Determine your IC's transient thermal behavior to prevent overheating

Use Spice to determine the thermal time constants of the IC package.

By Milind Gupta and Da Weng, PhD, Maxim Integrated Products -- EDN, January 6, 2010

This article presents a method for predicting thermal behavior in integrated circuits. After characterizing thermal behavior, we formulate a mathematical model that simulates transient temperatures within the chip. We introduce physical laws governing thermal behavior and evaluate them for use in the thermal-body models defined for an IC. Based on that analysis, we then propose an equivalent passive RC (resistor-capacitor) network for modeling an IC's transient thermal behavior. To illustrate an application of the proposed analysis, we derive such an RC network for an LED-driver chip. We conclude with insights on the use of this approach and some ways to speed the creation of the RC models.

You often need to know the thermal behavior of an IC, especially for the power-management ICs used in automotive applications. Operating an IC at 125ºC might trigger the thermal-shutdown circuitry or exceed the product's safe operating temperature. Without a definite method of analysis, you won't be able to offer a reliable circuit. When defining a new IC, you need a way to predict thermal shutdown or excessive die temperature.

For operation in dc mode, you can often determine the junction temperature using data-sheet parameters such as θ_JA and θ_JC. However, to find out how high the junction temperature will peak for modes other than dc—such as a power MOSFET driven by a PWM signal for controlling LEDs or a switching regulator—you need transient thermal data. Such data is not typically found in data sheets.

Laws of thermal dynamics

You can derive the required relations for temperature versus time for any object using Newton's law of cooling (Equation 1) and the law of conservation of non-latent energy (Equation 2):

(1)

where T_B is body temperature, T_A is ambient temperature, k_A is constant of proportionality (>0), and t is time.

(2)

where P is constant power generated or imparted to the body, m is mass of the body, and c is the specific heat capacity of the body. Combining these laws yields Equation 3:

(3)

For an IC, its data sheet normally lists thermal data for the package, such as θ_JA (thermal resistance). That lets you analyze the steady-state thermal equilibrium for a package, to see if it agrees with Equation 3:

(3a)

at the steady state. Therefore,

(4)

This is the same as Equation 5:

(5)

where T_B is the temperature inside the package and T_A is the ambient temperature outside. Thus,

(6)

Defining the chip as a thermal system

A clear definition of the system is very important because the thermal result depends on that definition. In the cross-section of a chip mounted on a PCB, we see at least three different materials in the path from die to environment: the die itself, the mold epoxy, and the package (). Thermal models are based on one of two cases of heat flow, depending on the location of the dominant heat source: from an external source to the die (when the external source is the dominant heat generator), and from the die to the environment (when the die is the dominant heat generator).

Heat flow from an external source to the chip

Consider the system of , which shows a uniform body gaining energy from a power source and losing energy to the environment. This system also models thermal transients in the chip for heat sources outside the package because heat reaches the internal die through the package and the mold compound. The package normally has a much higher thermal resistance than that of the die itself because the die has lots of metal on it. The die, therefore, tracks the package temperature with almost no lag, causing the chip to behave as a single body. We can define this single-body system immediately, using Equation 3. Solving for T_B, we have:

(7)

where k_o is the constant of integration, which is solved according to the initial conditions. In general, this equation is useful for defining the thermal transient of a chip when the heat source is outside the chip.

As an example, suppose we want to determine the thermal transient for a chip whose initial temperature is T_i. Substituting t=0 and T_B=T_i in Equation 7,

(8)

Thus,

(9)

Considering the special case for which T_i=T_A,

(10)

Using Equation 6, we can re-write equations 9 and 10 as equations 11 and 12.

(11)
(12)

Equations 11 and 12 are useful for predicting either package or die chip temperature when the heat-generating source is outside the package. One example could be a nearby high-current MOSFET that dissipates lots of heat.

When we know k_A and θ_JA we can calculate the temperature at different times, or if P is a complex function of time we can use software such as MATLAB to write a program that plots temperature as a function of time, using the above equations to evaluate temperature as a time-based simulation.

The θ_JA value is provided in data sheets, but for setups that impose conditions other than those of the JEDEC standard, the use of that value for these calculations may cause errors. JEDEC standard 51-3 states: "It should be emphasized that values measured with these test boards cannot be used to directly predict any particular system application performance, but are for the purposes of comparison between packages" (Reference 2). Thus, to properly estimate temperature, you should either measure θ_JA for the prototype board or estimate it directly as shown in subsequent sections.

Heat flow from die to environment

Consider the system of , in which a three-body system similar to a chip generates heat on the die and dissipates it through the epoxy and package to the environment. Body 1 is the die, body 2 is the epoxy, and body 3 is the chip package. To solve this system, you must define the equations for all three bodies.

For body 1:

(13)

For body 2:

(14)

For body 3:

(15)

Where T_B1, T_B2, T_B3 are the instantaneous temperatures of bodies 1, 2, and 3, P₁₂ is power in the form of heat, transferred from body 1 to body 2, and P₂₃ is power in the form of heat, transferred from body 2 to body 3. Power generated by the die minus power absorbed by the die is

(16)

Power received by the epoxy minus power absorbed by the epoxy is

(17)

Substituting equations 16 and 17 into equations 13, 14, and 15 yields

(18)
(19)
(20)

The solution of this system in equations 18, 19, and 20 is eased by the use of Laplace transforms. The form of the solution is

(21)

where θ₁₂ is thermal resistance from body 1 to body 2, θ₂₃ is the thermal resistance from body 2 to body 3, θ_3A is the thermal resistance from body 3 to the environment, T₁, T₂, T₃ are the constants of integration, and m₁, m₂, m₃ are functions of k₁, k₂, k₃.

Equation 21 predicts die temperature in a very accurate way when the die is generating power. To use this equation you must know all the constants of integration plus m₁, m₂, m₃, which are complicated functions whose solution is difficult. To avoid this, we utilize a tool for solving differential equations to which most circuit engineers have easy access: Spice. In the next section we propose a circuit modeled by similar differential equations, then simulate the circuit and read out temperatures from the simulation.

RC network models thermal-transient differential equations

You can model the differential equations (equations 18, 19, and 20) with a simple RC network that represents power generated on the die (). Initial voltages on the capacitors represent initial temperatures of the die (C₁), the epoxy (C₂), and the package (C₃). V_A represents the ambient temperature of the environment, and current I_s going into capacitor C₁ represents power generated on the die. The differential equations representing voltages on the capacitors are as follows: (22)
(23)
(24)

These three equations correspond to equations 18, 19, and 20, with the following substitutions of variables:

 (24a)

The capacitor voltages correspond directly to temperatures of the die, epoxy, and package. Any Spice package can simulate the RC circuit. When we know the proper values of R₁, R₂, R₃, C₁, C₂, and C₃ modeled for a particular chip, we can then simulate the circuit to directly read out die temperature as the voltage on capacitor C₁. To determine the passive component values for a particular chip (R₁, R₂, R₃, C₁, C₂, and C₃), we rearrange Equation 5 to obtain θ_JA, thermal resistance for the system, by measuring the die's steady-state final temperature:

(25)

where T_J is the steady-state junction temperature of the die, T_A is the ambient temperature, and P_G is power dissipated on the die.

Operating with the same dissipated power, P_G, as in step 1, you can create a data set for the chip's transient temperature variation by measuring the die temperature at different times starting at time 0. Then, based on the following constraint, do a curve-fitting exercise on the measured data to determine the values of R₁, R₂, R₃, C₁, C₂, and C₃:

(26)

Measuring the die temperature

There are a couple of practical methods to measure the die temperature of an integrated circuit (Reference 4). Here we will use the ESD diode forward-drop measurement method to get the chip temperature since it is easy and would not introduce a large amount of error. However, for a particular chip the die temperature measurement technique should be chosen carefully to make sure that the accuracy levels of the measurement remain within acceptable limits. Some steps and points to consider in summary are as follows (Reference 4):

• Make sure that the ESD diode chosen for measurement does not have any large parasitic resistance and a large current flow that would offset the diode drop read-out. It is best to discuss this with the IC manufacturer to determine the estimated maximum internal bond-wire and metallization resistance.

• Also make sure the ESD diode is near the hotspot of the chip or in the area where you are actually concerned with the temperature to provide better estimations of the temperature and deliver more accurate results.

• If choosing a FET's on-resistance as a temperature indicator, make sure at the measurement point that the FET is fully on and in the dropout mode.

To measure die temperature with the ESD diode forward-drop approach, we need a diode on the chip to which we can apply forward bias and measure its voltage. That can easily be done on most chips with an ESD diode connected between a pin and the supply voltage. Because measured data gives us the diode voltage, we must consider and include the relationship of a diode voltage with temperature (Reference 3).

Diode voltage decreases with a nearly constant slope and negligible deviation. If plotted with respect to temperature, it looks like . (T_A and V_DA are the ambient temperature and diode voltage at ambient temperature.) We therefore know one point on the graph and its slope. Slope can be derived by measuring the diode voltage at different temperatures in a temperature-controlled oven, or you can use a number like 2 mV/ºK, which is valid with minimal error for a wide range of diode currents (Reference 3). These numbers should apply to any other chip as well, but for accuracy it is always better to measure the slope associated with the current intended for biasing the diode. Any temperature can now be represented in terms of the diode voltage: (27)

where T is the temperature for which the diode voltage is V_D, and s is the slope of the graph (s&0). Substituting this expression in equations 11 and 12 yields the following:

(28)
(29)

Substituting also in equations 18, 19, and 20:

(30)
(31)
(32)

To apply our RC network properly in curve fitting the measured voltage-transient data for the diode, all we need to do is set the magnitude of the current source as

(33)

Because s&0, you can realize Equation 33 by reversing the current source direction and setting its magnitude to |sP_G|.

Experimental determination and verification of RC network

To demonstrate a practical application of the simulation model using the equations derived above, we note some experimental results using a linear LED driver. These chips operate up to 40V using few external components and supply an LED string with up to 200 mA (). Note that the IC can dissipate lots of heat if the internal MOSFET sees high current combined with a large dropout voltage (as it does when the LED string's forward voltage is low). The voltage across R_SENSE (between CS+ and GND) is regulated to 200 mV ±3.5%, which allows that resistor to set the LED current. The chip's DIM input provides a wide range of PWM dimming for the LEDs, and because it also withstands high voltages, it can connect directly to the IN pin.

To obtain a direct indication of the die temperature, we measure the forward-bias voltage of an internal ESD diode connected between the DIM and IN pins. This diode is biased at ~100 μA, causing its forward voltage to vary 2 mV/ºK, as can be confirmed by heating the part in a controlled-temperature oven. shows the setup for these experiments.

The 5V source and 56-kΩ resistor provide 100 μA for forward-biasing the ESD diode, and the driver is programmed to deliver 200 mA of output current for the LEDs.

In this state, the part carries a lot of current. Our ESD diode measurement is in the path of that measurement, so we will get some error due to the parasitic resistance of the bond wire and internal metallization. From the internal layout and bond-wire length calculation, the worst-case parasitic resistance is estimated to be 50 mΩ. With 200 mA this will cause an error of around 10 mV maximum in our diode reading and so our accuracy error will be larger than 5ºC. Additionally, the ESD diode on the die is placed adjacent to the on-chip power MOS device and thermal shutdown circuitry. This makes the diode the best indicator of that region's temperature.

The following section describes how you can use the test setup to capture thermal-transient diode voltages for use in the system-definition equations mentioned in equations 7 and 21.

System definition 1


To calculate k_A and θ_JA (for substitution in Equation 11), we heat the chip using a hot-air gun. The chip should be powered off because we don't want internal heat generation. Heating the part with a hot-air gun causes the temperatures of the package and die to rise together. You can monitor the die's temperature change by measuring the diode voltage on a scope ().

As we see in the scope output, the diode voltage decreases when we heat the chip, with an exponential rate of change as the equation predicts. Near the center of the curve the hot-air gun is switched off, causing the package and die to begin cooling and the diode voltage to rise, again following an exponential curve.

We don't know exactly how much heat is being imparted from the heat gun to the chip, so to avoid dealing with that unknown we first adjust Equation 28 to fit only the rising part of the curve, which represents a chip that is cooling after being heated (). This curve-fitting exercise lets us estimate the best value for k_A. Because the rising part of the curve represents cooling with no heat power transferred to the package, the package is simply cooling down, with P=0. Equation 28 therefore simplifies to

(34)

We know the values for V_DA (643 mV from the initial measurement at room temperature) and V_Di (the reading for t=0 reference), so to determine k_A, we must just adjust the equation so it includes a couple of readings on the rising curve. This exercise yields k_A=–0.0175. A graph of the readings (diode voltages in mV, with respect to time in seconds) and Equation 34 with the above k_A is shown in .

As we can see in the graph, Equation 34 closely follows the measured data for k_A=–0.0175. To verify that our equations are correct, we try to fit the falling curve on Equation 28 with the value we determined for k_A. The equation fits very accurately (). Thus, we see that the System 1 equation closely matches the experimental data.

System definition 2

Verification of the System 2 equations is more difficult. We must generate heat on the die, measure the die temperature using the diode forward voltage, and fit that temperature value to a simulated value for the C₁ voltage of the proposed RC network. This task is accomplished by writing a program using MATLAB.

It is important to record the thermal transient at a time for which the initial temperature of the whole chip is known so that we also know the initial capacitor voltages for solving the RC network. In the test setup (), we now turn on the current and capture the diode voltage on a scope ().

Similar transients are recorded for three different power-dissipation levels, and one curve is fitted to that data. The result of fitting from the first set of data, in which the power dissipation is 1.626W, is the circuit in . The graph in compares the measured and fitted data. Similarly, the graph in shows how the RC network fits the second set of readings (power dissipation of 2.02W), and the graph in shows how it fits the third set of readings (power dissipation of 1.223W).

These experimental results show that the measured results closely match the theoretical model. Such modeling is useful for simulating the transient temperature of an IC once you've modeled the RC network for that particular chip. The model can also be used for chips of similar size to determine their thermal characteristics during their definition phase. That capability can give you an idea about limitations of operation for the chip, which in turn helps you define the chip's operation modes to prevent overheating.

Conclusion

The above sections have outlined and described a way to model the thermal behavior of a chip as an RC network that can then be simulated easily using a Spice tool. To improve the accuracy of this model, you should take data at both extremes of power dissipation and for one level in the middle. Fitting the RC network to all three levels simultaneously makes the model usable for most practical power-dissipation levels. It also improves model accuracy when you measure the data at different ambient temperatures. These exercises should improve accuracy when necessary, but most applications do not require that you know the temperature with high accuracy. Applications and design engineers as well as system designers may find the method useful. To provide better and more detailed chip information, a company can create RC network models for its ICs and make them available with the chips' corresponding SPICE models.

References

Package Thermal Resistance Values (θJA, θJC) for Dallas Semiconductor Temperature Sensors, http://www.maxim-ic.com/appnotes.cfm/an_pk/3930. "Package Thermal Characteristics," Actel Application Note, http://www.actel.com/documents/Pack_Therm_AN.pdf. "What's All This VBE Stuff, Anyhow?" Bob Pease, National Semiconductor, http://www.national.com/rap/Story/vbe.html. "Hot, cold, and broken: thermal-design techniques," Paul Rako, technical editor, EDN, http://www.edn.com/article/CA6426879.html.