3
Original scientific paper
 MIDEM Society
Investigations on the Influence of Selected Factors 
on Thermal Parameters of Impulse-Transformers
Krzysztof Górecki, Krzysztof Górski, Janusz Zarębski
Gdynia Maritime University, Department of Marine Electronics, Gdynia, Poland
Abstract: In the paper the results of experimental investigations illustrating the influence of the selected factors on parameters of the 
thermal model of the transformer are presented. The form of this model and the applied method of measurements of the transformer 
structural components’ self and mutual transient thermal impedances are described. The influence of the selection of material of the 
core, its geometrical dimensions, spatial orientation, shape of the core, frequency of the primary winding current in the transformer 
and power lost  in this element on the considered thermal parameters of the transformer are discussed. An analytical formula 
describing the dependence of the considered transient thermal impedances on the internal temperature of the windings is proposed 
and verified experimentally. 
Keywords: thermal parameters;selfheating; impulse-transformers
Vplivi izbranih parametrov na termične parametre 
impulznih transformatorjev
Izvleček: Članek prikazuje rezultate eksperimentalnih raziskav vplivov izbranih faktorjev na parameter termičnega modela 
transformatorja. Opisan je model, uporabljene merilne metode in vzajemne tranzientne termične impedance. V smislu termičnih 
parametrov so obravnavni: izbira materiala jedra, geometrija, orientiranost, frekvenca toka primarnega navitja in izgube. Predlagana in 
eksperimentalno preverjena je analitična enačba odvisnosti tranzientne termične impedance od temperature navitja. 
Ključne besede: termični parametri; lastno segrevanje; impulzni transformator
* Corresponding Author’s e-mail:  k.gorecki@we.am.gdynia.pl
Journal of Microelectronics, 
Electronic Components and Materials
Vol. 47, No. 1(2017), 3 – 13
1 Introduction
Impulse-transformers are commonly used in switched-
mode power electronic converters [1, 2, 3, 4]. The con-
sidered elements have a simple construction - they 
consist of the ferromagnetic core and windings. The 
properties of both these components influence tem-
perature, the change of which causes changes of the 
value of technical parameters of the core and windings 
[4, 5]. Particularly, when the core temperature is higher 
than the Curie temperature, magnetic permeability 
of the ferromagnetic core decreases to the value near 
permeability of free air, and the transformer practically 
does not transfer energy from the input to the output. 
In turn, the excess of the admissible temperature of 
windings can cause destruction of isolation of winding 
wires [4, 5], leading to the short-circuit of turns. 
The temperatures of the core and windings of the 
transformer during its operation are higher than the 
ambient temperature as a result of self-heating in the 
core and in the winding [5-14], as well as mutual ther-
mal coupling between them. Calculating values of tem-
perature of structural components of the transformer 
demands using the thermal model. Such a model can 
have the form of the detailed model, making it possi-
ble to calculate time-space distribution of temperature 
in the considered element [9-12, 15], or the compact 
model - making possible calculations of waveforms of 
temperature of this element [6, 13, 14] or the selected 
structural components of the transformer [5, 7]. In the 
paper [13] many results of calculations of temperature 
distribution in a planar transformer obtained with the 
use of the finite element method (FEM) are presented. 
In these calculations uniform distributions of the pow-
4
er dissipated in the core and in the windings were as-
sumed. The results of measurements shown in this pa-
per prove that this assumption is fulfilled. 
In the case of the use of the compact thermal model it 
is indispensable to measure the transformer’s self and 
mutual transient thermal impedances. Thermal models 
presented in papers [5, 6, 7, 8, 13] are linear models, i.e. 
transient thermal impedances in these models neither 
depend on constructional factors nor on the power 
lost in the transformer. Yet, in papers [16, 17, 18] (for in-
stance) it was shown that thermal parameters of semi-
conductor devices strongly depended, among other 
things, on the power dissipated in them, the type of the 
case, or the manner of mounting the considered de-
vice. It can be expected that a similar influence will be 
observed in the case of magnetic elements, to which 
the transformer belongs.
In the paper the method to measure thermal param-
eters existing in the thermal model of the transformer 
and the results of measurements of these parameters 
illustrating the influence of the selected factors on ef-
ficiency of transformers cooling are presented. Trans-
formers with ring cores considered in the present pa-
per are characterized by uniform distribution of the 
magnetic force and magnetic flux density, similarly as 
transformers considered in paper [13]. Therefore, distri-
bution of power density in the core is also uniform.
In Section 2 the form of the thermal model of the trans-
former is presented and the transformer’s self and mu-
tual transient thermal impedances are defined. Section 
3 contains a description of the measuring method to 
measure the transformer’s self and mutual transient 
thermal impedances. In Section 4 the results of meas-
urements of thermal parameters of the selected con-
structions of transformers are presented and the analyt-
ical formula describing the influence of the dissipated 
power on the considered parameters is proposed and 
experimentally verified.
2 Thermal Model of Transformer 
The thermal model describes the dependence of the 
internal temperature of the electronic component on 
the power emitted in it. In the case of commonly used 
compact thermal models this dependence can be de-
scribed by means of the convolution integral [19, 20, 
21]. As it is shown, among other things, in papers [5, 7, 
22], in order to describe correctly thermal properties of 
magnetic elements it is indispensable to take into ac-
count both self-heating in the core and in windings, as 
well as mutual thermal coupling between them. Hence, 
the temperature of k-th winding can be described by 
the following formula
 
( ) ( ) ( )
( ) ( )∫
∑∫
−⋅+
−⋅+=
=
t
cthck
n
j
t
jthwkjawk
dtptZ
dtptZTtT
0
1 0
'
'
ττ
ττ +
 (1)
where Ta denotes the ambient temperature, k repre-
sents the name of the winding, pk(t) denotes the pow-
er dissipated in k-th winding of the transformer and 
Z’thwkj(t) represents the time derivative of the devices’ 
self (for k = j) or mutual (for k K j) transient thermal 
impedance between the windings of the transformer, n 
– the number of windings, Z’thck(t) – the time derivative 
of mutual transient thermal impedance between the 
core and the k-th winding, whereas pc(t) is the power 
dissipated in the core. 
In turn, the temperature of the core is given by the fol-
lowing formula 
 
( ) ( ) ( )
( ) ( )∫
∑∫
−⋅+
−⋅+=
=
t
cthc
n
j
t
jthcjac
dtptZ
dtptZTtT
0
1 0
'
'
ττ
ττ +
 (2)
where Z’thc(t) is the time derivative of the core’s self 
transient thermal impedance. 
The core’s self or mutual transient thermal impedance 
can be modelled using the classical formula [19, 23, 24] 
( ) 









−⋅−⋅= ∑
=
N
k thk
kthth
taRtZ
1
exp1
τ
 (3)
where Rth is thermal resistance, tthk denotes k-th ther-
mal time constant, ak is the ratio factor corresponding 
to this time constant, whereas N is the number of ther-
mal time constants. 
The presented thermal model, given by equations (1 – 
3) of the transformer can be described as the RC electri-
cal analog, like the linear thermal model presented in 
[5]. In the real situation, where the parameters existing 
in Eq. (3) depend on the value of the dissipated power, 
the nonlinear thermal model is needed. The simple 
form of this model for the electronic device non-cou-
pled thermally with any other device is proposed in 
[17]. In this model, the controlled voltage and current 
sources are used instead of RC elements. 
K. Górecki et al; Informacije Midem, Vol. 47, No. 1(2017), 3 – 13
5
3 Method to Measure Thermal 
Parameters of the Transformer 
The transformer’s self and mutual transient thermal im-
pedances of the selected magnetic devices were meas-
ured with the use of the method described in papers 
[25, 26]. This method is realised by means of the meas-
urement set presented in Fig.1. 
Figure 1: Measurement set for measuring thermal pa-
rameters of the transformer [26] 
The measurements are realised in two steps. The first 
step needs stimulations of the primary winding with 
a current step and the measurement of waveforms of 
the windings temperature and the core temperature 
by means of thermo-hunters (pyrometers with IR sen-
sors) until the thermally steady-state is achieved. In 
the considered measurement set the thermo-hunters 
of the type PT-3S by Optex are used [27]. The range of 
the measured temperature is from 0 to 200 oC and the 
resolution is equal to 0.1oC. These thermo-hunters are 
situated in the distance of 25 mm from the measured 
transformer. The area, on which the temperature is 
measured, has the form of a circle with the diameter 2.5 
mm. During measurements it was assumed that emis-
sivity of all components of the considered transformers 
has the constant value equal to 0.95. This value is close 
to typical values of emissivity of materials used to con-
struct transformers.
As it results from the Authors’ investigations presented 
in [26], in the area of a transformer, in which the wind-
ings are located, practically uniform distribution of the 
temperature is observed. Similarly, temperature distri-
bution of the core is quasi-uniform. Therefore, averag-
ing the temperature value in a circle of the diameter 
equal to 2.5 mm does not cause a visible measurement 
error. 
The waveforms of the core temperatures TC(t) and of 
the winding temperature TW(t) while heating the trans-
former are registered by the computer PC coupled 
with two thermo-hunters. By means of the voltme-
ter and the ammeter the values of the voltage on the 
primary winding V1 and the current of this winding I1 
are measured at the steady-state. The results of these 
measurements are used to calculate transient thermal 
impedance of the winding ZthW(t) and mutual transient 
thermal impedance between the core and the wind-
ings ZthWC(t) with the following formulas 
( ) ( )
11 IV
TtTtZ aWthW ⋅
−=     (4)
( ) ( )
11 IV
TtTtZ aCthWC ⋅
−=     (5)
where Ta is the ambient temperature. 
In the second step, the primary winding of the trans-
former is stimulated by a sinusoidal signal of frequency 
fs, whereas the temperature of the core is measured by 
the thermo-hunter. When the steady state is obtained, 
the hysteresis loop of the magnetising characteristic 
B(H) of the core is measured using the oscilloscope and 
next - transmitted to the computer (PC). 
The waveforms of the magnetic force H(t) and flux den-
sity B(t) are calculated using the following formulas
 ( ) ( )
Fe
C
Sz
tuCRtB
⋅
⋅⋅
=
2
    (6)
 ( ) ( )
Fel
tiztH 11 ⋅=     (7)
in which R and C denote resistance of the resistor and 
capacitance of the capacitor used in  Fig.1, respectively, 
z1 and z2 are numbers of turns in primary and second-
ary windings, respectively, SFe – the cross-section area 
of the core, lFe – magnetic path in the core, uC – voltage 
on the capacitor C, whereas i1 – the current of the pri-
mary winding. 
The area SH of the obtained hysteresis loop B(H) is given 
by following formula
∫= BdHSH      (8)
This integral is calculated using the Excel software and 
the method of numerical integration. The used values 
of frequency fs are in the range from 1 kHz to 100 kHz. 
Next, in the moment t = 0 the power supply of the pri-
mary winding is switched off and waveforms of the 
core temperature TC(t) are measured until the steady 
state is obtained. The transient thermal impedance of 
the core ZthC(t) is calculated using the following formula 
 
 
V
Thermo-hunters
A
Oscilloscope
OX
OY
Rw
R
C
Transformer
Vin
PC
K. Górecki et al; Informacije Midem, Vol. 47, No. 1(2017), 3 – 13
6
 ( ) ( ) ( )
HsC
CC
thC SfV
tTtTtZ
⋅⋅
−== 0    (9)
where VC represents volume of the core. In Eq. (9) only 
the power dissipated in the core is taken into account, 
whereas the power dissipated in the windings is omit-
ted. This is justified, if the value of the power dissipated 
in primary and secondary windings is much smaller 
than the power dissipated in the core. Such conditions 
are fulfilled when 
2
2
21
2
1 RIRIVfS CH ⋅+⋅>>⋅⋅                  (10)
where I1 and I2 are RMS values of primary and second-
ary windings currents, whereas R1 and R2 are resistanc-
es of these windings.
4 Investigation Results 
Using the method presented in Section 3, measure-
ments of thermal parameters of transformers contain-
ing ferromagnetic cores made of the powdered iron 
(RTP), the ferrite (RTF) and the nanocrystalline cores 
(RTN) were performed.
The first series of measurements was performed for 
ring cores of the dimensions RTP 26.9x14x11 made 
of material T106-26, RTN-26x16x12 made of material 
M-070, RTF-25x15x10 made of material F-867, called in 
the further part of this paper small ring cores. The sec-
ond series of measurements was made for cores made 
of the same ferromagnetic materials, but of the big-
ger dimensions: RTP 39.9x24.1x14.5 made of material 
T157-26, RTF 40x24x16 made of material F-867, called 
in the further part of this paper large ring cores and 
for the pot core B65701 -T1000-A48 of the diameter 30 
mm and heights 19 mm made of material N48. In Ta-
ble 1 the values of basic parameters of the considered 
ferromagnetic materials are collected. In this table Bsat 
denotes saturation flux density at the magnetic field 
strength Hsat, HC is the coercive field strength, TC – Curie 
temperature, whereas PV – relative core losses. 
In marking ring cores each number means the outside 
diameter, the inside diameter and the height of the 
core expressed in millimetres, respectively. The view 
of the ring core with its dimensions is shown in Fig. 2, 
whereas the view of the pot core is shown in Fig. 3. The 
temperature of windings of the transformer with the 
pot core can be measured using a hole in this core.  
Figure 2: View of the ring core with its dimensions  
On small ring cores two windings containing 22 turns 
were wound with copper wire in the enamel of the di-
ameter 0.8 mm. In turn, on large ring cores two wind-
ings containing 30 turns of copper wire in the enamel 
of the diameter 0.8 mm were wound. The transformer 
with the pot core contains two windings made with the 
same wire consisting of 22 turns. The views of trans-
formers with the ring core and with the pot core are 
shown in Fig. 4 and Fig. 5, respectively. 
As it is visible, the distance between turns in each wind-
ing is not constant. The temperature of these windings 
is measured for the area in which this distance is the 
smallest and much smaller the measuring spot of the 
used thermo-hunter.  
In the following figures the results of measurements 
of self and mutual transient thermal impedances Zth(t)
of the selected constructions of impulse transformers 
(solid lines) and approximation of these curves (dashed 
lines) with Eq. (3) are presented. The values of param-
eters of the model described with the equation (3) are 
estimated with the use of the method proposed in [5]. 
The value of Rth is estimated by averaging the wave-
Table 1: Values of basic parameters of the considered ferromagnetic materials
parameter T106-26 and T157-26 M-070 F-867 N48
manufacturer Micrometals Magnetec Feryster Epcos
Bsat [T] 1.38 1.2 0.5 0.42
Hsat [A/m] 19.9x103 590 966 1200
HC [A/m] 440 9 75 26
TC [oC] 750 600 215 170
PV [kW/m3] 180 @f=100 kHz 800 @f=100 kHz 400 @f=100 kHz -
 
K. Górecki et al; Informacije Midem, Vol. 47, No. 1(2017), 3 – 13
7
form of the considered transient thermal impedance 
in the steady state (typically for the last 100 seconds), 
whereas the values of the parameters ai and thermal 
time constants tthi are determined by the least square 
method. Starting from the longest thermal time con-
stant through approximation based on the formula [5]
 
( ) 















−⋅−−= ∑
−
=
1
1
exp)(1ln
i
j thj
j
th
th
i
ta
R
tZty
τ
    (11) 
the linear function given by
Figure 3: Cross-sections of the pot core B65701 -T1000-
A48 with its dimensions  
Figure 4: View of the transformer with the ring core
Figure 5: View of the transformer with the pot core
)ln()( i
thi
i a
tty +−=
τ                   (12) 
is used.
Attention is focused on the parameters describing self-
heating phenomenon in the core (ZthC(t)) and in the 
winding (ZthW(t)), as well as mutual thermal coupling 
between the core and the windings (ZthWC(t)). The mu-
tual thermal couplings between the windings are not 
taken into account.  
The values of parameters Rth, ai, tthi approximating the 
selected waveforms of the investigated transformers’ 
self and mutual transient thermal impedances are col-
lected in Tables 2 - 5.
Table 2: Parameters values of transient thermal imped-
ances in transformers with the small ring core RTP situ-
ated horizontally
parameter ZthW(t) ZthWC(t) ZthC(t) 
Rth [K/W] 22.15 18.12 25.39
a1 0.664 0.758 0.925
a2 0.206 0.242 0.068
a3 0.13 0.007
tth1 [s] 661.2 710.5 702.1
tth2 [s] 134.1 259 283
tth3 [s] 10 12.8
 
 
 
 
 
 
K. Górecki et al; Informacije Midem, Vol. 47, No. 1(2017), 3 – 13
8
Table 3: Parameters values of transient thermal imped-
ances in transformers with the big ring core RTP
orientation horizontally vertically
parameter ZthW(t) ZthWC(t) ZthW(t) ZthWC(t)
Rth [K/W] 13.5 11.1 11.1 9.01
a1 0.6 0.774 0.742 1
a2 0.192 0.226 0.118
a3 0.148 0.09
a4 0.06 0.05
tth1 [s] 1062.5 1078.4 680.1 654.9
tth2 [s] 470.5 459.4 114.8
tth3 [s] 30.9 19.4
tth4 [s] 5.64 14.7
Table 4: Parameters values of transient thermal imped-
ances in transformers with the small ring core RTF 
parameter ZthW(t) ZthWC(t) ZthC(t) 
Rth [K/W] 24.88 14.26 11.98
a1 0.651 1 0.92
a2 0.255 0.08
a3 0.094
tth1 [s] 474.1 449.5 483.4
tth2 [s] 126.8 53.1
tth3 [s] 9
Table 5: Parameters values of transient thermal imped-
ances in transformers with the big ring core RTF
orientation horizontally vertically
parameter ZthW(t) ZthWC(t) ZthW(t) ZthWC(t)
Rth [K/W] 14 7.59 13.42 6
a1 0.595 1 0.605 1
a2 0.225 0.275
a3 0.13 0.118
a4 0.05 0.002
tth1 [s] 918 1050.5 792.8 800
tth2 [s] 224.8 152.8
tth3 [s] 23.5 11.18
tth4 [s] 5.67 5.67
Comparing the data collected in the mentioned tables 
one can observe that the description of the considered 
transient thermal impedances demands the use of a 
different number of thermal time constants tthi. In the 
case of large ring cores RTP and RTF up to 4 thermal 
time constants appear in the description ZthW(t), while 
in the description ZthWC(t) for both ferrite cores RTF - just 
only one thermal time constant. In all the considered 
cases the prevailing meaning has the longest thermal 
time constant tth1. The corresponding to it weight- co-
efficient a1 assumes the values in the range from 0.595 
to 1. In turn, the values th1 are in the range from 471 s 
(for the small ring core RTF) to 1078 s for the large ring 
core RTP.
The presented below results of investigations illustrate 
the influence of geometrical dimensions of the core 
(Fig.6), its spatial orientation (Fig. 7), shape of the core 
(Fig. 8), current of the primary winding (Fig. 9), mate-
rial the core is made of (Fig. 10) and frequency of the 
current on the primary winding (Fig. 7) on waveforms 
of the transformers’ self and mutual transient thermal 
impedances. 
In Fig. 6 the measured and modelled with the Eq. (3) 
waveforms of transient thermal impedance of the 
winding ZthW(t) and mutual transient thermal imped-
ance between the winding and the core ZthWC(t) for 
transformers containing ring cores RTP (Fig.6a) or ring 
cores RTF (Fig. 6b) of different dimensions are present-
ed. The measurements were performed at the stimula-
tion of the primary winding with the direct current of 
the value equal to 9 A. 
Figure 6: Measured (solid lines) and modelled (dashed 
lines) waveforms of transient thermal impedances in 
transformers with ring cores RTP (and) and RTF (b) of 
different dimensions at the stimulation with the direct 
current of the primary winding and the horizontal ori-
entation of the core
 
 
 
0
5
10
15
20
25
1 10 100 1000 10000
t [s]
Z t
hW
(t)
 Z
th
W
C(
t) 
 [K
/W
] ZthWC(t) 
ZthW(t) RTP core situated horizontally
ZthW(t)  
ZthWC(t) 
`
small ring core
`
big ring core
 
 
 
0
5
10
15
20
25
30
1 10 100 1000 10000
t [s]
Z t
hW
(t)
 Z
th
W
C(
t) 
 [K
/W
]
RTF cores situated horizontally
ZthW(t) 
ZthW(t)  
ZthWC(t) ZthWC(t) 
ZthW(t)  
ZthWC(t) 
small ring core 
pot core 
 big ring core 
a)
b)
K. Górecki et al; Informacije Midem, Vol. 47, No. 1(2017), 3 – 13
9
As one can notice in Fig. 6, the process of heating the 
core and the winding of the transformer with the con-
sidered cores runs slowly. The time indispensable to 
obtain the steady state exceeds 3000 s for the small 
ring core, 4000 s - for the pot core and 5000s - for the 
large ring core. The value of transient thermal imped-
ances ZthW(t) is about 40% greater for the transformer 
with the small ring core RTP than for the transformer 
with the large ring core RTP (Fig. 6a). In the case of the 
transformer with the core RTF (Fig. 6a) one obtained 
greater by about 10% values of the considered param-
eter for transformers with ring cores made of the same 
material, while the transformer with the pot core shows 
average values of the time needed to settle the course 
between the values of this parameter corresponding 
to different measurements of ring cores. Waveforms 
of mutual transient thermal impedance between the 
winding and the core ZthWC(t) are late in relation to 
waveforms ZthW(t) by more than 20 s, and values ZthWC(t) 
at the steady-state are smaller than the value ZthW(t) at 
the steady-state. For transformers with cores RTP this 
difference is about 15%, and in the case of transform-
ers with cores RTF these differences are bigger and are 
about 50% for ring cores and about 30% for the sot 
core. For all the considered waveforms the very good 
agreement between the results of measurements and 
the calculations performed with the use of the consid-
ered model is obtained. 
In Fig. 7 waveforms ZthW(t) and ZthWC(t) for transform-
ers with large ring cores RTP (Fig. 7a) and RTF (Fig.7b) 
placed both horizontally and vertically at the stimula-
tion of the primary winding with the direct current of 
the value 9 A are presented. 
As one should expect the waveforms ZthW(t) and ZthWC(t) 
obtained in horizontal orientations lie above the wave-
forms obtained for the transformers situated vertically. It 
is the result of more efficient convection of heat for ele-
ments situated vertically, similarly to the situation with 
transistors mounted on any heat-sink situated vertically 
[9]. The vertical orientation of the considered elements 
quickens the perpendicular air flow along the sides of 
these elements, because the length of these sides is 
greater at this arrangement of the investigated element. 
The values of the considered transient thermal imped-
ances in the steady-state in the horizontal and vertical 
orientation differ from each other by about 15% for the 
transformer with the core RTP and by about 10% for the 
transformer with the core RTF. It is visible that the consid-
ered model approximates well the measured waveforms 
ZthW(t), whereas it is possible to observe divergences 
between the measured and approximated waveforms 
ZthWC(t), especially for small values of time t. 
In Fig.8 the influence of the selection of material of the 
ferromagnetic core on waveforms ZthW(t) and ZthWC(t) of 
transformers containing the small (Fig. 8a) or large (Fig. 
8b) ring core is illustrated. The measurements were 
performed at the stimulation of the primary winding 
with the direct current of the value 9A. 
As it can be noticed in Fig.8a, heat removal from the 
transformer containing the ferrite core (RTF) is the least 
efficient, while transformers with the powder core 
(RTP) and nanocrystalline core (RTN) have almost iden-
tical waveforms ZthW(t). It is worth noticing that dissipa-
tion of power of the same value in the winding causes 
a considerably higher temperature increase of the core 
RTP than the remaining cores, considered in this paper. 
This results from the greater value of thermal conduct-
ance of this material, which causes more efficient re-
moval of heat generated in the winding through the 
core. Therefore, in the transformer with this core con-
siderably smaller differences appear between tem-
peratures of the winding and of the core than for the 
remaining considered transformers. The qualitatively 
similar results were obtained for transformers contain-
ing the large ring core. In this case, increases of the 
Figure 7: Measured (solid lines) and modelled (dashed 
lines) waveforms of transient thermal impedances in 
the transformer with the core RTP (and) and RTF (b) at 
the stimulation with the direct current in the vertical 
and horizontal orientation of transformers
 
 
 
0
2
4
6
8
10
12
14
16
1 10 100 1000 10000
t [s]
Z t
hW
(t)
 Z
th
W
C(
t) 
 [K
/W
]
ZthW(t) 
RTP big ring core
core situated horizontally
core situated vertically
ZthWC(t) 
 
 
 
0
2
4
6
8
10
12
14
16
1 10 100 1000 10000
t [s]
Z t
hW
(t)
 Z
th
W
C(
t) 
 [K
/W
]
RTF big ring core
ZthW(t) 
ZthWC(t
)
core situated horizontally
core situated vertically
a)
b)
K. Górecki et al; Informacije Midem, Vol. 47, No. 1(2017), 3 – 13
10
winding and the core temperatures of the transformer 
with the core RTP differ only just by about 20%, and for 
the transformer with the core RTF – up to about 60%.
In Fig. 9 the influence of the current flowing through 
the primary winding of the transformer on waveforms 
of transient thermal impedance of the winding and 
mutual thermal impedance between the winding and 
the core was illustrated. The measurements were per-
formed for the transformer with the large ring core RTP 
situated horizontally for the current equal 7.35A and 
9.1A, respectively. 
As one can notice, with the current of the primary 
winding of the value 9.1A the values ZthW(t) and ZthWC(t) 
are by about 8% smaller than with the current equal to 
7.35A. The improvement of efficiency of cooling with 
an increase of the current of the winding results from 
an increase in the value of the power dissipated in this 
element, which causes a temperature rise of the wind-
ing. In turn, the temperature rise of the winding causes 
an increase in efficiency of heat convection. 
Fig. 10 illustrates the influence of the shape of the 
waveforms ZthC(t) of the transformer with the core RTF 
at the stimulation of the primary winding with the si-
nusoidal signal. In this case the power dissipated in the 
winding is negligible in relation to the power dissipat-
ed in the core. 
Figure 10: Measured (solid lines) and modelled 
(dashed lines) waveforms of transient thermal imped-
ance of the core of transformers with the core RTF at 
the stimulation with the sinusoidal current
In Fig. 10 distinct differentiation of the obtained wave-
forms ZthC(t) is visible. The large ring core assures the 
most efficient cooling and the small ring core - the least 
efficient. The average values ZthC(t) are obtained for the 
transformer with the pot core. The obtained values 
of the considered transient thermal impedance differ 
from each other even four times. 
In Fig. 11 the influence of frequency of the signal stimu-
lating the primary winding of the transformer with the 
large ring core RTP on the waveforms ZthC(t) and ZthCW(t) is 
illustrated. Investigations were made for the transformer 
placed horizontally at two values of frequency equal to 
25kHz and 75kHz, respectively. It is visible that an increase 
Figure 8: Measured (solid lines) and modelled (dashed 
lines) waveforms of transient thermal impedances of 
transformers with the small (a) or large (b) ring core 
made of different materials
 
 
 
0
5
10
15
20
25
30
1 10 100 1000 10000
t [s]
Z t
hW
(t)
 Z
th
W
C(
t) 
 [K
/W
]
 RTF
 RTP 
 RTN 
small ring core situatead horizontally
ZthW(t) 
ZthWC(t) 
 RTF
 
 
 
0
2
4
6
8
10
12
14
16
1 10 100 1000 10000
t [s]
Z t
hW
(t)
 Z
th
W
C(
t) 
 [K
/W
]
big ring core situatead horizontally
ZthW(t) 
ZthWC(t) 
RTP
RTF
a)
b)
Figure 9: Measured (solid lines) and modelled (dashed 
lines) waveforms of transient thermal impedances of 
transformers with the large ring core RTP at the stimu-
lation with the direct current of different values
 
 
0
2
4
6
8
10
12
14
16
1 10 100 1000 10000
t [s]
Z t
hW
(t)
 Z
th
W
C(
t) 
 [K
/W
]
big ring core RTP situated horizontally
ZthW(t) 
ZthWC(t) 
I1 = 7.35 A
I1 = 9.1 A
 
 
 
0
2
4
6
8
10
12
14
1 10 100 1000 10000
t [s]
Z t
hC
(t)
 [K
/W
]
pot core
RTF cores situated vertically small ring core
big ring core
K. Górecki et al; Informacije Midem, Vol. 47, No. 1(2017), 3 – 13
11
of frequency of the stimulating signal from 25 kHz to 75 
kHz causes an increase in the value of transient thermal 
impedances ZthC(t) and ZthCW(t) by about 10%.
Figure 11: Measured (solid lines) and modelled 
(dashed lines) waveforms of transient thermal imped-
ance of the core and mutual transient thermal imped-
ance between the core and the winding of the trans-
former with the small ring core RTP at the stimulation 
with the sinusoidal current of frequency equal to 25kHz 
and 75kHz 
The observed differences between waveforms of the 
considered thermal parameters can be caused not only 
by changes of the value of frequency, but also with 
changes of the power dissipated in the core at these 
frequencies (1 W at f = 25 kHz and 0.8 W at f = 75 kHz). 
Similarly, as this was shown in Figs. 9, an increase in the 
value of the power dissipated in the considered ele-
ment causes a decrease in the value of thermal param-
eters of the transformer. 
As one can notice in Figs. 6 - 11 for all the considered 
constructions of transformers and for all the considered 
cooling conditions the good agreement between the 
results of measurements and calculations performed 
with the use of Eq. (3) was obtained. However, as it is re-
sults among other things, from papers [16, 17] thermal 
resistance existing in this model is a decreasing func-
tion of the power dissipated in the modelled element. 
The decreasing dependence of thermal resistance on 
the dissipated power is a result of improved cooling of 
the transformer component with an increase in their 
temperature. As it is commonly known efficiency of 
convection and radiation increases with an increase in 
temperature of the heat source. Therefore, the depend-
ence of thermal resistance on internal temperature of 
the heating source (the core or the winding) can be ap-
proximated with the function of the form:
 







 −−⋅+=
p
ththth a
TTRRR 010 exp                 (13)
where T denotes temperature of the considered com-
ponent of the transformer, whereas Rth0, Rth1, T0 and ap 
are the model parameters. 
The correctness of the presented description was veri-
fied experimentally. For example, in Fig.12 calculated 
(lines) and measured (points) dependences of thermal 
resistance of the winding RthW and mutual thermal re-
sistance between the winding and the core RthWC on 
the dissipated power in the winding of the transformer 
with the big ring RTP core are shown. Solid lines cor-
respond to the transformer placed horizontally, and 
dashed lines - to the transformer situated vertically. 
Figure 12: Measured (points) and modelled (lines) de-
pendences of thermal resistance of the winding and 
mutual thermal resistance between the winding and 
the core on the dissipated power for the transformer 
with the big ring core RTP at the stimulation with the 
dc current 
As it is visible, both the considered dependences are 
monotonically decreasing functions. The good agree-
ment between the results of calculations and measure-
ments is obtained. This proves that the proposed de-
scriptions of the dependences RthW(p) and RthWC(p) are 
useful. 
Figure 13: Measured (solid lines) and modelled 
(dashed lines) waveforms of self transient thermal im-
pedance of the winding of the transformer with the big 
ring core RTP situated vertically at the stimulation with 
the direct current of different values
 
 
0
5
10
15
20
25
30
1 10 100 1000 10000
t [s]
Z t
hC
(t)
 Z
th
C
W
(t)
  [
K
/W
]
RTP small ring core situated horizontally ZthC(t)  
ZthCW(t) 
f = 25 kHz
f = 75 kHz
 
 
0
2
4
6
8
10
12
14
16
18
20
0 1 2 3 4 5 6
p [W]
R
th
W
 R
th
W
C 
[K
/W
] RthW
RthWC
 
 
 
0
2
4
6
8
10
12
14
16
18
0,1 1 10 100 1000 10000 100000
t [s]
Z t
hW
(t)
 [K
/W
]
big RTP core situated vertically
I1 = 4.2 A
I1 = 5.2 A
I1 = 7.5 A
I1 = 9.2 A
K. Górecki et al; Informacije Midem, Vol. 47, No. 1(2017), 3 – 13
12
The considered dependences were used to model wave-
forms of transient thermal impedances of the impulse 
transformer. In this model only the value of thermal re-
sistance is a function of the dissipated power given by 
Eq. (7), whereas the values of parameters ai and tthi exist-
ing in Eq. (3) are constant for the considered transformer. 
For example, in Fig.13 the waveforms of measured (solid 
lines) and calculated (dashed lines) transient thermal im-
pedance of the winding of the transformer with the big 
RTP core are presented at different values of the power 
dissipated in the primary winding. As it is visible, the 
good agreement between the results of calculations and 
measurements is obtained. 
The difference between the temperature inside the 
core and on its surface is caused only by a non-zero 
value of thermal conductance of material used to make 
the core. Of course, the temperature inside the core is 
the highest, when power losses in the core dominate. 
In turn, the temperature rise of the core as a result of 
thermal coupling between this core and the windings 
is higher on the surface of the core than in the middle 
of it. Therefore, it is difficult to formulate the universal 
dependence between the temperature inside the core 
and on its surface. It is possible only to estimate the dif-
ference between temperatures of the surface and the 
middle of the core assuming that the only mechanism 
of heat transfer is conduction, generation of heat ap-
pears in the infinitely thin ring situated in the middle 
of the core, and heat flux density has uniform distribu-
tion. With these assumptions, for the ring core of the 
outside diameter equal to 5 cm, the inside diameter 
equal to 3 cm and the height equal to 1 cm, made of 
powdered iron at the power dissipated in the core of 
the value equal to 1 W the maximum temperature dif-
ference between the middle of the core and its surface 
amounts to 14 K.
5 Conclusions 
In the paper the method to measure transient thermal 
impedances in the transformer and the results of meas-
urements, illustrating the influence of the selected fac-
tors on waveforms of these thermal parameters, are 
presented. The research done by the Authors proves 
that efficiency of cooling the structural components 
of the transformer can be characterised by means of 
parameters of the compact thermal model and that 
the waveforms of the considered thermal parameters 
change depending on many factors. 
For example, an increase in the value of the power 
dissipated in the winding by more than 30% causes a 
decrease of the waveforms ZthW(t) by several percent. 
In turn, enlargement of the diameter of the ring core 
by about 60% is effective with deterioration of thermal 
resistance by even about 50% and with extension of 
the indispensable time to obtain the steady state even 
by about 60%. For comparatively large sizes of the in-
vestigated elements, the time indispensable to obtain 
the thermally steady state reaches even 2 hours. The 
change of spatial orientation of the transformer (hori-
zontal or vertical orientation) causes a change of the 
considered parameters by even about 20%.
The essential meaning has also the material, the core is 
made of, because its thermal conductance influences 
the measured waveforms ZthW(t) and ZthWC(t) in an essen-
tial manner. The least differences between waveforms 
of the mentioned thermal parameters, not exceeding 
20%, were observed for transformers with cores RTP 
(characterised by high thermal conductance), and the 
greatest (by even above 50%) - for cores RTF.
The analytic description of dependences of self and 
mutual thermal resistances of the transformer on the 
power dissipated in it is proposed. It was shown experi-
mentally that the use of this dependence in the clas-
sical literature model of transient thermal impedance 
assured correct modelling of waveforms of transient 
thermal impedances of the transformer over a wide 
range changes of the current of the primary winding.
It is worth noticing that the classical literature descrip-
tion of transient thermal impedance with the proposed 
description of thermal resistance enables very good 
approximation of the measured waveforms of transient 
thermal impedance of the winding for all the consid-
ered transformers. One observes, however, essential 
differences between the measured and approximated 
waveforms of mutual transient thermal impedances 
between the winding and the core and transient ther-
mal impedances of the core for transformers contain-
ing ferrite or nanocrystalline cores. In the mentioned 
above transformers a large delay of the process of heat-
ing the core appears, which reaches even 100 s. The 
correct modelling of this delay demands correction in 
the thermal model of the transformer, which is at pre-
sent an objective of the Authors investigations. 
The results of investigations presented in this paper can 
be useful for constructors of impulse-transformers and 
constructors of switched-mode power supplies contain-
ing these transformers. These results will be of service also 
to the Authors as experimental material, indispensable to 
formulate the non-linear electro-thermal model of the im-
pulse-transformer. In this model the influence of internal 
temperature of the core and the windings, of dimensions 
of the core and its spatial orientation on thermal param-
eters of the transformer will take into account. 
K. Górecki et al; Informacije Midem, Vol. 47, No. 1(2017), 3 – 13
13
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Arrived: 31. 10. 2016
Accepted: 04. 01. 2017
K. Górecki et al; Informacije Midem, Vol. 47, No. 1(2017), 3 – 13