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A specifications-oriented initial design methodology for power transformers

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Abstract

Power transformers are classified among the most crucial power system components. Currently, the power transformer international competitive market drives manufacturers to adopt design strategies that lead to cost minimization while maintaining performance quality. Obviously, most of the aforementioned strategies rely on precise computation techniques involving electromagnetic field aspects for calculated refinements. In all cases, rough design detail estimation is indispensable to initiate the design refinement process and, in some cases, to address price quotation requests. This paper presents a fast specifications-oriented initial design methodology. Although the proposed methodology is analytical in nature it implicitly incorporates design parameter restrictions inferred from more sophisticated studies. Details of the proposed methodology in addition to a couple of design case studies are given in the paper.

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Abbreviations

a :

Spacing between low and high voltage windings

\(a_{c1}\) :

Primary winding conductor cross sectional area

\(a_{c2}\) :

Secondary winding conductor cross sectional area

B :

Maximum core flux density

b :

Primary or secondary winding thickness

\(C_{Fe}\) :

Specific core losses constant

D :

Core bounding diameter

f :

Supply frequency

\(H_W\) :

Window Height

\(I_{ph1}\) :

Primary phase current

\(I_{ph2}\) :

Secondary phase current

J :

Windings current density

\(K_c\) :

Gross area to maximum circular area ratio

\(K_f\) :

Laminations stack factor

\(K_H\) :

Winding height to window height ratio

\(K_W\) :

Window height to width ratio

\(l_{mt}\) :

Average windings turn length

\(l_{mt1}\) :

Average primary winding turn length

\(l_{mt2}\) :

Average secondary winding turn length

\(l_{WH}\) :

Windings height

\(N_{1}\) :

Primary winding number of turns

\(N_{2}\) :

Secondary winding number of turns

\(P_{cu}\) :

Copper losses

\(P_{cu-con}\) :

Termination connections losses

\(P_{cu-eddy}\) :

Windings eddy current losses

\(P_{cu-ohmic}\) :

Ohmic copper losses

\(P_{{F}_{e}}\) :

Core losses

\(P_{NL}\) :

No-load losses

\(P_{stray}\) :

Stray losses

S :

Rated Volt-Ampere

\(S_{W}\) :

Window space factor

\(V_{l1}\) :

Primary line voltage

\(V_{l2}\) :

Secondary line voltage

\(Vol_{cu}\) :

Overall copper volume

\(V_{ph1}\) :

Primary phase voltage

\(W_{Fe}(B,f)\) :

Specific core losses as a function of flux density and frequency

\(W_{W}\) :

Window width

X :

Equivalent leakage reactance per phase

\(\delta _{Fe}\) :

Steel laminations density

\(\rho _{cu}\) :

Specific resistivity of copper

\(\mu _{o}\) :

Permeability of free space

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Author information

Authors and Affiliations

  1. Electrical Power and Machines Department, Faculty of Engineering, Cairo University, Giza, 12613, Egypt

    A. A. Adly

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  1. A. A. Adly
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Correspondence to A. A. Adly.

Appendix 1: detailed worked example

Appendix 1: detailed worked example

A detailed worked example for the implementation of the proposed design methodology is provided in this Appendix. More specifically, a detailed worked example is given for a core-type, oil immersed, three-phase stack core power transformers whose material type is Armco Steel TRAN-COR-H0 CARLITE-3 [19].

1.1 Target specifications

$$\begin{aligned} \begin{array}{l} S=25\times 10^6\,\mathrm{VA},V_{l1} =66\,\mathrm{KV},V_{l2} =11\,\mathrm{KV, DYn11, number of stack core steps = 11} \\ f={50\,\mathrm{Hz}},X=10.48\,\%, P_{CU} =85.2\times 10^3\,\mathrm{W}, P_{NL} =15.5\times 10^3\,\mathrm{W.} \\ \end{array} \end{aligned}$$

1.2 Estimated design parameters

From [10, 11], it is reasonable to assume the following parameters:

$$\begin{aligned} K_f =0.95, S_W =0.2, K_H =0.9. \end{aligned}$$

Knowing that the number of stack core steps = 11, it can be deduced from [10, 11] that:

$$\begin{aligned} K_C =0.958. \end{aligned}$$

Referring to the core material data specifications [19], thus:

$$\begin{aligned} C_{Fe} =96\times 10^{-6}\,\mathrm{W/kg/sq.\,freq/sq.\,flux density,}\,\delta _{Fe} =7650\,\mathrm{kg/m}^{3}. \end{aligned}$$

From [14], it is reasonable to assume: \(K_W =2.28.\)

It is also well known that \(\mu _o =4\pi \times 10^{-7}\,\mathrm{H/m}\) and \(\rho _{cu} =2.1\times 10^{-8}\,\Omega \mathrm{m}.\)

1.3 Step-by-step design calculations and results

$$\begin{aligned} I_{ph1}= & {} {\frac{S}{3V_{ph1}}}={\frac{25\times 10^{6}}{3\times 66\times 10^{3}}}=126.2626\,\mathrm{A,}\\ I_{ph2}= & {} {\frac{S}{3V_{ph2}}}={\frac{25\times 10^{6}}{3\times ({11\times 10^{3}} / {\sqrt{3}})}}=1312.16\,\mathrm{A,} \\ X= & {} \left( {\frac{X~\%}{100}}\right) \frac{V_{ph1} }{I_{ph1} }=\left( {\frac{10.48}{100}}\right) \frac{66\times 10^3}{126.2626}={54.78106 }\Omega . \end{aligned}$$

From (13):

$$\begin{aligned} H_W= & {} \sqrt{\frac{864\rho _{cu} K_H K_W^2 I_{ph1}^2 }{11\pi f\mu _o S_W}\frac{X}{P_{cu} }} \nonumber \\= & {} \sqrt{\frac{864\times 2.1\times 10^{-8}\times 0.9\times ( {2.28})^2\times ( {126.2626})^2}{11\pi \times 50\times 4\pi \times 10^{-7}\times 0.2}\frac{{54.78106}}{85.2\times 10^3}}\nonumber \\= & {} 1.415522022\approx 1.42\mathrm{m}. \end{aligned}$$
(19)

From (14):

$$\begin{aligned} \alpha _1= & {} \frac{7.2\pi \rho _{cu} S_W H_W^2 }{4K_W P_{cu} }=\frac{7.2\pi \times 2.1\times 10^{-8}\times 0.2\times ( {1.415522022})^2}{4\times 2.28\times 85.2\times 10^3}\nonumber \\= & {} 2.4498\times 10^{-13}, \nonumber \\ \alpha _2= & {} \frac{64.8\pi \rho _{cu} S_W H_W^3 }{80K_W^2 P_{cu} }=\frac{64.8\pi \times 2.1\times 10^{-8}\times 0.2\times ( {1.415522022})^3}{80\times ( {2.28})^2\times 85.2\times 10^3}\nonumber \\= & {} 6.84423\times 10^{-14}, \nonumber \\ J= & {} \sqrt{\frac{1}{( {\alpha _1 D+\alpha _2 })}} =\sqrt{\frac{1}{(2.4498\times 10^{-13}D+6.84423\times 10^{-14})}}. \end{aligned}$$
(20)

From (15):

$$\begin{aligned} \beta= & {} \frac{1.3\pi C_{Fe} f^2\delta _{Fe} K_f K_c }{4P_{NL} }=\frac{1.3\pi \times 96\times 10^{-6}\times ({50})^2\times 7650\times 0.95\times 0.958}{4\times 15.5\times 10^3}\nonumber \\= & {} 0.11006857, \nonumber \\ \beta _1= & {} 6\beta =6\times 0.11006857=0.66041142, \nonumber \\ \beta _2= & {} \beta \left( {\frac{3K_W +4}{K_W }}\right) H_W \!=\!0.11006857\times \left( {\frac{3\times 2.28+4}{2.28}}\right) \times 1.415522022\nonumber \\= & {} 0.74075466, \nonumber \\ B= & {} \sqrt{\frac{1}{(\beta _1 D^3+\beta _2 D^2)}} =\sqrt{\frac{1}{(0.66041142D^3+0.74075466D^2)}}. \end{aligned}$$
(21)

From (16):

$$\begin{aligned} \begin{array}{ll} \gamma &{}=\left( {\frac{{3.33}\pi K_f K_c S_W f}{4K_W }}\right) H_W^2 \\ &{}=\left( {\dfrac{3.33\pi \times 0.95\times 0.958\times 0.2\times 50}{4\times 2.28}}\right) \times ( {1.415522022})^2=20.9180715. \\ \end{array} \end{aligned}$$

From (17):

$$\begin{aligned} \omega =\frac{S^2}{\gamma ^2}=\frac{( {{25}\times {10}^{6}})^2}{( {20.9180715})^2}=1.42836\times 10^{12}. \end{aligned}$$

From (18):

911

$$\begin{aligned}&(\alpha _1 \beta _2 +\alpha _2 \beta _1 )\omega \\&\quad =(2.4498\times 10^{-13}\times 0.74075466+6.84423\times 10^{-14}\times 0.66041142)\times 1.42836\times 10^{12} \\&\quad =0.323766, \\&(\alpha _1 \beta _2 +\alpha _2 \beta _1 )^2\omega ^2=0.104824, \\&4( {{1-}\alpha _1 \beta _1 \omega })\alpha _2 \beta _2 \omega =\left\{ {{\begin{array}{ll} {4\times (1-2.4498\times 10^{-13}\times 0.66041142\times 1.42836\times 10^{12})} \\ {\times 6.84423\times 10^{-14}\times 0.74075466\times 1.42836\times 10^{12}} \\ \end{array} }} \right\} \\&\quad ={0.222726,} \\&2( {{1-}\alpha _1 \beta _1 \omega })=2\times (1-2.4498\times 10^{-13}\times 0.66041142\times 1.42836\times 10^{12})={1.537819,} \\ \end{aligned}$$

Hence,

$$\begin{aligned} \begin{array}{ll} D&{}=\dfrac{(\alpha _1 \beta _2 +\alpha _2 \beta _1 )\omega +\sqrt{(\alpha _1 \beta _2 +\alpha _2 \beta _1 )^2\omega ^2+4( {{1-}\alpha _1 \beta _1 \omega })\alpha _2 \beta _2 \omega } }{2( {{1-}\alpha _1 \beta _1 \omega })} \\ &{}=\dfrac{0.323766+\sqrt{0.104824+0.222726} }{1.537819}=0.582699\mathrm{m}\\ \end{array} \end{aligned}$$
(22)

Substituting (19) and (22) into (20) and (21) we get:

$$\begin{aligned} J=\sqrt{\dfrac{1}{(2.4498\times 10^{-13}\times 0.582699+6.84423\times 10^{-14})}} =2176012\,\mathrm{A/m}^{2},\nonumber \\ \end{aligned}$$
(23)

and,

$$\begin{aligned} B=\sqrt{\frac{1}{(0.66041142\times (0.582699)^3+0.74075466\times (0.582699)^2)}} =1.617589\,\mathrm{T.}\nonumber \\ \end{aligned}$$
(24)

1.4 Comparison between actual and computed results

A comparison between actual and computed leading design parameters are given in the following table. These results demonstrate the capability of the proposed approach to offer a relatively accurate initial design details (Table 3).

Table 3 Comparison between the 25 MVA transformer actual and computed leading design parameters
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Adly, A.A. A specifications-oriented initial design methodology for power transformers. Energy Syst 8, 285–296 (2017). https://doi.org/10.1007/s12667-016-0197-5

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