# Thermal Memory Effects

As it is indicated by its name, the thermal memory effect is caused by the electrothermal coupling in the power transistor. It is a function of the power dissipated in the transistor, which directly affects the temperature of the transistor junction. As a result, the characteristics of the transistor in terms of gain and output power capability change versus these temperature variations. Given the fact that the temperature will vary more slowly than the amplitude of the signal variation, the thermal memory effect manifests and usually impacts the low frequency components of the signal below the MHz range. To analyze the thermal memory effect in transistors, one should first analyze the power dissipation and temperature change in the power amplifier circuit [23-26].

The power dissipation in a FET (Field Effect Transistor) operated in normal conditions (gate current equals to zero) is provided by:

where *v _{ds}(t*) is the drain-source voltage and

*i*is the drain current of the transistor.

_{ds}(t)In order to analyze the temperature variation in the transistor junction, thermal impedance, *Z _{th},* is defined as the ratio between the temperature rise and heat flow from the device. Figure 2.6a shows the heat dissipation in a power transistor, from the device chip to the heat sink passing through the package of the device and circuit board. Given that the heat dissipation from one stage to another is not instantaneous, a delay and discharging behavior can be modeled. These effects will result in a non-purely resistive thermal impedance model. In this model, thermal resistances,

*R*describe a steady-state behavior of the temperature while thermal capacitances,

_{th},*C*describe the dynamic behavior. Together, thermal resistors,

_{th},*R*and thermal capacitors,

_{th},*C*result in modeling temperature variation with a giving rising and falling constant

_{th},*R*

_{th}C_{th}.Using complex thermal impedances for each connection, this heat dissipation can be modeled by a set of lumped elements forming a low-pass filter topology as shown in Figure 2.6b. This modeling is in agreement with the expectations that we presented earlier in this section, which consists of the fact that thermal memory effect affects low frequency components of the signal. In practice, the low-pass filter topology will have a bandwidth varying between 100 kHz and 1 MHz depending on the nature of

**Figure 2.6 **Modeling of thermal memory effects in a power transistor. (a) Different temperatures definitions. (b) Circuit modeling of the temperature variation

the chip connection to the heat sink. The thermal memory effect then affects signal frequency components lower than 1 MHz.

Using the circuit modeling of Figure 2.6a, the temperature variation in the transistor junction can then be given by:

More precisely, the thermal impedance may vary versus frequency and the dissipated power has different frequency components. In fact, the dissipated power is the product of two signals around DC and the fundamental frequency. Such product will have components around DC, the envelope, the fundamental frequency, and second harmonic. The products around the fundamental frequency and the second harmonic are filtered out by the filter topology and only the DC and envelope components affect the temperature variation of the power transistor junction. Equation 2.20 can then be rewritten as:

where *f** _{1}* and f

_{2}are two different frequencies within the band of the modulated signal,

*Pdi*

_{ss}ipated( f*- f*

_{1}_{2}) is the part of the envelope component of the dissipated power corresponding the two different frequencies f

_{1}and f

_{2}.

*R*

_{th}(0

*Hz)*and

*Z*

_{th}(f

_{1}*- f*are the thermal resistor value at zero frequency and the thermal impedance at frequency (f

_{2})_{1}- f

_{2}); and

*p*(0

_{dissipated}*Hz)*is the DC component of the power dissipation.

The temperature variation expression of Equation 2.21 includes two terms. The first term is related to the DC dissipation and is frequency independent. The second term

**Figure 2.7 **Simplified transistor thermal modeling

is function of the envelope dissipation and is frequency dependent, which means that any change in the transistor characteristics due to the temperature variation results in frequency dependent effects or memory effects.

Much research work has investigated the temperature variation in the power transistor junction for different types of signals in order to model it and understand its effect on the generation of thermal memory effects. These activities contributed to proper modeling and linearization of the thermal memory effects. To understand better how the junction temperature of the power transistor varies as a function of the input signal and how this will affect the signal integrity, an analysis and modeling of the transistor thermal behavior in the presence of a pulsed signal is given next [27].

First, given the fact that the thermal constants related to the heat sink dissipation, *Rth_heat sink,* and *C _{th}__{heatsink}*, are too large compared to the package and chip thermal

^{constant}s ^{R}th_package^{, C}*th_package*^{, R}*th_chip*^{, and C}*th_chip ^{,}* the temp

^{erature of the heat sink}

*T _{th},* is almost equal to the ambient temperature - and hence it can be considered independent from signal variations. One can therefore ignore the effect of

*R*and

_{th}__{heatsink}*C*Moreover, in order to further simplify the analysis, one can model the joint effect of the heat dissipation in the chip and package by a set of an equivalent thermal resistor

_{th}__{heat s}i_{nk}.*R*and an equivalent thermal capacitor

_{th}*C*This results in simplifying the circuit in Figure 2.6b to the circuit in Figure 2.7. Using this simplified modeling circuit for the junction temperature variation, the relationship between the junction

_{th}.^{tem}p^{erature} *T _{JU}nction*

^{and the ambient tem}p

^{erature}

*Ta*

_{mb}ient^{is}g

^{iven b}y

^{:}

where *p _{dissipated}(t)* defined in Equation 2.19 can also be expressed as a function of the output power,

*P*and instantaneous power efficiency,

_{R}p__{oUt}(t),*n(t),*of the power amplifier by:

Equation 2.22 is a first order non-homogenous differential equation that has a general solution in the form of:

where *т = R _{th}C_{th}* and C

_{1}is a constant that can be determined by initial conditions.

**Figure 2.8 **Junction temperature variation for a step input. (a) Input signal variation versus time. (b) Corresponding junction temperature variation versus time

If the driving signal in the power amplifier is a step input signal, as shown in Figure 2.8a, the dissipated power follows a step signal shape as well (see Figure 2.8b) and can be given by:

In this case, it can be easily shown that, if *т << t _{o},* the junction temperature expression in Equation 2.24 becomes:

^{where T}*junction,L = ^{R}th^{P}L + ^{T}ambient*

^{and T}

*junction,H =*.

^{R}th^{P}H +^{T}ambientUsing similar reasoning, and by noting that the mathematical formulation will be the same independently from the sign of *(P _{H} - P_{L}),* it can be concluded that if the driving input of transistor is a pulsed signal with period

*T*as shown in Figure 2.9a, the junction temperature will have the form of Figure 2.9b.

_{o}>> т**Figure 2.9 **Junction temperature variation for a pulsed input. (a) Input signal variation versus time. (b) Corresponding junction temperature variation versus time

The junction temperature variation can be obtained from Equation 2.26 as follows:

where *AP _{disspated} = P_{H} - P_{L}* is the maximum variation in the instantaneous power dissipation in the transistor.

The variation in the junction temperature as a function of the signal level results in changes in the power amplifier complex gain, which result in distortion related to thermal memory effect. Indeed, Figure 2.10 shows the variation of the measured complex gain versus the junction temperature for a power amplifier using a 90-W LDMOS (Laterally Diffused Metal Oxide Semiconductor) transistor.

Higher temperatures result in lower gain. Therefore, in the case of a pulsed signal, when transiting from a low to a high level, the junction temperature is low and the gain is higher. During the high level cycle, the junction temperature rises exponentially and the gain drops accordingly. Similarly, when transiting from a high level to a low level, the junction temperature is high and the gain is low. During the low level

**Figure 2.10 **Complex gain variation as a function of the junction temperature. (a) Gain in dB. (b) Phase shift in degrees

cycle, the junction temperature drops exponentially and gain increases accordingly. This behavior is shown in Figure 2.11, which shows how the output power is distorted compared to the input power for a power amplifier with pulsed input. This distortion is caused uniquely by thermal memory effects.