6 Calculation of Load Loss

6.1 CALCULATION OF I 2R LOSS

The I 2 R loss is calculated by adding up the DC resistance losses of all windings. The phase current
corresponding to the required tap position shall be taken for the calculation. The losses need to be
calculated for the reference temperature, and the resistance values corresponding to the applicable
reference temperature shall be used. As per IEC 60076, the reference temperature is 75°C for oil-
filled transformers, whereas it is 85°C as per ANSI C57.12 standard.
The I2 R loss can be estimated by knowing the weight of conductor and the current density. The
calculation using resistance value shall be used for design, and the method as per this section shall
be used for quick checking/estimation only.

I 2 R = Kδ 2W ( Watts) (6.1)

where
K = 2.4 for copper
13.0 for aluminium
δ = current density (A/mm2)
W = bare conductor weight (kg)
The above formula is derived by the following equations:

ρl
I 2R = I 2 (6.2)
A

lAD = W

ρ = resistivity of conductor material

W = weight
l = length
D = density
A = area of conductor

W
∴l ∝
AD
Substituting in Equation (6.2)

I 2ρ W δ 2ρ W
I 2R ∝ 2
∝ = kδ 2W
AD D

When values of ρ and D are substituted for copper and aluminium, the values of K become 2.4 and
13, respectively, at 75°C.

6.2 CALCULATION OF EDDY CURRENT LOSSES AND STRAY LOSSES

Eddy current and stray losses are produced due to leakage fluxes in windings, bushing plates,
busbars, tanks, core channels and other metal parts in proximity of the leakage flux. The spatial

65
66 Power and Distribution Transformers

distribution of the axial and radial leakage flux is required for accurate calculation of these losses,
especially when the ampere-turn distribution is asymmetrical and when the leakage flux density is
higher. For transformers up to 10000 kVA, 33 kV class, the following formula can be used for the
estimation of eddy current and stray losses.

Total eddy current and stray loss = Eddy current loss in HV winding

+ eddy current loss in LV winding + Stray loss in bushing plate + stray loss in tank

+ Stray loss due to circulating current + Stray loss in channels + lead wire loss

+ miscellaneous stray loss

Total load loss = I 2R loss + windingeddy current loss + stray loss

6.2.1 EDDY CURRENT LOSS IN WINDING

ws = 9 f 2 B 2t 2 w × 10 –3 Watts for Copper winding ( for rectangular conductor ) (6.3)

ws = 9 f 2 B 2t 2 w × 0.49 × 10 –3 Watts for Copper winding ( for round conductor ) (6.4)

ws = 19 f 2 B 2t 2 w × 10 –3 Watts for Aluminium winding ( for rectangular conductor ) (6.5)

ws = 19 f 2 B 2t 2 w × 0.49 × 10 –3 Watts for Aluminium winding ( for round conductor ) (6.6)

where
Ws = stray loss in winding
f = frequency (Hz)
B = maximum value of axial leakage flux density (Tesla)

IT
B = 1.78 × 10 −3
Hav

I = rated phase current of winding

T = turns at rated tap of the same windings

HV axial height + LV axial height

Hav = Average axial height ( mm ) =
2

t = thickness of conductor (mm)

w = bare weight of winding (kg)

Note
1. Winding stray loss shall be calculated separately for HV and LV windings. If separate tap
winding is used, eddy current loss of tap winding shall be calculated using the maximum
value of leakage flux linkage to the tap coil.
2. Total weight of winding shall include the weight of tap conductors also which are inside
the leakage field. If taps are taken from inside layer/layers of winding, the complete turns
will be subjected to leakage flux, even if some turns are not carrying current.
Calculation of Load Loss 67

• Eddy current loss of winding (Alternate Formula 1 for copper winding)

2 3.84
 f 
Peddy = 3.1   
t 
 50   10 
( nr ml )2.07 (6.7)

where
f = frequency
t = conductor thickness (mm)
nr = number of parallels in radial direction
ml = number of layers of winding
(nrml) = total number of conductors in radial direction
I2 R = I2 R loss

Example

I2R loss = 19,214 W

f = 50 Hz
t = 3.2 mm
nr = 5
ml = 4
2 3.84
 50   3.2  19,214
Peddy = 3.1   
 50   10 
(5 × 4 )2.07 × (6.8)
100

= 3.1 × 0.01256 × 493.2 × 192.14 = 3690 watts

• Eddy current loss as % of I2R loss winding (Alternate Formula 2 for copper winding)

( IT )2 k t 2
WE =
(100.H )2 × 1.07 × P 2 (6.9)

WE = eddy current loss as % of I2R loss (copper winding)

I = phase current
T = turns/phase
K = 15 for normal spiral winding, transposed helical winding, disc winding
t = conductor thickness (mm)
P = current density (Amps/sq mm)
H = effective electrical height of winding (mm)

6.2.2 STRAY LOSS IN BUSHING PLATE

1.5
 f 
WBp = 0.045 I 1.25   ( Watts) (6.10)
 50 

WBp = stray loss on tank plate from bushing current

I = rated current of transformer through bushing (Amps)
When brass or stainless steel is used,
1.5
 f 
WBp = 0.045 I 1.25   × 0.5 ( Watts) (6.11)
 50 
where
f = rated frequency
68 Power and Distribution Transformers

6.2.3 STRAY LOSSES IN FLITCH PLATE (TIE PLATE)

• If tie rods are provided, these losses are not applicable
• If flitch plates are provided, losses are calculated as per the following formula:

Losses of steel flitch Plate = 2.6 × 10 4 W 2.4 B 2 (6.12)

W = width of flitch plate (m)

B = flux density in flitch plate (Tesla)
For stainless steel flitch plate

Losses = 1.44 × 10 5 W 3 B 2 ( kW/m ) (6.13)

The flux density in flitch plate is calculated as below:

B1 µ1
= (6.14)
B2 µ 2

B1 = flux density in core (Tesla)

B2 = Flux density in flitch plate (Tesla)
µ1 = relative permeability of core ≈ 5000
µ 2 = relative permeability of flitch plate ≈ 200
If core flux density is 1.7 T, the flux density on 100 mm wide steel plate is calculated as follows:

µ2 1.7 × 200
B2 = 1.7 × = = 0.068 Tesla
µ1 5000
W = 0.1 m
Length of tie plate = 1.1 m

Losses in Tie Plate = 2.6 × 10 4 × 0.12.4 × 0.0682 kW/m

= 0.4786 kW/m

For full length of tie plate loss = 1.1 × 0.4786 = 0.526 kW

This loss is due to the flux in the core. The losses due to the leakage flux will have to be added
to the above.

6.2.4 CIRCULATING CURRENT LOSS IN CONTINUOUS DISC WINDING

( KR )4  1 − 2 + 4  × I 2 R
1 5 4
Pcc = 2.4 × 10 6 × (6.15)
180  n n 

where
R = radial size of the insulated turns
= n(t + ti) in m
n = number of turns in radial direction
t = thickness of one conductor
ti = insulation thickness between conductors

hc  t  f
k = 0.48 × 10 −4
hw  t + ti  p (6.16)
Calculation of Load Loss 69

TABLE 6.1
Increase in Circulating Current Loss %
Number of Conductors
in Radial Direction/Disc Proportionate Increase
1 0
2 0
3 0.4938
4 0.7031
5 0.8064

p = resistivity of conductor material

hc = total height of copper in axial direction
hw = total winding height in axial direction

 hc   t 
  and   are space factors in axial and radial directions
 w
h  t + ti 

Example

f = 50 Hz
t = 2.4 mm
ti = 0.7 mm
hc
= 0.8
hw
ρ = 2.1 ×10 −8 Ω /m ( for copper )

∴ R = 8 ( 2.4 + 0.7) × 10 −3 = 0.0248 m

 2.4  50
K = 0.48 × 10 −4 0.8 ×  = 1.843
 2.4 + 0.7  2.1 × 10 −8

n=8

(1.843 × 0.0248)4  1 − 2 + 4  × I 2 R loss

( )
1 5 4
Pcc = 2.4 × 10 6 ×
180 8 8

(
= 0.053 × I 2 R loss )

Increase of circulating current loss depends on number of conductors in radial direction as given
in Table 6.1.

6.2.5 EMPIRICAL FORMULA FOR TANK LOSS CALCULATION

Empirical formula–1

 IT  2 
Tank Loss = 6.3 ×   × 10 −6 × ( LVOD + HL Gap) × ( LV RD + 3 × HL Gap + HV RD 

 h  
(6.17)
70 Power and Distribution Transformers

Where,
h = winding axial height (Electrical) (mm)
I = phase current (A)
T = Number of turns / phase
All dimensions are in mm
For corrugated fin type (with corrugations all round), multiply the above by 0.70.

Empirical formula–2
1.5
 f  I 2T 2
W = N ×  × 10 −4 (6.18)
 50  H
W = tank loss (watts)
N = number of winding faces exposed to tank
(N = 6 if winding to tank clearance is almost equal all around)
(N = 5 if tap changer is on short side, etc.)
I = phase current
T = turns/phase
H = window height
HV OD = outer diameter of HV winding
C = clearance from HV to tank
f = frequency

Empirical Formula 3
Stray loss in tank due to axial component of leakage flux {unshielded steel tank}

WT = 15 A   K 2φ 2 × 10 −10 watts at 50 Hz
L
 P

= 1.3 ×15A   K 2φ 2 × 10 −10 watts at 60 Hz

L
 P

where
Area of leakage flux flow 2
A= m /m
Tank perimeter
Area of leakage flux flow is calculated as below:

• Area of leakage flux flow is the shaded area in Figure 6.1. Ignore the area occupied by
OLTC (if any) and any major additions to tank size other than normal.

FIGURE 6.1 Area of leakage flux flow.

Calculation of Load Loss 71

Outer Winding

FIGURE 6.2 Parameters for tank loss calculation.

• Tank perimeter is the effective perimeter of the tank inside dimensions (mm). The param-
eters for tank loss calculations are shown in Figure 6.2.

L = H + 2B (mm)
H = winding axial height (mm)
B = average distance from the tank to the inside edge of outer winding
P = tank perimeter (mm)
K = 0.6
ɸmax = leakage flux/limb lines
N = turns in untapped winding
I = rated current in untapped winding

1.78 I N
Bmax = × 10 −3 Tesla
H
Leakage flux in lines is calculated as below
The mean circumference of the flux path has calculated from the parameters shown in Figure 6.3

Mean circumference of flux path = π  LV ID + 2 × WLV + Wg 

LV ID = Inside diameter of LV
WLV = Radial depth of LV
Wg = HV-LV gap

ϕ = Bmax × 10 2 × π [ LVID + 2WLV + Wg ] ×  WLV + Wg + WHV 

1 1
(6.19)
2 2 

Bmax

WLV
LV ID Wg WHV

FIGURE 6.3 Windings and HL gap.

72 Power and Distribution Transformers

6.2.6 LOSS ON TRANSFORMER TANK DUE TO HIGH-CURRENT BUSBARS

Figure 6.4 shows the parameters for calculations
W = 300 I2 dn
W = loss on tank due to current carrying busbar (watts (W)/m2)
I = current through busbar (kilo Amperes)
d = distance between busbar and tank (mm)
n = −0.4 for mild steel
= −0.8 for non-magnetic steel
The area of the tank affected by the proximity of busbar current is calculated as 1.5 × L × w
{L = length of busbar exposed to tank and w = width of busbar}

d
Bus bar

Steel Tank

FIGURE 6.4 Busbar clearance.

Example

Current = 2000 A (I = 2)
Distance between tank and busbar = 100 mm
Tank material is mild steel (n = −0.4)

W = 300 × 22 × 100 −0.4

= 190 W/m 2

a. If the tank material is stainless steel

W = 300 × 22 × 100 −0.8

= 30 W/m 2

b. If clearance is 50 mm with mild steel tank

W = 300 × 22 × 50 −0.4

= 251W/m 2

c. If clearance is 200 mm

W = 300 × 22 × 200 −0.4

= 144 W/m 2

6.2.7 EMPIRICAL FORMULA FOR CALCULATING TOTAL STRAY LOSS

2
 IT   b+c
Ws = K   D a + (6.20)
 1000 h   3
The parameters of formula 6.20 are shown in Figure 6.5.
Calculation of Load Loss 73

b a c h

FIGURE 6.5 Core coil assembly.

Ws = stray loss (watts)

K = 25 up to 20 MVA
40 25–40 MVA
50 50–100 MVA
a = H–L gap
b = radial depth of LV winding
c = radial depth of HV winding
D = mean diameter of leakage flux
(mean diameter of HL gap)
h = distance between core clamp flanges (if steel core clamp is used)
= window height (mm) (if steel core clamp is not used)
(All dimensions are in meters)

• If wooden core clamp is used and if top clamp ring is not steel, the stray loss is 70% of the
above value.
• If tank shield is used, the stray loss is 50% of the above value.

6.3 CALCULATION OF LOAD LOSS

W = 1.015 ( I 2 RLV + I 2 RHV + WsLV + WsHV + WB + WL + WCir + WContact + WMisc ) watts (6.21)

where
I 2 RLV = LV-I2 R loss (watts)
I 2 RHV = HV – I2 R loss (watts)
WsLV = LV eddy current loss (watts)
WsHV = HV eddy current loss (watts)
WB = bushing plate loss (watts)
WL = lead wire loss (watts)
W T = tank loss (watts)
WCir = circulating loss in parallel conductors
WContact = contact losses of busbar joints of high-current leads (watts)
(to be considered for current above 500 A)
Wmisc = Miscellaneous stray loss
74 Power and Distribution Transformers

Note
1. If LV leads carrying current in excess of 1000 A run in parallel with tank at close proxim-
ity such that the field strength on steel is in excess of 25 A/cm, the tank losses and miscel-
laneous losses will be higher than the above value.
2. Eddy current and stray loss calculations as per the section are not directly applicable for
transformers where harmonics are present. Examples are transformers for rectifier appli-
cation, drives and inverter-fed solar application; in such cases, the effect of harmonics on
eddy current and stray loss will have to be calculated.
3. Winding eddy current calculation and tank loss calculations as above assume that the
leakage flux density distribution is primarily axial where the ampere-turns are balanced.
If the ampere-turns are imbalanced due to the effect of asymmetry of windings, body tap-
ping especially from disc winding, the effect of radial component of leakage flux is to be
calculated. The flux density distribution (both axial and radial) will have to be calculated
for accurate calculations. When the asymmetry is high and accurate results are required,
FEM will be necessary.
4. For guaranteeing the losses, a safety margin based on previous test result is to be considered
in the design. This safety margin will change from manufacturer to manufacturer depending
on the production method and process stability. The average ratio of calculated value to the
measured results of several transformers will have to be calculated to fix a reliable factor.

Tables 6.2–6.4 show the measured results of load loss of a number of 3.15 MVA, 5 MVA and 8
MVA, 33/11 kV ONAN transformers.

TABLE 6.2
Measured Losses of 3.15 MVA Transformers
Load Stray Stray Load
LV HV Total I2R Total I2R Loss Loss Loss Loss
HV Resistance/ I2R LV I2R Amb. Losses Losses at at at at
Sl. Resistance/ Phase Losses Losses Temp at Amb. at 75°C Amb. Amb. 75°C 75°C
No Phase (Ω) (m/Ω) (W) (W) °C (W) (W) (W) (W) (W) (W)
1. 3574 94.5 10855 7749 28.5 18604 21887 20620 2016 1714 23601
2. 3479 94.4 10566 7741 29.0 13307 21497 20690 2016 2029 23601
3. 3568 94.57 10837 7755 24.0 18520 22167 20141 1621 1354 23521
4. 3547 94.17 10773 7722 25.5 18495 22009 19998 1503 1263 23272
5. 3463 93.93 10518 7701 26.5 19219 21593 20315 2096 1768 23366
6. 3486 94.97 10588 7788 27.5 18376 21701 20400 2024 1714 23415
7. 3479 95.23 10566 7809 26.5 18375 21783 20395 2020 1704 23487
8. 3446 93.57 10466 7673 26.5 18139 21717 20402 2263 1908 23626
9. 3566 96.48 10831 7912 30.5 18743 21884 20906 2063 1767 23651
10. 3589 95.95 10900 7865 27.0 15765 22206 20736 1968 1663 23869
11. 3645 96.55 11070 7917 31.5 18987 22086 20785 1798 1546 23632
12. 3573 93.3 10852 7651 25.5 18539 22062 20101 1562 1313 23375
13. 3540 95.33 10752 7817 26.0 18569 22055 20229 1660 1398 23453
14. 3462 92.63 10515 7596 20.5 18111 21974 20202 2081 1723 23697
15. 3537 94.93 10742 7785 25.0 18527 22090 20507 1980 1661 23751
16. 3553 97.93 10791 8031 26.0 18822 22356 20876 2054 1720 24085
17. 3481 93.5 10572 7667 25.0 18239 21746 20332 2093 1755 23501
18. 3579 94.97 10870 7788 32.0 18658 21663 20616 1958 1686 23349
19. 3664 96.87 11123 7944 32.0 19072 22143 20345 1273 1096 23239
Calculation of Load Loss 75

TABLE 6.3
Measured Load Loss of 5 MVA Transformer
Load Stray Stray Load
LV HV Total I2R Total I2R Loss Loss Loss Loss
HV Resistance/ I2R LV I2R Amb. Losses Losses at at at at
Sl. Resistance/ Phase Losses Losses Temp at Amb. at 75°C Amb. Amb. 75°C 75°C
No Phase (Ω) (m/Ω) (W) (W) °C (W) (W) (W) (W) (W) (W)
1. 2073 54.97 15901 11357 35.0 27258 31296 32555 5297 4614 35910
2. 2103 58.37 16093 12060 38.5 28153 31910 33313 5160 4552 36462
3. 2088 57.93 15978 11969 38.5 27947 31677 33510 5563 4903 36585
4. 2018 59.53 15442 12299 36.5 27844 31792 32779 4935 4322 36114
5. 2009 57.00 15373 11777 34.5 27505 31230 32014 4864 4228 35459
6. 2032 57.87 15549 11956 39.5 27505 31062 32613 5105 4523 35585
7. 1969 56.50 15067 11673 36.0 26740 30588 31810 5070 4432 35020
8. 1971 56.90 15082 11673 33.5 26755 30990 32103 5345 4632 35522
9. 2072 61.67 15855 12742 42.0 28597 32004 33301 4704 4203 36207
10. 2073 56.63 15863 11700 35.5 27563 31586 32929 5365 4682 36270
11. 1988 57.30 15212 11839 37.5 27051 30774 32136 5035 4470 35248
12. 1982 57.93 15167 11969 38.5 27136 30757 32273 5137 4532 35280
13. 1994 57.57 15259 11894 33.0 27153 31408 32203 5050 4366 35774
14. 1950 56.45 14922 11663 33.0 26635 30809 31698 5063 4377 35186
15. 19735 56.71 15102 11717 33.5 26819 30964 31838 5019 4347 35311
16. 19805 56.67 15155 11709 33.0 26864 31074 31858 4994 4317 35391
17. 1975 56.68 15113 11711 34.0 26824 30912 31572 4751 4123 35035
18. 2061 57.68 15771 11917 34.0 27688 31905 32512 4824 4186 36094
19. 1965 56.85 15037 11746 34.0 26883 30980 31554 4671 4053 35033
20. 1935 55.22 14807 11409 27.0 26067 30943 31923 5856 4949 35792
21. 1932 56.12 14784 11595 30.0 26479 30975 31918 5439 4649 35624
22. 2004 56.18 15335 11607 31.0 26942 31399 32319 5377 4613 36013

In Table 6.5, the averages of measured losses of the three ratings and the ratio of calculated
values to the measured figures are compared. The calculation methods of stray losses in this section
give actual results which are very close.
76 Power and Distribution Transformers

TABLE 6.4
Measured Load Loss of 8 MVA Transformer
Load Stray Stray Load
LV HV Total I2R Total I2R Loss Loss Loss Loss
HV Resistance/ I2R LV I2R Amb. Losses Losses at at at at
Sl. Resistance/ Phase Losses Losses Temp at Amb at 75°C Amb. Amb. 75°C 75°C
No Phase (Ω) (m/Ω) (W) (W) °C (W) (W) (W) (W) (W) (W)
1. 1.1295 32.19 22127 17026 36.5 38796 44297 48761 9965 8727 53024
2. 1.0765 31.575 21079 16701 31.5 37851 44029 47361 9510 8176 52205
3. 1.1015 31.407 21578 16612 27.0 38190 45187 47169 8979 7589 52776
4. 1.0975 31.657 21500 16744 28.5 38244 44993 47858 9614 8172 53165
5. 1.09955 32.095 21540 16976 31.5 38516 44803 48630 10114 8695 53498
6. 1.09315 31.775 21415 16807 31.5 38222 44461 48925 10706 9204 53665
7. 1.09055 31.89 21364 16867 30.0 38231 44723 49306 11075 9467 54190
8. 1.0788 31.297 21133 16554 28.0 37687 44422 49999 12312 10445 54567
9. 1.08165 31.532 21189 16678 30.0 38155 44634 50578 12423 10620 55254
10 1.07775 31.03 21113 16412 31.5 37880 44063 47686 9806 8430 52493
11. 1.0806 31.238 21169 16522 32.0 37762 43844 46411 8649 7449 51293
12. 1.08485 31.35 21252 16582 29.5 37906 44427 46905 8990 7679 52105
13. 1.08235 31.563 21203 16694 29.5 37542 44000 48267 10725 9151 53151

TABLE 6.5
Comparison of Measured and Calculated Values
Calc.
winding
Eddy Calc. Other Total Calc. Total Measured Standard Deviation
current Stray Stray Stray Losses (W) Calculated/ of Measured Value
kVA Losses (W) Losses (W) Losses (W) (Average Value) Measured for Stray Losses (W)
3150 1219 381 1600 1621 0.987 306
5000 3933 588 4521 4432 1.02 342
8000 8080 908 8988 8754 1.03 969