An Electrical Circuit based 3-D Steady State Thermal model of a Fan Cooled 60 Hz, 20 kW 3-Phase Plasma

Cutting Power Supply Transformer

Girish R. Kamath Hypertherm, Inc.

Etna Road Hanover, NH 03755

e-mail: girish.kamath@hypertherm.com

Abstract— This paper presents an electric circuit based 3-D steady state thermal model of a 3-phase, 60 Hz, 20 kW transformer operating under forced air cooled conditions. The model is developed using a cost effective approach and consists of current sources that represent the power loss, voltage sources for the temperature and resistors for the associated conduction and convective heat transfer terms to solve the underlying heat transfer equation. The paper includes a procedure for deriving these components. Simulation results using a standard circuit simulator are then compared with experimental data. These show that the trends in various temperatures predicted by simulation are in good agreement with those from experiment. Thus the model can be used to comparatively evaluate potential transformer designs during a design optimization process. Finally, the paper discusses limitations and advantages of the proposed method and identifies ways to further improve the model performance.

Keywords-thermal modeling; transformer; Forced air cooling; temperature rise; heat transfer coefficient; empirical correlations

I. INTRODUCTION A typical evaluation of a magnetic component such as a

transformer involves building and testing it to its specifications under standard operating conditions. This is because its power rating is mainly determined by temperature. This evaluation can be an iterative and lengthy process consuming valuable engineering resources, time and cost, especially for medium and large power transformers. Furthermore, recent increases in the prices of steel and copper have necessitated the need for an optimal design especially of medium and high power line frequency transformers.

Several approaches for estimating the temperature rise have been disclosed in the literature. In [1, 2], the transformer core and copper losses are theoretically estimated and the resulting temperature rise calculated under natural convection conditions. Here, the formulation for the convective mode of heat transfer is obtained using Newton’s law of cooling. The heat transfer coefficient formula use in [1] is successfully verified by means of FEA (Finite Element Analysis) in [3] and shown to be applicable only to pot cores. Hence, [2] uses a different heat transfer coefficient formula suitable for EFD cores. [2] also includes the effect of radiation heat transfer in

the temperature rise. [4, 5] use an equivalent electrical circuit based thermal model for estimating the temperature rise. The conduction heat transfer component is derived analytically for the 1-D case. [4] uses dimensionless empirical relations to theoretically derive the convective heat transfer coefficient in contrast to an FEA based approach that was verified by experiment in [5]. [6] adopts a numerical approach to develop a 1-D equivalent electrical circuit thermal model of a transformer. However, it does not include the convective heat transfer component. In all these cases, the analysis was applied to naturally air cooled low power high frequency transformers. Another approach is to obtain the resulting temperature rise by solving the underlying mass, force and energy balance equations using an FEA tool like in [3]. This approach yields more accurate results, especially in laminar flow situations. However, it is extremely expensive in terms of computing resources as well as cost of the tool itself.

This paper analyses the performance of a 3-phase, 60 Hz, 20 kW transformer used in a 130A, 150V output Plasma Cutting Power Supply with forced air cooling. The approach used is cost-effective in that it does not use an FEA tool. A 3-D electric circuit based thermal model is developed to solve the underlying heat transfer equation and determine the resulting temperature rise. The model consists of current sources for representing the power loss, voltage sources for the temperature and resistors for the associated conduction and convective heat transfer terms. Thus, the approach used in [6] in deriving the conduction heat transfer component is applied here to the 3-D case. The convective heat transfer coefficient term is obtained by using the dimensionless parameter correlations as shown in [4, 7, 8]. Air flow velocities required for this estimation are obtained by measurement. In addition, both the transformer core and copper loss values used in the model are obtained under the actual operating conditions. This helps improve the model accuracy and the circuit based representation enables it to be simulated in a standard circuit simulator like ORCAD. The quality of the model is then verified by comparing the simulation results with those from experiment. The simulation results show good agreement with experiment in so far as confirming the phase winding temperature trends is concerned. The temperature rise results show errors in the range 25-40%. Notwithstanding these errors, the method can be used to perform a comparative

0197-2618/07/$25.00 © 2007 IEEE 1773

evaluation of potential transformer alternatives at the design stage, saving time and cost in the design optimization process.

II. THEORETICAL BACKGROUND

A. Review of the heat transfer equation The method of using an electrical circuit based numerical

method for solving partial differential equations was first proposed by [9]. Its application to solving the heat transfer equation has been presented in standard heat transfer text books like [7, 8] and is presented below. Since the transformer is undergoing conduction heat transfer within and convection heat transfer on its surfaces, it is useful to review the underlying equations that govern these phenomena. For this, consider an isotropic solid body experiencing conduction and convection heat transfer and shown in Fig. 1(a). The conduction heat transfer component is formulated on the basis of Fourier’s law of conduction [7, 8]. The differential form of the heat transfer equation in steady state is shown in (1) and is obtained by applying the energy balance principle to a differential volume (δx, δy, δz) shown in Fig. 1(b).

0q)2z

T2

2y

T2

2x

T2k( =+

∂

∂+∂

∂+∂

∂ (1)

where, T – Temperature at (X, Y, Z) q - power generated in the element per unit volume k – thermal conductivity of the element

The body is also undergoing convection heat transfer on its top surface. This mode of heat transfer is formulated using Newton’s law of cooling and is shown in (2) below.

)T(Thq s'

∞−∗= (2)

where, q’ – heat flux in W/m2 Ts – body surface temperature T∞ - cooling fluid temperature h – convection heat transfer coefficient

Applying the energy balance equation to the surface results in the boundary condition shown in (3). This boundary condition constrains the heat transfer equation (1) at the said surface.

)T(T*hzT*k s

Zz∞

=−=

∂∂− (3)

B. Equivalent Electrical Circuit Representation of Heat Transfer Equation The solution to (1) subject to the boundary condition (3) is

obtained using the numerical method. For this, an approximate or finite difference version of (1) is used. This is shown in (4) below.

0QRδT

RδT

RδT

RδT

RδT

RδT

gz

z

z

z

y

y

y

y

x

x

x

x =++++++−

−

+

+

−

−

+

+

−

−

+

+ (4)

where,

-yy-xx Rz*x*k

y*0.5R;Rz*y*k

x*0.5R ==== ++ δδδ

δδδ

-zz Ry*x*k

z*0.5R ==+ δδδ

Qg – power generated in the element +xTδ , −xTδ , +yTδ , −yTδ , +zTδ , −zTδ - temperature

differences as shown in Fig. 1(b)

The approximate form of the boundary condition (3) is shown in (5) below.

thz

z

R)TT(

RT ∞

+

+ −= sδ

(5)

where,

y*x*h1R th δδ

=

Fig. 1(b) Close-up view of Differential Element

Fig. 1(a) Solid body undergoing conduction and convection heat

transfer

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An equivalent electrical circuit form of (4) subject to the boundary condition (5) is developed by representing Rx+, Ry+, Rz+, Rx-, Ry-, Rz-, and Rth as resistors and Q as a current source. These elements are connected to the node at the center of the element whose voltage is the resulting temperature T. Temperature T∞ of the cooling fluid is represented as a voltage source. Fig. 2 shows a schematic of this circuit representation.

III. TRANSFORMER THERMAL MODEL DERIVATION A thermal model of a 3-phase, 60 Hz, 20 kW transformer

operating under forced air cooled conditions is developed using the approach described above. Table 1 below summarizes the transformer electrical specifications.

TABLE I. TRANSFORMER SPECIFICATIONS SUMMARY

Winding Description Nominal Voltage (V) Nominal Current (A)

Primary -1 240 33 Primary-2 240 33 Secondary 208 72 Max flux density Bmax – 1.8 T Overload Power rating – 1.25*Nominal Power rating Input voltage variation: ±20% ½” wide ducts provided between Primary-1- Primary-2 and Primary-2 – Secondary windings for forced air cooling

A picture of the 3-phase transformer along with cross-sectional views of one phase of the transformer in the X-Y and Y-Z planes are shown in Figs. 3(a), (b) and (c)

A. Thermal Model derivation of the Transformer windings The winding thermal model is obtained by dividing each

winding layer into its four sides with each side replaced by an element of the same dimensions. An equivalent electric circuit of the type shown in Fig. 2 is then derived for this element. Thus each layer has four such circuits that are appropriately connected to each other to complete the circuit for that layer. The equivalent thermal circuit of once such layer with only the conduction heat transfer portion represented is shown in Fig. 4.

a) Conduction Heat Transfer Considerations In each circuit, resistors Rxcu, Rycu and Rzcu are the

conduction heat resistors of the copper sheet in the x, y and z directions. Similarly, Rxvr, Rxml represent the conduction heat transfer due to the varnish/conductor insulation and insulation tape layers. The resistor values are dependent on their respective material thermal conductivity values as well as thickness and cross-sectional areas. The thermal conductivity parameter values are summarized in Table 2. Figure 4 also includes the component values of a winding layer thermal circuit. The current source Qside in each circuit is the resistive power loss in that portion of the layer at room ambient. Table

Fig. 4 Winding layer Thermal Model – Conduction part only

Fig. 3(c) Phase Y Winding cross-sectional view in the Y-Z plane

Fig. 3(b) Phase Y Winding cross-sectional view in the X-Y plane

Fig. 2 Electrical Circuit representation of Heat transfer equation

Fig. 3(a) 3-phase transformer

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3 lists the winding dc resistances, currents and their respective copper loss values at 24.5kW load. Qside is then obtained using the values listed in this table.

TABLE II. SUMMARY OF MATERIAL THERMAL CONDUCTIVITIES

Material Thermal Conductivity (W/m*K)

Copper 393 Nomex-410 (winding insulation tape) 0.15 Varnish 0.35 Conductor enamel insulation 0.2 CRGO steel lamination (for the core) 22

TABLE III. WINDING COPPER LOSS AT 384V AND 24.5 KW LOAD

Winding Current (A) DC resistance (mΩ) Copper Loss (W)

Primary-1 44 21 40.6 Primary-2 44 27 52.3 Secondary 96.4 15 139.4

Further, the impact of temperature on Qside is accounted for by making it a function of the winding temperature as shown in (6) below.

20)*0.0068(1.6068T)*0.00068(1.6068*C)(20Q

(t)Qo

sideside +

+= (6)

where, Qside(200C) – Power loss at 20oC T – Average Element Temperature in oC b) Forced Convection Heat Transfer Considerations

The windings are fan cooled externally from the front and sides and internally through the ducts provided by spacers between windings as shown in Fig. 3(c). The heat transfer coefficients required for estimating the respective convection heat resistors are obtained by using the empirical relations listed in Table 5. The dimensionless parameters that constitute these relations are listed in (7)-(9). The relations are selected depending on whether the air flow is laminar or turbulent and also whether the flow is internal or external. The latter is determined visually. The former requires calculation of the Reynolds number as shown in (7) and application of the criterion listed in Table 4, column 1.

TABLE IV. LIST OF FORCED CONVECTION EMPIRICAL RELATIONS

Nature of Flow Empirical Relation Laminar (Re < 5*105) External Nu = 0.664*Re0.5*Pr0.333

Turbulent (Re > 5*105) External Nu = 0.037*Re0.8*Pr0.333

Laminar (Re < 2300) Internal

Nu =1.86*Gz0.333*(µair/µairTs)0.14 where Gz = Re*Pr*D/L; µairTs-air viscosity at surface temperature Ts

Turbulent (Re>2300) Internal Nu = 0.023*Re0.8*Pr0.4

air

airaire

*D*VR

µρ

= (7)

where Re – Reynold’s number Vair – air free flow velocity D – hydraulic diameter of the object ρair - density of air = 1.03 kg/m3

µair – viscosity of air = 2.04e-5 N.s/m2

air

p_airairr k

C*P

µ= (8)

where

Pr – Prandtl number

Cp_air – Air Specific heat capacity at constant pressure = 1009 J/Kg K

kair – air thermal conductivity = 0.03 W/m K

air

fu k

L*hN = (9)

where, Nu – Nusselt’s number L – characteristic length of object hf – forced convective heat transfer coefficient

A thermistor based air-flow meter[10] is used to measure the flow velocities at various locations on the transformer for this purpose. These measurements are conducted with only the forced cooling system in operation. The influence of unit operation at load on the air flow is not included in these measurements. The magnitude and direction of these velocities at various locations on the front and back of the transformer are shown in Figs. 5(a) and 5(b), respectively.

Fig. 5(a) Transformer Front Air flow velocities in m/s

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The various parameters of air used in the equations above assume an air film temperature Tf of 70oC and 1 atm pressure. The heat transfer coefficient hf is obtained from Nu and used to calculate Rth using (10).

A*h

1Rf

th = (10)

where,

A – Area over which the heat transfer coefficient was calculated

c) Consideration of Natural Convection Effects It is found that the impact of natural convection at certain

locations of the transformer especially at the back of the core and the windings cannot be ignored. The criterion used for making this evaluation is outlined in [8]. The procedure involves calculation of the Grashof number GrL at the location of interest. This dimensionless number is calculated using (11) below.

2air

2air

3s

rL*L*)T(T**g

Gµ

ρβ ∞−= (11)

where, g- gravity = 9.8 m/s2

1-

sK deg

0.5*)T(T1

∞+=β

Then, the ratio 2e

rL

RG

is calculated. For those cases where

both forced and natural convection effects need to be

considered 1RG

2e

rL ≈ . These two effects are quantitatively

combined by calculating an equivalent convection heat transfer resistor that is obtained from a parallel combination of the

individual heat transfer resistors. The benefits and drawbacks of this technique are discussed in greater detail in Section V.

The procedure for estimating the natural convection heat transfer coefficient hnat is identical to the one described for forced convection. The different dimensionless empirical relations and the criteria for their selection are listed in Table 5 below.

TABLE V. LIST OF NATURAL CONVECTION EMPIRICAL RELATIONS

Nature of Flow Empirical Relation Laminar (GrL*Pr < 109) External Nu = 0.59*( GrL*Pr )0.25

Turbulent (GrL*Pr > 109) External Nu = 0.1*( GrL*Pr )0.333

The equivalent heat transfer coefficient heq is then obtained as,

fnateq h

1h

1h1 += (12)

B. Thermal Model derivation of the Core The conduction and convection resistor components for the

core thermal model are derived in the manner as described above. Here, the core is divided into 13 sections. A core loss value of 80W is obtained from an open circuit test of the transformer at 384V, 60 Hz. The resulting thermal model for the core without the windings is shown in Fig. 6. The thermal model for the whole transformer is then obtained by appropriate interconnection of the core and winding models.

Fig. 5(b) Transformer Back Air flow velocities in m/s

Fig. 6 Core Thermal model

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IV. SIMULATION CIRCUIT AND RESULTS Schematics of the thermal model for one transformer phase

in the X-Y and Y-Z planes are shown in Figs. 6(a) and 6(b). The simulation temperatures are obtained by averaging the values obtained at the top, center and bottom of each winding.

Experimental readings are obtained under the following conditions:

• 384V, 60 Hz, 3-φ, -20% input voltage

• 24.5 kW load, +20% overload

• 20oC room ambient temperature

The temperatures are measured using thermocouples located at the center of one of the layers of each winding. The bar charts in Fig. 7 display the winding temperatures in each

phase from simulation and experiment. Both simulations and experiment show that:

• In all cases, in each phase the winding temperatures increase in the same winding sequence, i.e. Primary-1, Primary-2 and Secondary

• Among the phases, Phase Y has the highest winding temperature, while Phase R has the lowest

V. REMARKS ON COMPARATIVE EVALUATION The results show good agreement between simulation and

experiment when considering trends in the individual winding temperatures in all cases. This extends to the average phase temperatures as well.

Thus the model can be used to compare alternative

Fig. 6(a) Phase R of Transformer Thermal model in the X-Y Plane

Fig. 6(b) Phase R of Transformer Thermal model in the Y-Z Plane

Fig. 6 Circuit schematic of Phase R Winding Thermal model

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transformer designs during a design optimization phase. It also provides insights into temperature trends at other transformer locations that are not measured during the experiment.

However, errors in the range 25 – 40% for the winding temperatures indicate limitations of the model.

These can be attributed to:

• Variations in the air flow velocity measurements: The air flow measurements are taken at room ambient with only the cooling system under operation and not under actual operating conditions. For this reason, the impact

of temperature due to unit operation at load is not included in these readings. Time averaged velocity values are used to reduce the variations observed in the instantaneous velocity values.

• Errors in modeling natural convection effects: The model currently uses a crude approximation to represent the natural convection effects on external surfaces. Further, mixed convection effects in internal flow cases such as inside the winding ducts have not been considered in the model. However, empirical relations that involve combined convection effects for

Phase R Right Side Temperatures

121.5126.9

129.7

86.891.5

98.8

75

85

95

105

115

125

Primary-1 Primary-2 SecondaryWindings

deg.

C

simulation Phase R right side experiment- Phase R right side

Phase R left Side Temperatures

120.4

85.790.1

96.8

126128.6

75

85

95

105

115

125

Primary-1 Primary-2 SecondaryWindings

deg.

C

simulation Phase R left side experiment- Phase R left side

Fig. 7(a) Right Side Phase R Winding Temperatures Fig. 7(b) Left Side Phase R Winding Temperatures

Phase B right side temperatures

128

135.3138.2

94.6

100.7

110.7

75

85

95

105

115

125

135

Primary-1 Primary-2 SecondaryWindings

deg.

C

simulation Phase B right side experiment- Phase B right side

Phase B left side temperatures

127.7

133.7

92.395.6 97.8

136.6

75

85

95

105

115

125

135

Primary-1 Primary-2 SecondaryWindings

deg.

C

simulation Phase B left side experiment- Phase B left side

Fig. 7(e) Right side B-Phase Winding Temperatures Fig. 7(f) Left side B-Phase Winding Temperatures

Fig. 7 Winding Temperatures from Simulation and Experiment

Phase Y right side temperatures

128.4

136.17140.04

98.2105.6

111

75

85

95

105

115

125

135

145

Primary-1 Primary-2 Secondarywindings

deg.

C

simulation Phase Y right side experiment- Phase Y right side

Phase Y left side temperatures

127.7

90.9

97.8

107

135.3139.2

75

85

95

105

115

125

135

Primary-1 Primary-2 Secondarywindings

deg.

C

simulation Phase Y left side experiment- Phase Y left side

Fig. 7(c) Right Side Phase Y Winding Temperatures Fig. 7(d) Left Side Phase Y Winding Temperatures

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internal [7] and external flows [11, 12] are available. This indicates the model performance can be further improved by appropriate use of these relations.

• Radiation heat transfer considerations: The model does not include the effect of heat transfer by radiation. This is due to the lack of availability of appropriate material parameter values for making the estimation.

• Accuracy limits of empirical relations: The results obtained using empirical correlations are known to have errors in the range ±10% [7]. Further, these are obtained under controlled laboratory conditions. Air flow velocity measurements show the presence of significant cross flow velocities at the front of the core and the windings. The convective effect of these velocities is not considered in the model due to lack of availability of an empirical relation that suits the transformer geometry.

• Lack of availability of material parameter values like thermal conductivity and dimensions for materials such as conductor insulation, core material, bobbin and varnish: These values are generally not easily available. Hence, typical values are used instead and their dependence on temperature is not considered in the model.

• Granularity of the thermal model representation: The model is derived using a limited number of nodes. Increasing the number of nodes should help provide additional insights into temperature variations at different parts of the transformer at the cost of increased model complexity.

CONCLUSIONS This paper presents a cost-effective method of evaluating

the thermal performance of a 3-phase, 60 Hz, 20 kW Plasma Cutting Transformer. This involves deriving an equivalent electric circuit based 3-D thermal model to solve the underlying heat transfer equation. The procedure for developing the model is presented in the paper. A comparative evaluation shows that there is good agreement between simulation and experiment in estimating trends in the winding temperatures. However, there are errors in the range 25-40%. Factors influencing these errors are identified and methods to improve the model representation discussed. The model in addition to analyzing existing transformer designs can also be used to comparatively evaluate alternative transformer designs during the design optimization process.

ACKNOWLEDGMENT The author would like to thank his colleagues Norm

LeBlanc and Paul Tillman for their efforts in gathering and making available the experimental data required for this paper.

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