EIEN20 Design of Electrical Machines

28-JAN-2020EIEN20

Design of Electrical Machines

3. Equivalent circuits Formulation & implementation

Lund University / LTH / IEA / Avo Reinap / EIEN20 / 2020-01-28 2

L3: Equivalent circuit•

Equivalent

circuits

–

Lumped

Element Model–

Equivalent

Impedance

Transforms

•

Calculation example

in the first home assignment

•

Introduction

to

calculation

methods•

Equivalent circuit method

•

Magnetic equivalent equivalent

circuit•

Introduction to the second home assignment

W

W

W

https://en.wikipedia.org/wiki/Equivalent_circuits

https://en.wikipedia.org/wiki/Lumped-element_model

https://en.wikipedia.org/wiki/Equivalent_impedance_transforms


Model based observation

•

Changing the material between winding turns from air 0.03 W/mK

to impregnation 0.3 W/mK

•

200 turns, conductor diameter reduced from 1.0 to 0.8 mm•

Keeping the same current: 5A current density increases from 6.3…9.4 A/mm2


Model based observation•

Obviously, higher fill factor means lower resistance and resistive power losses

•

Shorter heat paths between conductors provide lower hotspot temperature

•

Impregnation improves heat conduction across the coil 0.44 0.48 0.52 0.56 0.6 0.64

70

80

90

100

110

120

pow

er lo

sses

P, [

W/m

]

fill factor, Kf [-]0.44 0.48 0.52 0.56 0.6 0.64

0

0.2

0.4

0.6

0.8

1

Tem

pera

ture

, [C

]

0.44 0.48 0.52 0.56 0.6 0.6460

80

100

120

pow

er lo

sses

P, [

W/m

]

fill factor, Kf [-]0.44 0.48 0.52 0.56 0.6 0.64

0

100

200

300

Tem

pera

ture

, [C

]

0.44 0.48 0.52 0.56 0.6 0.6460

80

100

120

pow

er lo

sses

P, [

W/m

]

fill factor, Kf [-]0.44 0.48 0.52 0.56 0.6 0.64

0

100

200

300

Tem

pera

ture

, [C

]


Previous lecture

•

Coil: L=0.02m, W=0.05m, J=2A/mm2, q=57600W/m3, amb

=20°C•

Analytic: max

=41.60°C (a rod) max

=63.20°C (a plate)•

Thermal EC: P=14.4W, Gconv

=0.7W/K, surf

=40.57°C, Gcond

=0.5+0.08W/K, max

=65.4°C •

Heat transfer FE: max

=57.76°C surf

=40.57°C


Equivalent thermal conductivity

•

Equivalent thermal conductivity of a coil or a winding is given by the filling factor of the conductor (copper)

wires and the thermal conductivity of the medium between the conductors

inscond

fcondfins

ins

f

cond

f

eff

kkLkLkLL

11

fcondfins

inscondeff kk

1

Home assignment

Analysis of heat transfer in a single- phase transformer


Goal and Geometric modeller•

Thermal analysis of a single phase shell type of transformer

–

Specify B find Pcore

–

Specify J and Pcoils

so that coil

< limit

–

Maximise I

where IJAe

and BAm

The proportions between the electric and magnetic circuit is changed: Ks=lslot/(lslot+lcore)


Thermal EC and topology matrix

1

23

5

4

6

9

7

10 11

12

8

12

34

56

7

89

10

11

12 13

1415

16

•

th=[1 1 2 la(1)*Ath(1)/Lth(1);•

2 2 3 la(2)*Ath(2)/Lth(2);

•

3 3 4 la(3)*Ath(3)/Lth(3);•

4 4 5 la(4)*Ath(4)/Lth(4);

•

5 1 6 la(5)*Ath(5)/Lth(5);•

6 2 7 la(6)*Ath(6)/Lth(6);

•

7 3 8 la(7)*Ath(7)/Lth(7); •

8 6 7 la(8)*Ath(8)/Lth(8);

•

9 7 8 la(9)*Ath(9)/Lth(9);•

10 8 4 la(10)*Ath(10)/Lth(10);

•

11 6 9 la(11)*Ath(11)/Lth(11);];•

nelm

= 11; % number of elements

•

ndof

= 9; % number of nodesElements: thermal conductivity, la [W/mK],Area, Ath [m2] and length Lth [m]


Thermal equivalent elements

•

Elements defined by: –

Thermal conductivity, la or λ

[W/mK],

–

Area, Ath

[m2] and length Lth

[m]•

Homogeneous bidirectional flux flow is assumed in a single heat conductivity element

Gth

•

The cross section variation and convection can be included

cool

el

iee

i

ei

eeth

AA

cl

AG

1 1

1

y

x

z

h

wl


Solving a TEC•

The node potential method

is used to calculate the

relations between the thermal conductivity (stiffness) matrix G, temperature (unknowns) vector and thermal flux input (load) vector Q

of known losses

G=Q•

Each and every thermal element Ge

is assembled

to a global matrix G with a connection between elements and boundaries

•

Unknowns (n) are calculated in respect to references

(r)

n

= Gnn-1(Qn - Knr

r

)


Initial data from EC and FE modelsKs [%]

wECM

[°C]JwECM

[A/mm2]cECM

[°C]wFEM

[°C]JwFEM

[A/mm2]cFEM

[°C]10 134.8 13.59 112.9 135.1 16.06 103.2

20 134.8 6.72 106.6 135.7 7.64 97.6

30 134.8 4.87 101.3 134.3 5.33 92.4

40 134.8 3.93 96.7 134.9 4.21 89.1

50 134.8 3.35 92.3 135.1 3.53 86.2

60 134.8 2.95 88.0 135.0 3.06 83.2

70 134.8 2.65 83.8 135.0 2.73 80.5

80 134.8 2.41 79.6 135.1 2.47 77.7

90 134.8 2.18 75.6 135.0 2.25 75.2


Observations•

The estimation error

between the thermal EC and FE

model is larger when the geometric proportions between the sides of the coil (coil width/coil length) is

bigger•

thermal EC shows a higher hotspot

temperature than

the thermal FE model for the same current

loading•

It is inconvenient to use the same thermal converging conditions

for a small un-proportional and a large

proportional coil (max 25 iterations in FEM vs

max 190 in TEC)

Equivalent Circuit Method

From physical

understanding

to mathematical formulation

and method

implementation


Physical understanding

Mathematical formulation

Test and measurements

Model Real device

Physical understanding•

Cause-effect relationship

•

mathematic description and measurement of physical phenomena in order to get a good understanding

•

material properties


Mathematic formulation•

Basic formulation for electromagnetic devices

•

Heat transfer

is described by heat equation

•

Electromagnetism by Maxwell’s equations

•

Electro-mechanism

by electromagnetic stress tensor or virtual work

Gauss’s Law, Heat transfer

Faraday’s law

Ampere’s circuital law

Gauss’s Law, Electricity

Gauss’s Law, Electricity

Magnetic stress per unit of area

Change of system energy

S m dstF

tBE

JH

D

q

0 B

mWF


Numerical modelling

•

Model

–

description

of system: process in a structure•

Modelling

–

study

of the behaviour

of the model

•

Numerical

modelling

–

handle

the complexity

of PDE

Method Finite element method (FEM)

Finite difference method (FDM)

Boundary element method (BEM)

Equivalent circuit method (ECM)

Point mirroring method (PMM)

Principle of discretisation

m1

m2

q

q*

Geometry approximation Extremely flexible Inflexible Extremely flexible Specific

geometries Simple

geometries

Non-linearity Possible Possible Troublesome Possible By constant factors

Computational cost High High High Very low Low


Equivalent element

Length

Area

Flow

Potential difference•

Definition of geometric and medium properties

•

Obey to a physical laws i.e. relations

•

Is equivalent to ‘physical reality’

i.e. has similar or

identical effect

areaklengthimpedance


Equivalent circuit relations

Relation Electrical circuit

Magnetic circuit

Thermal circuit

Cooling circuit

Potential U=E·l N·I=H·l =G·l P=·l

Flow I=J·A Φ=B·A Q=q·A Q=v·A

Conductive element

G=γ·A/l G=μ·A/l G=λ·A/l G=·A/l

Ohm’s Law U=I·R N·I=Φ·R =Q·R P=Q·R


Formulation of a transfer problem•

Heat source Q(x) [J/sth] per unit of time and length

•

The heat inflow H [J/s] at position x, and outflow H+dH

at position x+dx

•

Transfer problem is described by conservation equation

and constitutive

relationx

dx

L

H H+dH

Q(x)


Conservation equation•

The conservation equation

describes the balance of

the time independent heat flow

QdxdHdHHdxQH


Constitutive relation•

the heat flux

q [J/(m2s)]

is specified as the flow

through the cross-section area per unit time

•

constitutive relation

defines the heat flow inside the medium

AHq

dxdq


Heat transfer•

Heat transfer

problem according to potential and heat

source/sink

•

The stationary (i.e. time-independent) heat problem is described as a balance

between heat supply to the body

per unit of time and the amount of the heat leaving the body per unit of time.

0

Q

dxdA

dxd


Solution•

The differential equation

of the transfer balance is

solved for the finite size of volume which boundaries specify the flow through them.

•

In order to solve the heat transfer second-order differential equation, two boundary conditions

need to

be specified to the two ends of fin. •

At one of these ends either temperature

or flux

q

is

given.


Equivalent element•

Flux tube

•

The scalar potential difference (potential drop)

–

temperature–

voltage

–

magnetomotive

force

•

The ratio of potential difference to flux is a function of flux tube geometry and medium properties

x A(x+dx)L

q

(L)

A(x)

(0)

q

l

xAxcdxuluR

0

0


Similarities I

•

The heat transfer

problem describes the temperature distribution between a source Q

and coolant

•

The magnetostatic

problem specifies the magnetic potential Vm

according to the magnet flux Ψ.

•

The displacement u

is the unknown for elasticity problem with body forces b.

0

Q

dxdA

dxd

0

dx

dVA

dxd m

0

Q

dxdV

Adxd e

0

b

dxduAE

dxd


Similarities II•

The ability to conduct flow in different media and physical problem are defied with thermal conductivity λ, magnetic permeability μ

or elasticity E

and the

corresponding cross section A.

l

xAxdxR

0

l

m xAxdxR

0

l

xAxdxR

0

l

E xAxEdxR

0


Thermal equivalent circuit

•

Heat transfer in two different medium

that is described by heat conductivity k1

(W/K)

and k2

(W/K)•

elements are connected in series

along x-axis

•

The positive direction

of thermal flow Q (W)

is chosen to be in the direction of x-axis.

1 2 k1

Q1 Q2

2 3 k2

Q2 Q3x


Thermal flux•

The thermal flux q

within the element

is described in

accordance with a constitutive law i.e. a relation which describes how the material conducts heat.

•

The heat flow is represented as the heat fluxes (including external) acting on the element nodes.

121 kq

1212 kQ 2111 kQ


Matrix form of element•

According to balance equation, the sum of all the heat fluxes acting on the element nodes is equal to zero.

•

The characterization of one element

can be expressed in matrix forms.

•

Each element

independently in the system can be described in respect of the element relation.

2

1

2

1

11

11

QQ

kkkk

eee faK


Expanded element relation •

The expanded element relation

i.e. the first element

relation to the whole system in accordance with this example is:

000000

12

11

3

2

1

11

11

QQ

kkkk

eeee11 faK


The complete system•

The complete system equation of the entire thermal circuit is the sum

of the expanded element stiffness

relations of each element and load vector

in accordance with equilibrium conditions for the nodal points.

3

22

12

1

3

2

1

22

2211

11

0

0

QQQ

Q

kkkkkk

kk

fKa


Defining a heat transfer problem•

The determinant

of the assembled total

stiffness matrix K

is equal to zero. •

In order to obtain unique solution

for the

unknown temperature

and at least one node point has to be prescribed a priori.

•

The specification of the given temperature is an essential boundary condition, which prescribes the value of variable itself and is necessary in order to solve the system of equations.

•

The heat flux is a natural boundary condition and this specifies thermal insulation, the heat

flow into or out from the system.

MEC – Magnetic Equivalent Circuit

Example

based

on a electromagnetic circuit


From Real to Equivalent circuits

i

Origin of magnetic flux• Current carrying coils• Permanent magnets

Medium carrying the magnetic flux• magnetic core• Air-gap

Force of origin• Interaction• AttractionProblem solving domain

• Symmetry• Boundary


From Real to Equivalent circuits

i

Coil and Core• MMF source• Nonlinear reluctance of core Permanent magnet

• Remanence flux source• Inherent reluctance

Air-Gap• Parametric gap reluctance

Problem solving domain• Symmetry• Boundary

NiMMFAB

l)(

)()(

0 A

l

ABRR

lHAB

C

R


Example: electromagnetic circuit•

Electromagnet + a core with permanent magnet

•

‘magnetizing’

flux path through the core

•

Ampere’s circuit Law

•

Circuit consists of a soft magnetic core, a magnet and a coil.

iNlH

iNlHlHlH ggfefepmpm


Example: parameterization•

Geometry and materials are parameterized

•

magnetic field intensity H is replaced with flux density B

•

The same flux slows through the cross-section areas A

ggfefefe

pmpmrpm

HBHB

HBB

00

0

gfepmggg

fefefepmpmpm

AB

ABAB


Example: equivalent circuit•

After the first replacement

•

After the second replacement

•

Flux path in the same medium is summon up to a single element

iNlB

lB

lBB

gg

fefe

fepm

pm

rpm

000

pmpmpm

pmgfepm

fe

g

fefe

fe

pmpm

pm

Al

iN

Al

Al

Al

0

000


Example: Finite element model•

FE model

•

Comparison –

Bc=0.73T froth previous formulation

–

Bc=0.62T froth FE model

•

Difference is due to a leakage flux path


Formulating a MEC•

Assume

flux path (circular) and define elements

•

Three types of permeance

elements

Pm: –

nonlinear core elements Pm=f(B)

–

parametric gap elements Pm=f() –

leakage permeances.

•

The node potential method

is used to calculate magnetic scalar potential (unknowns) vector Vm

with

respect to the permeance

(stiffness) matrix G, and magnetic flux input (load) vector Ψ.

ΨGVm


Magnetic equivalent circuit•

Relative simple to ‘add’

leakage elements•

Circuit described in a topology matrix

•

Comparison –

Bc=0.73T from previous formulation

–

Bc=0.62T from FE model–

Bc=0.68T from MEC model


Sources•

The remanence

BR

of the permanent magnet and the cross-section area Apm

of the magnet determine the value of the source magnetic flux.

•

In order to consider the non-zero current in the armature coil the mmf

source

has to be added to the

algebraic equations for the node points.

pmR AB pmC

pmRpm lH

ABP

12121221 PNiGVV mm


Magnetic saturation•

The magnetic saturation is taken into account in nonlinear elements

Gm,

•

The permeability update bases on the previous

update and on the resent estimation from μ=f(B)

ababababba GFGuu

)(ψGu

eeeee AGuuB /21

ndof

e

ni

ndof

n

ni

ni

err

nelm

e

ei

nelm

e

ei

ei

err

u

uuu

B

BBB

1

2

1

2

1

1

2

1

2

1

e

ienl

ei

ei

enl

cB

11

3

3

10

10

err

err

u

B

Fpmmecl ,,, Mμ

40 60 80 100 120 140 160 180 2000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1

2

3

4

56

78910

field intensity, H [A/m]

flux

dens

ity, B

[T]

Home assignment

Analysis of electromagnetism in a single-phase transformer


Goal: Geometry vs Equivalent circuits

•

Same transformer as in the first assignment

–

Proportional core and slot for coils Ks=0.5

–

Current driven magnetic circuit =f(I)

–

Voltage driven electric circuit I=f(U)

–

Comparison between equivalent circuit and finite element method


Circuits & phasors•

Magnetisation

–

Resistance R1

and inductance L1

of primary coil–

Leakage and mutual inductance Lσ+Lm=L1

–

Core

losses

Rm–

Complex

current

Io

and

magnetising

flux Ψm•

Magnetically

coupled

–

Secondary

circuit–

EMF E2

=jωΨm•

Electrically

loaded

–

Max P2

power I2

=E2

/(Z2

+R)–

Primary

current

I1

=I2

+Io

Io

IoR

IoLΨm

E2 I2

I1


Equivalent circuit

•

The resulting equivalent system according to equations•

Corresponding components in phasor

diagram

201 nIII

222111122222 jXRIjXRnInUjXRIEU

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EIEN20 Design of Electrical Machines