**1. Introduction**

130 Power Quality Harmonics Analysis and Real Measurements Data

converters with controllable rectifying. This can be explained by the similar mechanism of

The results from the investigation can be used for more precise designing of LCC converters used as power supplies for electric arc welding aggregates, powerful lasers, luminescent

Al Haddad, K., Cheron, Y., Foch, H. & Rajagopalan, V. (1986). Static and dynamic analysis of

Bankov, N. (2009) Influence of the Snubbers and Matching Transformer over the Work of a

Cheron, Y., Foch, H. & Salesses, J. (1985). Study of resonant converter using power

Cheron, Y. (1989). *La commutation douce dans la conversion statique de l'energie electrique*, Technique et Documentation, ISBN : 2-85206-530-4, Lavoisier, France. Malesani, L., Mattavelli, P., Rossetto, L., Tenti, P., Marin, W. & Pollmann, A. (1995).

Jyothi, G. & Jaison, M. (2009). Electronic Welding Power Source with Hybrid Resonant

Liu, J., Sheng, L., Shi, J., Zhang, Z. & He, X. (2009). Design of High Voltage, High Power and

Ivensky, G., Kats, A. & Ben-Yaakov, S. (1999). An RC load model of parallel and series-

*on Power Electronics*, Vol. 14, No.3, (May 1999), pp. 515-521, ISSN: 0885-8993.

a series resonant converter operating above its resonant frequency, *Proceedings of* 

Transistor Resonant DC/DC Converter. *Elektrotehnika*&*Elektronika (Sofia, Bulgaria)*,

transistors in a 25-kW X-Rays tube power supply. *IEEE Power Electronics Specialists* 

Electronic Welder With High-Frequency Resonant Inverter. *IEEE Transactions on Industry Applications*, Vol. 31, No.2, (March/April 1995), pp. 273-279, ISSN: 0093-

Inverter, *Proceedings of 10th National Conference on Technological Trends (NCTT09)*,

High Frequency in LCC Resonant Converter. *Applied Power Electronics Conference and Exposition, APEC 2009. Twenty-Fourth Annual IEEE*, pp. 1034-1038, ISSN: 1048-

parallel resonant DC-DC converters with capacitive output filter. *IEEE Transactions* 

the rectifier operation in the investigated converter.

*SATECH'86*, pp.55-68, Boston, USA.

Vol. 44, No. 7-8, pp. 62-68, ISSN 0861-4717.

pp. 80-84, Kerala, India, 6-7 Nov 2009.

2334, Washington, USA, 15-19 Feb. 2009.

*Conference*, ESA Proceedings, 1985, pp. 295-306.

lamps etc.

**9. References** 

9994.

Power devices may fail catastrophically if the junction temperature becomes high enough to cause thermal runaway and melting. A much lower functional limit is set by temperature increases that result in changes in device characteristics, such as forward breakover voltage or the recovery time, and failure to meet device specifications.

Heat generation occurs primarily within the volume of the semiconductor pellet. This heat must be removed as efficiently as possible by some form of thermal exchange with the ambient, by the processes of conduction, convection or radiation.

Heat loss to the case and heat-sink is primarily by conduction. Heat loss by radiation accounts for only 1-2% of the total and can be ignored in most situations. Finally, loss from the heat-sink to the air is primarily by convection. When liquid cooling is used, the heat loss is by conduction to the liquid medium through the walls of the heat exchanger. Heat transfer by conduction is conveniently described by means of an electrical analogy, as it shows in Table 1.


Table 1. Thermal and electrical analogy

Thermal Analysis of Power Semiconductor Converters 133

1

The response of a single element can be extended to a complex system, such as a power semiconductor, whose thermal equivalent circuit comprises a ladder network of the separate

The transient response of such a network to a step of input power takes the form of a series of exponential terms. Transient thermal impedance data, derived on the basis of a step input of power, can be used to calculate the thermal response of power semiconductor devices for a variety of one-shot and repetitive pulse inputs. Further on, the thermal response for commonly encountered situations have been computed and are of great value to the circuit designer who must specify a power semiconductor device and its derating characteristics.

Figure 2 shows the rectangular pulse series and the equation (3) describes this kind of input

 

*P if nT t nT FM P t*

0 1

*if nT t n T*

,

(3)

1 *<sup>j</sup> <sup>t</sup> <sup>k</sup>*

(2)

*r C* means thermal time constant.

Fig. 1. Transient thermal equivalent circuit for power semiconductors

resistance and capacitance terms shown in Fig. 1.

**2.1 Rectangular pulse series input power** 

Fig. 2. Rectangular pulse series input power

The thermal response is given by the following equation,

where *j jj* 

power.

*thjCDC <sup>j</sup> <sup>j</sup> Z t re*

Taking into account the thermal phenomena complexity for power semiconductor devices it is very difficult to study the heating processes both in steady-state or transitory operating conditions, using the traditional analytical equations. The modeling concepts have their strength for different grades of complexity of the power circuit. It is important to achieve an efficient tradeoff between the necessary accuracy, required simulation speed and feasibility of parameter determination, (Kraus & Mattausch, 1998). Approaches to simulate these processes have already been made in earlier work. Numerical programs based on the method of finite differences are proposed in (Wenthen, 1970), or based on formulation of charge carrier transport equations, (Kuzmin et al., 1993). A physical model using the application of continuity equation for description of the carrier transport in the low doped layer of structures is proposed in (Schlogl et al., 1998). A simple calculation procedure for the time course of silicon equivalent temperature in power semiconductor components based on the previously calculated current loading is shown in (Sunde et al., 2006). In order to take into account the nonlinear thermal properties of materials a reduction method based on the Ritz vector and Kirchoff transformation is proposed in (Gatard et al., 2006).

The work described in (Chester & Shammas, 1993) outlines a model which combines the temperature dependent electrical characteristics of the device with its thermal response. The most papers are based on the thermal RC networks which use the PSpice software, (Maxim et al., 2000; Deskur & Pilacinski, 2005). In (Nelson et al., 2006) a fast Fourier analysis to obtain temperature profiles for power semiconductors is presented. Electro-thermal simulations using finite element method are reported in (Pandya & McDaniel, 2002) or combination with the conventional RC thermal network in order to obtain a compact model is described in (Shammas et al., 2002). Most of the previous work in this field of thermal analysis of power semiconductors is related only to the power device alone. But in the most practical applications, the power semiconductor device is a part of a power converter (rectifier or inverter). Hence, the thermal stresses for the power semiconductor device depend on the structure of the power converter. Therefore, it is important to study the thermal behaviour of the power semiconductor as a component part of the converter and not as an isolated piece. In the section 2, the thermal responses related to the junction temperatures of power devices have been computed. Parametric simulations for transient thermal conditions of some typical power rectifiers are presented in section 3. In the next section, the 3D thermal modelling and simulations of power device as main component of power converters are described.

#### **2. Transient thermal operating conditions**

The concept of thermal resistance can be extended to thermal impedance for time-varying situations. For a step of input power the transient thermal impedance, ZthjCDC(t), has the expression,

$$Z\_{t \text{bjCDC}}\left(t\right) = \frac{\Delta\theta\_{j\text{C}}\left(t\right)}{P} \tag{1}$$

where:

ZthjCDC(t) means junction-case transient thermal impedance; jC(t) – difference of temperature between junction and case at a given time t; P – step of input power.

The transient thermal impedance can be approximated through a sum of exponential terms, like in expression bellow,

Taking into account the thermal phenomena complexity for power semiconductor devices it is very difficult to study the heating processes both in steady-state or transitory operating conditions, using the traditional analytical equations. The modeling concepts have their strength for different grades of complexity of the power circuit. It is important to achieve an efficient tradeoff between the necessary accuracy, required simulation speed and feasibility of parameter determination, (Kraus & Mattausch, 1998). Approaches to simulate these processes have already been made in earlier work. Numerical programs based on the method of finite differences are proposed in (Wenthen, 1970), or based on formulation of charge carrier transport equations, (Kuzmin et al., 1993). A physical model using the application of continuity equation for description of the carrier transport in the low doped layer of structures is proposed in (Schlogl et al., 1998). A simple calculation procedure for the time course of silicon equivalent temperature in power semiconductor components based on the previously calculated current loading is shown in (Sunde et al., 2006). In order to take into account the nonlinear thermal properties of materials a reduction method based

on the Ritz vector and Kirchoff transformation is proposed in (Gatard et al., 2006).

power converters are described.

expression,

where:

P – step of input power.

like in expression bellow,

**2. Transient thermal operating conditions** 

ZthjCDC(t) means junction-case transient thermal impedance;

The work described in (Chester & Shammas, 1993) outlines a model which combines the temperature dependent electrical characteristics of the device with its thermal response. The most papers are based on the thermal RC networks which use the PSpice software, (Maxim et al., 2000; Deskur & Pilacinski, 2005). In (Nelson et al., 2006) a fast Fourier analysis to obtain temperature profiles for power semiconductors is presented. Electro-thermal simulations using finite element method are reported in (Pandya & McDaniel, 2002) or combination with the conventional RC thermal network in order to obtain a compact model is described in (Shammas et al., 2002). Most of the previous work in this field of thermal analysis of power semiconductors is related only to the power device alone. But in the most practical applications, the power semiconductor device is a part of a power converter (rectifier or inverter). Hence, the thermal stresses for the power semiconductor device depend on the structure of the power converter. Therefore, it is important to study the thermal behaviour of the power semiconductor as a component part of the converter and not as an isolated piece. In the section 2, the thermal responses related to the junction temperatures of power devices have been computed. Parametric simulations for transient thermal conditions of some typical power rectifiers are presented in section 3. In the next section, the 3D thermal modelling and simulations of power device as main component of

The concept of thermal resistance can be extended to thermal impedance for time-varying situations. For a step of input power the transient thermal impedance, ZthjCDC(t), has the

The transient thermal impedance can be approximated through a sum of exponential terms,

*thjCDC*

jC(t) – difference of temperature between junction and case at a given time t;

*Z t*

*jC*

*P*

*t*

(1)

$$Z\_{\rm thj; CDC}\left(t\right) = \sum\_{j=1}^{k} r\_j \left(1 - e^{-\frac{t}{r\_j}}\right) \tag{2}$$

where *j jj r C* means thermal time constant.

The response of a single element can be extended to a complex system, such as a power semiconductor, whose thermal equivalent circuit comprises a ladder network of the separate resistance and capacitance terms shown in Fig. 1.

Fig. 1. Transient thermal equivalent circuit for power semiconductors

The transient response of such a network to a step of input power takes the form of a series of exponential terms. Transient thermal impedance data, derived on the basis of a step input of power, can be used to calculate the thermal response of power semiconductor devices for a variety of one-shot and repetitive pulse inputs. Further on, the thermal response for commonly encountered situations have been computed and are of great value to the circuit designer who must specify a power semiconductor device and its derating characteristics.

#### **2.1 Rectangular pulse series input power**

Figure 2 shows the rectangular pulse series and the equation (3) describes this kind of input power.

$$P(t) = \begin{cases} P\_{\rm FM} & \text{if } \quad nT \le t \le nT + \theta, \\ 0 & \text{if } \quad nT + \theta < t \le (n+1)T \end{cases} \tag{3}$$

The thermal response is given by the following equation,

Thermal Analysis of Power Semiconductor Converters 135

*PFM t if nT t nT P t*

1 1

*<sup>T</sup> <sup>t</sup> <sup>k</sup> FM i T i i <sup>T</sup> <sup>i</sup> <sup>T</sup>*

1

*i*

*<sup>T</sup> <sup>t</sup> <sup>k</sup> FM i T i i <sup>T</sup> <sup>i</sup> <sup>T</sup>*

*<sup>e</sup> <sup>P</sup> <sup>T</sup>*

*i*

*e*

1

*i*

*<sup>e</sup> P T r T e if nT t n T*

1 ,

*PFM t if nT t nT P t*

 

1

*<sup>T</sup> <sup>t</sup> <sup>k</sup> FM i T i i T <sup>i</sup> <sup>T</sup>*

*i*

*T*

*e*

*i*

*e*

1

*r T if nT t n T*

0 1

*if nT t n T*

*i*

*T*

*i*

*e*

*e P T r tT e if nT t nT*

0 1

*if nT t n T*

*i*

*rtT e if nT t nT*

*T*

*i*

*e*

1 ,

1 ,

*i*

1

*i*

1

,

(7)

(9)

 

(10)

(8)

In the case when *n* , the thermal response will be,

1

1

**2.3 Decreasing triangle pulse series input power** 

Fig. 4. Decreasing triangle pulse series input power

At limit, when *n* , the thermal response will be:

1

1

*k*

*t*

*jC*

1 1

*FM i i i T <sup>i</sup> <sup>T</sup>*

*P T*

1

1

Figure 4 shows a decreasing triangle pulse series with its equation (9).

*t*

*jC*

$$\Delta\theta\_{j\subset\{n+1\}}(t) = \begin{cases} P\_{\rm{FM}} \sum\_{i=1}^{k} r\_i \left| 1 - \frac{e^{-\frac{t-nT}{T\_i}} \left(1 - e^{-\frac{(n+1)T}{T\_i}}\right)}{1 - e^{-\frac{T}{T\_i}}} \right| - \left(1 - e^{-\frac{nT}{T\_i}}\right) e^{-\frac{T-\theta}{T\_i}} \\\\ \frac{1 - e^{-\frac{T}{T\_i}}}{1 - e^{-\frac{T}{T\_i}}} \end{cases} \quad \text{if} \quad nT \le t \le nT + \theta,\tag{4}$$

$$P\_{\rm{FM}} \sum\_{i=1}^{k} r\_i e^{-\frac{(t-nT-\theta)}{T\_i}} \frac{1 - e^{-\frac{\theta}{T\_i}}}{1 - e^{-\frac{T}{T\_i}}} \quad \text{if} \quad nT + \theta < t \le (n+1)T$$

For a very big number of rectangular pulses, actually *n* , it gets the relation:

$$\Delta\theta\_{j\gets n}(t) = \begin{cases} P\_{\rm FM} \sum\_{i=1}^{k} r\_i \left(1 - e^{-\frac{t}{T\_i}}\right) \frac{1 - e^{-\frac{T-\theta}{T\_i}}}{1 - e^{-\frac{T}{T\_i}}} & \text{if} \quad nT \le t \le nT + \theta\_i \\\\ P\_{\rm FM} \sum\_{i=1}^{k} r\_i e^{-\frac{t-\theta}{T\_i}} \frac{1 - e^{-\frac{\theta}{T\_i}}}{1 - e^{-\frac{T}{T\_i}}} & \text{if} \quad nT + \theta < t \le (n+1)T \end{cases} \tag{5}$$

Therefore, the junction temperature variation in steady-state conditions will be,

$$\begin{split} \Delta\theta\_{f\gets\infty} &= \left(P\_{\rm F\gets\mathcal{U}} - \frac{\theta}{T}P\_{\rm F\gets\mathcal{U}}\right) \sum\_{i=1}^{k} r\_i - P\_{\rm F\gets\mathcal{U}} \sum\_{i=1}^{k} r\_i e^{-\frac{\theta}{T\_i}} \frac{1 - e^{-\frac{T-\theta}{T\_i}}}{1 - e^{-\frac{T}{T\_i}}} = P\_{\rm F\gets\mathcal{U}} \sum\_{i=1}^{k} r\_i \left[1 - \frac{\theta}{T} - e^{-\frac{\theta}{T\_i}} \frac{1 - e^{-\frac{T-\theta}{T\_i}}}{1 - e^{-\frac{T}{T\_i}}}\right] = \\ &= P\_{\rm F\gets\mathcal{U}} \sum\_{i=1}^{k} r\_i \left(\frac{1 - e^{-\frac{\theta}{T\_i}}}{1 - e^{-\frac{T}{T\_i}}} - \frac{\theta}{T}\right) \end{split} \tag{6}$$

#### **2.2 Increasing triangle pulse series input power**

A series of increasing triangle pulses is shown in Fig. 3 and the equation which describes this series is given in (7).

Fig. 3. Increasing triangle pulse series input power

*<sup>e</sup> P re e if nT t n T*

1

*i*

*i*

*e*

1 1 1

*jC FM FM i FM i T T FM i*

*e e P P r P re P r e T T*

A series of increasing triangle pulses is shown in Fig. 3 and the equation which describes

*i*

*<sup>T</sup> <sup>t</sup> <sup>k</sup> <sup>T</sup> T FM i T <sup>i</sup> <sup>T</sup>*

*i i i i*

*t nT n T nT T T T TT*

1

*i*

*P r if nT t nT*

<sup>1</sup> 1 1

<sup>1</sup> 1 ,

*i*

*<sup>e</sup> Pre if nT t nT*

*i*

*<sup>e</sup> P re if nT t n T*

<sup>1</sup> <sup>1</sup>

1 1 <sup>1</sup>

*i i*

*k k T T <sup>k</sup> T T*

*i i <sup>i</sup> T T*

1 1

1 ,

 

*i i*

 

*T T*

*i i*

*e e*

(5)

 

(6)

(4)

1

1 1

*e e ee*

1

1

1

this series is given in (7).

*<sup>e</sup> P r*

1

*k T FM i T <sup>i</sup> <sup>T</sup>*

*i*

*i*

 

*e*

*T*

**2.2 Increasing triangle pulse series input power** 

Fig. 3. Increasing triangle pulse series input power

1

*jC n*

1

*k*

1

*jC*

1

1

*<sup>e</sup> <sup>t</sup>*

1

*e*

*FM i <sup>T</sup> <sup>i</sup> <sup>T</sup>*

*<sup>e</sup> <sup>t</sup>*

*FM i <sup>T</sup> <sup>i</sup> <sup>T</sup>*

*i i i i*

For a very big number of rectangular pulses, actually *n* , it gets the relation:

1

*i*

Therefore, the junction temperature variation in steady-state conditions will be,

*<sup>t</sup> <sup>k</sup> <sup>T</sup> T FM i T <sup>i</sup> <sup>T</sup>*

*t nT n T <sup>k</sup> <sup>T</sup> T T*

$$P(t) = \begin{cases} t \frac{P\_{FM}}{\theta} & \text{if } \quad nT \le t \le \theta + nT \\ 0 & \text{if } \quad \theta + nT < t \le (n+1)T \end{cases} \tag{7}$$

In the case when *n* , the thermal response will be,

$$
\Delta\theta\_{j\subset n}(t) = \begin{cases}
\frac{P\_{FM}}{\theta} \sum\_{i=1}^{k} r\_i \left\{ t - T\_i \left[ 1 - \frac{1 - \left( 1 - \frac{\theta}{T\_i} \right) e^{-\frac{T\_i - \theta}{T\_i}}}{1 - e^{-\frac{T}{T\_i}}} e^{-\frac{t}{T\_i}} \right] \right\} & \text{if} \quad nT \le t \le \theta + nT\_s \\\\ \frac{P\_{FM}}{\theta} \sum\_{i=1}^{k} r\_i T\_i \frac{\frac{\theta}{T\_i} - e^{-\frac{\theta}{T\_i}} - 1}{1 - e^{-\frac{T}{T\_i}}} e^{-\frac{t - \theta}{T\_i}} & \text{if} \quad \theta + nT < t \le \{n + 1\}T
\end{cases} \tag{8}
$$

#### **2.3 Decreasing triangle pulse series input power**

Figure 4 shows a decreasing triangle pulse series with its equation (9).

Fig. 4. Decreasing triangle pulse series input power

$$P(t) = \begin{cases} 1 - t \frac{P\_{\rm FM}}{\theta} & \text{if } \quad nT \le t \le \theta + nT \\ 0 & \text{if } \quad \theta + nT < t \le (n+1)T \end{cases} \tag{9}$$

At limit, when *n* , the thermal response will be:

$$\Delta\theta\_{j\subset\kappa}(t) = \begin{cases} \frac{P\_{\rm FM}}{\theta} \sum\_{i=1}^{k} r\_i \left\{ (\theta - t) - T\_i \left[ -1 + \frac{\theta}{T\_i} - e^{-\frac{T-\theta}{T\_i}} \right] \begin{matrix} -\frac{t}{T\_i} \\ -1 + \frac{\theta}{T\_i} - e^{-\frac{T}{T\_i}} \end{matrix} \right\} & \text{if} \quad nT \le t \le \theta + nT\_i \\\\ \frac{P\_{\rm FM}}{\theta} \sum\_{i=1}^{k} r\_i T\_i \frac{1 - \left(1 + \frac{\theta}{T\_i}\right) e^{-\frac{\theta}{T\_i}}}{1 - e^{-\frac{T}{T\_i}}} & \text{if} \quad \theta + nT < t \le (n+1)T \end{cases} \tag{10}$$

Thermal Analysis of Power Semiconductor Converters 137

Fig. 6. Trapezoidal pulse series input power

*P t*

1

*e t*

1

1 2

1 2

this kind of waveform is given by (16).

**2.6 Partial sinusoidal pulse series input power** 

*jC <sup>t</sup> <sup>k</sup> P P <sup>T</sup>*

where:

At limit, *n* , the thermal response is given by,

1 2

*i*

1 2

*e*

*i*

1

*i i T <sup>i</sup> <sup>T</sup>*

*G*

*<sup>i</sup> <sup>T</sup>*

1

*FM FM FM*

1

*rT e if nT t n T*

2 1 1 2

*FPP PP e*

A partial sinusoidal pulse series waveform is shown in Fig. 7. The equation which describes

*GPP PP e*

*P P FM FM FM FM*

1 2 2 1

*P P FM FM FM FM*

*i i T FM i*

<sup>1</sup> <sup>1</sup>

*<sup>t</sup> <sup>k</sup> P P <sup>T</sup> FM FM*

*i*

*i*

1 21 ,

2 1

 

<sup>1</sup> ,

*<sup>F</sup> P P rT e P t T if nT t nT*

*<sup>t</sup> P PP if nT t nT*

 

0 1

*i*

*T T*

 

> 

*i*

*T*

(13)

(15)

(14)

*if nT t n T*

1 1,

*i i*

*T T*

*i i*

*T T*

1 1

#### **2.4 Triangle pulse series input power**

A series of triangle input power is shown in Fig. 5. The equation which describes this kind of series is given in (11).

Fig. 5. Triangle pulse series input power.

$$P(t) = \begin{cases} t\frac{P\_{\rm FM}}{\Theta} & \text{if } \quad nT \le t \le \Theta + nT \\ \left(2 - \frac{t}{\Theta}\right) P\_{\rm FM} & \text{if } \quad \theta + nT < t \le 2\Theta + nT \\ 0 & \text{if } \quad 2\theta + nT \le t \le (n+1)T \end{cases} \tag{11}$$

For junction temperature computation when *n* , the following relation will be used:

$$
\Delta\theta\_{\text{QCD}}(t) = \begin{cases}
\frac{P\_{\text{EM}}}{\theta} \sum\_{i=1}^{k} r\_{i} \left[ T\_{i} \frac{1 - 2e^{-\frac{T}{T\_{i}}} + e^{-\frac{T-\theta\theta}{T\_{i}}}}{1 - e^{-\frac{T}{T\_{i}}}} e^{-\frac{t}{T\_{i}}} + (t - T\_{i}) \right] & \text{if} \quad nT \le t \le \theta + nT\_{\text{r}} \\
\frac{P\_{\text{EM}}}{\theta} \sum\_{i=1}^{k} r\_{i} \left\{ -T\_{i} \frac{2 - e^{-\frac{\theta}{T\_{i}}} - e^{-\frac{T-\theta}{T\_{i}}}}{1 - e^{-\frac{T}{T\_{i}}}} e^{-\frac{t-\theta}{T\_{i}}} + \left[ (2\theta - t) + T\_{i} \right] \right\} & \text{if} \quad \theta + nT < t \le 2\theta + nT\_{\text{r}} \\
\frac{P\_{\text{EM}}}{\theta} \sum\_{i=1}^{k} r\_{i} T\_{i} \frac{T\_{i} - \left( \frac{2 - \theta}{T\_{i}} \right)}{1 - e^{-\frac{T}{T\_{i}}}} e^{-\frac{t - 2\theta}{T\_{i}}} & \text{if} \quad 2\theta + nT < t \le (n+1)T \\
\end{cases} (12)$$

#### **2.5 Trapezoidal pulse series input power**

Figure 6 shows a trapezoidal pulse series with the equation from (13).

#### Fig. 6. Trapezoidal pulse series input power

$$P(t) = \begin{cases} P\_{1FM} + \left(P\_{2FM} - P\_{1FM}\right)\frac{t}{\theta} & \text{if } \quad nT \le t \le \theta + nT\_s\\ 0 & \text{if } \quad \theta + nT < t \le (n+1)T \end{cases} \tag{13}$$

At limit, *n* , the thermal response is given by,

$$
\Delta\theta\_{j\gets n}(t) = \begin{cases}
\frac{1}{\theta} \sum\_{i=1}^{k} r\_i \left[ T\_i \frac{F\_{p\_1 p\_2}}{T} e^{-\frac{t}{T\_i}} + P\_{\text{IFM}} + \frac{P\_{2\text{FM}} - P\_{\text{IFM}}}{\theta} (t - T\_i) \right] & \text{if} \quad nT \le t \le \theta + nT\_i \\
\frac{1}{\theta} \sum\_{i=1}^{k} r\_i T\_i \frac{G\_{p\_1 p\_2}}{1 - e^{-\frac{T\_i}{T\_i}}} e^{-\frac{t - \theta}{T\_i}} & \text{if} \quad \theta + nT < t \le (n + 1)T
\end{cases} \tag{14}
$$

where:

136 Power Quality Harmonics Analysis and Real Measurements Data

A series of triangle input power is shown in Fig. 5. The equation which describes this kind

(12)

(11)

,

 

2 2 ,

*if nT t n T*

1 2 ,

2 1 *<sup>i</sup>*

*<sup>T</sup> e if nT t n T*

<sup>2</sup> <sup>2</sup> 2 ,

02 1

*i*

*<sup>P</sup> e e r T e t T if nT t nT*

*t*

*i*

*<sup>P</sup> e e t rT e tT if nT t nT*

*t*

For junction temperature computation when *n* , the following relation will be used:

2

 

*i i*

*i i T i*

 

*T*

 

 2

 

*T T*

*i*

*e*

*i i*

*i*

*t*

*t if nT t nT*

*<sup>t</sup> Pt P if nT t nT*

**2.4 Triangle pulse series input power** 

Fig. 5. Triangle pulse series input power.

1

 

1

1

**2.5 Trapezoidal pulse series input power** 

*k FM*

*<sup>P</sup> rT*

*FM*

2

*e*

1

*k T T FM T*

*<sup>i</sup> <sup>T</sup>*

*<sup>i</sup> <sup>T</sup>*

1

*k T T FM T jC i i T i*

*i*

*T*

*e*

 

*e*

*i*

Figure 6 shows a trapezoidal pulse series with the equation from (13).

1

1

*i i T <sup>i</sup> <sup>T</sup>*

*P*

*FM*

of series is given in (11).

$$\begin{aligned} F\_{P\_1 P\_2} &= P\_{2FM} - P\_{1FM} \left( 1 + \frac{\theta}{T\_i} \right) + \left[ P\_{1FM} - P\_{2FM} \left( 1 - \frac{\theta}{T\_i} \right) \right] e^{-\frac{T-\theta}{T\_i}} \\ G\_{P\_1 P\_2} &= P\_{1FM} - P\_{2FM} \left( 1 - \frac{\theta}{T\_i} \right) + \left[ P\_{2FM} - P\_{1FM} \left( 1 + \frac{\theta}{T\_i} \right) \right] e^{-\frac{\theta}{T\_i}} \end{aligned} \tag{15}$$

#### **2.6 Partial sinusoidal pulse series input power**

A partial sinusoidal pulse series waveform is shown in Fig. 7. The equation which describes this kind of waveform is given by (16).

Thermal Analysis of Power Semiconductor Converters 139

impedance offered by the device in this region of operation, is often sufficient to handle the

A transient thermal calculation even using the relation (2), is very complex and difficult to do. Hence, a more exactly and efficiently thermal calculation of power semiconductors at different types of input power specific to power converters, can be done with the help of

Further on, it presents the waveforms of input powers and junction temperatures of power semiconductors, diodes and thyristors, from different types of single-phase bridge rectifiers. Also, temperature waveforms in the case of steady state thermal conditions, are shown. Using PSpice software, a parametric simulation which highlights the influence of some

On ordinate axis, the measurement unit in the case of input power waveforms, is the watt, and in the case of temperatures, the measurement unit is the 0C, unlike the volt one that appears on graphics. This apparent unconcordance between measurement units is because thermal phenomena had been simulated using electrical circuit analogy. The notations on the graphics P1, P2 and P3 mean input powers and T1, T2 and T3 temperatures, respectively.

The waveforms of the input powers and junction temperatures of power diodes from the structure of a single-phase uncontrolled bridge rectifier are shown in the below diagrams.

 Time 0s 20ms 40ms 60ms 80ms 100ms

From the above graphics, Fig. 8, the input power variation P1, P2 and P3 with the load resistance values can be noticed. The increase of load values leads to small input power values, and finally, to the decrease of junction temperature magnitudes, T1, T2 and T3, Fig. 9, and also to the decrease of temperature variations. In the case of quasi-steady state thermal conditions, Fig. 10, there are a clearly difference between temperatures waveforms variation. Also, the time variations of temperature values are insignificantly. The maximum value of T1 temperature, Fig. 10, outruns the maximum admissible value for power semiconductor junction, about 1250C. Therefore, it requires an adequate protection for the power diode or

Fig. 8. Input power waveforms at load resistance variation with 10, 20, 50Ω

P3

P2

P1

**3. Thermal simulations of power semiconductors from rectifiers** 

parameter values upon temperature waveforms has been done.

power that is dissipated.

PSpice software and/or 3D finite element analysis.

**3.1 Single-phase uncontrolled bridge rectifier** 

V(MULT1:OUT)

increasing of load resistance.

0V

0.5KV

1.0KV

Fig. 7. Partial sinusoidal pulse series input power

$$P(t) = \begin{cases} P\_{\rm FM} \sin\left(\alpha t + \gamma\right) & \text{if} \quad nT \le t \le \theta + nT \\ 0 & \text{if} \quad \theta + nT < t \le (n+1)T \end{cases} \tag{16}$$

In order to establish the junction temperature when *n* , it will use the relation,

$$\Delta\theta\_{\text{QCD}}(t) = \begin{cases} P\_{\text{FM}} \left[ Z\sin\left(\alpha t + \gamma - \delta\right) - \sum\_{i=1}^{k} r\_i \left[ \sin\left(\gamma - \rho\_i\right) - \sin\left(\gamma - \rho\_i + \alpha\theta\right)e^{\frac{T-\theta}{T\_i}} \right] \frac{e^{-\frac{T}{T\_i}}}{\left[ \left(1 - e^{-\frac{T}{T\_i}} \right) \sqrt{1 + \left(\alpha T\_i\right)^2} \right]} \right] \tag{17} \\\\ \qquad\quad\quad\quad\quad\quad\quad \quad nT \le t \le \theta + nT\_r \\\\ P\_{\text{FM}} \sum\_{i=1}^{k} r\_i \left[ \sin\left(\alpha\theta + \gamma - \rho\_i\right) - \sin\left(\gamma - \rho\_i\right) e^{-\frac{\theta}{T\_i}} \right] \frac{e^{\frac{t-\theta}{T\_i}}}{\left[ \left(1 - e^{-\frac{T}{T\_i}} \right) \sqrt{1 + \left(\alpha T\_i\right)^2} \right]} \\\\ \qquad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad$$

where:

$$\text{ctg}\phi\_i = \frac{1}{\alpha T\_i}; \quad Z^2 = \sum\_{i=1}^k \left(r\_i \cos^2 \phi\_i\right)^2 + \sum\_{i=1}^k \left(\frac{r\_i}{2} \sin 2\phi\_i\right)^2; \quad \text{tg}\,\delta = \frac{\sum\_{i=1}^k \frac{r\_i}{2} \sin 2\phi\_i}{\sum\_{i=1}^k r\_i \cos^2 \phi\_i} \tag{18}$$

Extremely short overloads of the type that occur under surge or fault conditions, are limited to a few cycles in duration. Here the junction temperature exceeds its maximum rating and all operational parameters are severely affected. However the low transient thermal

Fig. 7. Partial sinusoidal pulse series input power

 

*jC*

where:

 *FM i <sup>i</sup> <sup>i</sup> <sup>T</sup> <sup>i</sup> <sup>T</sup>*

*<sup>e</sup> P r <sup>e</sup>*

*<sup>k</sup> <sup>T</sup> <sup>T</sup>*

 

*<sup>r</sup> ctg Z r tg <sup>T</sup>*

; cos sin 2 ;

*i i i i k*

sin sin

*if nT t n T*

 

*t if nT t nT*

 

,

sin sin sin

1

*i i i*

*i*

*i*

1

*r*

*i*

*k i*

*r*

cos

*i i*

(18)

(16)

*i*

(17)

1 1

*i*

*i*

*e T*

<sup>1</sup> <sup>2</sup>

*i*

*e T*

*i i*

*t*

1 1

1 1 2

*k k i*

*i*

sin , 0 1

*if nT t n T*

*<sup>t</sup> <sup>T</sup> <sup>k</sup> <sup>T</sup> <sup>T</sup>*

 

*P t if nT t nT FM P t*

 

 

In order to establish the junction temperature when *n* , it will use the relation,

 

*FM <sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>T</sup> <sup>i</sup> <sup>T</sup>*

*<sup>e</sup> PZ t r <sup>e</sup>*

<sup>1</sup> <sup>2</sup>

*i*

<sup>2</sup> <sup>2</sup> 2 2 <sup>1</sup>

2

sin 2 <sup>1</sup> <sup>2</sup>

Extremely short overloads of the type that occur under surge or fault conditions, are limited to a few cycles in duration. Here the junction temperature exceeds its maximum rating and all operational parameters are severely affected. However the low transient thermal

impedance offered by the device in this region of operation, is often sufficient to handle the power that is dissipated.

A transient thermal calculation even using the relation (2), is very complex and difficult to do. Hence, a more exactly and efficiently thermal calculation of power semiconductors at different types of input power specific to power converters, can be done with the help of PSpice software and/or 3D finite element analysis.
