Chapter 12

# Digital display thermometer

Take 15 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

If you have more questions, feel free to ask them on the forum.

The principle of a digital display thermometer is :

The principle of a digital display thermometer

The temperature probe is a platinum resistor, whose resistance varies with temperature according to the law :

Where is the temperature expressed in , the coefficient of temperature of the resistivity ( ), the resistance in  at the temperature and the resistance in at ( ).

This sensor is placed in one arm of a measuring bridge (Wheatstone bridge) fed by an ideal generator of direct current ( ).

Resistances , , and are assumed to be independent of temperature.

Wheatstone bridge

## Question

Determining the voltage  as a function of , , , and .

### Solution

The KVL applied to the loop defined in the figure above gives :

Yet, according to the rule of the current divider :

And :

Referring in expression of , we finally obtain :

## Question

What value does it give to  if the bridge is in equilibrium at , that is to say at that temperature ?

Keep this value for the following exercises.

### Solution

So that the bridge is in equilibrium at the temperature of , it is necessary that :

So :

## Question

The temperature may vary between and , determine in terms of temperature .

Make the numerical application for , , , , and .

Does linearly vary with temperature ?

### Solution

At any temperature (in ), platinum resistance can be expressed as :

The voltage can then be written :

Consequently, using the equilibrium condition of the bridge at :

Numerically, we get :

With in and in .

The voltage values for the conditions suggested in the statement are listed in the table below :

 Température Voltage (V)

It is found that in the range of considered temperatures, and that, therefore, : the voltage varies linearly with temperature.

## Question

The signal delivered is small, it is amplified. Which amplifier circuit do you use ? After amplification and shaping, the tension is obtained :

With in °C and in .

### Solution

A simple amplifier circuit can, for example, be a non-inverting amplifier circuit, made of an ideal operational amplifier and whose schema is recalled in the figure below, for which :

As part of this exercise, the affine variation between and is :

With in and in .

Non-inverting amplifier circuit

## Question

The voltage is applied to an analog-digital converter (ADC) successive approximation - bits - scale ( - ).

The ADC can encode the analog voltage into a number of binary digits ( bits).

The transfer characteristic is given in the figure below.

• Why do we use the base not base ? How many digital values ADC can it distinguish ?

• What is the minimum variation of so that the numerical value in base is changed by one unit, that is to say the most significant bit ?

• Deduce the minimum temperature variation that can be enjoyed with this arrangement.

### Solution

• The base is used here because it is, in general, used in electronic logic.

The number of digits, which can be equal to or . Therefore, may take values.

• The minimum voltage variation that can be indicated will be given by :

Which corresponds to a minimum change in temperature equal to :

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