Power Electronics
Psi or Theta: Which One Should You Choose?

Psi or Theta: Which One Should You Choose?

Many device data sheets now list both of these thermal characterization parameters, but to apply them accurately in power-supply designs, engineers must understand the subtle differences in how these terms are defined.

Over the years, standards organizations have undertaken numerous efforts to standardize the test methods that are used to characterize the thermal performance of semiconductor devices. Groups such as Semiconductor Equipment and Materials International (SEMI), the Electronic Industries Alliance (EIA) and the Joint Electron Device Engineering Council (JEDEC) have developed several standards or specifications that define methods of measuring a variety of thermal characterization parameters.

For example, the SEMI organization drafted standards for measuring the thermal resistance of ceramic packages, integrated circuit packages and semiconductor packages under different environmental and test conditions. These standards defined the familiar theta (θ) terms for thermal resistance such as θJA and θJC.

But recognizing certain limitations of these standards, EIA and JEDEC later came along and defined their own standards for measuring the performance of semiconductor devices. In the process, they clarified the applicability and narrowed the scope of the existing theta terms for thermal resistance, while also creating a new set of thermal characterization parameters, symbolized by the Greek letter psi (Ψ). As with theta, the psi terms usually carry subscripts that reflect measurement conditions.

As the EIA/JEDEC standards describe, the psi and theta terms are related but have different meanings and implications for use. Now that both terms are appearing more frequently on device data sheets, it's imperative that system designers understand the distinctions between the terms and how these terms are defined, so that they understand how the device vendors are characterizing their parts.

This article will first review, in some detail, the nuances of the standards themselves. It will then illustrate, by using a relatively simple system model, why these seemingly subtle differences in definitions between theta and psi are, in fact, profound. Armed with that knowledge, designers can make more accurate predictions about the thermal performance of the semiconductor devices in their systems.

Thermal Parameters Defined

Many of the now-familiar thermal-resistance parameters were originally defined by the SEMI organization. Three standards in particular are worth noting. In quoting excerpts from these standards, I have italicized certain passages to highlight some of their limitations.

  1. SEMI G30-88 “Test Method for Junction-to-Case Thermal Resistance Measurements of Ceramic Packages.” This test method deals only with junction-to-case or mounting surface measurements of thermal resistance and limits itself to heatsink and fluid bath testing environments. The heatsink mounting method for measuring junction-to-case thermal resistance is a conservative measure of the package's ability to transfer heat to the ambient environment, because heatsinking is provided only on one side of the package, whereas the fluid bath mounting method has the potential for equally cooling both sides of the package.

  2. SEMI G38-0996 “Test Method for Still- and Forced-Air Junction-to-Ambient Thermal Resistance Measurements of Integrated Circuit Packages.” This test method deals only with junction-to-ambient measurements of thermal resistance and limits itself to still- and forced-air convection testing environments.

  3. SEMI G68-0996 “Test Method for Junction-to-Case Thermal Resistance Measurements in an Air Environment for Semiconductor Packages.” The measurement results are usually different from the results obtained by testing in the fluid bath environment described in SEMI G30-88 and in SEMI G43-87 “Test Method for Junction-to-Case Thermal Resistance Measurements of Molded Plastic Packages,” not summarized here.

From these definitions, it should be clear that it is very difficult to define a thermal parameter that's going to apply to a package under all circumstances. Yet, for some reason, device manufacturers — even the ones following these standards — tend to gloss over this fact in their data sheets.

However, in the industry at large there was a recognition that the SEMI standards were inadequate. Consequently, in the early 1990s, EIA/JEDEC (www.jedec.org), the developer of standards for the solid-state industry, drafted its own comprehensive thermal standards.

As can be seen in the following excerpt from JESD51-2 “Integrated Circuits Thermal Test Method Environmental Conditions — Natural Convection (Still Air),” published in 1995, JEDEC recognized the same issues as SEMI:

“The purpose of this document is to outline the environmental conditions necessary to ensure accuracy and repeatability for a standard junction-to-ambient (θJA) thermal resistance measurement in natural convection. The intent of θJA measurements is solely for a thermal performance comparison of one package to another in a standardized environment. This methodology is not meant to and will not predict the performance of a package in an application-specific environment.”

So, how did JEDEC improve the situation? In my view, the most significant improvement was the definition of some new terms. In particular, farther into JESD51-2, § 4.3 states:

“… The junction-to-top center-of-package thermal characterization parameter, ΨJT, is calculated using the following equation: ΨJT = (TJss - TTss)/ PH. … The relationship between the junction-to-ambient thermal resistance, θJA , and the junction-to-top center-of-package thermal characterization parameter, ΨJT , is described by: θJA = ΨJT + ΨTA , where ΨTA equals thermal characterization parameter from top surface of package-to-air (°C/W). … The thermal characterization parameters, Ψ

It took another decade, but the logical extension of Ψ to other points of interest was codified in 2005, when JEDEC published JESD51-12 “Guidelines for Reporting and Using Electronic Package Thermal Information,” and in which can be found, among other statements:

“… The purpose of the JESD51 standards is to compare the thermal performance of various packages under standardized test conditions. While standardized thermal test information cannot apply directly to the many specific applications, the standardized results can help compare the relative thermal performance of different packages. A more meaningful comparison is possible if the test conditions are understood along with the factors affecting package thermal performance. Brief discussions of key topics are included in this guideline.”

Within the guidelines, it is noted that ΨJB was not included in the natural convection standard because it had not yet been defined, but that it may be added in the future. JESD51-12 further went on to clarify some existing terminology, in particular θJC and θJB:

“The conduction thermal resistances θJCx and θJB are measured with nearly all of the component power dissipation flowing through either the top or the bottom of the package. The values may be useful for comparing packages but the test conditions don't generally match the user's application. … θJCx is the junction-to-case thermal resistance. The ‘x’ indicates the case surface where TCase is measured and through which the heat is forced to flow during the θJCx measurement, ‘top’ for the top surface or ‘bot’ for the bottom surface. … Ideally, during θJCx measurement, close to 100% of the power flows from the junction to the ‘x’ case surface. The θJCx nomenclature is used to avoid the confusion associated with qJC. Historically, the θJC case surface is defined as the ‘outside surface of the package (case) closest to the chip mounting area when that same surface is properly heat sunk.’ This could be either the top or bottom surface, but it is not always clear which surface was used when a θJC value is reported.”

In § 5.2.3 of the JESD51-12 Guidelines, ΨJT and ΨJB thermal characterization parameters are discussed:

“The thermal characterization parameters ΨJT and ΨJB are measured by suppliers at the same time and in the same environments as θJA or θJMA. Users can apply the Ψ equations to estimate the component junction temperature in their application by measuring a component temperature in the application environment and using the appropriate Ψ thermal characterization parameter. This estimated junction temperature can then be compared with a junction temperature specification. A component power estimate is required. … Using ΨJT or ΨJB values together with package top or board temperature measurements in a system requires good temperature measurement technique, comparable to that used when the supplier measured ΨJT or ΨJB. … When a heatsink or added heat spreader is present, neither ΨJT nor ΨJB can be used to estimate the junction temperature. It can be approximated using θJCtop , measuring the heatsink temperature in the application as close to the package interface as possible, and accounting for the temperature difference across the heatsink to case interface. Alternatively, some suppliers may provide a junction-to-sink ΨJS thermal parameter that may be used analogously to ΨJT , recognizing that the ΨJS value is dependent on the package-to-heatsink interface.”

A Closer Examination

Fig. 1 and Fig. 2 introduce a very generic “thermal system” that illustrates what's different between the definitions of θ and Ψ. Referring to Fig. 1, if you can define the path along which heat flows and quantify the heat along that path, only then can you call it θ. With this restriction, it may be seen that the traditional definitions of θJA and θJC are applicable. Regarding θJA, all the heat that can flow must originate at the junction and end up at ambient (even if the path isn't pinned down, at least you know the heat can't sneak away to somewhere not encompassed by the system). Regarding θJC, you can presume that something approaching 100% of total device heat is flowing out through the heatsink, which it is supposed to do. Of course, you have to know what location on the package, exactly, is meant by the word “case” in the specifications.

On the other hand, referring to Fig. 2, if all you can do is measure the total device power and you have no idea of how much of it flows up, down, out the leads or through the air gap (pretty typical in the semiconductor packaging thermal lab), yet you really would like to measure the temperature at one or more points around the boundary of the package (which is actually pretty easy), then you can't call it a θ, but must instead call it a Ψ. Thus, JEDEC's new terms ΨJT , ΨTA, ΨJL and ΨJB are all concessions to the reality that it's easy to measure temperature and really difficult to measure heat flux — especially on miniscule semiconductor devices.

Still, you may be asking, why is this distinction between θ and Ψ so important? Take a look at the simple four-resistor package model shown in Fig. 3. I've used θ's here for the four individual resistors, because I'm defining them to be true thermal path resistances. (In particular, I'm using θJCtop as the true path resistance from the junction to the top of the package, where it's exposed to the air; in a moment, I'll relate it to the original JEDEC-defined ΨJT to represent the characterization parameter to the top center of the plastic case.)

As you see, what I've really done is to create a two-resistor “compact thermal model” of the package, along with two external resistances that digest the entire external system behavior into a very minimal set of parameters.

Taking the system as a whole, the overall thermal resistance, junction to ambient, can be written as follows:

It takes a little more effort, but if you work it out, you'll also be able to derive the following:

What this says, if you think about it, is that even when the two-package thermal resistances are absolutely and truly constant (the two-parameter compact thermal model values θJB and θJCtop), the corresponding Ψ values ΨJB and ΨJC are not constants (unless the overall ratio of the two individual paths happens to remain constant).

To put some meat onto this skeleton, look at Fig. 4, Fig. 5 and Fig. 6. All I've done to generate these graphs is hold θJB and θJCtop constant, and let θBA and θCtopA vary. (For one scenario in each graph, I let the ratio of θCtopA to θBA be fixed — and called the plot “var airflow.” In the other scenario, I held θCtopA constant and allowed θBA to vary by itself — and called the plot “var brd only.”)

Now you might not have as much as three orders of magnitude of possible variation in your particular application board or airflow. But that's not totally outrageous over all possible users and applications of a particular device. The point is that ΨJT , in particular, can easily vary by 300% in a variable airflow situation and by more than 1000% when the board resistance alone changes by a factor of 1000.

Unfortunately, ΨJT is the value very often reported by another name in data sheets in which vendors haven't been careful to define their terms; through oversight or unfamiliarity with the newer JEDEC standards, it might even be called θJC. So if you wondered what that ΨJT parameter meant when you saw it recently, now you know. And if you aren't sure how your device supplier measured and reported its data sheet thermal values, now you should care!

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