# How to measure temperature in integrated systems

As processor speeds increase and draw more power, thermal management becomes an increasingly important issue within all devices where microcontrollers are used. As thermal management becomes more significant, the techniques used become even more complex and conditional. However, when broken down to the basics, thermal management still boils down to measuring temperature and driving cooling mechanisms such as fans, hydraulic cooled systems and clock throttling.

The more accurately a temperature can be measured, the more finely tuned the cooling mechanisms can be driven. This is beneficial in saving power (for driving cooling mechanisms), increasing performance and providing a cleaner acoustic environment.

Temperature measurement methods in integrated systems are performed in a variety of ways depending on the requirements of the system. Some examples of these methods include the thermocouple, thermistor, and the most commonly used solution, thermal diodes. A thermocouple is a loop of metal with known thermal properties. A current is passed through the loop and the voltage is measured thus measuring the impedance of the loop. As a result, as the temperature increases the impedance changes.

A device very similar to a thermocouple is the thermistor. This device is constructed in such a way that its resistance is directly linear with temperature. In order to accurately measure the temperature using a thermistor, the current flowing into the device must be known and the voltage measured across the device must be very accurate.

The use of thermocouples and thermistor circuits to measure temperature is practical and inexpensive. In many applications, they are sufficient. However, because they do not easily lend themselves to delta measurements, the accuracy to which they can measure is limited.

A slightly more complex but very common method of measuring temperature is to take advantage of the forward voltage (V_{F}) behavior of a diode with respect to temperature. At any given current, the forward voltage (V_{F}) of a diode is given by equation 1 (see addendum).

A problem with measuring V_{F} directly is that the I_{S} term is highly temperature dependent, which means that an equation V_{F}(T) does not have a closed form solution and becomes very difficult to predict. Additionally, generating a precise current source that does not vary with power supply, processing variations and temperature is very difficult.

Because of the nature of logarithms, if V_{F} is measured at two separate temperatures, the difference between the two measurements will be linearly temperature dependent and the I_{S} terms will cancel. In addition, the linear term is a function of a ratio of currents that are relatively straightforward to implement and independent of the operating conditions, which can adversely affect a single current source. Equations 2 - 4 (see addendum) illustrate how this delta V_{F} term is determined with equation 4 representing the voltage that is commonly evaluated when performing temperature measurements.

Figure 1 shows the ideal relationship between Temperature and Delta V_{F}. In this curve, the η factor is exactly equal to 1.000 and the ratio of I_{F2} / I_{F1} is equal to 17. Easily illustrated, this ratio is commonly used in temperature monitors to achieve good matching characteristics.

*Figure 1 - Delta V*

_{F}vs. Temperature
The vast majority of thermal sensing diodes consist of either diode connected discrete transistors (such as a 2N3904), or the base-emitter junction of substrate transistors. The basic principles of operation for using transistors instead of diodes still apply and the diode equations above apply to transistors with V_{F} replaced by V_{BE} (or V_{EB} for a PNP type device) and I_{F} replaced by I_{C}.

For first order operation, equation 5 (see addendum) shows the relationship that is measured to detect temperature. From equation 4 (see addendum), a 1°C change in temperature corresponds to an approximate 244μV change in V_{F}. This very small value is often multiplied and then measured by an Analog to Digital converter (A/D converter) and then transmitted over a number of communications protocols to a microcontroller.

Any errors that occur in measuring this small voltage will contribute greatly to the final measured temperature. There are four primary sources of error that will affect this ΔV_{BE} number and, ultimately, the temperature reported. These sources of error are series resistance, the ideality (η) term of the diode equation, noise injection and non-ideal beta of the transistors.

Series resistance affects the measurement of the ΔV_{BE} in a direct way adding an offset that will be dependent upon the differential in absolute currents used to measure the temperature. If this series resistance is not ignored, equation 5 becomes equation 6 (see addendum).

As implied by equation 6, this series resistance produces an offset in the temperature reading that varies according to the value of the series resistance. Figure 2 shows the temperature error at a single temperature with respect to this series resistance.

*Figure 2 - Temperature Error vs. Series Resistance at 0°C*

The ideality factor term, η, is based on physical properties of the transistor construction. This term is a constant for any particular device though it can vary between individual devices. The ideality factor term contributes to error when it is not equal to 1.000, or to whatever ideality factor that the temperature monitor has been configured for. In equation 5, the temperature measurement is inversely proportional to this term and, if it varies from ideal (or expected), it will create a multiplicative term that is not constant with temperature.

Below, Figure 3 shows the effective error that is induced by various ideality factor mismatches.

*Figure 3 - Temperature Error Due to Ideality Factor (η)*

Noise injection is more difficult to quantify in absolute terms. The V_{BE} measurement is generally done at a specific frequency and passed through a low pass filter. The majority of noise injected onto diode lines is attenuated, however, because low pass filters are not ideal, the residue of noise creates a DC offset. Because the A/D converter samples the DC term of the V_{BE} measurement, some of the noise cannot be removed by common mode rejection and hence creates an offset in the temperature measurement. The magnitude of the noise error depends on the profile of the noise that is injected, as well as the filtering capabilities of the measurement circuitry.

The final source of error is introduced because most measurement diodes are, in actuality, transistors, or V_{BE} junctions. In a diode, the I_{F} term sets the forward voltage V_{F}. When using a diode connected transistor, or a substrate V_{BE} junction, the current that is being driven into the diode, and the current that is set at an exactly known ratio, is the emitter current, whereas the collector current determines the V_{BE} voltage. The relationship between emitter current and collector current is well known and is where beta comes into effect. Figure 4 shows several common configurations for connecting transistors as thermal diodes.

*Figure 4 - Diode-connected transistor configurations*

The absolute value of beta is not important and will not contribute to a measurement error. What is important is that beta is not a constant value over process variations or operating conditions. Beta changes with temperature, but more importantly, it will change as a result of the collector current that is in the transistor. This delta changes the current ratio of the measurement and thus contributes to offset in a similar way as the ideality factor term η.

As shown in equation 8 (see addendum), if β2 = β1 = β, then the second term in the natural log in the denominator is equal to 1 and equation 8 becomes equal to equation 5. Likewise, as β increases, the effect of a mismatch in beta becomes smaller. In many discrete transistor devices, beta is on the order of 50 to 100, however in substrate PNP devices, the magnitude of beta is often less than 1 and may be as low as 0.25.

As can be seen in Figure 5, beta variation of only 1 can cause as much as a 0.75 degree error. In very small geometry devices, beta can vary as much as 35% over the range of currents used.

*Figure 5 - Temperature Error vs. Beta variation*

Because each of these error sources is independent, each must be addressed independently to fully correct the measured temperature. As can be seen, even a small mismatch or error can lead to substantial error in the measured temperature. This error requires that the thermal management systems must have more margin-for-error, which in turn leads to less optimal cooling solutions. Reducing or eliminating the primary sources of error in temperature measurement can increase power savings efficiency of the thermal management systems.

**Addendum / Equation Guide**

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

**Where:**

V_{F} = forward voltage across the diode

V_{BE} = base-emitter voltage of the transistor

I_{F} = Current forced into the diode

I_{E} = Current forced into emitter of the transistor

I_{S} = Saturation current of the diode or transistor

η = Ideality factor of the diode (nominally 1.00)

k = Boltzmann's constant = 1.381e-23

q = electron charge = 1.602e-19

T = Temperature in Kelvin = Temperature in C + 273.15

**About the Author:**

*Christopher Fischbach began working as an Architecture Engineer for SMSC in 2003. He has spent the last 6 years working in the semiconductor industry in various technical and design positions at Gain Technology and then SMSC. Chris holds a BS degree in electrical engineering from the University of Arizona. *

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