PSD Surge Current Dependence on Current Impulses

 

Simulation of surge current dependence on length and number of forward current impulses flowing through a power semiconductor device

 

Introduction

The surge current value is one of the most important characteristics of a power semiconductor device (PSD), which determines its output capacity. The surge current is usually interpreted as the maximum amplitude of the forward current sinusoidal pulse, that when passed through the PSD (without application of reverse current), the definite classification parameters of the device do not go beyond their limit. Passing current of greater amplitude through the device results in irreparable breakdown of the PSD and its further destruction, owing to overheating.

At present, the surge current value is estimated on the basis of computation of a PSD structure's temperature ^[1-3]. Temperature is determined by solving of a heat conduction equation. With that, numerical methods ^[1-4] are prevalent as the analytical solving of the above-mentioned equation involves the use of several simplifying assumptions and does not guarantee the desired accuracy.

It is necessary to note that numerical methods of surge currents calculation mean also a computational burden. That is why, as a rule, calculations of such kind are made for single-shot pulses of a current of defined width. Usually this width equals 10 ms, corresponding to a commercial current frequency of 50 Hz. Increase in number of forward current impulses flowing through the PSD, and also their width change, result in exponentially rapid rise of counting time since in numerical calculations of surge currents, iterative procedures are used. In this connection, the search for analytical expressions adequately showing surge current's dependence on width and number of forward current impulses flowing through a PSD is an urgent question.  

In this artiсle, the above mentioned expressions are obtained using the thermodynamic approach to solving of the PSD heating problem. The conformity of expressions is tested by way of comparing the theoretical calculations with computer simulation results.

Surge Current Dependence on Length of Forward Current Single Pulse Flowing Through a PSD

The quantity of heat Q₁(t), which is excreted at passing of forward current single-impulse through PSD with the length t, is spent for the device heating and is partially diffused in surroundings as a consequence of its cooling.

Therefore,

$$\begin{equation} Q_1 (\tau ) = Q_T( \tau ) + \Delta Q (\tau) \end{equation}$$

where Q_T(t) is the quantity of heat, the excretion of which results in increase of PSD temperature, and ΔQ(t) is the quantity of heat lost at cooling of the device during flowing current pulse through it.

The equation (1) completely determines the surge current dependence on the length of the forward current single-pulse of arbitrary shape. Thus, while getting the explicit mode of this dependence, difficulties are connected primarily with the fact that electro- and thermophysical characteristics of a PSD to a certain extent depend on temperature, and consequently, as it increases they will be changed with the course of time t.

To simplify the problem, it must be taken into consideration that the temperature variation characteristic time is more than pulse length t. As the process of heat conduction has a diffusive nature, the speed of current change is usually more than the speed of the device temperature variation. Thus its electro- and thermophysical parameters vary more slowly than the current. Therefore, to a first approximation it is possible to consider their values as constants and proportional to the values under the initial temperature of the device.

Hereinafter we will assume that the initial temperature of the PSD is equal to the temperature T₀ of the surroundings, and after the current pulse advances in any point of the device, for example, in its silicon structure, the critical temperature T_С, is reached, and the device is failed (i.e. the current pulse height equals to the surge current value I_TSM for the given device). In this case, the quantities of heat Q_T(t) and ΔQ(t) can be expressed by experimentally measured device characteristics - the heat capacity C₀ of silicon structure and the thermal resistance R_th .

$$\begin{equation} Q_T(\tau ) = \alpha C_0 \Delta T\end{equation}$$

$$\begin{equation} \Delta Q (\tau ) = \beta \frac{\Delta T}{R_{th}} \tau \end{equation}$$

where ΔT = T _C - T₀ , and aspect ratios α and β represent adjusted parameters of the simulation. For the calculation of heat Q ₁(τ), the following is taken into consideration: with the current pulse advancing, the main power is dissipated on the linear section of the E-I curve of the PSD. Due to this, it is reasonable to consider that

$$\begin{equation} Q_1(\tau) \approx \int_0^\tau [V_0 + RI (t)] I(t)dt \end{equation}$$

where V₀ is the cut off voltage of the linear E-I curve of the PSD, R is its differential resistance, I(t) is the current strength flowing through the device. Assuming that V₀ and R are constants, for a sine pulse

$$\begin{equation} I(t) = I_{TSM} sin \Bigg ( \frac{\pi t}{\tau} \Bigg ) \end{equation}$$

Integrating (4), we get:

$$\begin{equation} Q_1 (\tau) \approx \frac {2 \gamma}{\pi} V_0 I_{TSM} \tau + \frac{1}{2} RI_{TSM}^2 \tau \end{equation}$$

where γ is an adjustable parameter of the model. Expressions (2), (3), and (6) subject to equality (1) result in the following surge current dependence on the pulse length τ:

$$\begin{equation} I_{TSM} (\tau ) = \frac{2 \gamma V_0}{\pi R} \Bigg ( \sqrt{\frac{\alpha \pi ^2 R \Delta T}{2 \gamma_0^2R_{th}} \Bigg ( \frac{C_0R_{th}}{\tau} + \frac{\beta}{\alpha} \Bigg ) +1} -1 \Bigg ) \end{equation}$$

The equation obtained indicates that in the region of small γ (at γ << C₀ R_th )

$$\begin{equation} I_{TSM} (\tau) \approx \sqrt{\frac{2 \alpha C_0 \Delta T}{R \tau}} \end{equation}$$

i.e. I_TSM (τ) ~ 1/ √ τ . This case associates the sine pulse with such a high frequency advancing through the device with the fact that there is no time to remove the heat produced by the device - ΔQ( τ ) ≈ 0 . In the region of low frequency impulses the value of the surge current tends to a constant limit

$$\begin{equation} I_{TSM} (\infty) \approx \sqrt{\frac{2 \beta \Delta T}{RR_{th}} } \end{equation}$$

This  limiting value of current conforms to a situation where practically all power excreted at current pass is dispersed in the surroundings.

Note, that formulas (8) and (9) help us determine parameters α and β by experiment.

The results of theoretical calculations of the formula (7) for a thyristor and a diode, are shown in Figure 1 (continuous curves). The comparison of these results with computer simulation results, based on the numerical solving of a thermal conductivity equation (see Figure 1 and 2), is the evidence of satisfactory consistency of the evolved theory with the data from computer experiments.

Figure 1. Surge current dependence on forward current single pulse length for a thyristor

Figure 2. Surge current dependence on forward current single pulse length for a thyristor

The difference between the theoretical results for the thyristor in our example and the data provided by the computer simulation in the region of small γ can be explained by the fact that for thyristors in the high frequency region, their breakover ^[1-3] essentially influences their surge current value.

For theoretical calculations it is assumed that the parameters α, β, and γ have the same value for both types of devices. Therefore, in both cases the theory satisfactorily describes computer simulation results at γ = 0, as in the expression (6), given large values of surge currrents, the first term (linear with respect to current) may be neglected. The theoretical curves shown in Figure 1 correspond in particular to this parameter γ. Thus for a PSD with a high level of surge current, its dependence on forward current single-pulse length is described, to a high degree of accuracy, using the following expression.

$$\begin{equation} I_{TSM} (\tau) \approx \sqrt{\frac {2 \alpha \Delta T}{RR_{th}} \Bigg ( \frac{C_0R_{th}}{\tau} + \frac {\beta}{\alpha}} \Bigg ) \end{equation}$$

Surge Current Dependence on Number of Forward Current Impulses Flowing Through a PSD

The following recurrence relation forms the basis of calculation of surge current dependence on the number N of forward current pulses flowing through PSD:

$$\begin{equation} NQ_1 (N) = Q_1 (1) + (N-1) \Delta Q_1 (N) \end{equation}$$

where Q₁(1) is the quantity of heat which should be exerted at passing of a single current pulse for the temperature at the breakdown point of the PSD to be increased from the initial value T₀ to the critical value T_С; Q₁ (N) is the quantity of heat which should be exerted at passing of one pulse through the device if the number of these pulses is N; Q₁(N) is the quantity of heat which is lost by cooling of the device during the time of two successive current pulses. Using the same assumptions about electro- and thermophysical parameters and of the E-I curve of the device, as in the previous section, it is easy to understand that in case of pulses of sinusoid form (5), the values Q₁(1) and Q₁(N) will be described by expressions which are similar to formula (6):

$$\begin{equation} Q_1 (N) \approx \frac{2 \gamma}{\pi } V_0 \tau I_{TSM} (N) + \frac{1}{2} R \tau I_{TSM}^2 (N) \end{equation}$$

$$\begin{equation} Q_1 (1) \approx \frac{2 \gamma}{\pi } V_0 \tau I_{TSM} (1) + \frac{1}{2} R \tau I_{TSM}^2 (1) \end{equation}$$

Hence, neglecting items which are linear, and taking equation (11) into consideration we come to the following surge current dependence on the number of forward current impulses flowing through PSD:

$$\begin{equation} I_{TSM} (N) \approx \sqrt{ \frac{I_{TSM}^2 (1)}{N} + \frac{2(N-1) \Delta Q_1 (N)}{NR \tau}} \end{equation}$$

where I_TSM (1) is the surge current value for a single pulse, which can be estimated by formula (10). Further simplification of the formula (14) is connected with the search of explicit expression for a heat Q₁(N), disseminated in surroundings during two successive current pulses. It is apparent a-priori, that this value will be connected to the same thermo-physical characteristics of the device, as the value ΔQ( τ ), discussed in the previous section:

$$\begin{equation} \Delta Q_1 (N) \sim \frac {\Delta T}{R_{th}} . 2 \tau \end{equation}$$

However, apart from it, the heat ΔQ₁(N) will depend also from N, as with the increase in number of current pulses at a single impulse passing through the device it will be heated progressively less, and consequently, the heat output, proportional to temperature gradient, will decrease.  It is reasonable to presume, that the dependence ΔQ₁(N) has the following powermode appearance:

$$\begin{equation} \Delta Q_1 (N) \sim \frac{1}{N^v} \end{equation}$$

where v is an adjustable parameter of the simulation (v > 0). Then taking into account (15) we come to following equation:

$$\begin{equation} \Delta Q_1 (N) = \beta \frac{\Delta T}{R_{th}N^v} 2 \tau\end{equation}$$

where β is an aspect ratio (adjustable parameter of the simulation), which should be close to value β used in the previous section. Subject to equation (17) for the surge current dependence on the number of forward current impulses flowing through PSD, we finally have:

$$\begin{equation} I_{TSM} (N) \approx \sqrt{\frac{I_{TSM}^2 (1)}{N} + 4\beta \frac {(N-1)}{N^{v+1}} \frac{\Delta T}{RR_{th}}}\end{equation}$$

Theoretical calculations results carried out using the formula (18) for a thyristors Т243-500 and a diode DA343-630-34 (continuous curves), and computer simulation results, based on the numerical solving of a heat conduction equation (markers) are shown in Figure 3. The comparison of these results is evidence of the satisfactory consistency of the theory and the data collected during the simulation.

Figure 3. The surge current dependence on the number N of forward current impulses flowing through a PSD

In the calculations it was presumed that parameters b and n  have the same value for both types of devices. The curves shown in the figure conform to value n = 0,38.

Note, the formula (18) allows us to calculate "the ideal longevity" of the device at forward current sinusoid impulses of specified length and amplitude passing through it. In fact, if it is considered that the environment is invariable, the failure in the device operation in this case will be connected with its gradual heating to a critical temperature as a result of a cumulative effect. Considering, that the amplitude of impulses conforms to surge current I_TSM(N), using expression (18) it becomes possible to determine the number N of current pulses which will pass through the device before it will fall out of action. Thus, when one knows a pulse length t, it is easy to evaluate time Δt_max, which should pass before the failure of a device: Δt_max » 2tN. Solving of this task is very simple if current pulses amplitude I_max is of much less the value I_TSM (1), defined by expression (7) or (10). In this case the number N of current pulses which will pass through the device before it will fall out of action is high  (N >> 1). Therefore, allowing that 0 < n < 1 and on the basis of the formula (18), it is possible to note down the following:

$$\begin{equation} I_{max} \equiv I_{TSM} (N) \approx \sqrt{\frac{4 \beta}{N^v} \frac{\Delta T}{RR_{th}}}\end{equation}$$

Hence

$$\begin{equation} (N) \approx \Bigg (\frac{4 \beta \Delta T}{I_{max}^2RR_{th}} \Bigg ) ^{1/v} \end{equation}$$

and "the ideal longevity" of the device

$$\begin{equation} \Delta t_{max} \approx 2 \tau . \Bigg ( \frac{4\beta \Delta T}{I_{max}^2 RR_{th}} \Bigg )^{1/v} \end{equation}$$

Conclusion

The results obtained in this article demonstrate that there are simple analytical expressions that adequately describe surge currents depedence on length and the number of forward pulses flowing through a PSD. For all types of devices, these expressions have the unified view with the factors dependent on electro and thermophysical characteristics of a certain device. So, for PSDs with high-level values of surge currents, their dependence on a single-pulse length τ is determined by the expression:

$$\begin{equation} I_{TSM} (\tau) \approx \sqrt{\frac{A}{\tau} + B}\end{equation}$$

where

$$\begin{equation} A = \frac{2\alpha C_0 \Delta T}{R} , B = \frac {2\beta \Delta T}{RR_{th}} \end{equation}$$

The surge current dependence on the number of pulses going through these devices looks like:

$$\begin{equation} I_{TSM} (N) \approx \sqrt{\frac{I_{TSM}^2 (1)}{N} + 2B \frac {(N-1)}{N^{v+1}}}\end{equation}$$

where 0 ≤ v ≤ 1 and I_TSM(1) is the value (22) of surge current for a single pulse.

 

List of Literature
1. V. Grigorenko, P. Dermenzhi, V. Kuzmin, T. Mnatsakanov. Simulation and design automation of power semiconductor devices – М.: Energoatomizdat, 1988. p. 280
2. A. Rabinerson, G. Ashkinazi. Load conditions of power semiconductor devices. – М.: Energia, 1976. p. 296
3. V. Bardin. Reliability of power semiconductor devices. – М.: Energia, 1978. p. 96
4. B. Polsky. Numerical simulation of semiconductor devices. – Riga: Zinatne, 1986. p.168

 

For more information, please read:

Modeling of Power Semiconductor Devices

Approaches to Mounting Power Semiconductor Devices

Device Failure due to Electrical and Thermal Conditions

Power Electronics Design, Simulation, and Analysis Tools