Estimation models with scaling laws and linear regression

Written by Marc Budinger, INSA Toulouse, France

The estimation models calculate the component characteristics requested for their selection without requiring a detailed design. Scaling laws are particularly suitable for this purpose. This notebook illustrates the approach with IGBT component.

IGBT modules

Validation of the obtained scaling laws is realized thanks linear regression on catalog data.

The following article give more details and other models for power converters:

Giraud, X., Budinger, M., Roboam, X., Piquet, H., Sartor, M., & Faucher, J. (2016). Optimal design of the Integrated Modular Power Electronics Cabinet. Aerospace Science and Technology, 48, 37-52.

Scaling laws

Assumptions and notation

The IGBT chip or MOSFET are constituted of many arranged structures called “unit cells”. The more the cells are provided, the lower the on-state voltage will be. The number of cells is assumed to be proportional to the current rating. We seek to establish the scaling laws for the same level of maximum voltage and the same technology. The thicknesses of the different layers (silicon, insulation, copper plate, …) will therefore be assumed to be constant.

Notation: The x* scaling ratio of a given parameter is calculated as \(x^*=\frac{x}{x_{ref}}\) where \(x_{ref}\) is the parameter taken as the reference and \(x\) the parameter under study.

Thermal resistance

Exercice : Propose a scaling law which links thermal resistance \(R_{th}\) to rated current \(I_0\). Estimate the thermal resistance for a 300 A component knowing the following reference component:
\(I_{0,ref} = 100 A\)
\(R_{th_JC,ref} = 0.27 °C/W\)

I0_ref=100.  # [A]
R_th_jc_ref=.27 # [°C/W]

print("The estimated thermal resistance is: %.2f °C/W" %(R_th_jc_ref*(300/I0_ref)**(-1)))
print("The estimation model is: R_th_jc = %.2e.I0^(-1)" % (R_th_jc_ref*(1/I0_ref)**(-1)))

The estimated thermal resistance is: 0.09 °C/W
The estimation model is: R_th_jc = 2.70e+01.I0^(-1)

Validation with a data plot

We will compare the scaling law with a plot on catalog data.

Import data

The first step is to import catalog data stored in a .csv file. We use for that functions from Panda package (with here an introduction to panda).

# Panda package Importation
import pandas as pd

# Read the .csv file with bearing data
path='https://raw.githubusercontent.com/SizingLab/sizing_course/main/laboratories/Lab-watt_project/assets/data/'
df = pd.read_csv(path+'DataIGBT.csv', sep=';')
# Print the head (first lines of the file)
df.head()

	REF	Emax	I0	Eon_off	Vcc_ref	Vf0	Rf0	Qrr	Rth_jc
0	SKMy50GB063D	600	50	4.3	300	0.9	15.0	3.2	0.50
1	SKMy50GB063D	600	75	5.5	300	0.9	13.3	3.7	0.35
2	SKMy50GB063D	600	100	7.0	300	0.9	10.0	6.0	0.27
3	SKMy50GB063D	600	200	18.5	300	0.9	5.5	12.7	0.14
4	SKMy50GB063D	600	300	27.0	300	0.9	3.7	18.0	0.09

Plot data

We can now compare graphically the catalog data with scaling law:

• blue points, catalog data

• red line, scaling law

• red point, the reference for scaling law.

For the plot, we use the matplotlib package.

import numpy as np
import matplotlib.pyplot as plt

# Add 1 mm diameter point
I0=np.linspace(10, 300, num=50)

# Scaling law
Rth=R_th_jc_ref*(I0/I0_ref)**(-1)

# plot
h, ax = plt.subplots(1,1,sharex=True)
ax.plot(I0, Rth, '--r',df['I0'],df['Rth_jc'],'.b', I0_ref, R_th_jc_ref, '.r')
ax.set_ylabel('Thermal resistance (junction to case) [°C/W]')
ax.set_xlabel('Rated current [A]')
ax.grid()

Other scaling laws

The following table summarize the scaling laws which can be used for the sizing code.

Capacitor

	Scaling laws	References
Definition parameter :
Diameter	\(D\)	100 mm
Height	\(H\)	155 mm
Capacitance	\(C^*= D^{*2}*H^{*}\)	\(1000\mu F\)
Max voltage	\(V^*=1\)	800 V
Resistance ESR	\(R_s^*=D^{*-2}\)	\(3.2 m\Omega\)
Thermal resistance	\(R_{th}^*=D^{*-2/3} H^{*-1/3}\)	\(3°C/W\)
Wheight	\(M^*=D^{*2} H^*\)	1500 g

IGBT

	Scaling laws	References
Definition parameter :
Current	\(I\)	\(80 A\)
Maximum voltage	\(V_{max}^* = 1\)	\(900 V\)
Voltage drop	\(V_0^*=1\)	\(1 V\)
Dynamic resistance	\(R_0^*=I^{*-1}\)	20 m\({\Omega}\)
Commutation losses	\((E_{on}+E_{off})^*=I^* E^*\)	\(8.2 mJ\) for \(E=450 V\)
Thermal resistance	\(R_{th_JC}^*=I^{*-1}\)	\(0.30 °C/W\)

Diode (Current evolution equal to IGBT current)

	Scaling laws	References
Definition parameter :
Current	\(I\)	\(41 A\)
Maximum voltage	\(V_{max}^* = 1\)	\(900 V\)
Voltage drop	\(V_0^*=1\)	\(1 V\)
Dynamic resistance	\(R_0^*=I^{*-1}\)	\(15 m\Omega\)
Commutation losses	\(t_{rr}I_{RM}^{*}=I^*E^*\)	\(17.2 mJ\) for \(E=600 V\)
Thermal resistance	\(R_{th_JC}^{*}=I^{*-1}\)	\(0.47 °C/W\)

Exercice: Calculate the characteristics of a 50mm diameter and 50mm height capacitor.