Patent Publication Number: US-2017351279-A1

Title: Electronic device and processor control method

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application is based upon and claims the benefit of priority of the prior Japanese Patent Application No. 2016-113787, filed on Jun. 7, 2016, the entire contents of which are incorporated herein by reference. 
     FIELD 
     The embodiments discussed herein relate to an electronic device and a processor control method. 
     BACKGROUND 
     Currently, electronic devices, such as a smartphone and a tablet terminal, are becoming widespread and are making progress in multi-functionalization and performance improvement. The heat generation amount of a processor provided in an electronic device increases with such multi-functionalization and performance improvement. On the other hand, it is not easy to increase the cooling capacity of the electronic device, due to shape restriction. Hence, when the processor operates under a high load for a long time, the processor is not cooled enough, and heat is transferred from the processor to a housing surface so as to increase a surface temperature, in some cases. 
     As a clock frequency becomes higher, the heat generation amount of the processor becomes larger, in many cases. In a conceivable solution, the processor is configured to be operable at a plurality of clock frequencies, and the electronic device restricts the clock frequency in such a manner that the surface temperature does not exceed a threshold value. However, the processor heat transfers to the housing surface after a delay, and even if the clock frequency is rapidly reduced, the surface temperature does not stop rising immediately. Thus, when there is a possibility that the surface temperature starts rising and exceeds the threshold value, the electronic device restricts the clock frequency before the surface temperature exceeds the threshold value, in some cases. 
     For example, a dynamic temperature control device that controls the clock frequency of the processor is proposed. The proposed dynamic temperature control device includes a sensor that measures a heat source temperature, another sensor that measures a circumference temperature, and a controller that restricts the clock frequency of the processor. The controller estimates the highest temperature expected to be reached when the current clock frequency continues, from the heat source temperature, the circumference temperature, and a predetermined calculation formula. When the estimated highest reached temperature exceeds a threshold value, the controller lowers the clock frequency to a level at which the highest reached temperature does not exceed the threshold value. 
     See, for example, Japanese Laid-open Patent Publication No. 2012-74064. 
     In a steady state in which the surface temperature is stabilized by the restriction of the clock frequency, the surface temperature converges around the threshold value, and the clock frequency of the processor is expected to be at a certain level corresponding to the threshold value of the surface temperature. However, there is a problem of how to control the clock frequency during a transient state from the temperature rising start to the steady state of the surface temperature. 
     A heat capacity exists between the processor and the housing surface, and thus it is possible that, even if the clock frequency is not rapidly reduced to a level corresponding to the steady state, the surface temperature does not exceed the threshold value. On the contrary, it is concerned that the processor performance decreases excessively, if the clock frequency is immediately reduced to the level corresponding to the steady state when a possibility of the surface temperature exceeding the threshold value is detected, as in the technology described in above Japanese Laid-open Patent Publication No. 2012-74064. 
     SUMMARY 
     According to one aspect, there is provided an electronic device including: a memory; a temperature sensor; a housing that contains the temperature sensor; and a controller connected to the memory, contained in the housing, including a processor capable of operating at a plurality of clock frequencies, and configured to perform a procedure including: calculating a temperature index value of a temperature of a surface of the housing by using the temperature sensor, and calculating a prediction reached value that the temperature index value is predicted to reach a certain amount of time or more later from change of the calculated temperature index value, calculating a rate parameter indicating a target of a decrease rate for decreasing the prediction reached value, based on the prediction reached value calculated at a certain time point and a threshold value of the temperature index value, and controlling a clock frequency of the processor before the certain amount of time elapses, based on the calculated rate parameter and correspondence information that associates a plurality of target values of the prediction reached value and the plurality of clock frequencies. 
     The object and advantages of the invention will be realized and attained by means of the elements and combinations particularly pointed out in the claims. 
     It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory and are not restrictive of the invention. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  illustrates an example of an electronic device of a first embodiment; 
         FIG. 2  is a block diagram illustrating exemplary hardware of a portable terminal device; 
         FIG. 3  is a block diagram illustrating exemplary hardware of a design apparatus; 
         FIG. 4  illustrates an exemplary arrangement of heat sources and temperature sensors; 
         FIG. 5  illustrates an example of a heat circuit model that exhibits a transient response of heat conduction; 
         FIG. 6  illustrates an example of heat conduction from a plurality of heat sources; 
         FIG. 7  is a graph illustrating first exemplary change of relative temperatures of a plurality of temperature sensors; 
         FIG. 8  is a graph illustrating second exemplary change of relative temperatures of a plurality of temperature sensors; 
         FIG. 9  illustrates an example of an intermediate parameter table; 
         FIG. 10  illustrates an example of a parameter table; 
         FIG. 11  is a graph indicating an estimation example of relative temperatures of a plurality of heat sources; 
         FIG. 12  illustrates an example of a past data table; 
         FIG. 13  is a graph indicating an estimation example of contribution to a housing surface from heat sources; 
         FIG. 14  is a graph indicating an estimation example of a surface temperature; 
         FIG. 15  illustrates an example of a correspondence information table; 
         FIG. 16  is a block diagram illustrating exemplary functions of a portable terminal device and a design apparatus; 
         FIG. 17  is a flowchart illustrating an exemplary procedure of parameter decision; 
         FIG. 18  is a flowchart illustrating an exemplary procedure of heat source control; 
         FIG. 19  is a graph illustrating exemplary change of T sat (t) and T surf-1 (t) corresponding to a plurality of values of α; 
         FIG. 20  is a graph illustrating exemplary change of T sat (t) and T surf 1 (t) when a rate parameter is not used; and 
         FIG. 21  illustrates a control example of a surface temperature of a housing. 
     
    
    
     DESCRIPTION OF EMBODIMENTS 
     Several embodiments will be described below with reference to the accompanying drawings, wherein like reference numerals refer to like elements throughout. 
     First Embodiment 
     A first embodiment will be described.  FIG. 1  illustrates an example of an electronic device of the first embodiment. 
     The electronic device  1  of the first embodiment has a housing surface  2  which is touched by a person. The electronic device  1  is a portable terminal device, such as a smartphone, a mobile phone, a tablet terminal, and a notebook computer. The electronic device  1  restricts a clock frequency of a processor  4   a  to prevent the temperature of the housing surface  2  from rising too high, in some cases. 
     The electronic device  1  includes a temperature sensor  3 , a control unit  4 , and a memory unit  5 . The temperature sensor  3 , the control unit  4 , and the memory unit  5  are located inside the housing of the electronic device  1 , for example. The temperature sensor  3 , the control unit  4 , and the memory unit  5  may be located on a substrate inside the housing. The control unit  4  includes the processor  4   a . The memory unit  5  may be integrated in the control unit  4 . 
     The temperature sensor  3  is a sensor device that measures a temperature at a position where the temperature sensor  3  is located. The temperature sensor  3  is a thermistor, for example. The electronic device  1  may be provided with a plurality of temperature sensors located at different positions, including the temperature sensor  3 . 
     The processor  4   a  is a computing device capable of operating at a plurality of clock frequencies. The processor  4   a  is a central processing unit (CPU), a digital signal processor (DSP), or a graphics processing unit (GPU), for example. The control unit  4  may include another processor and may include an application specific electronic circuit, such as an application specific integrated circuit (ASIC) and a field programmable gate array (FPGA). The control unit  4  controls the clock frequency of the processor  4   a  by using the processor  4   a , another processor, or another electronic circuit. When the processor  4   a  or another processor is used, the control unit  4  may execute a processor control program that describes a below procedure. The processor control program is stored in the memory unit  5  or another memory device. 
     The memory unit  5  stores correspondence information  5   a . The memory unit  5  may be a volatile semiconductor memory, such as a random access memory (RAM), or may be a non-volatile storage, such as a flash memory. The correspondence information  5   a  will be described later. 
     The control unit  4  calculates a temperature index value of the temperature of the housing surface  2  by using the temperature sensor  3 . The temperature index value indicates a temperature difference between the temperature of the housing surface  2  and a reference temperature, for example. The reference temperature is an outside air temperature around the electronic device  1 , for example. Note that the electronic device  1  may include a plurality of heat sources including the processor  4   a . In that case, the control unit  4  may separately estimate a temperature difference generated by heat generation of the processor  4   a , which is part of the temperature difference between the temperature of the housing surface  2  and the reference temperature, by using a plurality of temperature sensors of different positions. 
     The control unit  4  calculates a prediction reached value  6  that the temperature index value is predicted to reach a certain amount of time or more later, from temporal change of the calculated temperature index value. The prediction reached value  6  indicates the highest value that is reached when a current change tendency of the temperature index value continues, for example. For example, the control unit  4  calculates the prediction reached value  6  from the temperature index value at a certain time point, an increase amount of the temperature index value per unit time, and a predetermined estimation equation. The control unit  4  may further use a delay parameter (for example, a thermal time constant) indicating delay of heat transfer from the processor  4   a  to the housing surface  2  in order to calculate the prediction reached value  6 . 
     The control unit  4  calculates a rate parameter  8  indicating a target decrease rate for decreasing the prediction reached value  6  on the basis of the prediction reached value  6  calculated at a certain time point and a threshold value  7  of the temperature index value. The control unit  4  may calculate the rate parameter  8  only when the prediction reached value  6  calculated at the certain time point is larger than the threshold value  7 . The rate parameter  8  is a time constant indicating a rate at which the prediction reached value  6  converges to the threshold value  7 , for example. The control unit  4  may further use the delay parameter indicating delay of heat transfer from the processor  4   a  to the housing surface  2  in order to calculate the rate parameter  8 . For example, the control unit  4  calculates a decrease rate a by dividing the threshold value  7  by the prediction reached value  6  and calculates the rate parameter  8  which is a product of α and the thermal time constant from the processor  4   a  to the housing surface  2 . 
     The threshold value  7  is a temperature index value corresponding to an allowable heat generation amount for the processor  4   a . For example, the control unit  4  calculates the temperature of the housing surface  2  at a certain time point and decides the threshold value  7  on the basis of the temperature difference (increase margin amount) between a predetermined allowable temperature and the calculated temperature of the housing surface  2  and the temperature index value at the certain time point. When the electronic device  1  includes a plurality of heat sources, the control unit  4  decides the threshold value  7  by adding the increase margin amount to a temperature difference generated by heat generation of the processor  4   a , for example. 
     The control unit  4  controls the clock frequency of the processor  4   a  before a certain amount of time elapses, on the basis of the correspondence information  5   a  and the rate parameter  8 . The correspondence information  5   a  associates a plurality of target values of the prediction reached value  6  with a plurality of clock frequencies of the processor  4   a . A certain target value is reached by the temperature index value when the processor  4   a  continues operating at a certain clock frequency, for example. The correspondence information  5   a  may be generated in advance and written into the electronic device  1 . The control unit  4  lowers the upper limit of the clock frequency of the processor  4   a  in a phased manner such that the prediction reached value  6  decreases at a decrease rate according to the rate parameter  8  with elapse of time, for example. For example, the control unit  4  decides the target value of the prediction reached value  6  of a predetermined time later on the basis of the rate parameter  8 , and sets the upper limit at the clock frequency associated with the decided target value. 
     Here, when time elapses sufficiently and the electronic device  1  reaches a steady state, it is expected that the temperature of the housing surface  2  stops rising, and that the calculated temperature index value converges to the threshold value  7 . In this case, it is expected that the prediction reached value  6  converges to the threshold value  7 , and that the clock frequency of the processor  4   a  or its upper limit converges to a level corresponding to the threshold value  7 . On the other hand, in a transient state in which the temperature of the housing surface  2  is rising, the clock frequency of the processor  4   a  is restricted in accordance with a decrease rate indicated by the rate parameter  8  and does not necessarily decrease to the level corresponding to the threshold value  7  rapidly. 
     According to the electronic device  1  of the first embodiment, the temperature index value of the temperature of the housing surface  2  is calculated by using the temperature sensor  3 , and the prediction reached value  6  which the temperature index value is predicted to reach a certain amount of time or more later is calculated. The rate parameter  8  indicating the target of the decrease rate at which to decrease the prediction reached value  6  is calculated on the basis of the prediction reached value  6  calculated at a certain time point and the threshold value  7 . The clock frequency of the processor  4   a  is controlled on the basis of the calculated rate parameter  8  and the correspondence information  5   a  that associates a plurality of target values of the prediction reached value  6  with a plurality of clock frequencies. 
     Thereby, the clock frequency is needless to be restricted excessively as compared with a case in which the clock frequency is rapidly decreased to the level corresponding to the threshold value  7  when the prediction reached value  6  exceeds the threshold value  7 . That is, when the heat capacity between the processor  4   a  and the housing surface  2  is taken into consideration, even if the clock frequency exceeds the level corresponding to the threshold value  7  temporarily in the transient state, the subsequent temperature index value does not exceed the threshold value  7  in some cases, and thus there is much time to converge the clock frequency. The restriction of the clock frequency is made gradual by utilizing this transient state, and thereby the clock frequency is needless to be restricted excessively, and the performance of the processor  4   a  is prevented from decreasing. 
     Second Embodiment 
     Next, a second embodiment will be described.  FIG. 2  is a block diagram illustrating exemplary hardware of a portable terminal device. 
     The portable terminal device  100  of the second embodiment has a surface which is touched by a user. The portable terminal device  100  is a smartphone, a mobile phone, a tablet terminal, or a notebook computer, for example. Note that the portable terminal device  100  corresponds to the electronic device  1  of the first embodiment. 
     The portable terminal device  100  includes a control unit  111 , a RAM  112 , a non-volatile memory  113 , a wireless interface  114 , a display  115 , a touch panel  116 , an image capturing device  117 , a speaker  118 , a microphone  119 , and a medium reader  120 . Also, the portable terminal device  100  includes an electric power control unit  121 , temperature sensors  122   a ,  122   b ,  122   c , and  122   d , a battery  123 , and a charge circuit  124 . 
     The control unit  111  controls the portable terminal device  100 . The control unit  111  includes CPUs  111   a  and  111   b , a DSP  111   c , and a GPU  111   d.    
     Each of the CPUs  111   a  and  111   b  is a processor that includes a computation circuit that executes commands of programs. Each of the CPUs  111   a  and  111   b  loads at least a part of the programs and data stored in the non-volatile memory  113  into the RAM  112 , and executes the programs. Each of the CPUs  111   a  and  111   b  may include a plurality of CPU cores. A procedure of the second embodiment may be executed in parallel by using the CPUs or the CPU cores. 
     The DSP  111   c  processes digital signals. For example, the DSP  111   c  processes a transmission signal transmitted from the wireless interface  114  and a reception signal received by the wireless interface  114 . Also, for example, the DSP  111   c  processes an audio signal output to the speaker  118  and an audio signal input from the microphone  119 . The GPU  111   d  processes an image signal. For example, the GPU  111   d  generates an image to be displayed on the display  115 . 
     The RAM  112  is a volatile semiconductor memory that temporarily stores the programs executed by the CPUs  111   a  and  111   b  and the data used in computation. This data includes the correspondence information. The correspondence information associates a plurality of target values of a prediction reached value which the temperature index value of the housing surface temperature of the portable terminal device  100  is predicted to reach a certain amount of time or more later, with a plurality of clock frequencies of the processor. The processor is one or more of the CPUs  111   a  and  111   b , the DSP  111   c , and the GPU  111   d . Note that the portable terminal device  100  may include a memory of a type other than the RAM  112  and may include a plurality of memories. 
     The non-volatile memory  113  is a non-volatile memory device that stores software programs, such as an operating system (OS), middleware, and application software, as well as data. The programs include a surface temperature calculation program for estimating the surface temperature of the portable terminal device  100  and a processor control program for controlling the clock frequency of the processor. The non-volatile memory  113  is a flash memory or a solid state drive (SSD), for example. Note that the portable terminal device  100  may include a non-volatile memory device of another type, such as a hard disk drive (HDD). 
     The wireless interface  114  is a communication interface for communicating with other communication devices, such as a base station and an access point, via a wireless link. Note that the portable terminal device  100  may include a wired interface for communicating with other communication devices, such as a switch and a router, via a wired cable. The wireless interface  114  includes an electric power amplifier  114   a  that amplifies a transmission signal. The electric power amplifier  114   a  is sometimes simply referred to as an amplifier and is sometimes referred to as a high power amplifier (HPA). 
     The display  115  displays an image in accordance with a command from the control unit  111 . For example, a liquid crystal display (LCD), an organic electro-luminescence (OEL) display, or the like is used as the display  115 . 
     The touch panel  116  is located over the display  115 . The touch panel  116  detects touch operation of the user on the display  115 . The touch panel  116  detects a position touched by a finger or a touch pen and reports the detected position to the control unit  111 . For example, a matrix switch method, a resistance film method, a surface acoustic wave method, an infrared light method, an electromagnetic induction method, an electrostatic capacitance method, or the like is used as a position detection method. Note that the portable terminal device  100  may include another input device, such as a keypad. For example, the keypad includes one or two or more input keys. The keypad detects user&#39;s pressing of an input key and reports the pressed input key to the control unit  111 . 
     The image capturing device  117  captures a still image or a moving image. For example, a charge coupled device (CCD) sensor, a complementary metal oxide semiconductor (CMOS) sensor, or the like is used as an image sensor. The image capturing device  117  stores image data of a captured image in the RAM  112  or the non-volatile memory  113 . 
     The speaker  118  acquires an electrical signal as an audio signal from the control unit  111  and reproduces sound by converting the electrical signal to a physical signal. For example, when the user talks through a telephone, voice of an opposite-end person and background noise are reproduced. The microphone  119  converts a physical sound signal to an electrical signal and outputs the electrical signal as an audio signal to the control unit  111 . For example, when the user talks through a telephone, voice of the user and background noise are input from the microphone  119 . 
     The medium reader  120  is a reader device that reads programs and data recorded in a storage medium  120   a . The storage medium  120   a  is, for example, a flash memory, a magnetic disk such as a flexible disk (FD) and an HDD, an optical disc such as a compact disc (CD) and a digital versatile disc (DVD), a magneto-optical disk (MO), or the like. The medium reader  120  stores the programs and data read from the storage medium  120   a  in the RAM  112  or the non-volatile memory  113 . 
     The electric power control unit  121  controls electric power consumption of the control unit  111  and the wireless interface  114  by changing operation levels of the control unit  111  and the wireless interface  114 . As the operation level becomes higher, the electric power consumption becomes larger, and the heat generation amount becomes larger. For example, the electric power control unit  121  changes the clock frequency of the CPUs  111   a  and  111   b . As the clock frequency becomes higher, the computing powers of the CPUs  111   a  and  111   b  becomes greater, and the electric power consumption and the heat generation amount become larger. Also, for example, the electric power control unit  121  changes the communication rate of the wireless interface  114 . As the communication rate becomes higher, the electric power consumption and the heat generation amount of the electric power amplifier  114   a  become larger. 
     Each of the temperature sensors  122   a ,  122   b ,  122   c , and  122   d  measures a temperature at a position where the temperature sensor is located. Thermistors are used as the temperature sensors  122   a ,  122   b ,  122   c , and  122   d , for example. The temperature sensor  122   a  is located near the CPU  111   a . The temperature sensor  122   b  is located near the charge circuit  124 . The temperature sensor  122   c  is located near the electric power amplifier  114   a . The temperature sensor  122   d  is located near the battery  123 . The temperature sensors  122   a ,  122   b ,  122   c , and  122   d  report measured temperatures to the control unit  111 . 
     The battery  123  is a secondary battery that is chargeable and dischargeable repeatedly. The charge circuit  124  accumulates electrical energy in the battery  123 . The battery  123  supplies the accumulated electrical energy to components of the portable terminal device  100 . For example, the battery  123  supplies the electrical energy to the CPUs  111   a  and  111   b  and the wireless interface  114 . The charge circuit  124  acquires the electrical energy from an external power supply located outside the portable terminal device  100  and charges the battery  123  with the electrical energy. The charge circuit  124  charges the battery  123  with the electrical energy when the portable terminal device  100  is connected to the external power supply. 
     Here, it is possible that the user touches the surface of the portable terminal device  100 , and thus it is preferable that the surface temperature of the portable terminal device  100  do not become too high. Thus, the portable terminal device  100  estimates the surface temperature by using the temperature sensors  122   a ,  122   b ,  122   c , and  122   d . An estimation equation for estimating the surface temperature is generated by a design apparatus in advance and is stored in the portable terminal device  100 . 
       FIG. 3  is a block diagram illustrating exemplary hardware of a design apparatus. The design apparatus  200  of the second embodiment generates an estimation equation used in estimating the surface temperature of the portable terminal device  100 . The estimation equation generated by the design apparatus  200  is stored in the non-volatile memory  113  of the portable terminal device  100  in advance. Note that the estimation equation may be transmitted from the design apparatus  200  or another device to the portable terminal device  100  via a network. The design apparatus  200  may be a client apparatus, such as a client computer operated by the user, and may be a server apparatus, such as a server computer. The design apparatus  200  includes a CPU  211 , a RAM  212 , an HDD  213 , an image signal processing unit  214 , an input signal processing unit  215 , a medium reader  216 , and a communication interface  217 . 
     The CPU  211  is a processor that includes a computation circuit that executes commands of programs. The CPU  211  loads into the RAM  212  at least a part of programs and data stored in the HDD  213  and executes the programs. The RAM  212  is a volatile semiconductor memory for temporarily storing the programs executed by the CPU  211  and the data that the CPU  211  uses in computation. The HDD  213  is a non-volatile memory device that stores software programs, such as an OS, middleware, and application software, and data. Note that the design apparatus  200  may include a memory device of another type, such as a flash memory and an SSD. 
     The image signal processing unit  214  outputs an image to a display  221  connected to the design apparatus  200  in accordance with a command from the CPU  211 . The input signal processing unit  215  acquires an input signal from an input device  222  connected to the design apparatus  200  and outputs the input signal to the CPU  211 . The input device  222  may be a pointing device, such as a mouse, a touch panel, and a touch pad, a keyboard, a remote controller, a button switch, or the like. Also, a plurality of types of input devices may be connected to the design apparatus  200 . 
     The medium reader  216  is a reader device that reads programs and data recorded in a storage medium  223 . The storage medium  223  is a magnetic disk such as a flexible disk and an HDD, an optical disc such as a CD and a DVD, a magneto-optical disk, or a semiconductor memory, for example. The medium reader  216  stores the programs and data read from the storage medium  223  in the RAM  212  or the HDD  213 , for example. 
     The communication interface  217  is connected to a network  224  and communicate with other devices via the network  224 . The communication interface  217  may be a wired communication interface connected by a cable and a communication element such as a switch, or may be a wireless communication interface connected by a wireless link with a base station. 
     Next, heat transfer in the portable terminal device  100  will be described.  FIG. 4  illustrates an exemplary arrangement of the heat sources and the temperature sensors. 
     The portable terminal device  100  includes a housing  101  and a substrate  102 . The housing  101  surrounds the components of the portable terminal device  100 . The outside of the housing  101  may be touched by the user. The substrate  102  is located inside the housing  101 . At least some of the components of the portable terminal device  100  are located on the substrate  102 . Lines (for example, copper lines) that electrically connect two or more components are formed on the substrate  102 . 
     The CPU  111   a , the charge circuit  124 , and the electric power amplifier  114   a  are located on the substrate  102 . The CPU  111   a , the charge circuit  124 , and the electric power amplifier  114   a  are components that generate comparatively large amounts of heat and may be referred to as heat sources. The temperature sensors  122   a ,  122   b , and  122   c  are located on the substrate  102  beside these heat sources. The temperature sensor  122   a  is at the closest position to the CPU  111   a  among the heat sources. The temperature sensor  122   b  is at the closest position to the charge circuit  124  among the heat sources. The temperature sensor  122   c  is at the closest position to the electric power amplifier  114   a  among the heat sources. 
     Also, the portable terminal device  100  includes the temperature sensor  122   d  that measures a reference point temperature. The temperature sensor  122   d  is sufficiently away from the heat sources, i.e., the CPU  111   a , the charge circuit  124 , and the electric power amplifier  114   a  as compared with the temperature sensors  122   a ,  122   b , and  122   c.    
     The temperature change of the heat sources takes a sufficiently longer delay time to influence the measured temperature of the temperature sensor  122   d  than to influence the measured temperatures of the temperature sensors  122   a ,  122   b , and  122   c . That is, a thermal time constant, described later, of the temperature sensor  122   d  is sufficiently larger than those of the temperature sensors  122   a ,  122   b , and  122   c . The temperature sensor  122   d  may be located on the substrate  102  or may be located at a site other than the substrate  102 . In the second embodiment, the temperature sensor  122   d  is located near the battery  123 , for example. 
     The heat of the CPU  111   a  transfers to the temperature sensors  122   a ,  122   b , and  122   c  via the substrate  102 . The temperature sensor  122   a  is closest to the CPU  111   a , and thus the heat of the CPU  111   a  has the largest influence on the measured temperature of the temperature sensor  122   a . In the same way, the heat of the charge circuit  124  transfers to the temperature sensors  122   a ,  122   b , and  122   c  via the substrate  102 . The temperature sensor  122   b  is closest to the charge circuit  124 , and thus the heat of the charge circuit  124  has the largest influence on the measured temperature of the temperature sensor  122   b . The heat of the electric power amplifier  114   a  transfers to the temperature sensors  122   a ,  122   b , and  122   c  via the substrate  102 . The temperature sensor  122   c  is closest to the electric power amplifier  114   a , and thus the heat of the electric power amplifier  114   a  has the largest influence on the measured temperature of the temperature sensor  122   c.    
     Also, the heat of the CPU  111   a , the charge circuit  124 , and the electric power amplifier  114   a  transfers to the surface of the housing  101  via a tangible object or a space that exists between the substrate  102  and the housing  101 . The position of the highest surface temperature on the surface of the housing  101  depends on a combination of the heat generation amounts of the CPU  111   a , the charge circuit  124 , and the electric power amplifier  114   a . Thus, the position of the highest surface temperature is not fixed but changes according to a situation of how the components are used. 
     Note that the CPU  111   a  corresponds to the processor  4   a  of the first embodiment. The temperature sensor  122   a  corresponds to the temperature sensor  3  of the first embodiment. 
     Here, first, a heat transfer model in the case of only one heat source on the substrate  102  will be described. In this case, the heat generation amounts of the charge circuit  124  and the electric power amplifier  114   a  are small enough to be disregarded, and the heat source is the CPU  111   a  only, for example. 
     The heat of the CPU  111   a  of the heat source transfers to the temperature sensor  122   a . The heat gradually transfers from the CPU  111   a  to the temperature sensor  122   a , and thus a transient response as a non-steady state occurs in the measured temperature of the temperature sensor  122   a . Also, the heat of the CPU  111   a  of the heat source transfers to the surface of the housing  101 . The heat gradually transfers from the CPU  111   a  to the surface of the housing  101 , and thus a transient response as a non-steady state occurs in the surface temperature of the housing  101 . Here, a heat transfer model that considers both of the transient response from the CPU  111   a  to the temperature sensor  122   a  and the transient response from the CPU  111   a  to the housing  101  will be considered. 
     The measured temperature of the temperature sensor  122   a  is defined as formula (1). T sensor1 (s) is obtained by converting a relative temperature of the temperature sensor  122   a  relative to the reference point temperature of the temperature sensor  122   d  to a frequency domain by using Laplace transform. The relative temperature is calculated by subtracting the measured temperature (reference point temperature) of the temperature sensor  122   d  from the measured temperature of the temperature sensor  122   a . In the following, a variable in the time domain is sometimes denoted by “t”, and a variable in the frequency domain is sometimes denoted by “s”. T cpu (s) is obtained by Laplace transforming the relative temperature of the CPU  111   a . H(s) is a transfer function in the frequency domain for converting the relative temperature of the CPU  111   a  to the relative temperature of the temperature sensor  122   d . H(s) is a transfer function that considers the transient response, and includes h and τ h  as parameters. The h is a heat transfer coefficient, and τ h  is a thermal time constant. The values of h and τ h  are calculated in advance by using the design apparatus  200 . For example, h is equal to 0.500, and τ h  is equal to 40. 
     
       
         
           
             
               
                 
                   
                     
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                           ) 
                         
                       
                        
                       
                         
                           T 
                           cpu 
                         
                          
                         
                           ( 
                           s 
                           ) 
                         
                       
                        
                       
                           
                       
                        
                       where 
                        
                       
                           
                       
                        
                       
                         H 
                          
                         
                           ( 
                           s 
                           ) 
                         
                       
                     
                     = 
                     
                       h 
                       
                         1 
                         + 
                         
                           s 
                            
                           
                               
                           
                            
                           
                             τ 
                             h 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
       FIG. 5  illustrates an example of a heat circuit model that exhibits a transient response of heat conduction. The transfer function H(s) is derived from the heat circuit model illustrated in  FIG. 5 . This heat circuit model is provided between the CPU  111   a  and the temperature sensor  122   a . The heat circuit model includes a thermal resistor  21 , a thermal capacitor  22 , and a thermal amplifier  23 . The value of the thermal resistor  21  is R h . The value of the thermal capacitor  22  is C h . The product of R h  and C h  is the thermal time constant τ h . The amplification factor of the thermal amplifier  23  is h. An input of the thermal resistor  21  is an input of the heat circuit model. An output of the thermal resistor  21  is connected to an input of the thermal capacitor  22  and an input of the thermal amplifier  23 . An output of the thermal capacitor  22  is connected to an earth. An output of the thermal amplifier  23  is an output of the heat circuit model. 
     When the input of the thermal resistor  21  changes from “0” to “1” rapidly, the output of the thermal amplifier  23  changes from “0” to “h”. However, the output of the thermal amplifier  23  does not change from “0” to “h” rapidly but changes gradually. The delay time that it takes for the output of the thermal amplifier  23  to get close to “h” sufficiently since its start of change is the thermal time constant τ h . Thus, the measured temperature of the temperature sensor  122   a  becomes higher as the heat transfer coefficient h becomes larger, and becomes lower as the heat transfer coefficient h becomes smaller. Also, when the heat generation amount of the CPU  111   a  changes, the change rate of the measured temperature of the temperature sensor  122   a  becomes faster as the thermal time constant τ h  becomes smaller, and becomes slower as the thermal time constant τ h  becomes larger. 
     The heat transfer from the CPU  111   a  to the housing  101  may be modeled by using the same heat circuit model as the heat transfer from the CPU  111   a  to the temperature sensor  122   a . Note that values of parameters, such as a heat transfer coefficient and a thermal time constant, differ between the former and the latter. 
     The surface temperature of the housing  101  is defined as formula (2). T surface (s) is obtained by converting a relative temperature of the housing  101  relative to the reference point temperature to the frequency domain by Laplace transform. T cpu (s) is obtained by Laplace transforming the relative temperature of the CPU  111   a . G(s) is a transfer function in the frequency domain for converting the relative temperature of the CPU  111   a  to the relative temperature of the housing  101 . G(s) is a transfer function that considers the transient response, and includes g and τ g  as parameters. The g is a heat transfer coefficient, and τ g  is a thermal time constant. The values of g and τ g  are calculated in advance by using the design apparatus  200 . For example, g is equal to 0.425, and τ g  is equal to 85. 
     
       
         
           
             
               
                 
                   
                     
                       T 
                       surface 
                     
                      
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         G 
                          
                         
                           ( 
                           s 
                           ) 
                         
                       
                        
                       
                         
                           T 
                           cpu 
                         
                          
                         
                           ( 
                           s 
                           ) 
                         
                       
                        
                       
                           
                       
                        
                       where 
                        
                       
                           
                       
                        
                       
                         G 
                          
                         
                           ( 
                           s 
                           ) 
                         
                       
                     
                     = 
                     
                       g 
                       
                         1 
                         + 
                         
                           s 
                            
                           
                               
                           
                            
                           
                             τ 
                             g 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     The surface temperature of the housing  101  is defined as formula (3) from formula (1) and formula (2). T surface (s) in formula (3) is a relative temperature of the housing  101  in the frequency domain. Thus, the surface temperature of the housing  101  at a certain time point is defined as formula (4) by using inverse Laplace transform. T surface (t) is a surface temperature of the housing  101  at a time point t. T sensor2 (t) is a reference point temperature at a time point t which is measured by the temperature sensor  122   d . T surface (t) is calculated by adding T sensor2 (t) to a value obtained by inverse Laplace transforming T surface (s). 
     
       
         
           
             
               
                 
                   
                     
                       T 
                       surface 
                     
                      
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           G 
                            
                           
                             ( 
                             s 
                             ) 
                           
                         
                         
                           H 
                            
                           
                             ( 
                             s 
                             ) 
                           
                         
                       
                        
                       
                         
                           T 
                           
                             sensor 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                          
                         
                           ( 
                           s 
                           ) 
                         
                       
                     
                     = 
                     
                       
                         g 
                         h 
                       
                        
                       
                         
                           1 
                           + 
                           
                             s 
                              
                             
                                 
                             
                              
                             
                               τ 
                               h 
                             
                           
                         
                         
                           1 
                           + 
                           
                             s 
                              
                             
                                 
                             
                              
                             
                               τ 
                               g 
                             
                           
                         
                       
                        
                       
                         
                           T 
                           
                             sensor 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                          
                         
                           ( 
                           s 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       T 
                       surface 
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ℒ 
                         
                           - 
                           1 
                         
                       
                        
                       
                         ( 
                         
                           
                             g 
                             h 
                           
                            
                           
                             
                               1 
                               + 
                               
                                 s 
                                  
                                 
                                     
                                 
                                  
                                 
                                   τ 
                                   h 
                                 
                               
                             
                             
                               1 
                               + 
                               
                                 s 
                                  
                                 
                                     
                                 
                                  
                                 
                                   τ 
                                   g 
                                 
                               
                             
                           
                            
                           
                             
                               T 
                               
                                 sensor 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                              
                             
                               ( 
                               s 
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         T 
                         
                           sensor 
                            
                           
                               
                           
                            
                           2 
                         
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     Formula (4) is transformed to formula (5) by developing the inverse Laplace transform into a difference equation. The first term of the right side of formula (5) is the relative temperature of the temperature sensor  122   a , which is obtained by subtracting the reference point temperature from the measured temperature of the temperature sensor  122   a . The second term of the right side is the relative temperature of time Δt before of the temperature sensor  122   a , which is obtained by subtracting the reference point temperature of time Δt before from the measured temperature of time Δt before of the temperature sensor  122   a . At is a measurement cycle and is set to approximately 10 seconds, for example. t−Δt means the last measurement time. 
     
       
         
           
             
               
                 
                   
                     
                       
                         T 
                         surface 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           a 
                           0 
                         
                          
                         
                           ( 
                           
                             
                               
                                 T 
                                 
                                   sensor 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             - 
                             
                               
                                 T 
                                 
                                   sensor 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           a 
                           1 
                         
                          
                         
                           ( 
                           
                             
                               
                                 T 
                                 
                                   sensor 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                                
                               
                                 ( 
                                 
                                   t 
                                   - 
                                   
                                     Δ 
                                      
                                     
                                         
                                     
                                      
                                     t 
                                   
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 T 
                                 
                                   sensor 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                                
                               
                                 ( 
                                 
                                   t 
                                   - 
                                   
                                     Δ 
                                      
                                     
                                         
                                     
                                      
                                     t 
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                       - 
                       
                         
                           b 
                           1 
                         
                          
                         
                           ( 
                           
                             
                               
                                 T 
                                 surface 
                               
                                
                               
                                 ( 
                                 
                                   t 
                                   - 
                                   
                                     Δ 
                                      
                                     
                                         
                                     
                                      
                                     t 
                                   
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 T 
                                 
                                   sensor 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                                
                               
                                 ( 
                                 
                                   t 
                                   - 
                                   
                                     Δ 
                                      
                                     
                                         
                                     
                                      
                                     t 
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           T 
                           
                             sensor 
                              
                             
                                 
                             
                              
                             2 
                           
                         
                          
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     
                       where 
                        
                       
                           
                       
                        
                       
                         a 
                         0 
                       
                     
                     = 
                     
                       
                         
                           g 
                           
                             h 
                              
                             
                                 
                             
                           
                         
                          
                         
                           
                             
                               Δ 
                                
                               
                                   
                               
                                
                               t 
                             
                             + 
                             
                               2 
                               
                                 τ 
                                 h 
                               
                             
                           
                           
                             
                               Δ 
                                
                               
                                   
                               
                                
                               t 
                             
                             + 
                             
                               2 
                               
                                 τ 
                                 g 
                               
                             
                           
                         
                          
                         
                             
                         
                          
                         
                           a 
                           1 
                         
                       
                       = 
                       
                         
                           g 
                           
                             h 
                              
                             
                                 
                             
                           
                         
                          
                         
                           
                             
                               Δ 
                                
                               
                                   
                               
                                
                               t 
                             
                             - 
                             
                               2 
                               
                                 τ 
                                 h 
                               
                             
                           
                           
                             
                               Δ 
                                
                               
                                   
                               
                                
                               t 
                             
                             + 
                             
                               2 
                               
                                 τ 
                                 g 
                               
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     
                       b 
                       1 
                     
                     = 
                     
                       
                         g 
                         
                           h 
                            
                           
                               
                           
                         
                       
                        
                       
                         
                           
                             Δ 
                              
                             
                                 
                             
                              
                             t 
                           
                           - 
                           
                             2 
                             
                               τ 
                               g 
                             
                           
                         
                         
                           
                             Δ 
                              
                             
                                 
                             
                              
                             t 
                           
                           + 
                           
                             2 
                             
                               τ 
                               g 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     The third term of the right side is the relative temperature of time Δt before of the housing  101 , which is obtained by subtracting the reference point temperature of time Δt before from the surface temperature of time Δt before of the housing  101 . The fourth term of the right side is the reference point temperature measured by the temperature sensor  122   d . The first term of the right side is multiplied by a coefficient a 0 ; the second term of the right side is multiplied by a coefficient a 1 ; and the third term of the right side is multiplied by a coefficient b 1 . The a 0 , a 1 , and b 1  are defined as in formula (5) by using h, g, τ h , and τ g . Thereby, the surface temperature of the housing  101  is estimated by using the measured temperatures of the temperature sensors  122   a  and  122   d , the measured temperatures of the last time of the temperature sensors  122   a  and  122   d , and the estimated value of the surface temperature of the last time of the housing  101 . 
     Next, the above heat transfer model is applied to a case including a plurality of heat sources.  FIG. 6  illustrates an example of heat conduction from a plurality of heat sources. 
     The Laplace-transformed relative temperature of the CPU  111   a  is T h1 (s). The heat of the CPU  111   a  transfers to the temperature sensor  122   a  via a transfer function H 11 (s), transfers to the temperature sensor  122   b  via a transfer function H 12 (s), and transfers to the temperature sensor  122   c  via a transfer function H 13 (s). In the same way, the Laplace-transformed relative temperature of the charge circuit  124  is T h2 (s). The heat of the charge circuit  124  transfers to the temperature sensor  122   a  via a transfer function H 21 (s), transfers to the temperature sensor  122   b  via a transfer function H 22 (s), and transfers to the temperature sensor  122   c  via a transfer function H 23 (s). The Laplace-transformed relative temperature of the electric power amplifier  114   a  is T h3 (s). The heat of the electric power amplifier  114   a  transfers to the temperature sensor  122   a  via a transfer function H 31 (s), transfers to the temperature sensor  122   b  via a transfer function H 32 (s), and transfers to the temperature sensor  122   c  via a transfer function H 33 (s). 
     The Laplace-transformed relative temperature of the temperature sensor  122   a  is T s1 (s). T s1 (s) is a combination of contribution from the CPU  111   a , contribution from the charge circuit  124 , and contribution from the electric power amplifier  114   a . Thus, T s1 (s) is defined as formula (6). H ij (s) is a transfer function that expresses heat transfer from an i-th heat source to a j-th temperature sensor. h ij  is a heat transfer coefficient from an i-th heat source to a j-th temperature sensor, and τ hij  is a thermal time constant from an i-th heat source to a j-th temperature sensor. The values of h ij  and τ hij  are calculated in advance by using the design apparatus  200 . 
     
       
         
           
             
               
                 
                   
                     
                       
                         T 
                         
                           s 
                            
                           
                               
                           
                            
                           1 
                         
                       
                        
                       
                         ( 
                         s 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             H 
                             11 
                           
                            
                           
                             ( 
                             s 
                             ) 
                           
                         
                          
                         
                           
                             T 
                             
                               h 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                            
                           
                             ( 
                             s 
                             ) 
                           
                         
                       
                       + 
                       
                         
                           H 
                           12 
                         
                          
                         
                           
                             T 
                             
                               h 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                            
                           
                             ( 
                             s 
                             ) 
                           
                         
                       
                       + 
                       
                         
                           
                             H 
                             13 
                           
                            
                           
                             ( 
                             s 
                             ) 
                           
                         
                          
                         
                           
                             T 
                             
                               h 
                                
                               
                                   
                               
                                
                               3 
                             
                           
                            
                           
                             ( 
                             s 
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         where 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         H 
                         ij 
                       
                        
                       
                         ( 
                         s 
                         ) 
                       
                     
                     = 
                     
                       
                         h 
                         ij 
                       
                       
                         1 
                         + 
                         
                           s 
                            
                           
                               
                           
                            
                           
                             τ 
                             hij 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     In the same way, the Laplace-transformed relative temperature of the temperature sensor  122   b  is T s2 (s). T s2 (s) is a combination of the contribution from the CPU  111   a , the contribution from the charge circuit  124 , and the contribution from the electric power amplifier  114   a . The Laplace-transformed relative temperature of the temperature sensor  122   c  is T s3 (s). T s3 (s) is a combination of the contribution from the CPU  111   a , the contribution from the charge circuit  124 , and the contribution from the electric power amplifier  114   a . Thus, T s1 (s), T s2 (s), and T s3 (s) are expressed in a matrix form as in formula (7). A matrix H is a matrix of a transfer function. The element of i-th row and j-th column of the matrix H is H ij (s) that indicates heat transfer from an i-th heat source to a j-th temperature sensor. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               T 
                               
                                 s 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                              
                             
                               ( 
                               s 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               T 
                               
                                 s 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                              
                             
                               ( 
                               s 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               T 
                               
                                 s 
                                  
                                 
                                     
                                 
                                  
                                 3 
                               
                             
                              
                             
                               ( 
                               s 
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       H 
                        
                       
                         [ 
                         
                           
                             
                               
                                 
                                   T 
                                   
                                     h 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   T 
                                   
                                     h 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                   
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   T 
                                   
                                     h 
                                      
                                     
                                         
                                     
                                      
                                     3 
                                   
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     = 
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   H 
                                   11 
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   H 
                                   12 
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   H 
                                   13 
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   H 
                                   21 
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   H 
                                   22 
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   H 
                                   23 
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   H 
                                   31 
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   H 
                                   32 
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   H 
                                   33 
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 
                                   T 
                                   
                                     h 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   T 
                                   
                                     h 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                   
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   T 
                                   
                                     h 
                                      
                                     
                                         
                                     
                                      
                                     3 
                                   
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     Also, the heat of the CPU  111   a  transfers to the surface of the housing  101  via a transfer function G 1 (s). The heat of the charge circuit  124  transfers to the surface of the housing  101  via a transfer function G 2 (s). The heat of the electric power amplifier  114   a  transfers to the surface of the housing  101  via a transfer function G 3 (s). 
     The Laplace-transformed relative temperature of the surface of the housing  101  is T surf (s). T surf (s) is a combination of the contribution from the CPU  111   a , the contribution from the charge circuit  124 , and the contribution from the electric power amplifier  114   a . Thus, T surf (s) is defined as formula (8). G i (s) is a transfer function that expresses the heat transfer from an i-th heat source to the housing  101 . g i  is a heat transfer coefficient from an i-th heat source to the housing  101 , and τ gi  is a thermal time constant from an i-th heat source to the housing  101 . The values of g i  and τ gi  are calculated in advance by using the design apparatus  200 . 
     
       
         
           
             
               
                 
                   
                     
                       T 
                       surf 
                     
                      
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             G 
                             1 
                           
                            
                           
                             ( 
                             s 
                             ) 
                           
                         
                          
                         
                           
                             T 
                             
                               h 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                            
                           
                             ( 
                             s 
                             ) 
                           
                         
                       
                       + 
                       
                         
                           
                             G 
                             2 
                           
                            
                           
                             ( 
                             s 
                             ) 
                           
                         
                          
                         
                           
                             T 
                             
                               h 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                            
                           
                             ( 
                             s 
                             ) 
                           
                         
                       
                       + 
                       
                         
                           
                             G 
                             3 
                           
                            
                           
                             ( 
                             s 
                             ) 
                           
                         
                          
                         
                           
                             T 
                             
                               h 
                                
                               
                                   
                               
                                
                               3 
                             
                           
                            
                           
                             ( 
                             s 
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         where 
                          
                         
                           
 
                         
                          
                         
                           
                             G 
                             i 
                           
                            
                           
                             ( 
                             s 
                             ) 
                           
                         
                       
                     
                     = 
                     
                       
                         g 
                         i 
                       
                       
                         1 
                         + 
                         
                           s 
                            
                           
                               
                           
                            
                           
                             τ 
                             gi 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     Formula (8) may be expressed in a matrix form. When T h1 (s), T h2 (s), and T h3 (s) are developed by using formula (7), T surf (s) is expressed as the product of G i (s), the inverse matrix of H, and T sj (s) as in formula (9). That is, the relative temperatures of the CPU  111   a , the charge circuit  124 , and the electric power amplifier  114   a  are estimated from the relative temperatures of the temperature sensors  122   a ,  122   b , and  122   c  and the inverse matrix of H. Moreover, the surface temperature of the housing  101  is estimated from the relative temperatures of the CPU  111   a , the charge circuit  124 , and the electric power amplifier  114   a , G 1 (s), G 2 (s), and G 3 (s). 
     
       
         
           
             
               
                 
                   
                     
                       T 
                       surf 
                     
                      
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     G 
                                     1 
                                   
                                    
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     G 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     G 
                                     3 
                                   
                                    
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         T 
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 
                                   T 
                                   
                                     h 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   T 
                                   
                                     h 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                   
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   T 
                                   
                                     h 
                                      
                                     
                                         
                                     
                                      
                                     3 
                                   
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     = 
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     G 
                                     1 
                                   
                                    
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     G 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     G 
                                     3 
                                   
                                    
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         T 
                       
                        
                       
                         
                           H 
                           
                             - 
                             1 
                           
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     T 
                                     
                                       s 
                                        
                                       
                                           
                                       
                                        
                                       1 
                                     
                                   
                                    
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     T 
                                     
                                       s 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                     
                                   
                                    
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     T 
                                     
                                       s 
                                        
                                       
                                           
                                       
                                        
                                       3 
                                     
                                   
                                    
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     The inverse matrix of H is defined as formula (10). Note that the parameter s of each transfer function is omitted in formula (10). Δ is an eigenvalue and is equal to (H 11 H 22 −H 12 H 21 )H 33 +(H 13 H 21 −H 11 H 23 )H 32 +(H 12 H 23 −H 13 H 22 )H 31 . 
     
       
         
           
             
               
                 
                   
                     H 
                     
                       - 
                       1 
                     
                   
                   = 
                   
                     
                       1 
                       Δ 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   H 
                                   22 
                                 
                                  
                                 
                                   H 
                                   33 
                                 
                               
                               - 
                               
                                 
                                   H 
                                   23 
                                 
                                  
                                 
                                   H 
                                   32 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   H 
                                   13 
                                 
                                  
                                 
                                   H 
                                   32 
                                 
                               
                               - 
                               
                                 
                                   H 
                                   12 
                                 
                                  
                                 
                                   H 
                                   33 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   H 
                                   12 
                                 
                                  
                                 
                                   H 
                                   23 
                                 
                               
                               - 
                               
                                 
                                   H 
                                   13 
                                 
                                  
                                 
                                   H 
                                   22 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   H 
                                   23 
                                 
                                  
                                 
                                   H 
                                   31 
                                 
                               
                               - 
                               
                                 
                                   H 
                                   21 
                                 
                                  
                                 
                                   H 
                                   33 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   H 
                                   11 
                                 
                                  
                                 
                                   H 
                                   33 
                                 
                               
                               - 
                               
                                 
                                   H 
                                   13 
                                 
                                  
                                 
                                   H 
                                   31 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   H 
                                   11 
                                 
                                  
                                 
                                   H 
                                   23 
                                 
                               
                               - 
                               
                                 
                                   H 
                                   13 
                                 
                                  
                                 
                                   H 
                                   21 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   H 
                                   21 
                                 
                                  
                                 
                                   H 
                                   32 
                                 
                               
                               - 
                               
                                 
                                   H 
                                   22 
                                 
                                  
                                 
                                   H 
                                   31 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   H 
                                   12 
                                 
                                  
                                 
                                   H 
                                   31 
                                 
                               
                               - 
                               
                                 
                                   H 
                                   11 
                                 
                                  
                                 
                                   H 
                                   32 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   H 
                                   11 
                                 
                                  
                                 
                                   H 
                                   22 
                                 
                               
                               - 
                               
                                 
                                   H 
                                   12 
                                 
                                  
                                 
                                   H 
                                   21 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     Here, the transfer function H ij (s) in formula (10) includes the thermal time constant τ hij . Hence, there is a problem that the calculation amount of the product of the inverse matrix of H and the relative temperatures of the temperature sensors  122   a ,  122   b , and  122   c  becomes enormous, and that the load of the portable terminal device  100  becomes too large. When T surf (s) is calculated by formula (9), sixth order filter calculation is performed 9 times. In general, when there are n (n is an integer equal to or larger than 2) sets of heat sources and temperature sensors, 2n order filter calculation is performed n 2  times. 
     Thus, in the second embodiment, the portable terminal device  100  approximately calculates T surf (s) by a method of a smaller calculation amount. Specifically, the portable terminal device  100  uses transfer functions that do not consider the transient response on the substrate  102 , when estimating the heat source temperatures of the CPU  111   a , the charge circuit  124 , and the electric power amplifier  114   a  from the measured temperatures of the temperature sensors  122   a ,  122   b , and  122   c . Then, the portable terminal device  100  uses transfer functions that consider the transient response from the substrate  102  to the housing  101  as well as the transfer delay on the substrate  102 , when estimating the surface temperature of the housing  101  from the heat source temperatures of the CPU  111   a , the charge circuit  124 , and the electric power amplifier  114   a . This approximation calculation utilizes the following characteristics of heat transfer. 
       FIG. 7  is a graph illustrating first exemplary change of the relative temperatures of the temperature sensors. A graph  31  indicates change of the relative temperatures (dT) of the temperature sensors  122   a ,  122   b , and  122   c  relative to the reference point temperature. A curving line  31   a  indicates the change of the relative temperature of the temperature sensor  122   a  located near the CPU  111   a . A curving line  31   b  indicates the change of the relative temperature of the temperature sensor  122   b  located near the charge circuit  124 . A curving line  31   c  indicates the change of the relative temperature of the temperature sensor  122   c  located near the electric power amplifier  114   a.    
     As illustrated in the graph  31 , when the charge circuit  124  starts operating, the relative temperatures of the temperature sensors  122   a ,  122   b , and  122   c  start rising immediately. Also, when the charge circuit  124  stops operating, the relative temperatures of the temperature sensors  122   a ,  122   b , and  122   c  start falling immediately. In the same way, when the CPU  111   a  starts operating, the relative temperatures of the temperature sensors  122   a ,  122   b , and  122   c  start rising immediately. Also, when the CPU  111   a  stops operating, the relative temperatures of the temperature sensors  122   a ,  122   b , and  122   c  start falling immediately. 
     As described above, the changes of the relative temperatures of the temperature sensors  122   a ,  122   b , and  122   c  are of a comparatively small delay and a fast response rate in relation to the changes of the heat source temperatures. That is, the thermal time constants of the measured temperatures of the temperature sensors  122   a ,  122   b , and  122   c  are smaller than that of the surface temperature of the housing  101 . Also, variation of response rates is small in the temperature sensors  122   a ,  122   b , and  122   c . This is because a substance having a small heat resistance, such as a copper line, is used in the substrate  102 , and the heat transfer from the heat sources to the temperature sensors  122   a ,  122   b , and  122   c  is fast. 
     Thus, the inverse matrix of H is approximately broken down as in formula (11). This means that h ij  which is the numerator of the transfer function H ij (s) and 1+sτ hij  which is the denominator of the transfer function H ij (s) are separated from each other. The operator of the right side of formula (11) is a Hadamard product which differs from a usual matrix product, and is an operator for calculating a product for each corresponding matrix element. The second term of the right side is an inverse matrix of a matrix that includes heat transfer coefficients h ij  and does not include a thermal time constant, and is used in estimating the heat source temperatures. The first term of the right side is a matrix that includes one thermal time constant for each heat source, and is used in estimating a surface temperature from the heat source temperatures. 
     
       
         
           
             
               
                 
                   
                     H 
                     
                       - 
                       1 
                     
                   
                   ≈ 
                   
                     
                       [ 
                       
                         
                           
                             
                               1 
                               + 
                               
                                 s 
                                  
                                 
                                     
                                 
                                  
                                 
                                   τ 
                                   
                                     h 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               1 
                               + 
                               
                                 s 
                                  
                                 
                                     
                                 
                                  
                                 
                                   τ 
                                   
                                     h 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               1 
                               + 
                               
                                 s 
                                  
                                 
                                     
                                 
                                  
                                 
                                   τ 
                                   
                                     h 
                                      
                                     
                                         
                                     
                                      
                                     3 
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                     ∘ 
                     
                       
                         [ 
                         
                           
                             
                               
                                 h 
                                 11 
                               
                             
                             
                               
                                 h 
                                 12 
                               
                             
                             
                               
                                 h 
                                 13 
                               
                             
                           
                           
                             
                               
                                 h 
                                 21 
                               
                             
                             
                               
                                 h 
                                 22 
                               
                             
                             
                               
                                 h 
                                 23 
                               
                             
                           
                           
                             
                               
                                 h 
                                 31 
                               
                             
                             
                               
                                 h 
                                 32 
                               
                             
                             
                               
                                 h 
                                 33 
                               
                             
                           
                         
                         ] 
                       
                       
                         - 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     Estimating heat source temperatures by using the second term means assuming a steady state without considering a transient response of a non-steady state. Thereby, the calculation amount for estimating the heat source temperatures is reduced significantly. The heat source temperatures estimated by using the second term disregard the delays of the heat transfer from the heat sources to the temperature sensors  122   a ,  122   b , and  122   c , and thus are equivalent to the heat source temperatures of a predetermined time before rather than the heat source temperatures of a time at which the temperatures are measured by the temperature sensors  122   a ,  122   b , and  122   c . Thus, the first term is incorporated in the transfer function for estimating the surface temperature from the heat source temperatures. This means that the estimated delays of the heat source temperatures are considered in estimating the surface temperature. 
     Formula (11) uses a thermal time constant τ h1  corresponding to the CPU  111   a , a thermal time constant τ h2  corresponding to the charge circuit  124 , and a thermal time constant τ h3  corresponding to the electric power amplifier  114   a . Note that the thermal time constants on the substrate  102  are small, and thus the same value may be used for τ h1 , τ h2 , and τ h3 . The value of τ hi  is calculated in advance by using the design apparatus  200 . 
     According to the above approximation calculation, the relative temperatures T h1 (s), T h2 (s), and T h3 (s) of the heat sources are calculated by formula (12). The matrix H* is an approximate transfer function on the substrate  102 . An element of i-th row and j-th column of the matrix H* is h ij  which indicates heat transfer that disregards a thermal time constant from an i-th heat source to a j-th temperature sensor. The inverse matrix of H* corresponds to the second term of the right side of formula (11). The inverse matrix of H* is defined as formula (13). Δ is an eigenvalue and is equal to (h 11 h 22 −h 12 h 21 )h 33 +(h 13 h 21 −h 11 h 23 )h 32 +(h 12 h 23 −h 13 h 22 )h 31 . The transient response is not considered, and thus the calculation amount of the product of the inverse matrix of H* and T s1 (s), T s2 (s), and T s3 (s) decreases more significantly than the calculation amount of the product of the inverse matrix of H and T s1 (s), T s2 (s), and T s3 (s). 
     
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 T 
                                 
                                   h 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 T 
                                 
                                   h 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 T 
                                 
                                   h 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                 
                               
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           H 
                           
                             * 
                             
                               - 
                               1 
                             
                           
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     T 
                                     
                                       s 
                                        
                                       
                                           
                                       
                                        
                                       1 
                                     
                                   
                                    
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     T 
                                     
                                       s 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                     
                                   
                                    
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     T 
                                     
                                       s 
                                        
                                       
                                           
                                       
                                        
                                       3 
                                     
                                   
                                    
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                       = 
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     h 
                                     11 
                                   
                                 
                                 
                                   
                                     h 
                                     12 
                                   
                                 
                                 
                                   
                                     h 
                                     13 
                                   
                                 
                               
                               
                                 
                                   
                                     h 
                                     21 
                                   
                                 
                                 
                                   
                                     h 
                                     22 
                                   
                                 
                                 
                                   
                                     h 
                                     23 
                                   
                                 
                               
                               
                                 
                                   
                                     h 
                                     31 
                                   
                                 
                                 
                                   
                                     h 
                                     32 
                                   
                                 
                                 
                                   
                                     h 
                                     33 
                                   
                                 
                               
                             
                             ] 
                           
                           
                             - 
                             1 
                           
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     T 
                                     
                                       s 
                                        
                                       
                                           
                                       
                                        
                                       1 
                                     
                                   
                                    
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     T 
                                     
                                       s 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                     
                                   
                                    
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     T 
                                     
                                       s 
                                        
                                       
                                           
                                       
                                        
                                       3 
                                     
                                   
                                    
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   
                     H 
                     
                       * 
                       
                         - 
                         1 
                       
                     
                   
                   = 
                   
                     
                       1 
                       Δ 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   H 
                                   22 
                                 
                                  
                                 
                                   H 
                                   33 
                                 
                               
                               - 
                               
                                 
                                   H 
                                   23 
                                 
                                  
                                 
                                   H 
                                   32 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   H 
                                   13 
                                 
                                  
                                 
                                   H 
                                   32 
                                 
                               
                               - 
                               
                                 
                                   H 
                                   12 
                                 
                                  
                                 
                                   H 
                                   33 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   H 
                                   12 
                                 
                                  
                                 
                                   H 
                                   23 
                                 
                               
                               - 
                               
                                 
                                   H 
                                   13 
                                 
                                  
                                 
                                   H 
                                   22 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   H 
                                   23 
                                 
                                  
                                 
                                   H 
                                   31 
                                 
                               
                               - 
                               
                                 
                                   H 
                                   21 
                                 
                                  
                                 
                                   H 
                                   33 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   H 
                                   11 
                                 
                                  
                                 
                                   H 
                                   33 
                                 
                               
                               - 
                               
                                 
                                   H 
                                   13 
                                 
                                  
                                 
                                   H 
                                   31 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   H 
                                   11 
                                 
                                  
                                 
                                   H 
                                   23 
                                 
                               
                               - 
                               
                                 
                                   H 
                                   13 
                                 
                                  
                                 
                                   H 
                                   21 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   H 
                                   21 
                                 
                                  
                                 
                                   H 
                                   32 
                                 
                               
                               - 
                               
                                 
                                   H 
                                   22 
                                 
                                  
                                 
                                   H 
                                   31 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   H 
                                   12 
                                 
                                  
                                 
                                   H 
                                   31 
                                 
                               
                               - 
                               
                                 
                                   H 
                                   11 
                                 
                                  
                                 
                                   H 
                                   32 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   H 
                                   11 
                                 
                                  
                                 
                                   H 
                                   22 
                                 
                               
                               - 
                               
                                 
                                   H 
                                   12 
                                 
                                  
                                 
                                   H 
                                   21 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     Then, according to the above approximation calculation, the relative temperature T surf (s) of the housing  101  is calculated by formula (14) from the estimated T h1 (s), T h2 (s), and T h3 (s). T surf (s) is expressed as a sum of products of G* i (s) and T hi (s). G* i (s) is a transfer function that expresses the heat transfer from an i-th heat source to the housing  101 . g i  is a heat transfer coefficient from an i-th heat source to the housing  101 ; τ hi  is a thermal time constant on the substrate  102  of an i-th heat source; and τ gi  is a thermal time constant from an i-th heat source to the housing  101 . As described above, the thermal time constant τ hi  on the substrate  102  is incorporated in the transfer function for estimating the surface temperature from the heat source temperatures, in order to reflect the estimated delays of the heat source temperatures. 
     
       
         
           
             
               
                 
                   
                     
                       
                         T 
                         surf 
                       
                        
                       
                         ( 
                         s 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     G 
                                     1 
                                     * 
                                   
                                    
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     G 
                                     2 
                                     * 
                                   
                                    
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     G 
                                     3 
                                     * 
                                   
                                    
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         T 
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 
                                   T 
                                   
                                     h 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                 
                                  
                                 
                                   ( 
                                   s 
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     Next, a decision method of parameters by the design apparatus  200  will be described. The design apparatus  200  decides the heat transfer coefficients h 11 , h 12 , h 13 , h 21 , h 22 , h 23 , h 31 , h 32 , and h 33  on the substrate  102 . For that purpose, temperatures measured when each heat source is operated independently are acquired by using an actual device of the portable terminal device  100  or sample implementation. 
     Specifically, the CPU  111   a  is operated alone while the charge circuit  124  and the electric power amplifier  114   a  are stopped, in order to acquire the measured temperatures of the temperature sensors  122   a ,  122   b ,  122   c , and  122   d . The design apparatus  200  calculates the relative temperatures of the temperature sensors  122   a ,  122   b , and  122   c  by subtracting the reference point temperature of the temperature sensor  122   d  from the measured temperatures of the temperature sensors  122   a ,  122   b , and  122   c . Then, the design apparatus  200  calculates h 21  and h 31  that most appropriately express the relationship between the relative temperature of the temperature sensor  122   a  and the relative temperatures of the temperature sensors  122   b  and  122   c , by fitting in the time domain. h 11  is “1.0”. Note that h 11 , h 12 , and h 13  may be calculated with reference to directly measured heat source temperature of the CPU  111   a , instead of calculating h 21  and h 31  with reference to the relative temperature of the temperature sensor  122   a.    
     In the same way, the charge circuit  124  is operated alone while the CPU  111   a  and the electric power amplifier  114   a  are stopped, in order to acquire the measured temperatures of the temperature sensors  122   a ,  122   b ,  122   c , and  122   d . The design apparatus  200  calculates h 12  and h 32  that most appropriately express the relationship between the relative temperature of the temperature sensor  122   b  and the relative temperatures of the temperature sensors  122   a  and  122   c , by fitting in the time domain. h 22  is “1.0”. Also, the electric power amplifier  114   a  is operated alone while the CPU  111   a  and the charge circuit  124  are stopped, in order to acquire the measured temperatures of the temperature sensors  122   a ,  122   b ,  122   c , and  122   d . The design apparatus  200  calculates h 13  and h 23  that most appropriately express the relationship between the relative temperature of the temperature sensor  122   c  and the relative temperatures of the temperature sensors  122   a  and  122   b , by fitting in the time domain. h 33  is “1.0”. 
     That is, the heat transfer coefficients h 11 , h 12 , h 13 , h 21 , h 22 , h 23 , h 31 , h 32 , and h 33  are decided as in formula (15). T s1 (t), T s2 (t), and T s3 (t) are the measured temperatures of the temperature sensors  122   a ,  122   b , and  122   c  at a time point t. T rt (t) is the measured temperature of the temperature sensor  122   d  at the time point t and is the reference point temperature at the time point t. T h1 (t), T h2 (t), and T h3 (t) are the heat source temperatures at the time point t. As an example, h 11  is equal to 1.0; h 12  is equal to 0.6; h 13  is equal to 0.6; h 21  is equal to 0.7; h 22  is equal to 1.0; h 23  is equal to 0.4; h 31  is equal to 1.0; h 32  is equal to 0.3; and h 33  is equal to 1.0. 
     
       
         
           
             
               
                 
                   
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       FIG. 8  is a graph illustrating second exemplary change of the relative temperatures of the temperature sensors. A graph  32  indicates the relative temperatures of the temperature sensors  122   a ,  122   b , and  122   c , when the charge circuit  124  is operated alone while the CPU  111   a  and the electric power amplifier  114   a  are stopped. A curving line  32   a  indicates the relative temperature of the temperature sensor  122   a . A curving line  32   b  indicates the relative temperature of the temperature sensor  122   b . A curving line  32   c  indicates the relative temperature of the temperature sensor  122   c . The temperature sensor  122   b  is closest to the charge circuit  124 , and thus the relative temperature of the temperature sensor  122   b  is higher than those of the temperature sensors  122   a  and  122   c.    
     When fitting is performed in the time domain, the difference between the relative temperature of the temperature sensor  122   b  multiplied by h 12  and the relative temperature of the temperature sensor  122   a  becomes smallest with h 12 =0.6. Thus, h 12  is decided to be 0.6. In the same way, with h 32 =0.3, the difference between the relative temperature of the temperature sensor  122   b  multiplied by h 32  and the relative temperature of the temperature sensor  122   c  becomes smallest. Thus, h 32  is decided to be 0.3. 
     Next, the design apparatus  200  decides the heat transfer coefficients g 1 , g 2 , and g 3 , the thermal time constants τ h1 , τ h2 , and τ h3 , and the thermal time constants τ g1 , τ g2 , and τ g3 . For that purpose, temperatures at various positions are measured when each heat source is operated independently, by using an actual device of the portable terminal device  100  or sample implementation. Specifically, the heat source temperatures of the CPU  111   a , the charge circuit  124 , and the electric power amplifier  114   a  are measured. Also, a position at which the surface temperature is to be estimated on the surface of the housing  101  is selected, and the surface temperature of the selected position is measured. Two or more positions may be selected as the positions at which the surface temperatures are to be estimated. Also, the reference point temperature measured by the temperature sensor  122   d  is acquired. 
     The design apparatus  200  calculates the relative temperatures of the heat sources by subtracting the reference point temperature from the respective heat source temperatures. The design apparatus  200  converts time-series data of the relative temperatures of the heat sources to frequency-domain data by Laplace transform. Also, the design apparatus  200  calculates the relative temperature of the housing  101  by subtracting the reference point temperature from the surface temperature. The design apparatus  200  converts time-series data of the relative temperature of the housing  101  to frequency-domain data by Laplace transform. Then, the design apparatus  200  decides the heat transfer coefficients g 1 , g 2 , and g 3 , the thermal time constants τ h1 , τ h2 , and τ h3 , and the thermal time constants τ g1 , τ g2 , and τ g3  by fitting in the frequency domain. A least squares method is utilized for the fitting in the frequency domain, for example. 
     That is, the heat transfer coefficients g 1 , g 2 , and g 3 , the thermal time constants τ h1 , τ h2 , and τ h3 , and the thermal time constants τ g1 , τ g2 , and τ g3  are decided as in formula (16). T surf (t) is the surface temperature at a time point t. T rt (t) is the measured temperature of the temperature sensor  122   d  at the time point t and is the reference point temperature at the time point t. T h1 (t), T h2 (t), and T h3 (t) are the heat source temperatures at the time point t. 
     
       
         
           
             
               
                 
                   
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     In this way, the heat transfer coefficients h 11 , h 12 , h 13 , h 21 , h 22 , h 23 , h 31 , h 32 , and h 33 , the heat transfer coefficients g 1 , g 2 , and g 3 , the thermal time constants τ h1 , τ h2 , and τ h3 , and the thermal time constants τ g1 , τ g2 , and τ g3  are decided. 
       FIG. 9  illustrates an example of an intermediate parameter table. The design apparatus  200  generates an intermediate parameter table  41 . The intermediate parameter table  41  associates parameter names and their values. The parameters indicated by the parameter names include the heat transfer coefficients h 11 , h 12 , h 13 , h 21 , h 22 , h 23 , h 31 , h 32 , and h 33 , the heat transfer coefficients g 1 , g 2 , and g 3 , the thermal time constants τ h1 , τ h2 , and τ h3 , and the thermal time constants τ g1 , τ g2 , and τ g3 . These parameters are primary parameters included in the transfer function and may be referred to as intermediate parameters. The parameters included in the estimation equation for estimating the surface temperature are secondary parameters derived from these intermediate parameters. The values decided by the above method are associated with the parameter names. 
       FIG. 10  illustrates an example of a parameter table. The design apparatus  200  generates a parameter table  42  on the basis of the intermediate parameter table  41 . The parameter table  42  associates parameter names and values. The parameters indicated by the parameter names include h* 11 , h* 12 , h* 13 , h* 21 , h* 22 , h* 23 , h* 31 , h* 32 , h* 33 , a 01 , a 11 , b 11 , a 02 , a 12 , b 12 , a 03 , a 13 , and b 13 . These parameters are derived from the intermediate parameters of the intermediate parameter table  41 . The parameters of the parameter table  42  are stored in the portable terminal device  100 . 
     h* 11 , h* 12 , h* 13 , h* 21 , h* 22 , h* 23 , h* 31 , h* 32 , and h* 33  are the matrix elements of aforementioned formula (13). h* ij  appears at i-th row and j-th column of the inverse matrix of H*. h* 11 , h* 12 , h* 13 , h* 21 , h* 22 , h* 23 , h* 31 , h* 32 , and h* 33  are derived from the heat transfer coefficients h 11 , h 12 , h 13 , h 21 , h 22 , h 23 , h 31 , h 32 , and h 33 . a 01 , a 11 , b 11 , a 02 , a 12 , b 12 , a 03 , a 13 , and b 13  are parameters included in the difference equations of the transfer functions G* 1 (s), G* 2 (s), and G* 3 (s), as described later. a 01 , a 11 , b 11 , a 02 , a 12 , b 12 , a 03 , a 13 , and b 13  are derived from the heat transfer coefficients g 1 , g 2 , and g 3 , the thermal time constants τ h1 , τ h2 , and τ h3 , and the thermal time constants τ g1 , τ g2 , and τ g3 . 
     Next, an estimation method of a surface temperature by the portable terminal device  100  will be described. The portable terminal device  100  acquires the measured temperatures of the temperature sensors  122   a ,  122   b ,  122   c , and  122   d . Then, the portable terminal device  100  calculates the relative temperatures of the temperature sensors  122   a ,  122   b , and  122   c  by subtracting the reference point temperature of the temperature sensor  122   d  from the measured temperatures of the temperature sensors  122   a ,  122   b , and  122   c . The portable terminal device  100  estimates the relative temperatures of the CPU  111   a , the charge circuit  124 , and the electric power amplifier  114   a  from the relative temperatures of the temperature sensors  122   a ,  122   b , and  122   c  by formula (17). In this case, h* 11 , h* 12 , h* 13 , h* 21 , h* 22 , h* 23 , h* 31 , h* 32 , and h* 33  are used as the parameters. The estimation of the relative temperatures of the heat sources is performed in the time domain. Thus, the relative temperatures of the CPU  111   a , the charge circuit  124 , and the electric power amplifier  114   a  at a time point are estimated, each time the measured temperatures of the temperature sensors  122   a ,  122   b ,  122   c , and  122   d  are acquired. 
     
       
         
           
             
               
                 
                   
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       FIG. 11  is a graph indicating an estimation example of the relative temperatures of the heat sources. The graph  33  indicates the relative temperatures of the CPU  111   a , the charge circuit  124 , and the electric power amplifier  114   a  which are estimated from the graph  31  of  FIG. 7 . A curving line  33   a  indicates an estimated value of the relative temperature of the CPU  111   a . A curving line  33   b  indicates an estimated value of the relative temperature of the charge circuit  124 . A curving line  33   c  indicates an estimated value of the relative temperature of the electric power amplifier  114   a.    
     The portable terminal device  100  estimates the surface temperature at a predetermined position on the surface of the housing  101  on the basis of the estimated relative temperatures of the CPU  111   a , the charge circuit  124 , and the electric power amplifier  114   a . When two or more positions on the surface of the housing  101  are selected, the portable terminal device  100  estimates the surface temperatures of the selected positions. In that case, different transfer functions G* 1 (s), G* 2 (s), and G* 3 (s) are used, depending on the position for estimating the surface temperature. That is, the values of the parameters a 01 , a 11 , b 11 , a 02 , a 12 , b 12 , a 03 , a 13 , and b 13  are prepared for each position to estimate the surface temperature. 
     The surface temperature is defined as formula (18). T surf (t) is the surface temperature at a time point t. T surf-1 (t) is an increase amount of the surface temperature by the contribution from the CPU  111   a , that is, the heat generation of the CPU  111   a  at the time point t. T surf-2 (t) is an increase amount of the surface temperature by the contribution from the charge circuit  124 , that is, the heat generation of the charge circuit  124  at the time point t. T surf-3 (t) is an increase amount of the surface temperature by the contribution from the electric power amplifier  114   a , that is, the heat generation of the electric power amplifier  114   a  at the time point t. The surface temperature at the time point t is obtained by adding the contribution from the CPU  111   a , the contribution from the charge circuit  124 , and the contribution from the electric power amplifier  114   a  to the reference point temperature at the time point t. 
     
       
         
           
             
               
                 
                   
                     
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     The contribution from the CPU  111   a  at the time point t is equivalent to what the product of G* 1 (s) and T h1 (s) is inverse Laplace transformed. The contribution from the charge circuit  124  at the time point t is equivalent to what the product of G* 2 (s) and T h2 (s) is inverse Laplace transformed. The contribution from the electric power amplifier  114   a  at the time point t is equivalent to what the product of G* 3 (s) and T h3 (s) is inverse Laplace transformed. The inverse Laplace transform of the product of G* i (s) and T hi (s) is developed into calculation in the time domain as in formula (19), by using difference equation. 
         T   surf-i ( t )= a   0i ( T   hi ( t )− T   rt ( t ))+ a   1i ( T   hi ( t−Δt )− T   rt ( t−Δt ))− b   1i   T   surf-i ( t−Δt )   (19)
 
     T surf i (t) is contribution from an i-th heat source at a time point t. The first term of the right side of formula (19) is a relative temperature at the time point t of an i-th heat source. The second term of the right side is a relative temperature of time Δt before of the i-th heat source. Δt is a measurement cycle and is set at approximately 10 seconds, for example. t−Δt means the last measurement time. The third term of the right side is an estimated value of the contribution from the i-th heat source of time Δt before. 
     The first term is multiplied by a coefficient a 0i ; the second term is multiplied by a coefficient a 1i ; and the third term is multiplied by a coefficient b 1i . a 0i , a 1i , and b 1i  are defined as formula (20) by using g i , τ hi , and τ gi . Thereby, T surf 1 (t), T surf 2 (t), and T surf 3 (t) of formula (18) are calculated. 
     
       
         
           
             
               
                 
                   
                     
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     The relative temperature calculated at the last time and the contribution estimated at the last time are recorded for each heat source, in order to use the difference equation of formula (19).  FIG. 12  illustrates an example of a past data table. The portable terminal device  100  includes a past data table  43 . The past data table  43  associates heat sources, data names, and values. The heat sources include the CPU  111   a , the charge circuit  124 , and the electric power amplifier  114   a . The data names indicate data including a relative temperature of a heat source of the last time and estimated contribution from a heat source of the last time. That is, 6 values are recorded in the past data table  43 . The portable terminal device  100  updates the past data table  43 , each time the measured temperatures from the temperature sensors  122   a ,  122   b ,  122   c , and  122   d  are acquired and the surface temperature is estimated. 
       FIG. 13  is a graph indicating an estimation example of the contribution to the housing surface from the heat sources. The graph  34  indicates contribution to increasing surface temperature from the CPU  111   a , the charge circuit  124 , and the electric power amplifier  114   a , which is estimated from the graph  33  of  FIG. 11 . A curving line  34   a  indicates contribution to the housing  101  from the CPU  111   a . A curving line  34   b  indicates contribution to the housing  101  from the charge circuit  124 . A curving line  34   c  indicates contribution to the housing  101  from the electric power amplifier  114   a.    
       FIG. 14  is a graph indicating an estimation example of the surface temperature. The graph  35  indicates an estimated value of the surface temperature, an actual surface temperature, and a difference between the both. A curving line  35   a  indicates the estimated value of the surface temperature. The estimated value of the surface temperature is obtained by adding the contribution of the CPU  111   a , the charge circuit  124 , and the electric power amplifier  114   a  indicated in the graph  34 , to the reference point temperature. A curving line  35   b  indicates the surface temperature measured by using an actual device of the portable terminal device  100  or sample implementation. A curving line  35   c  indicates a difference between the curving line  35   a  and the curving line  35   b . According to this experimental example, the absolute value of the difference is within approximately 1° C. as indicated by the curving line  35   c.    
     The portable terminal device  100  controls the heat sources on the basis of the estimated surface temperature. First, the portable terminal device  100  calculates the maximum value (highest prediction reached value) that T surf-1 (t) is predicted to reach a certain amount of time or more later, from the temporal change of T surf-1 (t) which is the increase amount of the surface temperature due to the heat generation of the CPU  111   a . Note that T surf-1 (t) corresponds to the temperature index value of the first embodiment. 
     In the following, the temporal change of T surf-1 (t) is expressed by a slope dT surf-1 (t) of T surf-1 (t) at a time point t. T surf 1 (t) and dT surf 1 (t) are expressed as formula (21) generally. T sat (t) is the highest prediction reached value of T surf-1 (t) calculated at the time point t. The highest prediction reached value is the maximum value of T surf-1 (t) estimated under the assumption that the change tendency of the last T surf-1 (t) continues. τ g1  is a thermal time constant from the CPU  111   a  to the surface of the housing  101 . 
     
       
         
           
             
               
                 
                   
                     
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     T sat (t) is expressed by following formula (22), which is obtained from formula (21). The portable terminal device  100  calculates T sat (t) by formula (22). dT surf-1 (t) is calculated by dividing the difference between T surf 1 (t) and T surf-1 (tp) at a time point tp that is a certain amount of time (for example, 1 second, 10 seconds, etc.) before the time point t, by the certain amount of time, for example. When the certain amount of time is 1 second, dT surf-1 (t)=T surf-1 (t)−T surf-1 (tp) is established. When the certain amount of time is Δt seconds (for example, 10 seconds), dT surf 1 (t)=(T surf 1 (t)−T surf 1 (t−Δt))/Δt is established. Note that, when the above time point tp is 1 second before the time point t and the aforementioned Δt (measurement cycle) is 10 seconds, T surf-1 (tp) is provisionally calculated at the time point tp in order to obtain dT surf-1 (t). 
         T   sat ( t )= T   surf-1 ( t )+ dT   surf-1 ( t )τ g1    (22)
 
     Next, the portable terminal device  100  calculates a rate parameter indicating a target of a decrease rate for decreasing T sat (t) on the basis of the calculated T sat (t) and a threshold value for T surf-1 (t). 
     The threshold value is a temperature index value corresponding to the heat generation amount that is allowable for the CPU  111   a . For example, the portable terminal device  100  calculates T surf (t) which is the surface temperature of the housing  101  at the time point t as described above, and decides as the threshold value a value obtained by adding T surf-1 (t) to the temperature difference (increase margin amount) between a predetermined allowable temperature (for example, 45° C.) and the calculated T surf (t). In the following, the threshold value at a certain time point t is denoted by T max (t). Note that, when the surface temperature is estimated for two or more points on the housing  101 , the maximum surface temperature is selected. 
     The rate parameter is a time constant indicating a rate at which T sat (t) converges to T max (t). The portable terminal device  100  calculates a decrease rate α by dividing T max (t) by T sat (t) and calculates a rate parameter ατ g1  which is a product of α and the thermal time constant τ g1  from the CPU  111   a  to the surface of the housing  101 . That is, the time constant indicating the rate for decreasing T sat (t) to T max (t) is set to τ g1 ×T max (t)/T sat (t). 
     For example, when T sat (t) estimated at a time point t is 40° C., T max (t) of the time point t is 20° C., and τ g1  is 100 seconds, ατ g1 =0.5×100=50 seconds is established. Note that the time constant indicates a time that it takes for the change amount to reach (1-e −1 )≈63.2% of the target value, for example. Note that the portable terminal device  100  calculates the rate parameter, only when T sat (t) calculated at the time point t is larger than T max (t). 
     Next, the portable terminal device  100  controls the clock frequency of the CPU  111   a  on the basis of the rate parameter and correspondence information that associates a plurality of target values of T sat (t) and a plurality of clock frequencies of the CPU  111   a . A certain target value corresponds to the highest reached value reached by T surf-1 (t) when the CPU  111   a  continues operating at a certain clock frequency, for example. The correspondence information is stored in the RAM  112  or the non-volatile memory  113  as a correspondence information table described below, for example. 
       FIG. 15  illustrates an example of a correspondence information table. The correspondence information table  44  associates a plurality of target values of T sat (t) and a plurality of clock frequencies of the CPU  111   a.    
     For example, with a target value=5° C., the CPU  111   a  operates at a clock frequency 1.0 GHz. With a target value=10° C., the CPU  111   a  operates at a clock frequency 1.4 GHz. With a target value=15° C., the CPU  111   a  operates at a clock frequency 1.8 GHz. With a target value=20° C., the CPU  111   a  operates at a clock frequency 2.0 GHz. Note that, when the clock frequency in the CPU  111   a  is selectable more finely, it is possible to set the target value more finely. 
     This correspondence information table  44  may be generated in advance by the portable terminal device  100 , for example. The portable terminal device  100  sets T surf-1 (tx) obtained at a time point tx at which the CPU  111   a  has operated for a certain amount of time or more at a certain clock frequency, in the above correspondence information table  44  as the target value corresponding to the certain clock frequency. 
     The portable terminal device  100  decides the target value of T sat (t) of a predetermined time later on the basis of the rate parameter with reference to the above correspondence information table  44 , and sets the clock frequency associated with the decided target value as the upper limit of the clock frequency of the CPU  111   a , for example. 
     For example, the portable terminal device  100  sets the upper limit of the clock frequency of the time point t or later in such a manner that T sat (t) calculated at the time point t decreases to (T sat (t)−T max (t)) exp(−Δtx/ατ g1 )+T max (t) in Δtx seconds. 
     The portable terminal device  100  decides the target value equivalent to (T sat (t)−T max (t)) exp(−Δtx/ατ g1 )+T max (t) from the correspondence information table  44  and sets the clock frequency corresponding to the target value as the upper limit. When the correspondence information table  44  does not include a target value identical with (T sat (t)−T max (t)) exp (−Δtx/ατ g1 )+T max (t), the clock frequency corresponding to the closest target value is set as the upper limit, for example. 
     Note that Δtx is Δt multiplied by x (x is an integer equal to or greater than 1), for example. For example, x is equal to 2, and Δtx is equal to 20 seconds. When the value of x is large, an attenuating amount of T sat (t) becomes large, and a smaller upper limit is set for the clock frequency. This reduces the risk of the surface temperature of the housing  101  exceeding an allowable temperature. However, when x is too large, the upper limit of the clock frequency is restricted excessively. 
     The above process is performed every Δt, and the upper limit of the clock frequency of the CPU  111   a  is lowered in a phased manner such that the highest prediction reached value T sat  attenuates in accordance with the rate parameter calculated at each time. 
     Note that the portable terminal device  100  may switch on and off the charge circuit  124  on the basis of the estimated surface temperature. For example, when the estimated value of the surface temperature exceeds a predetermined allowable temperature, the portable terminal device  100  stops charging by turning off the charge circuit  124 . Also, the portable terminal device  100  may restrict the communication rate of the wireless interface  114  on the basis of the estimated surface temperature. For example, when the estimated value of the surface temperature exceeds a predetermined allowable temperature, the portable terminal device  100  reduces the communication rate of the wireless interface  114 . 
     Next, functions of the portable terminal device  100  and the design apparatus  200  will be described.  FIG. 16  is a block diagram illustrating exemplary functions of the portable terminal device and the design apparatus. 
     The portable terminal device  100  includes a parameter memory unit  131 , a past data memory unit  132 , a correspondence information memory unit  133 , a temperature measurement unit  134 , a surface temperature estimation unit  135 , and a heat source control unit  136 . The parameter memory unit  131 , the past data memory unit  132 , and the correspondence information memory unit  133  are implemented by using a memory region allocated in the RAM  112  or the non-volatile memory  113 , for example. The temperature measurement unit  134 , the surface temperature estimation unit  135 , and the heat source control unit  136  are implemented by using program modules executed by the CPU  111   a  or the CPU  111   b , for example. 
     The parameter memory unit  131  stores the parameter table  42  illustrated in  FIG. 10 . The parameter table  42  may be stored in the parameter memory unit  131  at the time of production or shipment of the portable terminal device  100 . Also, the parameter table  42  may be stored in the storage medium  120   a  and distributed to the portable terminal device  100 . Also, the parameter table  42  may be distributed to the portable terminal device  100  from a server apparatus via a wireless network. 
     The past data memory unit  132  stores the past data table  43  illustrated in  FIG. 12 . The correspondence information memory unit  133  stores the correspondence information table  44  illustrated in  FIG. 15 . The correspondence information table  44  may also be stored in the correspondence information memory unit  133  at the time of production or shipment of the portable terminal device  100 . Also, the correspondence information table  44  may be stored in the storage medium  120   a  and distributed to the portable terminal device  100 . Also, the correspondence information table  44  may be distributed to the portable terminal device  100  from the server apparatus via the wireless network. 
     The temperature measurement unit  134  periodically acquires the measured temperatures from the temperature sensors  122   a ,  122   b ,  122   c , and  122   d  at time intervals Δt (for example, Δt=10 seconds). The surface temperature estimation unit  135  periodically estimates the surface temperature of the housing  101  at time intervals Δt. Specifically, the surface temperature estimation unit  135  acquires the measured temperatures of the temperature sensors  122   a ,  122   b ,  122   c , and  122   d  from the temperature measurement unit  134 . The surface temperature estimation unit  135  calculates the relative temperatures of the heat sources on the basis of the measured temperatures of the acquired temperature sensors  122   a ,  122   b ,  122   c , and  122   d  and the parameter values stored in the parameter memory unit  131 . 
     Also, the surface temperature estimation unit  135  reads out the past data from the past data memory unit  132 . Then, the surface temperature estimation unit  135  estimates the surface temperature at a predetermined position on the surface of the housing  101  on the basis of the relative temperatures of the heat sources, the past data, and the parameter values stored in the parameter memory unit  131 . The surface temperature estimation unit  135  is also able to calculate the surface temperatures at a plurality of positions by using different parameter values. The surface temperature estimation unit  135  updates the past data stored in the past data memory unit  132 , by using a surface temperature estimation result. 
     The heat source control unit  136  controls components, which are heat sources, of the portable terminal device  100  on the basis of the surface temperature estimated by the surface temperature estimation unit  135 . When the surface temperature estimation unit  135  periodically calculates a plurality of surface temperatures, the heat source control unit  136  selects the maximum surface temperature from among the calculated surface temperatures. The heat source control unit  136  compares the surface temperature acquired from the surface temperature estimation unit  135  or the selected surface temperature with a predetermined threshold value. When the surface temperature exceeds the threshold value, the heat source control unit  136  restricts operation levels of the heat sources. 
     For example, by the aforementioned method, the heat source control unit  136  lowers the upper limit of the clock frequency of the CPU  111   a  in a phased manner such that the highest prediction reached value T sat  attenuates in accordance with the rate parameter. Also, for example, the heat source control unit  136  reduces the communication rate of the wireless interface  114 . Also, for example, the heat source control unit  136  stops charging by the charge circuit  124 . The CPU  111   a  instructs the electric power control unit  121  to control the operation levels of the CPU  111   a  and the wireless interface  114 , for example. 
     The design apparatus  200  includes an intermediate parameter memory unit  231 , a parameter memory unit  232 , a temperature data acquisition unit  233 , and a transfer function generation unit  234 . The intermediate parameter memory unit  231  and the parameter memory unit  232  are implemented by using a memory region allocated in the RAM  212  or the HDD  213 , for example. The temperature data acquisition unit  233  and the transfer function generation unit  234  are implemented by using program modules executed by the CPU  211 , for example. 
     The intermediate parameter memory unit  231  stores the intermediate parameter table  41  illustrated in  FIG. 9 . The intermediate parameter table  41  is generated by the design apparatus  200 . The parameter memory unit  232  stores the parameter table  42 . The parameter table  42  is generated by the design apparatus  200  on the basis of the intermediate parameter table  41 . The generated parameter table  42  may be stored in the non-volatile memory  113  of the portable terminal device  100  at the time of production or shipment of the portable terminal device  100 . Also, the generated parameter table  42  may be stored in the storage medium  120   a  and may be distributed via a network. 
     The temperature data acquisition unit  233  acquires various types of temperatures measured by using an actual device or sample implementation of the portable terminal device  100 . The measured temperature may be input into the design apparatus  200  by a user of the design apparatus  200 . Also, the measured temperature may be directly acquired from the actual device or the sample implementation of the portable terminal device  100  connected to the design apparatus  200 . 
     The transfer function generation unit  234  decides the values of the intermediate parameters included in the transfer functions by a fitting method, such as a least squares method, by using various types of measured temperatures acquired from the temperature data acquisition unit  233 . The transfer function generation unit  234  stores the calculated values of the intermediate parameters in the intermediate parameter memory unit  231 . Then, the transfer function generation unit  234  derives the values of the parameters included in the estimation equation for estimating the surface temperature, from the values of the intermediate parameters. The transfer function generation unit  234  stores the calculated parameter values in the parameter memory unit  232 . Note that one or two or more positions for estimating the surface temperature are designated by the user of the design apparatus  200 . 
       FIG. 17  is a flowchart illustrating an exemplary procedure of parameter decision. 
     (S 10 ) The transfer function generation unit  234  accepts designation of one or two or more positions (estimation points) on the housing surface for estimating the surface temperature, from the user of the design apparatus  200 . 
     (S 11 ) The temperature data acquisition unit  233  acquires heat source temperatures T hi (t), measured temperatures T sj (t) measured by the temperature sensors near the heat sources, a reference point temperature T rt (t), and a surface temperature T surf (t) of the designated estimation point. At this time, each of the heat sources are operated independently, so that the above various temperatures are measured in temporal sequence. 
     (S 12 ) The transfer function generation unit  234  calculates transfer functions H* ij =h ij  from the heat sources to the temperature sensors, by using the measured temperatures T sj (t) and the reference point temperature T rt (t). The transfer function H* ij =h ij  is calculated in the time domain by aforementioned formula (15). That is, the transfer function generation unit  234  calculates differences between the measured temperatures T sj (t) and the reference point temperature T rt (t) for the temperature sensors, and acquires heat transfer coefficients h ij  of the smallest difference among the temperature sensors. 
     (S 13 ) The transfer function generation unit  234  calculates an inverse function H* ij   1 =h* ij  of the transfer function H* ij =h ij . The inverse function is calculated from h ij  by aforementioned formula (13). 
     (S 14 ) The transfer function generation unit  234  calculates transfer functions G* i (s) from the heat sources to the estimation point on the housing surface, by using the heat source temperatures T hi (t), the reference point temperature T rt (t), and the surface temperature T surf (t). That is, the transfer function generation unit  234  decides the heat transfer coefficients g i  and the thermal time constants τ hi , τ g1i , and τ g2i  included in the transfer functions G* i (s). The transfer functions G* i (s) are calculated by a fitting method, such as a least squares method, in the frequency domain by aforementioned formula (16). That is, the transfer function generation unit  234  calculates and Laplace transforms the differences between the heat source temperatures T hi (t) and the reference point temperature T rt (t) for the heat sources, and calculates and Laplace transforms the differences between the surface temperatures T surf (t) and the reference point temperature T rt (t). The transfer function generation unit  234  acquires the heat transfer coefficients g i  and the thermal time constants τ hi , τ g1i , and τ g2i  of the smallest difference of formula (16). 
     (S 15 ) The transfer function generation unit  234  develops the transfer functions G* i (s) into difference equations. That is, the transfer function generation unit  234  calculates a 0i , a 1i , a 2i , b 1i , and b 2i  of aforementioned formula (20) on the basis of the heat transfer coefficients g i  and the thermal time constants τ hi , τ g1i , and τ g2i . 
     (S 16 ) The transfer function generation unit  234  sets the values of h* ij  calculated in step S 13  in the parameter table  42 . Also, the transfer function generation unit  234  sets the values of a 01 , a 11 , a 21 , b 11 , and b 21  calculated in step S 15  in the parameter table  42 . 
       FIG. 18  is a flowchart illustrating an exemplary procedure of heat source control. 
     (S 20 ) The temperature measurement unit  134  acquires the temperatures T sj (t) measured by the temperature sensors  122   a ,  122   b , and  122   c  near the heat sources. Also, the temperature measurement unit  134  acquires the reference point temperature T rt (t) measured by the temperature sensor  122   d.    
     (S 21 ) The surface temperature estimation unit  135  calculates the relative temperatures T hi (t)−T rt (t) of the heat sources by using the measured temperatures T sj (t), the reference point temperature T rt (t), and the inverse function H* ij   −1 =h* ij  of the transfer function H* ij . The relative temperatures of the heat sources are calculated in the time domain by aforementioned formula (17) by using the values of h* ij  set in the parameter table  42 . That is, the surface temperature estimation unit  135  calculates the differences between the measured temperatures T sj (t) and the reference point temperature T rt (t) for the temperature sensors, and calculates T hi (t)−T rt (t) by product-sum operation with h* ij . 
     (S 22 ) The surface temperature estimation unit  135  reads out the past data from the past data table  43 . That is, the surface temperature estimation unit  135  reads out the relative temperatures (differences between the heat source temperatures and the reference point temperature) calculated time Δt before for the heat sources. Also, the surface temperature estimation unit  135  reads out the estimated contribution to the housing surface which is calculated time Δt before, for each heat source. 
     (S 23 ) The surface temperature estimation unit  135  estimates the surface temperature T surf (t) from the relative temperatures T hi (t)−T rt (t) of the heat sources calculated in step S 21 , by using the difference equations. The surface temperature T surf (t) is calculated by aforementioned formula (18). That is, the surface temperature estimation unit  135  calculates the estimated contribution T surf-i (t) at a time point t, from the relative temperature at the time point t calculated in step S 21  and the past data read out in step S 22 , by aforementioned formula (19) for each heat source. At this time, the surface temperature estimation unit  135  uses the values of a 0i , a 1i , and b 1i  set in the parameter table  42 . Then, the surface temperature estimation unit  135  calculates the surface temperature T surf (t) by adding the estimated contribution T surf-i (t) of the heat sources to the reference point temperature T rt (t) at the time point t. 
     (S 24 ) The surface temperature estimation unit  135  records the relative temperatures of the heat sources which are calculated in step S 21  in the past data table  43  as the relative temperatures of time Δt before. Also, the surface temperature estimation unit  135  records the estimated contribution of each heat source which is calculated in step S 23  in the past data table  43  as the estimated contribution of time Δt before. The above steps S 22  to S 24  are executed for each estimation point, when two or more estimation points for estimating the surface temperature are designated. 
     (S 25 ) The heat source control unit  136  calculates a highest prediction reached value T sat (t) of the contribution to the surface temperature from the CPU  111   a  in accordance with formula (22). Also, the heat source control unit  136  calculates a threshold value T max (t) that is allowable for the contribution to the surface temperature from the CPU  111   a . Note that the surface temperature estimation unit  135  may calculate T surf-1 (t−1) one second before the calculation of T sat (t). Also, the heat source control unit  136  may calculate T sat (t) by using T surf-1 (t−Δt) calculated Δt seconds before. 
     (S 26 ) The heat source control unit  136  compares the calculated highest prediction reached value T sat (t) and the calculated threshold value T max (t) to determine whether T sat (t) exceeds T max (t). If T sat (t) exceeds T max (t), the process proceeds to step S 27 . In other cases, the process proceeds to step S 29 . 
     (S 27 ) The heat source control unit  136  calculates the decrease rate α=T max (t)/T sat (t) and then calculates the rate parameter ατ g1  which is a product of a and the thermal time constant τ g1  from the CPU  111   a  to the surface of the housing  101 . 
     (S 28 ) The heat source control unit  136  lowers the upper limit of the clock frequency on the basis of the rate parameter and the correspondence information table  44  illustrated in  FIG. 15 . The set upper limit is lower than the highest clock frequency that the CPU  111   a  is able to reach physically. Thereby, the CPU  111   a  is able to change the clock frequency only within a range of the set upper limit or below, in order to restrict the computing power and the heat generation amount of the CPU  111   a . Note that the clock frequency of the CPU  111   a  dynamically changes according to the load of the CPU  111   a , for example. 
     The heat source control unit  136  sets the upper limit of the clock frequency, in such a manner that T sat (t) at a time point t decreases to (T sat (t)−T max (t)) exp(−Δtx/ατ g1 )+T max (t) in Δtx seconds. For example, when Δt=10 seconds and x=2, the upper limit of the clock frequency is set in such a manner that T sat (t) becomes (T sat (t)−T max (t)) exp(−20/ατ g1 )+T max (t) in 20 seconds. In this case, the heat source control unit  136  selects the closest value to (T sat (t)−T max (t)) exp(−20/ατ g1 )+T max (t), among the target values that are set in the correspondence information table  44 , for example. Then, the heat source control unit  136  sets the upper limit at the clock frequency corresponding to the selected target value. 
     As described above, the clock frequency of the CPU  111   a  is restricted to reduce the heat generation amount. Note that the heat source control unit  136  may further control other heat sources, such as the charge circuit  124  and the electric power amplifier  114   a , to reduce the heat generation amount. Then, the process proceeds to step S 30 . 
     (S 29 ) The heat source control unit  136  abolishes the upper limit setting of the clock frequency of the CPU  111   a . Thereby, the upper limit of the clock frequency of the CPU  111   a  becomes identical with the highest clock frequency that is achievable physically. The CPU  111   a  is able to freely change the clock frequency within a physically achievable range, and the computing power and the heat generation amount of the CPU  111   a  are not restricted. Note that the heat source control unit  136  may make the upper limit of the clock frequency higher than that of the present moment, instead of abolishing the upper limit setting of the clock frequency. 
     (S 30 ) The temperature measurement unit  134  and the surface temperature estimation unit  135  wait for the time Δt (for example, 10 seconds). After the time Δt elapses, one cycle of heat source control ends, and the process is executed again from step S 20 . 
     By the above process, T sat (t) and T surf-1 (t) change as in the following, for example. 
       FIG. 19  is a graph illustrating exemplary change of T sat (t) and T surf-1 (t) corresponding to a plurality of values of α. In the following, the values of T sat (t) and T surf-1 (t) are normalized in such a manner that the highest prediction reached value calculated at a certain time point becomes 1.0, in some cases. 
     In the graph  36 , a curving line  36   a  indicates change of the normalized T sat (t) attenuated from 1 to α=0.75. A curving line  36   b  indicates change of the normalized T sat (t) attenuated from 1 to α=0.5. A curving line  36   c  indicates change of the normalized T sat (t) attenuated from 1 to α=0.25. 
     Also, in the graph  36 , a curving line  36   d  corresponds to the curving line  36   a  and indicates change of normalized T surf-1 (t) when α=0.75. A curving line  36   e  corresponds to the curving line  36   b  and indicates change of normalized T surf-1 (t) when α=0.5. A curving line  36   f  corresponds to the curving line  36   c  and indicates change of normalized T surf-1 (t) when α=0.25. 
     By the aforementioned control of the clock frequency, T sat (t) attenuates at a decrease rate based on ατ g1  and converges to α. Along with this, T surf-1 (t) converges to α. As described below, T sat (t) is controlled to be attenuated at the rate of the time constant ατ g1 , so that it is possible to increase the upper limit of the clock frequency of the CPU  111   a  up to a limit that prevents overshoot of T surf-1 (t). 
       FIG. 20  is a graph illustrating exemplary change of T sat (t) and T surf-1 (t) when the rate parameter is not used. In a graph  37 , curving lines  37   a  and  37   b  indicate change of the normalized T sat (t) attenuated from 1 to α=0.5. α=0.5 corresponds to the threshold value T max (t). The decrease rate of T sat (t) indicated by the curving line  37   a  is slower than the decrease rate indicated by the rate parameter ατ g1 . On the other hand, the decrease rate of T sat (t) indicated by the curving line  37   b  is faster than the decrease rate indicated by the rate parameter ατ g1 . Also, curving lines  37   c  and  37   d  indicate the change of normalized T surf-1 (t). The curving line  37   c  corresponds to the curving line  37   b , and the curving line  37   d  corresponds to the curving line  37   a.    
     When the clock frequency is decreased rapidly, the decrease rate of T sat (t) becomes fast like the curving line  37   b , and the increase rate of T surf-1 (t) becomes too slow like the curving line  37   c . Hence, despite the margin to the threshold value, the CPU  111   a  is operated at a clock frequency that is too low, possibly decreasing the performance of the CPU  111   a . On the other hand, when the clock frequency is decreased at a slow rate, the decrease rate of T sat (t) becomes slow like the curving line  37   a , and the increase rate of T surf-1 (t) becomes fast like the curving line  37   d . Hence, “overshoot”, in which T surf 1 (t) temporarily exceeds the threshold value like the curving line  37   d , occurs in some cases. 
     In contrast, as illustrated in  FIG. 19 , the performance of the CPU  111   a  is maximized within a range in which the surface temperature does not overshoot, by lowering the upper limit of the clock frequency in a phased manner such that T sat (t) decreases at a specific rate. The inventors of the present application discovered that the performance of the CPU  111   a  is maximized when the time constant for decreasing T sat (t) is ατ g1 . Thus, with the portable terminal device  100 , it is possible to maximize the performance of the CPU  111   a  to the limit that prevents the overshoot of the surface temperature. 
       FIG. 21  illustrates a control example of the surface temperature of the housing. The vertical axis represents a relative temperature dT [° C.] which is an increase amount relative to a reference point temperature, and the horizontal axis represents time t [sec]. In a graph  38 , a curving line  38   a  indicates temporal change of the highest prediction reached temperature when the clock frequency of the CPU  111   a  is not restricted (for example, when the CPU  111   a  continues operating at the highest clock frequency). A curving line  38   b  indicates temporal change of the highest prediction reached temperature when the clock frequency is restricted by the aforementioned method. A curving line  38   c  indicates temporal change of the contribution to the surface temperature from the CPU  111   a  when the clock frequency is restricted by the aforementioned method. 
     In the example of  FIG. 21 , the reference point temperature is 30° C. Also, the allowable surface temperature of the housing  101  is 45° C., and the threshold value of the contribution to the surface temperature from the CPU  111   a  is 15° C. When the load of the CPU  111   a  increases rapidly, it is expected that the contribution to the surface temperature from the CPU  111   a  increases to 40° C. shortly (the highest prediction reached temperature is 40° C.), like the curving line  38   a . If this goes on, the surface temperature of the housing  101  exceeds the allowable temperature. 
     In contrast, the heat source control unit  136  restricts the clock frequency in such a manner that T sat (t) decreases like the curving line  38   b , and thereby the surface temperature of the housing  101  converges to 15° C. like the curving line  38   c . Also, the performance of the CPU  111   a  is prevented from deteriorating due to excessive frequency restriction, by setting the time constant for decreasing T sat (t) to ατ g1 . 
     As described above, in the portable terminal device  100  of the second embodiment, the temperature difference T surf-1 (t) generated by the heat generation of the CPU  111   a , which is part of the difference between the surface temperature of the housing  101  and the reference temperature, is calculated as a temperature index value of the surface temperature of the housing  101 . Then, T sat (t) is calculated as the highest prediction reached value that T surf-1 (t) is predicted to reach over time. Also, the rate parameter ατ g1  indicating the target of the decrease rate for decreasing T sat (t) is calculated on the basis of T sat (t) calculated at a time point t and the threshold value T max (t). Then, the upper limit of the clock frequency of the CPU  111   a  is restricted on the basis of the calculated ατ g1  and the correspondence information that associates a plurality of target values of T sat (t) and a plurality of clock frequencies. 
     Thereby, the clock frequency is needless to be restricted excessively as compared with a case in which the upper limit of the clock frequency is lowered rapidly to a level corresponding to T max (t) when T sat (t) exceeds T max (t). That is, if the heat capacity between the CPU  111   a  and the surface of the housing  101  is taken into consideration, even when the clock frequency temporarily exceeds the level corresponding to T max (t) in a transient state, the subsequent T surf 1 (t) does not exceed T max (t) in some cases, and there is much time to converge the clock frequency. The restriction of the clock frequency is made gradual by utilizing this transient state, and thereby the clock frequency is needless to be restricted excessively, and the performance of the CPU  111   a  is prevented from decreasing. 
     Note that, as described above, the information processing of the first embodiment is achieved by causing the electronic device  1  to execute a program. The information processing of the second embodiment is achieved by causing the portable terminal device  100  and the design apparatus  200  to execute programs. 
     The programs may be recorded in a computer-readable storage medium (for example, the storage medium  120   a  or  223 ). The storage medium may be a magnetic disk, an optical disc, a magneto-optical disk, or a semiconductor memory, for example. The magnetic disk includes an FD and an HDD. The optical disc includes a CD, a CD-R(Recordable)/RW(ReWritable), a DVD, and a DVD-R/RW. The programs may be stored in a portable storage medium for the purpose of distribution. In that case, the programs may be copied and executed in another storage medium (for example, the non-volatile memory  113  or the HDD  213 ) from the portable storage medium. 
     In one aspect, the performance of the processor is prevented from deteriorating in temperature control. 
     All examples and conditional language provided herein are intended for the pedagogical purposes of aiding the reader in understanding the invention and the concepts contributed by the inventor to further the art, and are not to be construed as limitations to such specifically recited examples and conditions, nor does the organization of such examples in the specification relate to a showing of the superiority and inferiority of the invention. Although one or more embodiments of the present invention have been described in detail, it should be understood that various changes, substitutions, and alterations could be made hereto without departing from the spirit and scope of the invention.