外文翻譯--基于單片機的低成本教學數(shù)字溫度控制器

1、<p><b>　　外文翻譯原文</b></p><p>　　Teaching digital control using a low-cost microcontroller-based temperature control kit</p><p>　　Abstract The design of a low-cost digital tempera

2、ture control kit is described. The system enables the students to implement various control strategies using a microcontroller. The kit is intended to be helpful in a control laboratory as a complement to the digital con

3、trol system theory taught to undergraduate students. </p><p>　　Keywords microcontroller-based control; teaching digital control; temperature control</p><p>　　With the availability of low-cost

4、computers and microcontrollers, digital control has gained popularity and most current control systems are based on digital techniques. The same is true for simulation. Digital simulation techniques have replaced analogu

5、e simulators. These simulators are in the form of interactive computer packages used in the industry, in research laboratories, colleges and universities. Early simulation packages such as ACSL were designed for large ma

6、inframe computers and only</p><p>　　Simulation is an invaluable tool in teaching the theory of control system. For example, the student can plot the accurate root-locus of a complex System in a matter of a f

7、ew minutes rather than spending several hours. Similarly, the time and frequency responds of a system can very easily be plotted with the aid of simulator package. Although the simulators are very useful tools they are n

8、ot same as real world solutions. There are also cases in which computer models may be inappropriate, or the s</p><p>　　One of the problems with commercially available physical laboratory experiments is that

9、the experimental kits are usually very expensive, especially when a number of similar kits are purchased for teaching purpose. Such kits may also require frequent calibration and maintenance services as a result of compo

10、nent failures and ageing. Laboratory kits also do become obsolete quite rapidly as new products are developed.</p><p>　　This paper describes a low-cost temperature control kit which is designed and used in t

11、he engineering teaching laboratories of Near East University. The kit is based on the popular PIC16F877 model microcontroller, manufactured by Microchip Inc. The overall cost of the kit is less than $200, which is well b

12、elow the cost of similar commercially available educational temperature control kits. The design, modeling and digital control of the kit are described in detail.</p><p>　　Temperature control kit</p>

13、<p>　　Educational temperature control kits are not new. Many companies manufacturing laboratory kits also offer some kind of general process control or temperature control kits.TCL-1 by Kuruganti is a temperature co

14、ntrol loop trainer which is intended to show how the temperature in a heat exchanger can be controlled. TCL-1, also by Kuruganti, is an on/off-based temperature control teaching kit. PROCON by Feedback Instruments is a p

15、rogress control system, which includes rigs for level, flow, temperatur</p><p>　　Near East University offers undergraduate and graduate level engineering courses and control engineering is one of these topic

16、s, which is taught for one semester. There are no practical experiments and students have been using the MATLAB package to design, simulate and test their control theory. It was felt necessary to provide some practical e

17、xperiments to the students as a way of supporting the theoretical concepts taught in the classroom. The main reason to design a control kit rather than to</p><p>　　The block diagram of the digital temperatur

18、e control kit is shown in Fig.1. The working principle of this experiment consists of heating the water in a small container using a low-voltage electric heating element and a simple MOSFET-based power controller circuit

19、. A temperature sensor is immersed into the water whose output signal is sent to a PIC-type microcontroller. This signal is compared with a reference temperature signal and a PID controller algorithm is implemented by th

20、e microcontroller</p><p>　　This process is used to teach the following important concepts to the students:</p><p>　　·Modeling and identification of a real physical process</p><p&

21、gt;　　1、Using the Ziegler-Nichols tuning method</p><p>　　2、Using microcontrollers in process automation</p><p>　　3、Developing and experimenting with digital PID controllers.</p><p>　

22、　Figure 2 shows a picture of the prototype experiment kit. The kit is rather simple, consisting of only low cost materials. A round plastic container is used to store the water. The heater element and the sensor are imme

23、rsed in this container. The temperature is sensed using a low cost semiconductor sensor, which is protected inside a glass tube. The heating element is the type which is used in camping and other outdoor activities in or

24、der to warm up liquid in a cup, for example for making coffee</p><p>　　Fig 2: The temperature control kit</p><p>　　Figure 3 shows the electrical circuit diagram of the kit .The circuit is rather

25、 simple, consisting of only a few parts .LM35DZ is the analog semiconductor temperature sensor , PIC16F877 is the microcontroller ,and IRL1004 is a power MOSFET switch ,used to drive the heater element .</p><

26、p>　　The temperature sensor </p><p>　　The temperature sensor used in the experiment is a 3-pin semiconductor sensor with an output voltage directly proportional to the temperature. The output of the sensor

27、 is connected to one of the A/D converter inputs of the microcontroller.</p><p>　　There was the option of using a digital output sensor ,but they are usually more expensive and it was also felt necessary to

28、use an analog sensor and teach the students the practical applications of A/D converters.</p><p>　　The microcontroller</p><p>　　In order to lower costs ,we needed a microcontroller with a built-

29、in A/D converter. Process control algorithms require the use of floating-point arithmetic and as a result,a microcontroller with a large date memory was also required. Author requirement to lower the cost was a built-in

30、pulse width modulated (PWM) output ,which was used to drive the heater circuitry linearly .The development of floating-point arithmetic routines is very complex using the assembly language and it was decided to pro</p

31、><p>　　The PIC16F877 microcontroller satisfied all of our requirements. This is one of the most popular microcontrollers used in industry and it offers the following features:</p><p>　　1、8K x14 fla

32、sh program memory</p><p>　　2、368 bytes RAM memory</p><p>　　3、256 bytes EEPROM memory</p><p>　　4、8x10 bit A/D converters</p><p>　　5、Pulse width modulated (PWM)output<

33、/p><p>　　6、High-level language support</p><p>　　The FEDC compiler was used for program development. This is a Windows-based low-cost compiler for the PIC family of microcontrollers. The compiler of

34、fers a large number of standard C library functions, including support for floating-point arithmetic.</p><p>　　The heater driver</p><p>　　An IRL1004 power MOSFET switch is used to drive the heat

35、er element. This MOSFET can dissipate up 200W when mounted on a suitable heat sink. The heating element is connected to the drain pin of the MOSFET and the gate input is controlled from the microcontroller (see Fig.3).&l

36、t;/p><p>　　Large industrial temperature control systems are based on a.c. power control techniques using thyristors and triacs and appropriate theory is given to the students on this topic.</p><p>

37、<b>　　Modelling</b></p><p>　　The system can be approximated to a first-order system with a time lag. A simplified mathematical model of the overall system can be derived as described here.</p&g

38、t;<p>　　Mathematical model of the tank</p><p>　　The heat-balance equation for the tank can be written as:</p><p>　　Heat input to the system=heat increase in the system +heat losses</p&

39、gt;<p><b>　　If we let</b></p><p>　　m1=mass of water inside the tank</p><p>　　m2=mass of the water</p><p>　　c1=specific heat capacity of the water</p><p

40、>　　c2= specific heat capacity of the tank</p><p>　　Ignoring the heat loss through the walls of the tank and the heat capacities of the heater element and the mixer ,we can write the following equation &l

41、t;/p><p>　　Heat increase in the tank=(m1*c1+m2*c2) </p><p>　　Heat loss from the tank= h*A*(T-Ta)</p><p>　　Where Ta is the ambient temperature, A is the tank top area, and h is a consta

42、nt, which depends on the surface and the ambient temperature.</p><p>　　Thus, the heat input to the system is</p><p>　　E=(m1*c1+m2*c2) + h*A*(T-Ta) （1）</p><p&

43、gt;　　If we assume that the ambient temperature is constant, and let</p><p><b>　　Tq=T-Ta</b></p><p>　　We can write equation (1) as:</p><p>　　E=(m1*c1+m2*c2) + h*A*Tq</

44、p><p>　　Or, letting k1= m1*c1+m2*c2 and k2= h*A</p><p>　　= （2）</p><p>　　Which is a first order system with time constant k1/k2.</p><p&g

45、t;　　Mathematical model of the heater</p><p>　　The relationship between the applied voltage and energy generated by an electrical heating element is non-linear.</p><p>　　In this experiment this l

46、inearised by driving the heater from a pulse width modulated (PWM) signal. A pulse width modulated signal is generated from the microcontroller as shown in Fig4 where M and S are the mark and the space of the waveform, a

47、nd</p><p>　　T is the period ,i.e. T=M+S. This waveform is used to control a power MOSFET switch where the heater element is connected as the load of this device.</p><p>　　The r.m.s. value of the

48、 current through the heater can be calculated as</p><p><b>　　Or,</b></p><p><b>　?。?）</b></p><p>　　Fig.4 PWM heater waveform </p><p>　　Assumi

49、ng the heating element has a pure resistance ,R ,the average power delivered to the heater can be calculated as:</p><p>　　If we let then</p><p>　　=

50、 (4)</p><p>　　Equation (4) shows that the average power delivered to the load is linearly proportional to the on-time (M) of the signal.We will call M the duty cycle of the waveform.</p><p&g

51、t;　　The frequency of the waveform must be well above the closed-loop bandwidth of the control system so that the process is only affected by the mean level of the waveform. In this project, we will assume a frequency of

52、1kHz,i.e. the period is 1ms.</p><p>　　In this project,</p><p>　　Thus, the transfer function of the heater is, form equation (4):</p><p><b>　　Or ,</b></p><p>

53、;<b>　　(5)</b></p><p>　　Where is in watts and M is in seconds.</p><p>　　Equation (5) shows the linear relationship between the duty cycle of the applied signal and the average power

54、generated by the heater.</p><p>　　Mathematical model of the temperature sensor </p><p>　　The temperature sensor is a semiconductor device with a linear voltage-temperature relationship specifi

55、ed as 10mV/, i.e.</p><p><b>　　(6)</b></p><p>　　Where is the sensor output voltage in volts , and T is the temperature in .</p><p>　　Experiment example </p><

56、p>　　Identification of the system </p><p>　　The dynamic behavior of the system is identified using non-parametric , by using a reaction curve method .For this , the feedback loop is opened , and a step PWM

57、 input is applied to the heater driver by the microcontroller, The temperature of the water in the tank is then measured and recorded every second by connecting the output of the sensor to the voltage input of DrDaq har

58、dware and Picolog software . Both of these products are manufactured by Pico Technology . DrDaq is a small card which i</p><p>　　voltage ,humidity ,and temperature .Picolog software runs on a PC and can be u

59、sed to record the measurements of the DrDaq card in real time .The software includes a graphical option that enables the measurements to be plotted.</p><p>　　A Ziegler-Nichols tuning method is then used to i

60、dentify the system , as shown in Fig 5 .The open-loop system transfer function was found to be </p><p><b>　?。?）</b></p><p>　　The system has a large time lag (180 seconds) and a time

61、constant of 1800 seconds .</p><p>　　Choosing a controller algorithm</p><p>　　The PID algorithm was selected as the controller since it is probably the most extensively used method in industrial

62、process control applications .A large number of references can be found which describe the continuous and digital forms of this controller , its performance evaluation ,implementation and auto-tuning forms .</p>&

63、lt;p>　　Fig 5: using the Ziegler-Nichols method to find system parameters</p><p>　　The transfer function of the standard PID algorithm is ：</p><p>　　The block diagram of the continuous PID con

64、troller is shown in Fig 6 ,where , is the proportional gain ， is the integral time constant， is the derivative time constant ，u（t）is the controlling parameter，e（t） is the windage of the controlled parameter y（t）and the

65、 given parameter .</p><p>　　Fig 6: Block diagram of continuous PID controller</p><p>　　In the s-domain , the PID controller can be written as </p><p><b>　?。?）</b></p&

66、gt;<p>　　The discrete form of the PID controller can be derived by finding the z-transform of equation (9):</p><p><b>　?。?0）</b></p><p>　　Equation (10) is usually written as :

67、</p><p><b>　?。?1）</b></p><p><b>　　where：</b></p><p>　　PI controller</p><p>　　The Ziegler-Nichols parameters for a PI controller are :</p>

68、<p><b>　　and </b></p><p>　　Taking a sampling time of T=20s ,a and b in equation (10) are calculated to be a=10.9 ,b=0.37 </p><p>　　The PI algorithm implemented on the microcont

69、roller is the following :</p><p><b>　　BEGIN</b></p><p>　　Read a and b parameters of the controller</p><p>　　Read MAX and MIN</p><p>　　Read the set-point tem

70、perature</p><p>　　Initialise the A/D converter </p><p>　　DO FOREVER</p><p>　　Read the set-point r（KT）</p><p>　　Read water temperature y（KT）</

71、p><p>　　Calculate error e（KT）= r（KT）- y（KT）</p><p>　　Calculate proportional term q（KT）=a* e（KT）</p><p>　　Calculate integral term p（KT）=b* e（KT）+ p（KT-T）

72、</p><p>　　Calculate output u（KT）=p（KT）+q（KT）</p><p>　　If u（KT）>MAX</p><p>　　p（KT）=p（KT-T）</p><p><b>　　u（KT）=MAX</b></p><p>

73、;　　else if u（KT）<MIN</p><p>　　p（KT）=p（KT-T）</p><p><b>　　u（KT）=MIN</b></p><p><b>　　end if</b></p><p>　　Save for next cycle</p><p>

74、;　　p（KT-T）=p（KT）</p><p>　　e（KT-T）=e（KT）</p><p>　　Wait for next cycle</p><p><b>　　END DO</b></p><p><b>　　END</b></p><p>　　This algor

75、ithm was implemented using C language. The controller output is limited to be within MIN and MAX in order to avoid integral saturation.</p><p>　　The response of the system with the PI controller is shown in

76、Fig 7 .In this example, the set point was 30 and the temperature reached this value with no overshoot and no steady-state errors. One of the advantages of using the Ziegler-Nichols method is that it yields a satisfactor

77、y response with little effort .</p><p>　　Fig 7: Response of the system with PI controller</p><p>　　Some computer packages such as the ExperTune by Top Control run on a PC ,analyse a system in r

78、eal time and provide an optimum set of PID controller parameters .With the availability of such packages it should take much less time to tune a PID controller satisfactorily.</p><p>　　Conclusion</p>

79、<p>　　The design of a low-cost digital temperature control kit has been described .The aim in designing this kit was to teach engineering students the practical applications of the theory they are taught in the clas

80、sroom.</p><p>　　The kit is designed using standard low-cost components which are readily available in most electronic component shops .Another advantage of the kit is that it enables the students to experime

81、nt and learn the microcontrollers which are used extensively in most intelligent electronic control projects .The kit is complemented with a laboratory manual which is written to help the students follow the experiments

82、in an orderly way.</p><p>　　Other control algorithms and design procedures such as state-space techniques can be developed for the kit. It is also hoped to develop other automation kits in the near future ,s

83、uch as level control systems ,flow control systems ,servo control systems and so on .</p><p>　　References</p><p><b>　　外文翻譯中文</b></p><p>　　基于單片機的低成本教學數(shù)字溫度控制器</p>&

84、lt;p>　　摘要：這種低成本的數(shù)字溫度控制器的設計可以被這樣描述。由于使用了單片機，這一系統(tǒng)可以使學生能夠控制各種各樣的溫度。這一器件在控制實驗室是非常有用的，對于學生可以作為數(shù)字控制系統(tǒng)理論的一個補充。</p><p>　　關鍵詞：基于單片機控制；教學數(shù)字控制；溫度控制</p><p>　　伴隨著低損耗計算機和單片機的有效性、實用性，數(shù)字控制已經(jīng)變得越來越受歡迎，當前最流行的控制

85、都是以數(shù)字技術為基礎的。對于仿真器來說也是如此。數(shù)字模擬技術已經(jīng)取代了相類似的仿真器。 這些仿真器是以交互式的盒裝計算機的形式應用于工業(yè)，研究實驗室以及各種大學院校。早期的盒裝仿真器像ACSL是為了大型的中央處理器而設計的，因此只有那些大型的機構組織才有能力購買和使用這樣的盒裝仿真器。目前，像TUTSIM，20-sim，program CC，VisSim，Extend和MATLAB這樣的軟件包在計算機的平臺上是非常有用的，絕大多數(shù)的大學

86、院校都把這些軟件的花費作為其中的一項預算。其中一些軟件包（像MATLAB和program CC）現(xiàn)在也已經(jīng)提供了較低花費的版本，學生能都購買并且在自己的電腦上使用這些軟件，而不僅僅是在學校的實驗室里。</p><p>　　在控制系統(tǒng)的理論教學中仿真器是一種非常有用的工具。比如，學生能夠在幾分鐘的時間內(nèi)就可以精確地指出一個復雜系統(tǒng)的關鍵部位，而不是幾個小時的時間。類似地，當你使用了一個仿真器的幫助功能時，你就能很容

87、易地知道一個系統(tǒng)的相應時間和頻率。盡管仿真器是一個非常有用的工具，但是他們和現(xiàn)實社會中的器件還是存在一定得差別的。計算機中的一些例子也許并不適宜的或者是系統(tǒng)太復雜而不能用計算機中的數(shù)學等式來描述。根據(jù)作者的經(jīng)驗，當學生們能夠親眼看到實驗的各種現(xiàn)象時，他們會更好地學習工程課題。仿真器仍然會在分析的最初水平上被使用但是這并不能取代現(xiàn)實中的物理實驗。仿真器應該作為一種互補的工具而不是工程課題中的唯一工具。</p><p&g

88、t;　　物理實驗室里的實驗設備的一個主要問題是價格通常是很貴的，尤其是教學上的一些相類似的器件。一些儀器是需要經(jīng)常校準和像儀器中元器件的損壞或者到達使用期限錢的一些保養(yǎng)服務。隨著新產(chǎn)品的不斷產(chǎn)生，實驗室中的儀器更新是相當快的。</p><p>　　本文描述的是一種低損耗的溫度控制儀器，這種儀器是在東方的一所大學的工程教學實驗室設計出來，正在被使用中。這種儀器是基于當前非常流行的PIC16F877的單片機模型，是由

89、一家集成電路公司制造的。這種儀器的總共花費不到200美元，價格遠遠低于市場上相類似的教學溫度控制器件。本文將詳細描述這種溫度控制器件的設計，模型和數(shù)字控制。</p><p><b>　　溫度控制器件</b></p><p>　　教學溫度控制器件現(xiàn)在已經(jīng)非常普遍。許多實驗室器件的制造廠商也會提供一些全程控制或者是溫度控制器件。TCL-1可以說是一個溫度控制程序的培訓者，

90、它打算呈現(xiàn)一個加熱系統(tǒng)中的溫度是怎么樣被控制的。TCL-1當然也是基于溫度控制教學器件的。PROCON是一個溫度控制系統(tǒng)，它包括了高度，流量，溫度和PH值控制等裝置。在這里，溫度控制系統(tǒng)是采用了流動的水和簡易的PID控制來實現(xiàn)的。Elettronica公司制造了G34/EV，這G34/EV是基于PID控制的教學溫度控制單元。這種單元可以直接與計算機相連接，并且由PID控制器，電源放大器和溫度傳感器組成。</p><p

91、>　　Near East 大學為工程課程和控制工程的學生提供了一個學期的時間來學習這一課題。沒有現(xiàn)實的實驗，學生們就用MATLAB軟件包來進行設計，仿真和測試他們的控制理論。在教室里為學生提供一些實用的實驗是非常有必要的，它可以為理論概念的學習提供一種支持。設計控制器而不是去市場山購買控制器的主要原因是因為花費。過程控制在自動控制工程中是一個非常重要的領域，于是就決定做一個基于單片機的數(shù)字溫度控制實驗的。實驗室器件的一個目標是價格

92、便宜，但是在工業(yè)試驗中能夠確保使用的可靠性。在不久的將來，我希望有更多的教師和學生能夠參與到控制試驗中來。</p><p>　　圖1顯示的是數(shù)字溫度控制器的原理圖。這一實驗的工作原理是在一個小容器里使用低壓電加熱元件和一個簡單的基于MOSFET的電源控制電路來加熱水。將溫度傳感器放在水中，它可以將外界的信號傳送到直插式的單片機中。這一信號會和參考溫度信號進行比較，然后單片機會啟動PID算法會完成所要求的溫度控制。

93、</p><p>　　圖1：溫度控制器的原理圖</p><p>　　這一過程可以用來把以下重要的概念教給學生：</p><p>　　物理過程的模型和等效性</p><p>　　使用Ziegler-Nichol的方法</p><p>　　在自動化控制過程中單片機的使用</p><p>　　PID控

94、制器的發(fā)展和實驗</p><p>　　圖2顯示的實驗控制器的模型圖。這一器件非常簡單，僅有低成本的材料組成。圓形塑料容器是用來儲存水的。加熱元件和傳感器浸入容器中。溫度傳感器是用低成本的半導體元件做成的，它能夠被內(nèi)部的玻璃管保護。這種加熱器件被用于野營或者是其他戶外活動，能夠用來加熱杯子里的液體，比如沖咖啡。這種加熱設備的電壓是12V，通過的電流是10A，并且能夠提供120W的功率。實驗室電源供應商提供如此高功率

95、的通常價格比較昂貴，因此標準的350W的計算機電源被花費不多于50美元的電源取代了。在實驗器件中使用低壓有一種優(yōu)勢，系統(tǒng)是安全的因為沒有了電震的風險。</p><p>　　圖2：溫度控制器的實物圖</p><p>　　圖3顯示的是溫度控制器的電路圖。這電路圖是非常的簡單，僅僅有幾部分組成。LM35DZ是模擬的半導體溫度傳感器，PIC16F877是微控制器，IRL1004是MOSFET電源開

96、關，它能夠啟動加熱器件。</p><p>　　圖3：溫度控制器的電路原理圖</p><p><b>　　溫度傳感器：</b></p><p>　　試驗中使用的溫度傳感器是3個引腳的半導體傳感器，它有一個直接與溫度相對應的輸出電壓。傳感器的輸出端與微控制器中的A/D轉換器的一個輸入引腳相連接。</p><p>　　數(shù)字輸出

97、傳感器的使用通常是可以選擇的，但是它們一般都比較貴，因此使用模擬的傳感器和教學生A/D轉化器的實際應用是很有必要的。</p><p><b>　　微型控制器：</b></p><p>　　為了能使成本降到最低，我們需要一個微型控制器來建立一個A/D轉化器。過程控制算法要求使用浮點算術以及用它作為結果。因此需要大的數(shù)據(jù)存儲的微型控制器。低成本的另一個要求是建立一個脈寬輸

98、出調(diào)節(jié)（PWM），它能夠線性啟動加熱電路。浮點算術的發(fā)展歷程是非常復雜的，它需要使用的是匯編語言，因此決定用高級語言來編寫微型控制器中的程序，這種語言同樣支持浮點算術。</p><p>　　PIC16F877微型控制器符合我們的所有要求。在工業(yè)生產(chǎn)中，它是最受歡迎的微型控制器之一，因為它具有以下特征：</p><p>　　1、8Kx14的閃存</p><p>　　2

99、、368bytes的RAM存儲</p><p>　　3、256bytes的EEPROM存儲</p><p>　　4、8x10位的A/D轉換器</p><p>　　5、脈寬輸出調(diào)節(jié)（PWM）</p><p><b>　　6、支持高級語言</b></p><p>　　FEDC放大器程序擴大。這是一種基

100、于Windows的低成本放大器，它是微型控制器中的一種。這放大器能夠提供許多標準的功能，比如支持浮點算法。</p><p><b>　　啟動加熱裝置</b></p><p>　　IRL1004 MOSFET電源開關是用來啟動加熱裝置的。當加熱裝置的溫度達到合適的下降點時，這種MOSFET能夠驅散高達200W的能量。加熱元件可以和MOSFET的引腳相連接，而輸入端是由微

101、型控制器控制的（見圖3）。</p><p>　　工業(yè)上大的溫度控制系統(tǒng)都是基于交流電的控制技術，它們采用電子半導體閘流管和本文中教給學生的一些準確的理論。</p><p><b>　　模型</b></p><p>　　這一系統(tǒng)和先前的系統(tǒng)幾乎是一樣的。整個系統(tǒng)的簡單的數(shù)學模型會在這里進行詳細地描述。</p><p>&l

102、t;b>　　容器的數(shù)學模型:</b></p><p>　　容器的熱平衡等式如下：</p><p>　　系統(tǒng)的輸入熱量=系統(tǒng)增加的熱量+系統(tǒng)損失的熱量</p><p><b>　　假如我們設定：</b></p><p>　　m1 =容器中水的質量</p><p><b>

103、　　m2=容器的質量</b></p><p><b>　　c1=水的比熱</b></p><p><b>　　c2=容器的比熱</b></p><p>　　不考慮通過容器壁，加熱裝置的加熱能力散失的熱量，我們可以列出以下等式：</p><p>　　容器中增加的熱量=(m1*c1+m2*c

104、2)</p><p>　　容器中損失的熱量=h*A*(T-Ta)</p><p>　　在這里Ta是周圍環(huán)境的溫度，A是容器頂部的面積，h是常量，它決定于表面和外界環(huán)境的溫度。</p><p>　　因此，系統(tǒng)的輸入熱量為：</p><p>　　E=(m1*c1+m2*c2)+ h*A*(T-Ta)

105、 （1）</p><p>　　假如我們假定周圍環(huán)境的溫度是一個常量，那么</p><p><b>　　Tq=T-Ta</b></p><p>　　我們就可以把等式（1）寫為：</p><p>　　E=(m1*c1+m2*c2)+ h*A*Tq</p><p>　　或者是，我們令k1= m1*c

106、1+m2*c2 ，k2= h*A</p><p>　　則= （2）</p><p><b>　　加熱器的數(shù)學模型：</b></p><p>　　放大器的電壓和電熱器件產(chǎn)生的能量兩者之間的關系是非線性的。在這個實驗中，通過脈寬調(diào)節(jié)輸出的信號可以使放大器的電壓和電熱器產(chǎn)生的能