are set with

parameters c1_rho=0.080e-6 (for FeNi20Mn6) and c2_rho=0.076e-6 (for FeNi36). Total

resistance of circuit breaker, passing through

Figure 6 Part of bimetal strip electrical PEEC [R] model in form of the LTspice netlist. Circuit

elements around first six nodes are shown.



Figure 7 Bimetallic strip electrical PEEC model, in circuit with other lumped elements of electrical circuit.

Netlist of the PEEC model for all three domains (electrical, thermal, and mechanics, including interactions

between domains) is contained inside included netlist Bimetal_PEEC_0071_el_th_me.net. Contact controlled

from behavioral trip mechanism model is connected between points CB_FC and CB_MC, i.e. Fixed Contact

and Moving Contact of the Circuit Breaker. CB_T1 and CB_T2 are external terminals of the Circuit Breaker.

Switch is included only symbolically, whereas its function is contained in the moment when Power_Source is

switched on. Parameter lines define electrical parameters for two conductors.

bimetallic strip of thermal tripping and coil of magnetic tripping, is expected to be

negligible. It will only sense current determined by power source and load, shown in Figure 7.

Other parameters will determine physical dimensions of bimetallic strip, and adjustment of

these parameters will have influence on nominal circuit breaker current.

3.2. Bimetallic Strip Thermal PEEC Model and Coupling with Electrical PEEC

Model

Thermal modeling will use same geometry and meshing used for electrical PEEC model.

They can have different meshing, with condition in current version that cells in one domain

must be multiple of cells in other domains [20]. Model type generated for thermal domain is

[R, C]. For structural similarity with electrical PEEC, square basis functions were used for

derivation of this model. Analogies used are: temperature with electric potential, and heat

flow with electric current. Each node will have one capacitor (3), modeling thermal capacity,

and resistors between neighboring nodes (2), modeling thermal conductivities, as shown if

Figure 3(b), [36], [17], [20]. Boundary conditions are applied at boundary nodes [20], [25].

Conduction boundary conditions are used at both ends, where bimetallic strip is in contact

with conductors of circuit breaker electric path. At other boundary nodes of bimetallic strip

convection boundary conditions are applied. Energy transfer by radiation [37] is not modeled.

Thermal model includes heat source at all nodes (4), modeling electrical heat loss (6). Figure

8 shows part of bimetal strip thermal PEEC [R, C] model in form of the LTspice netlist.

Circuit elements around one corner node and one internal node are shown. Each node has one

thermal capacitor cCth* (* stands for node reference designator of form bbeesswwhh,

explained in Figure 5). Number of thermal resistors rRth* depends on number of neighboring

nodes. For 2D meshing and with square basis functions it is two for corner nodes, three for

edge nodes, and four for nodes inside the body. One controlled current generator bIth* will

serve as heat source, modeling power losses (6) in parts of several electrical resistors inside

corresponding volume (4). Last element in netlist for each node will have thermal resistor



Model


rRth*_amb driven by ambient temperature NTamb that models convection boundary

condition [20]. Thermal PEEC model is also using parameterized model, as for

Figure 8 Part of bimetal strip thermal PEEC [R, C] model in form of the LTspice netlist. Circuit elements

around one corner node and one internal node are shown. Each node has one thermal capacitor. Number of

thermal resistors depends on number of neighboring nodes. For 2D meshing and with square basis functions it is

two for corner nodes, three for edge nodes, and four for nodes inside the body. One current generator will serve

as heat source, modeling power losses in parts of several electrical resistors inside corresponding volume. Last

element in netlist for each node will have thermal resistor driven by ambient temperature that models convection

boundary condition.

electrical PEEC case. Final parameter values for simulation are set in LTspice with dot

command param.

Figure 9 shows thermal circuit with bimetal thermal PEEC model and other lumped

thermal elements. Parameter lines define thermal parameters for two conductors, convection

coefficients, and ambient temperature. NTcond1 and NTcond2 are temperatures of

conductors that are in direct contact with two sides of the bimetallic strip. NTamb is node

that drives convection thermal conductors. Model was validated in two canonical problems

that have theoretical solution [38].

Figure 9 Thermal circuit with bimetal thermal PEEC model and other lumped thermal elements for thermal

response under step change of boundary temperature at one side. Parameter lines define thermal parameters for

two conductors, convection coefficients, and ambient temperature. NTcond1 and NTcond2 are temperatures of

Bimetallic Strip Thermal PEEC

Model



conductors that are in direct contact with two sides of the bimetallic strip. NTamb is node that drives convection

thermal conductors.

First problem was response of thermal model alone by simulating response at fast change

in temperature of one side conductor that is in direct contact with bimetallic strip, from 20 ºC

to 30 ºC at NTcond2 in this case. Results of thermal response for bimetallic strip thermal

PEEC model are shown in Figure 10(a). Figure 10(a) shows linear steady-state temperature

distribution over strip length, from 20 ºC to 30 ºC with 1 ºC increase at every next node, being

the same to response predicted theoretically. Response is dominated from conductive heat

transfer between strip and edge conductors. It is negligibly influenced by ambient temperature

and thermal convection coefficient [37] (for practical range for them). Second problem

validates coupled electrical and thermal PEEC models, in testing model response under

uniform volumetric heat generation [38], [20]. Results of response are given in Figure 10(b),

for step change in load current from 0 to 10 A at t = 1 s. Temperature reaches maximum value

45.1324 ºC at middle of the strip. Theoretically predicted value is 45.1416 ºC, resulting in

0.02 % error. For theoretical calculation real load current was used, by taking into account

bimetallic strip resistance, otherwise error was 0.076 %. Temperature distribution is

symmetrical on two sides of the strip. That is why Figure 10(b) shows only 6 traces: traces

form bottom to up for nodes 1 to 6 in length direction. Traces for nodes 7 to 11 are overlapped

with traces for nodes 5 to 1. Time constant of the response at point in the middle is

τ = 16.78 s, as indicated in Figure 10(b) and inset with cursor data. Temperature profile in

length direction for different currents through circuit breaker is given in Figure 11, and

follows quadratic temperature profile [38]. Base current is 10 A (lowest trace, 'o'). Other

traces are for currents higher than base current for factors in the legend: 1.35, 1.50, 1.75, and

2.00.

Figure 10 Thermal response of bimetallic strip thermal PEEC model. (a) Response to step change in a

temperature from 20 ºC to 30 ºC of one conductor in direct contact with one side of bimetallic strip.

Other conductor and ambient temperature are at 20 ºC. Figure shows linear temperature distribution

over strip length, from 20 ºC to 30 ºC with 1 ºC increase at every next node. (b) Response to step

(a)

(b) τ = 16.78 s

Time (s)



Model


change in load current from 0 to 10 A at t = 1 s. Time constant of the response at point in the middle is

τ = 16.78 s.

Stepped trace in Figure 11 is temperature profile approximated with square basis functions

that were used for derivation of thermal PEEC method [20], [25], but in finite elements

method usually higher order basis functions are used, starting from piecewise linear [31],

[30], [19], [8].

Figure 11 Temperature profile in length direction for different electric currents through bimetallic

strip of circuit breaker. Base current is 10 A (lowest trace, 'o'). Other traces are for currents higher than

base current for factor in the legend: 1.35, 1.50, 1.75, and 2.00. Stepped trace is temperature profile

approximated with square basis functions that were used for derivation of thermal PEEC method.

3.3. Bimetallic Strip Mechanics PEEC Model and Coupling with Thermal PEEC

Model

Quasi-static [K] model (5) was used for building mechanics domain PEEC model for

bimetallic strip. Basic building elements are beam-columns in length and height direction.

Meshing used for mechanics domain was same one used for electrical and thermal domains.

Beam-column circuit model from Figure 4(b) is built as LTpice subcircuit model, as shown in

Figure 12(a). Each cell type from four present in current model has one subcircuit, but only

two of them are shown in Figure 12(a), for full cell size in x-direction (BeamColumnThX1)

and for full cell size in y-direction (BeamColumnThY1). Subcircuit block contains

interactions with thermal model, represented with current sources bI4 and bI5. Bimetallic strip

model was composed from 2D arrangement of Beam-columns elements. Part of LTspice

netlist for mechanics PEEC model is given in Figure 12(b). Coefficient of thermal expansion

and Young’s modulus for low and high expansion layers [39], [40], including cell dimensions,

were set with parameters lines inside netlist. Bimetallic strip is in cantilevered configuration.

Fixed boundary conditions for clamping bimetallic strip at one side are set inside netlist (three

zeros following spice element xBCX0201010101). Other boundary nodes of mechanics PEEC

model are left free, with one node as exception (Nme0101110103y) where interaction with

thermal tripping unit is implemented.

Mechanics PEEC model and interactions with thermal PEEC model are validated by

checking displacement and actuating force of free end tip (node Nme0101110103y) of the

bimetallic strip caused by temperature change of uniformly heated bimetallic strip [09], [40],

[41]. Uniform heating was obtained at steady state by imposing step temperature change at

two thermal conductive boundaries of thermal PEEC model, voltage sources V3 and V4 in

Figure 9 set as STEP type stepping from 20 °C to 30 °C. Theoretical values for deflection use



one constant in expression that is 0.5 [40] or 0.53 [41] giving corresponding deflections for

10 °C temperature change of 185.37 μm and 196.492 μm. Deflection obtained from

simulation with PEEC model was 185.34905 μm, with relating errors being -0.01127% and -

5.671%. Deflection is used for calculation of actuating mechanical force at free end tip [40],

[41].

Figure 12 Part of bimetallic strip mechanics PEEC [K] model in form of the LTspice netlist. a)

Subcircuit definitions for Beam-Column elements in length and height dimension. b) Using Beam-

Column subcircuits for building mechanics PEEC model. Bimetallic strip is in cantilevered

configuration, where three zeros in the first xBCX element are used to set boundary conditions at the

side where it is clamped to fixed conductor. Other nodes are left mechanically free. One node at other

end is used outside of the mechanics PEEC model for interacting with circuit breaker tripping unit,

node Nme0101110103y (not shown in this figure).

For two previous values of deflection calculated force is 1.0917 N and 1.1572 N, while

force obtained with simulation was 1.12767 N, and corresponding errors being equal to

+3.2943% and -2.5526%. Since force error are close with each other but of opposite sign,

value of constant around middle of two (0.5165) will give (close to) zero predicted error for

actuating force and deflection error of about -3.2055%. Practical applications usually will use

some form of adjusting operation point either caused from small mismatch between design

and physical implementation, or caused by span range for physical properties of materials

used to build bimetallic strip.

3.4. Bimetallic Strip Multiphysics PEEC Model and Lumped Circuit Model of

Thermal Tripping Unit

Circuit breaker thermal tripping unit commonly use bimetallic strip as current sensing and

actuation part for switching off the circuit in case of over-currents, as shown in Figure 1.

Bimetallic strip has two phases of action, free displacement from resting position until the

point it touches tripping mechanism, and exerting force to some moveable part (Slider and

Latch Bar in Figure 1) until it unlatches tripping unit. This form of acting is behaviorally

modeled with lumped circuit element, current source B3, as shown in Figure 13(a). It will not

(b)

(a)



Model


exert any force (analogous to current) to bimetallic strip until it passes the gap width, with

constant 0.0005 setting the gap to 0.5 mm. Model will behave as stiff spring after passing the

gap, with constant 100000 used to set spring stiffness. This elastic touching point will also

make simulation better conditioned. Spring force (current of B3) will serve as input for

Tripping Mechanism’s lumped model, and will control the state of the circuit breaker contact,

modeled with voltage controlled switch S1 (converted to current controlled switch by using

B1 as current-to-voltage converter). When force exceeds the set point (equal to sum of

threshold voltage Vt and hysteresis voltage Vh, being 0.4 + 0.6 = 1.0 V corresponding to 1.0

Newton force) it will switch of circuit breaker contact. Values are selected such that not to

allow automatic reclosure when bimetal cools down. Switch S1 uses reverse functionality

(normally closed), reflected as reversed values for on and off resistances at Ron and Roff

parameters of BiSw model. One example of activation of thermal tripping is given in

Figure 13(b). To speed-up the simulation time thermal capacitor values were decreased by

1000 times, causing dynamical changes to happen 1000 times faster and milliseconds would

correspond to seconds.

Figure 13 Lumped models for mechanical parts of the circuit breaker (a) and waveforms during one tripping

case (b). Red curve gives load current I(Rload). It is switched on by switch and off by over-current tripping unit.

Green curve shows bimetallic strip temperature at middle point, node nth0101060101 at thermal PEEC model.

The temperature increases rapidly in presence of over-load currents and falls when current is interrupted after

tripping. Position of deflected bimetallic strip tip, node nme0101110103y at mechanics PEEC model, is given

with pink curve. It increases (in negative direction) until it touches tripping mechanism, set at 500 μm (500 μV

in circuit model), and remains almost flat (action of stiff spring model) until tripping and cooling of bimetallic

strip, where it will show returning to resting no current state. Actuating force of bimetallic strip to tripping

mechanism is given with blue curve. If over-current is above some value force will increase until it overcomes



holding force, set at 1 N (1 A in circuit model), causing tripping and interruption of load current. Inset at bottom-

right represents cursor window giving over-current reaction time and bimetallic strip temperatures at rest and at

tripping moment.

When power source is switched on (at 1 s) load current I(Rload) will start to flow, red

curve at bottom plot. Temperature of bimetallic strip will rise, green curve V(nth0101060101)

in Figure 13(b). Free side of the bimetallic strip will start to deflect freely until the set point,

pink curve V(nme0101110103y) in Figure 13(b), with voltage in μV corresponds to deflection

in μm. When the Slider touches Latch Bar (at time ~4.13 s) position of the bimetallic strip

will remain unchanged, seen as flat part of last curve (small deviation from flat indicate

compression of spring model, and amount of deviation is determined by stiffness of the

spring). At the same time bimetallic strip will start to generate increasing force on Latch Bar,

blue curve I(B3). When this force overcomes friction between tip of the Extension Bar and

Latch Bar (set at 1 N by parameters of S1), unlatching and disconnection of the circuit will

happen (at time ~5.5 s). In real circuit breaker force would fall to zero. In model current is

interrupted, but force will return to zero in proportion with cooling of bimetallic strip (time

range from ~5.5 s to ~9.95 s). This does not pose any problem for simulation of circuit

breaker thermal tripping functional behavior, but during this time attempt to manually reclose

the circuit breaker may fail. After ~9.95 s time cooling of bimetallic strip will continue

according to its effective time constant (about 18 s) and position of its tip is shown to return to

resting no current state, green curve in Figure 13(b). Manual closing from ~9.95 s after should

be successful, with tripping time in case of persistent over-current condition being shorter the

closer to this time the closing happens. Reaction time of tripping was ~4.5 s after the power

source was switched on, seen at cursor window at inset in Figure 13(b). Inset shows also

temperature of the middle point of metallic strip (expected to be the hottest point) at ambient

temperature at rest (20 °C) and at tripping moment (~69.1 °C).

4. CONCLUSIONS

Multiphysics partial element equivalent circuit (muphyPEEC) method was used to model

bimetallic strip based thermal tripping of the circuit breaker. Interaction of this model with

other parts of the circuit breaker system in all three physical domains: electric, thermal, and

mechanics, is done with lumped models in circuit solver LTspice from Linear Technology.

Partial element equivalent circuit (PEEC) models for three domains (electrical, thermal and

mechanics) and interactions between domains (electrical-to-thermal and thermal-to-

mechanics) are validated with problems that have analytical solutions. Mechanics PEEC

model shows specific attributes compared to electromagnetic (electrical) and thermal PEEC

models, by having nodes with many degrees of freedom (three for planar case treated in this

paper), but preserves PEEC cell topology. Models are convenient for integrating them with

lumped elements and behavioral lumped models, and solving them simultaneously. These

PEEC-circuit integrations were shown by setting-up working condition at (natural) electrical

domain, by setting-up boundary conditions and excitations at thermal domain, and setting-up

boundary conditions and interacting with behavioral model of circuit breaker tripping

mechanism at mechanics model. Parametric multiphysics PEEC models provided flexible

solution for exploring different conditions for the same PEEC model, without repeating all

steps of PEEC model building. Similar bimetallic strip multiphysics PEEC model can be used

for other applications where bimetallic strip is used as part of the system.



Model


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Documents

ELECTRICAL-THERMAL-MECHANICS MODELING OF CIRCUIT …€¦ · circuit breakers use two types of tripping, thermal tripping for slight increase of load demand, and magnetic tripping