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Source: Vishay SMBJ36A-E3/52 datasheet

I understand the standoff voltage of the TVS diode. The breakdown voltage is said to be the point at which it starts flowing significant current, around 1mA. So what does the Max Breakdown voltage concept explain? Does our TVS diode burn when it constantly rises slightly above the maximum breakdown voltage?

Can it withstand this voltage continuously within the max and min voltage range?

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The breakdown voltage is expressed as a range because somewhere in this range the diode will start to carry significant current. Think of it as a tolerance band.

The voltage will not rise above the maximum breakdown voltage. If it does you are already exceeding the maximum rated power of the device by far.

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    \$\begingroup\$ A minor tweak: it will [rise above], at elevated temperature, including due to self-heating; the given figure is at 25°C. (The amount of change, over rated temperature, is usually on par with or smaller than the tolerance range, so this isn't very useful, just interesting.) \$\endgroup\$ Commented Mar 15 at 13:57
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    \$\begingroup\$ If it does you are already exceeding the maximum rated power of the device by far <-- fundamentally not true. You should look at a data sheet for a TVS properly and realize that the table of numbers is for a specific transient surge type (i.e. 10/1000). Faster transients that can generate voltages higher than the specified maximum breakdown voltage are easily permitted. It's all about energy handling and, a shorter pulse can be tolerated even if it results in a larger peak breakdown voltage. I mean, look at the clamp voltage; it is much higher than the breakdown voltage. \$\endgroup\$
    – Andy aka
    Commented Apr 5 at 11:43
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So what does the Max Breakdown voltage concept explain?

It tells you what the maximum terminal voltage that may be seen at the test current (\$I_T\$) for any particular device in a batch you tested. The minimum breakdown voltage tells you that for the test current \$I_T\$, the voltage might be somewhat lower.

In other words, it tells you something about the "spread" of breakdown voltages in a batch of items. Think of it as a product guarantee.

Does our TVS diode burn when it constantly rises slightly above the maximum breakdown voltage?

The only guarantee here is that if the voltage rises to the minimum breakdown voltage, none of the TVS diodes in the batch will burn. This is because (for the bottom device in your list, the power will be 12.2 volts x 1mA = trivially small.

However, if you have a device that has an actual breakdown voltage of 12.2 volts (the minimal in the data sheet) but tested it at 13.5 volts, it could easily burn. Even a device that was nominal i.e. 12.85 volts, it may still burn at 13.5 volts.

Can it withstand this voltage continuously within the max and min voltage range?

No. Here is what I estimate the SMBJ11 will be be subject to if it had a nominal breakdown voltage of 12.85 volts and was subject to an over-voltage of 13.5 volts: -

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As you can see it will dissipate 54 watts and burn inside 1 second: -

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Image modified from data sheet

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