Aerospace and Electronic Systems - December 2018 - 31

Sathik et al.

ΔT j =

Pdiss ∗ t
 zth ( t )

(8)

where pdiss is power dissipation inside the power module. From
(8), it is evident that change in thermal impedance has an effect
on change in temperature distribution across the power module.
Therefore, it is important to include the variation of transient thermal impedance while estimating the junction temperature of the
power module.
To calculate the junction temperature of the power module, the
power dissipation pulses are shortened into rectangular pulses with
the same period and the peak level. The junction temperature of the
power device is then calculated by relating it to the superposition
theorem assuming that the temperature of the junction relates to the
power dissipation over a pulse train. Hence, the temperature across
the power module at each switching period is expressed as:

Conduction Loss
The power dissipated during the turn-on period of the power converter directly depends on the applied pulse width modulation
(PWM) technique. Based on the commonly used PWM function,
the energy dissipated during conduction period Pcon, is derived as:
Pcon =

π
3Vceoic 1 + M SVM ( t )
sin (ωt ) d (ωt )
2
π 0

π
3i 2 R 1 + M SVM ( t ) 2
+   c ce 
sin (ωt ) d (ωt )
2
π 0

+
+

(14)

π
3Vdoid 1 − M SVM ( t )
sin (ωt ) d (ωt )
2
π 0

π
3id2 Rd 1 − M SVM ( t ) 2
sin (ωt ) d (ωt )
π 0
2

1. Temperature during turn on rise in period:

(

T j ( ton ) = Pavg Rth − Pavg zth( ton ) + zth( ton ) Eswon + Eswdoff

)

ton + Tc

(9)

2. Temperature during conduction period:
T j (tcon ) = Pavg Rth − Pavg zth(tcon ) +  Pcons zth(tcon )

(

+ Pcons zth(tcon +ton ) Eswon / ton − zth( tcon ) Eswon + Eswdoff

)t

on

(10)

+ Tc

In (14), M is the applied modulation index, ω is the angular
frequency of load current's, and Rde, Rce ,Vdo, Vceo are the conduction
resistances and fixed threshold voltages of the diode and IGBT,
respectively. For space vector modulation, the modulation function
is a combination of sine and cosine function, and it can be derived
by applying Fourier sequence [29]:
M SVM ( t ) = M sin (ωt )

3. Temperature during turn off fall in period:
T j t = Pavg Rth − Pavg zth t +  zth t Eswioff / ton
( off )
( off )
( off )

(

+ Pcons zth t +t + zth t +t +t Eswon + Eswdoff
( con off )
( con on off )

(

+ zth t +t Eswon + Eswdoff
( con off )

)

+

)

ton

(11)

ton − Pcons zth t + Tc
( off )

where Rth is the thermal resistance during the steady-state period,
and zth(ton ), zth(tcon ), and zth( t ) are thermal impedances during transient
off
periods such as conduction, turn-on rise, and turn-off fall periods,
respectively. They can be estimated by the following expression,

t 
zth ( ton ) =  zth t p * on  ,
t p 


( )

( )

(12)


t 
zth ( tcon ) =  zth t p * con   
t p 


( )

where zth( t ) is the transient thermal impedance for a single switchp
ing pulse width (tp), and it is extracted from the device thermal
impedance curve by applying an exponential fitting approach, as
follows:

( )

(

)

(

)

rthi t p =  p ∗ exp q * t p − r exp − s ∗ t p 


where p, q, r, and s are the fitting parameters.
DECEMBER 2018

)

(

) 

(15)




where k is the integer numbers.
Energy dissipated during the conduction period is expressed
as:
PconS =

(

)

1
Vceoic
30cos Φ − cos3Φ 
6 + M 3π cos Φ + Rceic2  − M

12π
45 3π
8

+

(

)

1
Vdoid
30cos Φ − cos3Φ 
6 − M 3π cos Φ + Rd id2  − M

12π
45 3π
8


(16)

where Φ is the angle between the modulation function of the fundamental component and the load current.


t 
zth toff =  zth t p * off  , 
t p 


( )

(

3 3 ∞  sin (12k + 3) ωt sin (12k + 3) ωt
−

8π k =0  18k 2 + 9k + 1
18k 2 + 27 k + 10


(13)

Switching Energy
In this work, the switching energy curve has been used to estimate
the actual switching losses across the power module. Generally, device manufacturers specify the switching energy curve of their product and it has two values; one is rated, and the other one is operational. For instance, the switching energy dissipated during the turn-off
fall period (Eswoff-rat) given in the manufacturer's specification sheet
stands for the rated condition of an IGBT. But working turn-off
switching energy (Eswoff) might be different from rated switching
energy (Eswoff-rat) for the maximum number of times. Therefore, this
article calculates switching energy dissipation by considering rated
and actual operating of a power device [30] and it is given as:

IEEE A&E SYSTEMS MAGAZINE

31
Aerospace and Electronic Systems - December 2018

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