Imagine having to attenuate a fairly high power signal, high enough in power that one single attenuator pad can't handle the power dissipation requirement. We can use a cascade of attenuators, but we want to spread the burden, so to speak, so that each attenuator in the cascade will dissipate an equal share of the total power dissipation.
The following derivation shows how to calculate the dB values for this:
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A while back, we looked at how the specific heat of a heat sink mass can be used to examine the thermal rise time of an eight pound mass. We will now rescale that to a one pound mass, just to make these numbers a little easier to deal with, as we examine one additional issue regarding thermal rise, the time.

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We have all kinds of information about finding maximum the temperatures that some particular things will get to, things like power supply heat sinks for example, but it may also be useful to look of how long will it take the temperature to get to that calculated extreme value.
With the specific heat of that particular something, we can make an estimate of the time constant of that something undergoing a thermal excursion:
Note: In this example, we look up the specific heat of aluminum at http://hyperphysics.phy-astr.gsu.edu/hbase/tables/sphtt.html where we find the value as 0.900 joules / °C / gram.
We examine in this example, a real life situation I once came across:
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Did you ever wonder how quickly or slowly a TO39 transistor package can change temperature? Maybe you've asked yourself at one time or another just how many seconds you have to shut off power before the device is destroyed? Here's a quick look at that question.
A type 2N2219 transistor is the guinea pig of this experiment. The transistor is wired up as shown here and its base to emitter voltage (Vbe) is measured at time intervals of ten seconds as the transistor heats up.

The value of Vbe is known to vary inversely with the transistor's temperature. Therefore, using a temperature coefficient of -2 mV per °C, we may estimate the temperature at each time of Vbe measurement and then plot that data.
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Switching power MOSFETs have a thermal runaway mechanism. A MOSFET's on-resistance rises with rising temperature and if the drain to source current doesn't materially change in response to that, a vicious cycle can occur. A rising Rdson leads to rising temperture which leads to rising Rdson which leads to rising temperature which leads to .......
You can test for this without having to burn anything by measuring MOSFET case temperature. Stasrt by measuring temperature at regular intervals as the devices get hot. Usually intervals of ten seconds are a good choice.
Once the MOSFETs have reached some high temperature, you turn off operating power AND you KEEP ON measuring tempertures as the MOSFETs cool down. Keep using the same time interval and keep on making the measurements until the MOSFET temperature has pretty much returned to the starting value.
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